% The following TeXfile comprises part of

%     MEASURE THEORY, by D.H.Fremlin
%          (University of Essex, Colchester, England;
%                   e-mail  fremdh@essex.ac.uk)

% Please contact me with any difficulties not covered by the remarks
% below.

% The file is set up for processing by plain TeX in the presence of
% files  amstex.tex ,  amssym.tex  and  amsppt.sty  suitably supported by the
% appropriate fonts.   If these are unavailable, a readable version
% can be obtained by removing the comment symbol  %  from the line

%        %   \amstexsupportedfalse

% below.   Some systems do not allow access to files  amstex.tex  and
% amsppt.sty , but instead employ a command "amstex <filename.tex>";
% for such systems it is sometimes right to delete or disable the
% commands

%        \input amstex
%        \documentstyle{amsppt}

% (I have little experience with this and should welcome comments.)

% Various formats of output are obtainable.   The principal variations
% are

%    1. "Standard" output uses the command
%               \smallprint
% and gives output adapted to A4 paper (21 cm x 29.7 cm).

%    2.  Using the command
%               \USsmallprint
% gives output adapted to 8.5" x 11" .

%    3.  Using the command
%               \noproofs
% produces an abridged version, in which proofs are omitted.

%    4.   Using the command
%               \resultsonly
% produces a super-abridged version, omitting comments and exercises
% as well as proofs.

% These commands are grouped together at the end of the introductory
% definitions;  they may be activated by deleting the initial comment
% symbols  % .

% Any diagrams included will be called for by inputting postscript files.
% (You can look for these by hunting for   \picture  in the code below.)
% To get these you will need the relevant  .ps  files with the expected
% names  ( \picture{mt133g}{...}  demands the file  mt133g.ps , for instance).


%(If you can read TeX fluently, you may wish to fast-forward to a line
%beginning   %file starts here   or   \frfilename
%if one of these is present.)

%TeXfiles with this heading are designed for processing with AmSTeX,
%accessed through TeXfiles  amstex.tex (with amssym.tex  and
% amsppt.sty  and possibly  amsppt.sti)  during compilation.   
%If these are unavailable, a readable
%version can (I hope) be produced by deleting the  %  before
% \amstexsupportedfalse  below.

%Some special difficulties arise with the importation of  .ps  files
%in diagrams.   Look for  \picture  below for notes on these.

%The following \count registers are used for special purposes:
%   \count1=\stylenumber  to specify style
%   \count2=\footnotenumber  to count footnotes

%AmSTeX declaration
\newif\ifamstexsupported
     \amstexsupportedtrue
%     \amstexsupportedfalse               %use when AMSTeX unavailable
\newif\ifDHFamstexloaded        %for detection of a non-standard AMSTeX
     \DHFamstexloadedfalse    
\newif\ifDHFamspptloaded        
     \DHFamspptloadedfalse    
\newif\iffourteensavailable
     \fourteensavailabletrue
    \fourteensavailablefalse
\newif\ifathome
     \athometrue     %to cope with a quirk     of my son's Ghostscript
     \athomefalse
\newif\iffirstpage
     \firstpagetrue

%Style assignments
\countdef\stylenumber=1
\countdef\footnotenumber=2
\def\fullsize{\magnification=\magstep1
     \def\pagewidth{390pt}
     \def\pageheight{9truein}
     \hsize\pagewidth
     \vsize\pageheight
     \stylenumber=1}
\def\smallprint{\magnification=\magstep0
     \overfullrule0pt
     \def\pagewidth{468pt}
     \def\pageheight{9truein}
     \hsize\pagewidth
     \vsize\pageheight
     \stylenumber=2}
\def\printersize{\magnification=\magstep1
     \baselineskip24pt
     \overfullrule0pt
     \def\pagewidth{330pt}
     \def\pageheight{8.5truein}
     \hsize\pagewidth
     \vsize\pageheight
     \stylenumber=3}
\def\refsize{\magnification=\magstep1
     \overfullrule0pt
     \def\pagewidth{390pt}
     \def\pageheight{9truein}
     \hsize\pagewidth
     \vsize\pageheight
     \stylenumber=4}
\def\rrversion{\magnification=\magstep1
     \overfullrule0pt
     \def\pagewidth{330pt}
     \def\pageheight{9truein}
     \hsize\pagewidth
     \vsize\pageheight
     \stylenumber=5}
\def\bigsize{\magnification=\magstep2
     \overfullrule0pt
     \def\pagewidth{325pt}
     \def\pageheight{9truein}
     \hsize\pagewidth
     \vsize\pageheight
     \stylenumber=6}
\def\bbigsize{\magnification=\magstep3
     \overfullrule0pt
     \def\pagewidth{271pt}
     \def\pageheight{9truein}
     \hsize\pagewidth
     \vsize\pageheight
     \stylenumber=7}
\def\USsmallprint{\magnification=\magstep0
     \overfullrule0pt
     \def\pagewidth{482pt}
     \def\pageheight{8.7truein}
     \hsize\pagewidth
     \vsize\pageheight
     \stylenumber=8}
\def\halfmagsize{\magnification=\magstephalf
     \def\pagewidth{427pt}
     \def\pageheight{9truein}
     \hsize\pagewidth
     \vsize\pageheight
     \stylenumber=9}
           % register  stylenumber  is used to control the output format

%addresses, letterheads
\def\deptaddress{\hskip 3truein Department of Mathematical Sciences\par
   \noindent e-mail: {\tt fremdh}\hskip 2.9truein\today\par
   \medskip}
\long\def\univaddress#1{\hskip 3truein Department of Mathematical Sciences\par
   \hskip 3truein University of Essex\par
   \hskip 3truein Colchester CO4 3SQ\par
   \hskip 3truein England\par
   %\hskip 3truein van Vleck Hall\par
   %\hskip 3truein 480 Lincoln Drive\par
   %\hskip 3truein Madison\par
   %\hskip 3truein WI 53706-1388, U.S.A.\par
   \hskip 3.2truein#1\par
   \noindent e-mail: {\tt david\@fremlin.org}
       %{\tt fremlin\@math.wisc.edu}
   \bigskip}
\long\def\homeaddress#1{\hskip 3truein 25 Ireton Road\par
   \hskip 3truein Colchester CO3 3AT\par
   \hskip 3truein England\par
   \hskip 3.2truein#1\par
   \noindent e-mail: {\tt david\@fremlin.org}
   \bigskip}
\long\def\letterhead#1#2#3{\univaddress{#1}
   {\parindent=0pt
    #2
   }
   \medskip

   \noindent Dear #3,
     }
\def\shortaddress{\noindent Departmentof Mathematical Sciences, University of Essex,
   Colchester CO4 3SQ, England\par
   \noindent e-mail: david\@fremlin.org}
\def\valediction#1#2{\bigskip
   \hskip 3truein #1,
   \vskip 1truein
   \hskip 3.3truein #2
   \bigskip}

%picture importation
   %It is clear that there is no general standard.   The syntax I
   %use when calling for a  .ps  file is
   %    \def\Caption{<...>}
   %    \picture{<filename>}{<dim>}
   %which is supposed to import  <filename>.ps  at height  <dim> ,
   %centered, and adding a one-line centered caption  <...> .
   %The basic macro listed in  \def\picture  is that which
   %used to work at Essex.   I now generally use the macro in
   % \def\picturewithpsfig , which requires a previous input
   % psfig.tex  or  psfig.sty ;
   %if you interpolate  \atUEssex  early in your file it will
   %give you the latter.
   %   I find that  psfig  is erratic in its centering, especially
   %under magnification, so I have written a routine
   % \centerpicturewithpsfig  (which may be called from
   % \centerpicture ) with the syntax
   %    \centerpicture{<filename>}{<dim>}{<dim>}
   %the first dimension being width and the second height.

   % A second package is called for by  \usegraphicx .   This requires
   % miniltx.tex  and  graphicx.sty ;  miniltx.tex  is currently (April 2016)
   % available from
   % http://mirror.ox.ac.uk/sites/ctan.org/macros/plain/graphics/miniltx.tex
   % The advantage of this is that it seems to allow manipulation of the
   % dvi>pdf stage using  dvipdfmx .

\def\picture#1#2{\bigskip
  \bigskip
  \immediate\write0{importing diagram #1.ps at height #2
         with dvitops}
  \vbox{\vskip #2
    \special{dvitops: import #1.ps \pagewidth #2}
    \Centerline{\Caption}
    }
   \medskip
   \def\Caption{}}
%an alternative version, without caption
\def\centerpicture#1#2#3{\bigskip
  \bigskip
  \immediate\write0{importing diagram #1.ps at width #2
         and height #3 with dvitops}
  \vbox{\vskip #3
    \special{dvitops: import #1.ps \pagewidth #3}
  }}
%to put pictures on same line
\def\sideshiftedpicture#1#2#3#4{\bigskip
   \bigskip
   \immediate\write0{importing diagram #1.ps at height #4
        and width #3 with dvitops}
   \vbox{\special{dvitops:  import #1.ps #3 #4}}
   \hskip #2 plus 0pt minus 0pt
   }
\def\startsideshiftedpicture{}
\newcount\scaleup
\newcount\scaledown
\scaleup=1
\scaledown=1
\def\scalesideshiftedpicture#1#2#3#4{
  \dimen4=#4
  \dimen3=#3
  \multiply\dimen4 by\scaleup
  \divide\dimen4 by \scaledown
  \multiply\dimen3 by\scaleup
  \divide\dimen3 by \scaledown
  \sideshiftedpicture{#1}{#2}{\dimen3}{\dimen4}}

\def\picturewithpsfig#1#2{\bigskip
   \immediate\write0{importing diagram #1.ps at height #2 with psfig}
   \vbox{\hsize\pagewidth
   \centerline{\psfig{figure=#1.ps,height=#2}}
   \medskip
   \centerline{\Caption}}
   \medskip
   \def\Caption{}}
\def\centerpicturewithpsfig#1#2#3{
  \immediate\write0{importing diagram #1.ps at width #2
         and height #3 with psfig}
  \dimen 3 = #2
  \dimen 4 = #3
  \dimen 2 = \pagewidth
  \advance\dimen2 by -\dimen3
  \divide\dimen2 by 2
  \discrversionB{\shrinkonestep}{}
  \discrversionH{\shrinkonestep}{}
  \discrversionF{\enlargeonestep}{}
  \discrversionG{\enlargeonestep\enlargeonestep}{}
  \dimen 6 = \dimen4
  \divide\dimen6 by 3
  \vskip\dimen6
  \ifdim\dimen2>0\hskip\dimen2\fi
     \vbox{\psfig{file=#1.ps,width=\dimen3,height=\dimen4}}
  \discrversionG{\vskip-\dimen6}{}
  }
\def\enlargeonestep{\dimen5 = 5truein
   \dimen6 = 7truein
   \ifdim\dimen3<\dimen5
      \ifdim\dimen4<\dimen6
         \divide\dimen3 by 5
         \multiply\dimen3 by 6
         \divide\dimen4 by 5
         \multiply\dimen4 by 6
      \fi\fi}
\def\shrinkonestep{\divide\dimen3 by 6
         \multiply\dimen3 by 5
         \divide\dimen4 by 6
         \multiply\dimen4 by 5}
\def\sideshiftedpicturewithpsfig#1#2#3#4{
   \immediate\write0{importing diagram #1.ps at height #4
      and width #3 with psfig}
   \dimen 4 = #4
   \dimen 3 = #3
   \discrversionB{\shrinkonestep}{}
   \discrversionH{\shrinkonestep}{}
   \discrversionF{\enlargeonestep}{}
   \discrversionG{\enlargeonestep\enlargeonestep}{}
%   \vbox{\psfig{figure=#1.ps,height=\dimen4,width=\dimen3}}
   \psfig{figure=#1.ps,height=\dimen4,width=\dimen3}
   \hskip #2 plus 0pt minus 0pt}

\def\picturewithgraphicx#1#2{\bigskip
   \immediate\write0{importing diagram #1.ps at height #2 with graphicx}
   \vbox{\hsize\pagewidth
   \centerline{\includegraphics[height=#2]{#1.ps}}
   \medskip
   \centerline{\Caption}}
   \medskip
   \def\Caption{}}
\def\centerpicturewithgraphicx#1#2#3{
  \immediate\write0{importing diagram #1.ps at width #2
         and height #3 with graphicx}
  \dimen 3 = #2
  \dimen 4 = #3
  \dimen 2 = \pagewidth
  \advance\dimen2 by -\dimen3
  \divide\dimen2 by 2
  \discrversionB{\shrinkonestep}{}
  \discrversionH{\shrinkonestep}{}
  \discrversionF{\enlargeonestep}{}
  \discrversionG{\enlargeonestep\enlargeonestep}{}
  \dimen 6 = \dimen4
  \divide\dimen6 by 3
  \vskip\dimen6
  \ifdim\dimen2>0\hskip\dimen2\fi
     \vbox{\includegraphics[width=\dimen3,height=\dimen4]{#1.ps}}
  \discrversionG{\vskip-\dimen6}{}
  }
\def\enlargeonestep{\dimen5 = 5truein
   \dimen6 = 7truein
   \ifdim\dimen3<\dimen5
      \ifdim\dimen4<\dimen6
         \divide\dimen3 by 5
         \multiply\dimen3 by 6
         \divide\dimen4 by 5
         \multiply\dimen4 by 6
      \fi\fi}
\def\shrinkonestep{\divide\dimen3 by 6
         \multiply\dimen3 by 5
         \divide\dimen4 by 6
         \multiply\dimen4 by 5}
\def\sideshiftedpicturewithgraphicx#1#2#3#4{
   \immediate\write0{importing diagram #1.ps at height #4
      and width #3 with graphicx}
   \dimen 4 = #4
   \dimen 3 = #3
   \discrversionB{\shrinkonestep}{}
   \discrversionH{\shrinkonestep}{}
   \discrversionF{\enlargeonestep}{}
   \discrversionG{\enlargeonestep\enlargeonestep}{}
%   \vbox{\includegraphics[height=\dimen4,width=\dimen3]{#1.ps}}
   \includegraphics[height=\dimen4,width=\dimen3]{#1.ps}
   \hskip #2 plus 0pt minus 0pt}

%alternative picture importation code;  I've been told about this
  %but have not used it successfully myself
%\def\Caption{}
%\input epsf
%\epsfverbosetrue
%\def\picture#1#2{\bigskip
%    \bigskip
%    \immediate\write0{importing diagram #1.ps at height #2 with epsf}
%    \epsfysize=#2
%    \centerline{\epsffile{#1.ps}}
%    \medskip
%    \Centerline{\Caption}
%    \medskip
%    }

%in addition it may be necessary to amend the imported  .ps  file
%by replacing the   %%BoundingBox   header by  %%BoundingBox:

\def\atUEssex{
   \input psfig.sty
   \def\picture##1##2{\picturewithpsfig{##1}{##2}}
   \def\sideshiftedpicture##1##2##3##4{
     \sideshiftedpicturewithpsfig{##1}{##2}{##3}{##4}}
   \def\centerpicture##1##2##3{
       \centerpicturewithpsfig{##1}{##2}{##3}}
   \def\startsideshiftedpicture{
       \sideshiftedpicture{empty}{1pt}{1pt}{1pt}}
   }
\def\usegraphicx{
   \input miniltx
   \def\Gin@driver{dvips.def}
   \input graphicx.sty
   \def\picture##1##2{\picturewithgraphicx{##1}{##2}}
   \def\sideshiftedpicture##1##2##3##4{
     \sideshiftedpicturewithgraphicx{##1}{##2}{##3}{##4}}
   \def\centerpicture##1##2##3{
       \centerpicturewithgraphicx{##1}{##2}{##3}}
   \def\startsideshiftedpicture{
       \sideshiftedpicture{empty}{1pt}{1pt}{1pt}}
}


%special fonts
\def\biggerscriptfonts{\scriptfont0=\eightrm
     \scriptfont1=\eighti
     \scriptfont2=\eightsy
     \scriptfont3=\eightex
     \scriptfont\bffam=\eightbf
     \scriptfont\msbfam=\eightmsb
     \scriptfont\eufmfam=\eighteufm
     }
\def\Loadcyrillic{\ifamstexsupported
      \input cyracc.def
      \font\tencyr=wncyr10
      \def\cyr{\tencyr\cyracc} \def\itcyr{\tenitcyr\cyracc}
    \else\def\cyr{\smc}\def\cdprime{u\i}\def\Cdprime{UI}\fi}
%http://wkweb3.cableinet.co.uk/linux/LDP/HOWTO/Cyrillic-HOWTO-7.html
%cyrillic.tex

\def\Loadfourteens{\iffourteensavailable
       \font\fourteenrm=cmr14
       \font\fourteenbf=cmbx14
       \font\fourteenit=cmsl14
   \else\font\fourteenrm=cmr10 scaled \magstep2
      \font\fourteenbf=cmbx10 scaled \magstep2
      \font\fourteenit=cmsl10 scaled \magstep2
   \fi}
\def\smallerfonts{\eightpoint
   \textfont\msbfam=\eightmsb
   \textfont\eufmfam=\eighteufm
   \textfont\bffam=\eightbf
   \textfont\eurmfam=\eighteurm
   \textfont\eusmfam=\eighteusm
   }
\font\twelveex=cmex10 scaled \magstep1


%Copyright declarations
\newcount\oldcopyrightdate
\newcount\newcopyrightdate
\oldcopyrightdate=0
\def\copyrightdate#1{\newcopyrightdate=#1
       \ifnum\newcopyrightdate=\oldcopyrightdate\else
       \oldcopyrightdate=#1
       \oldfootnote{}{\eightrm \copyright\ #1 D.~H.~Fremlin}
       \immediate\write0{Copyright #1}
       \fi}

%hyphenation
\hyphenation{equi-ver-id-ical}

%miscellaneous definitions
\def\Aut{\mathop{\text{Aut}}}
\def\backstep#1{\hskip -0.#1em plus 0 pt minus 0 pt}
\def\Bang{\vthsp\hbox{\bf X\hskip0.07em\llap X\hskip0.07em\llap X\rm}}
\def\BanG{\Bang{\hskip 1.111pt plus 3.333pt minus 0pt}}
%\def\BbbN{\mathchoice{\hbox{$\Bbb N$\hskip0.02em}}
%  {\hbox{$\Bbb N$\hskip0.02em}}
%  {\hbox{$\scriptstyle\Bbb N$\hskip0.04em}}
%  {\hbox{$\scriptscriptstyle\Bbb N$\hskip0.04em}}}
%\def\BbbQ{\mathchoice{\hbox{$\Bbb Q$\hskip0.02em}}
%  {\hbox{$\Bbb Q$\hskip0.02em}}
%  {\hbox{$\scriptstyle\Bbb Q$\hskip0.04em}}
%  {\hbox{$\scriptscriptstyle\Bbb Q$\hskip0.04em}}}
%\def\BbbR{\mathchoice{\hbox{$\Bbb R$\hskip0.02em}}
%  {\hbox{$\Bbb R$\hskip0.02em}}
%  {\hbox{$\scriptstyle\Bbb R$\hskip0.04em}}
%  {\hbox{$\scriptscriptstyle\Bbb R$\hskip0.04em}}}
%  %do not use with $\Bbb R^X$
%\def\BbbZ{\mathchoice{\hbox{$\Bbb Z$\hskip0.02em}}
%  {\hbox{$\Bbb Z$\hskip0.02em}}
%  {\hbox{$\scriptstyle\Bbb Z$\hskip0.04em}}
%  {\hbox{$\scriptscriptstyle\Bbb Z$\hskip0.04em}}}
\def\BbbN{\openBbb{N}}
\def\BbbQ{\openBbb{Q}}
\def\BbbR{\openBbb{R}}
  %do not use with $\Bbb R^X$  do not use in $_{\Bbb R^r}$
\def\BbbZ{\openBbb{Z}}
\def\Bbbone{\hbox{1\hskip0.13em\llap{1}}}
\def\biggerint{\textfont3=\twelveex\intop\nolimits}
  % you can get this size into a displayed formula with
  %  $$\hbox{$\biggerint_{\Cal S}\pmb{u}\,d\pmb{v}$}$$
  % but it may then seem cramped
  % or try \hbox{$\biggerint_{\Cal S}\,$}
\def\biggeriint{\textfont3=\twelveex\iint}
  % remarkable fact:  this seems to infect all \int within a given
  % $..$ ; but  $\biggerint$$\int$ seems to leave the $\int$ normal
  % really  \textfont3=\twelveex  is more reliable, since  \biggerint
  % as written gets turned off rather easily (e.g. by  \comment,
  % or by an extra  {...})
\def\bover#1#2{{{#1}\over{#2}}}
\def\Bover#1#2{\backstep4\hbox{
     \biggerscriptfonts${\shortstrut#1}\over{\shortstrut#2}$}}
  % when there are superscripts in the denominator, as in \Bover1{2^m}
  % or  \Bover1{\sqrt{2\pi}^r} , these can sometimes come out with better
  % horizontal alignment if you use  \Bover1{{\sshortstrut 2}^m}  or
  % \Bover1{{\sqrt{2\pi}}^r}
\def\bsover#1#2{\textstyle\bover{#1}{#2}}
\def\bsp{/\penalty-100}
  % \penalty seems to behave oddly in a \tt string in \fullsize mode
\def\Caption{}
\def\Center#1{

   \Centerline{#1}

   \noindent}
\def\Centerline#1{\vskip 5pt
     {\def\int{\text{}$$\textfont3=\twelveex\intop\nolimits}
      \def\dashint{\text{}$$\textfont3=\twelveex
         \mskip2mu\vrule height2.65pt width4.5pt depth-2.35pt\mskip-11mu
	 \intop\nolimits\mskip-2mu}
      \def\idashint{\text{}$$\int\mskip-7mu\dashint}	 
      \def\dashiint{\text{}$$\dashint\mskip-6mu\int}	 
      \def\dashidashint{\text{}$$\dashint\mskip-6mu\dashint}	 
      \def\underlineint{
           \text{}$$\textfont3=\twelveex\underline{\intop\nolimits}}
      \def\overlineint{
           \text{}$$\textfont3=\twelveex\overline{\intop\nolimits}}
      \def\iint{\text{}$$\textfont3=\twelveex
           \intop\nolimits\hskip-0.5em\intop\nolimits}
      \def\sum{\text{}$$\sumop}
      \def\prod{\text{}$$\prodop}
      \def\bigcap{\text{}$$\bigcapop}
      \def\bigcup{\text{}$$\bigcupop}
   \centerline{#1}}\vskip 5pt}
   \mathchardef\bigcapop="1354
   \mathchardef\bigcupop="1353
   \mathchardef\prodop="1351
   \mathchardef\sumop="1350
   %in  \Centerline{$\overline{}$, $\cmmnt{}$} must use \sumop etc
\def\cf{\mathop{\text{cf}}}
\def\circumflex#1{{\accent"5E #1}}
\def\closeplus{\hbox{+}}
\long\def\cmmnt#1{\ifresultsonly\else{#1}\fi}
\def\coint#1{\left[#1\right[}
\def\curl{\mathop{\text{curl}}}
\def\dashint{\mathchoice{-\mskip-20mu\intop\nolimits}
   {{\vrule height2.75pt width4.5pt depth-2.45pt\mskip-10mu\intop\nolimits}}
      {-\mskip-16.5mu\intop\nolimits} %not yet used
      {-\mskip-16.5mu\intop\nolimits}} %not yet used
\def\idashint{\mathchoice{\int\mskip-6mu\dashint}
      {\int\mskip-5mu\dashint}
      {\int\mskip-5mu\dashint} %not yet used
      {\int\mskip-5mu\dashint}} %not yet used
\def\dashiint{\mathchoice{\dashint\mskip-10mu\int}
      {\dashint\mskip-6mu\int}
      {\dashint\mskip-6mu\int} %not yet used
      {\dashint\mskip-6mu\int}} %not yet used
\def\dashidashint{\mathchoice{\dashint\mskip-10mu\dashint}
      {\dashint\mskip-6mu\dashint}
      {\dashint\mskip-6mu\dashint} %not yet used
      {\dashint\mskip-6mu\dashint}} %not yet used
\def\dashidashiint{\mathchoice{\dashint\mskip-8mu\dashint\mskip-10mu\int}
      {\dashint\mskip-4mu\dashint\mskip-8mu\int} 
      {\dashint\mskip-6mu\int} %not yet used
      {\dashint\mskip-6mu\int}} %not yet used
\def\dashiidashint{\mathchoice{\dashint\mskip-10mu\int\mskip-8mu\dashint}
      {\dashint\mskip-4mu\dashint\mskip-8mu\int} 
      {\dashint\mskip-6mu\int} %not yet used
      {\dashint\mskip-6mu\int}} %not yet used
\def\idashidashint{\mathchoice{\int\mskip-8mu\dashint\mskip-8mu\dashint}
      {\dashint\mskip-6mu\int} %not yet used
      {\dashint\mskip-6mu\int} %not yet used
      {\dashint\mskip-6mu\int}} %not yet used
\def\diam{\mathop{\text{diam}}}
\def\discrpage{\ifnum\stylenumber=1{\frnewpage}  %redefined in mt.tex
           \else\ifnum\stylenumber=3{\frnewpage}
           \else\ifnum\stylenumber=5{\frnewpage}
           \else\bigskip\fi\fi\fi}
\long\def\discrversionA#1#2{\ifnum\stylenumber=1#1\else#2\fi}
\long\def\discrversionB#1#2{\ifnum\stylenumber=2#1\else#2\fi}
\long\def\discrversionC#1#2{\ifnum\stylenumber=3#1\else#2\fi}
\long\def\discrversionD#1#2{\ifnum\stylenumber=4#1\else#2\fi}
\long\def\discrversionE#1#2{\ifnum\stylenumber=5#1\else#2\fi}
\long\def\discrversionF#1#2{\ifnum\stylenumber=6#1\else#2\fi}
\long\def\discrversionG#1#2{\ifnum\stylenumber=7#1\else#2\fi}
\long\def\discrversionH#1#2{\ifnum\stylenumber=8#1\else#2\fi}
           % facility for special interpolations in particular output
           % versions
\def\displaycause#1{\noalign{\noindent (#1)}}
\def\diverg{\mathop{\text{div}}}
\def\dom{\mathop{\text{dom}}}
\def\eae{=_{\text{a.e.}}}

\def\esssup{\mathop{\text{ess sup}}}
\mathchardef\Exists="3239

\def\family#1#2#3{\langle#3\rangle_{#1\in#2}}
\def\familyi#1#2{\family{i}{#1}{#2}}
\def\familyiI#1{\familyi{I}{#1}}

\def\frfilename#1{\ifnum\stylenumber=1{\noindent filename #1}
               \else\ifnum\stylenumber=6{\noindent{\fiverm filename #1}}
               \else\ifnum\stylenumber=7{\noindent{\fiverm filename #1}}
               \else\fi\fi\fi}
\def\Footnote#1#2{\footnote"#1"{#2}}
\mathchardef\Forall="3238
\def\frakc{\frak c\phantom{.}}
\def\frnewpage{\vskip \pageheight plus 0pt minus\pageheight
           \eject
           \gdef\topparagraph{}
           \gdef\bottomparagraph{}
           }
\def\frsmallcirc{{\hskip 1pt plus 0pt minus 0pt}
     {\raise 1pt\hbox{$\smallcirc$}}{\hskip 1pt plus 0pt minus 0pt}}

\def\geae{\ge_{\text{a.e.}}}
\def\grad{\mathop{\text{grad}}}
\def\grheada{{\bf ($\pmb{\alpha}$)}}
\def\grheadb{{\bf ($\pmb{\beta}$)}}
\def\grheadc{{\bf ($\pmb{\gamma}$)}}
\def\grheadd{{\bf ($\pmb{\delta}$)}}
\def\grheade{{\bf ($\pmb{\epsilon}$)}}
\def\grheadz{{\bf ($\pmb{\zeta}$)}}
\def\grv#1{{\accent"12 #1}}

\def\hfillall{\hskip\pagewidth plus0pt minus\pagewidth}

\def\Imag{\mathop{\Cal I\text{m}}}
\long\def\inset#1{{\narrower\narrower{\vskip 1pt plus 1pt minus 0pt}
     #1
     {\vskip 1pt plus 1pt minus 0pt}}}
\def\Int#1#2{\displaystyle{\int_{#1}^{#2}}}
\def\interior{\mathop{\text{int}}}

\def\leae{\le_{\text{a.e.}}}
\long\def\leaveitout#1{}
\def\LLcorner{\mathchoice{\,\hbox{\vrule height7pt width0.5pt depth0pt
       \vrule height0.5pt width5pt depth0pt}\,}
   {\,\hbox{\vrule height7pt width0.5pt depth0pt
       \vrule height0.5pt width5pt depth0pt}\,}
   {\,\hbox{\vrule height5pt width0.3pt depth0pt
       \vrule height0.3pt width4pt depth0pt}\,}
   {\,\hbox{\vrule height4pt width0.25pt depth0pt
       \vrule height0.25pt width3pt depth0pt}\,}
  }
\def\Ln{\mathop{\text{Ln}}}
\def\Loadfont#1{\font#1}
\def\loibr{\hbox{\hskip0.1em]\hskip0.05em}}
\def\Matrix#1{\pmatrix#1\endpmatrix}
\def\Mu{\text{M}}
\def\noamslogo{\def\amslogo{}}
\def\normalsubgroup{\vartriangleleft}
\def\Nu{\text{N}}
\def\ocint#1{\left]#1\right]}
\def\ofamily#1#2#3{\langle#3\rangle_{#1<#2}}
\let\oldcases=\cases
\let\oldfootnote=\footnote
\def\On{\mathop{\text{On}}}
\def\ooint#1{\left]#1\right[}
\def\openBbb#1{\Bbb#1\mskip1mu}
\def\pd#1#2{\bover{\partial#1}{\partial#2}}
\def\Pd#1#2{\Bover{\partial#1}{\partial#2}}
\def\pdd#1#2{{{\partial^2#1}\over{\partial#2^2}}}
\def\Pdd#1#2{\Bover{\partial^2#1}{\partial#2^2}}
\def\pde#1#2#3{{{\partial^2#1}\over{\partial#2\partial#3}}}
\def\Pde#1#2#3{\Bover{\partial^2#1}{\partial#2\partial#3}}
\def\Prf{\vthsp\hbox{\bf P\hskip0.07em\llap P\hskip0.07em\llap P\rm}}
\def\pushbottom#1{\hbox{\vrule height0pt depth#1 width0pt}}
\def\Qed{\vthsp\hbox{\bf Q\hskip0.07em\llap Q\hskip0.07em\llap Q\rm}}
\def\QeD{\Qed{\hskip 1.111pt plus 3.333pt minus 0pt}}
\def\Quer{\vthsp\hbox{\bf ?\hskip0.05em\llap ?\hskip0.05em\llap
   ?\thinspace\rm}}
\def\query{\discrversionA{\immediate\write0{query}
    \global\advance\footnotenumber by 1
    \oldfootnote{$^{\the\footnotenumber}$}{query}}{}}
\def\Real{\mathop{\Cal R\text{e}}}
\def\restr{\mathchoice
  {\hbox{$\restriction$}\mskip 1mu}
  {\hbox{$\restriction$}\mskip 1mu}
  {\hbox{$\scriptstyle\restriction$}\mskip 1mu}
  {\hbox{$\scriptscriptstyle\restriction$}\mskip 1mu}}
\def\restrp{\mathchoice
  {\hbox{$\restriction$}\mskip 2mu}
  {\hbox{$\restriction$}\mskip 2mu}
  {\mskip-1mu\hbox{$\scriptstyle\restriction$}\mskip 2mu}
  {\mskip-1mu\hbox{$\scriptscriptstyle\restriction$}\mskip 1mu}}
    %use in \restrp V \restrp W \restrp Y
    % \restrp\Cal P \restrp\Bbb R \restrp\frak
    % \restrp\Omega \restrp\Tau \restrp\Upsilon
    % \restrp[ \restrp\widehat{}
\def\Rho{\text{P}}
\def\roibr{\hbox{\hskip0.05em[\hskip0.1em}}
\def\RoverC{\hbox{\biggerscriptfonts${{\Bbb R}\atop{\Bbb C}}$}}
\def\sequence#1#2{\family{#1}{\Bbb N}{#2}}
\def\sequencen#1{\sequence{n}{#1}}
\def\setover#1#2{\overset{#1}\to{#2}}
\def\shortstrut{\hbox{\vrule height7.5pt depth2.5pt width0pt}}
\def\sshortstrut{\hbox{\vrule height5.8pt depth2.5pt width0pt}}
\def\smallcirc{{\scriptstyle\circ}}
\def\ssbullet{{\scriptscriptstyle\bullet}}
\def\symmdiff{\triangle}
\def\thicken#1{#1\hskip0.03em\llap #1\hskip0.03em\llap
   #1\hskip0.03em\llap #1}
\def\Tau{{\text{T}}}
\def\tbf#1{\text{\bf #1}}
\def\terror#1#2#3{#1:  `#2' for `#3'}
\def\today{\number\day.\number\month.\number\year}
\def\UEMDRR{University of Essex Mathematics Department Research Report}
\def\UEMDWP{University of Essex Mathematics Department Working Paper}
\def\versiondate#1{
    \immediate\write0{Version of #1}
    \def\vsdate{#1}
               \ifnum\stylenumber=1\hfill Version of #1
                              \medskip
               \else\ifnum\stylenumber=2\hfill Version of #1
                              \medskip
               \else\ifnum\stylenumber=6\hfill{\fiverm Version of #1}
                              \medskip
               \else\ifnum\stylenumber=7\hfill{\fiverm Version of #1}
                              \medskip
               \else\ifnum\stylenumber=8\hfill Version of #1
                              \medskip
               \else\fi\fi\fi\fi\fi}
\def\versionhead{\ifnum\stylenumber=3 \hfill PRINTER'S VERSION

                        \bigskip
                   \else\ifnum\stylenumber=4 \hfill REFEREE'S VERSION

                        \bigskip
                   \else \fi\fi}
\def\versionref{\ifnum\stylenumber=1\workingref
           \else\ifnum\stylenumber=2\workingref
           \else\ifnum\stylenumber=8\workingref
           \else\printerref\fi\fi\fi}
\def\vthsp{\hskip0.001em}

%Missing AmSTeX can be coped with by using the following lines:

\ifamstexsupported\else
   \immediate\write0{AmSTeX-unsupported version of  fremtex.tex }
   \fourteensavailablefalse
   \def\@{@}
   \def\\{\cr}
   \def\AmSTeX{AmS\TeX}
   \def\Bbb#1{\underline{\hbox{\bf #1}}}
   \def\biggerscriptfonts{}
   \def\bumpeq{=_{\text{approx}}}
   \def\Cal#1{{\cal#1}}
   \def\curlyeqprec{\preceq}
   \def\eightit{\it}
   \def\eightrm{\rm}
   \def\eightsmc{\smc}
   \def\eurm#1{\underline{\cal#1}}
   \def\Footnote#1#2{\footnote#1#2}
   \def\frac#1#2{{#1}\over{#2}}
   \def\frak#1{{\hbox{\bf #1\hskip0.1em\llap #1}}}
   \def\frnewpage{\vfill\eject}
   \def\iint{\int\!\!\!\int}
   \def\iiint{\int\!\!\!\int\!\!\!\int}
   \def\iiiint{\int\!\!\!\int\!\!\!\int\!\!\!\int}
   \def\llcorner{\lfloor}
   \def\loadeurm{}
   \def\loadeusm{}
%   \def\loadeusmx#1{\def\eusm#1{\underline{\cal#1}}}
   \def\ltimes{|\hskip-0.5em\times}
   \def\eusm#1{\underline{\cal#1}}
   \def\ltimes{|\times}
   \def\Loadfont#1{}
   \def\Matrix#1{\pmatrix{#1}}
   \def\noamslogo{}
   \def\nrightarrow{\not\rightarrow}
   \def\preccurlyeq{\preceq}
   \def\query{\discrversionA{\immediate\write0{query}
                             \footnote{$^*$}{query}}{}}
   \def\restriction{|}
   \def\rtimes{\times|}
   \def\setover#1#2{#2^{#1}}
   \def\shortstrut{\strut}
   \def\smallerfonts{\eightrm}
   \def\smallfrown{\wedge}
   \def\smc{}
   \def\spcheck{\check{\phantom{x}}}
   \def\sphat{\hat{\phantom{x}}}
   \def\square{\mathchoice\sqr64\sqr64\sqr43\sqr33}
   \def\sqr#1#2{{\vcenter{\vbox{\hrule height.#2pt
          \hbox{\vrule width.#2pt height#1pt \kern#1pt
                  \vrule width.#2pt}
          \hrule height.#2pt}}}}
   \def\pmb#1{\underline{#1}}
   \def\text#1{\hbox{#1}}
   \def\therefore{\ssbullet\raise.9ex\hbox{$\ssbullet$}\ssbullet}
   \def\twelvebf{\bf}
   \def\vartriangleleft{\triangleleft}
   \def\Vdash{|\vdash}
   \def\vDash{|=}
   \fi

%End of lines for missing AmSTeX

\ifamstexsupported\input amstex
%              \documentstyle{amsppt}
               \input amsppt.sty
               \PSAMSFonts
	       \tenpoint
               %\font\eightmsb=msym8
               \font\eightmsb=msbm10
               \font\eighteufm=eufm10 at 8 pt
               \font\eighteurm=eurm8
               \font\eighteusm=eusm8
               \font\varseveneufm=eufm10 at 7 pt
               \scriptfont\eufmfam=\varseveneufm
   \def\.#1{{\accent"5F #1}}
   \def\=#1{{\accent"16 #1}}
   \gdef\nopagenumbers{\NoPageNumbers}
   \else\fi

\newcount\footmarkcount
\ifDHFamstexloaded
    \loadmsBm
    \def\eightBbb#1{{\fam\msBfam#1}}
    \loadmsFm
    \def\eightfrak#1{{\fam\msFfam#1}}
  \else
    \def\eightBbb#1{{\scriptstyle\Bbb#1}}
    \def\eightfrak#1{{\scriptstyle\frak#1}}
  \fi
\ifathome\athome\fi

%interword glue (\tenrm standard):
%  (space) 3.333 + 1.667 - 1.111
%  (,)     3.333 + 2.083 - 0.889
%  (.)     4.444 + 5.    - 0.370

%very odd fact:  \discrversionA{2.}{}  doesn't work,
%   \discrversionA{ 2.}{} does

%The false line feed introduced by my Fortran programs, leading to
%a surplus  ae  at the top of the next page, seems to be associated
%with  Fk(\032)3081  or something similar in the PostScript file.


\immediate\write0{Extract from MEASURE THEORY,
   by D.H.Fremlin, University of Essex, Colchester.
   This material is copyright.   It is issued under the terms of the
   Design Science License as published in
   http://dsl.org/copyleft/dsl.txt.   This is a development version and the
   source files are not permanently archived.
   For further information contact david\@fremlin.org.}

%hyphenations
\hyphenation{Archi-me-dean}

%new style
\def\smallprintersize{\magnification=\magstep0
     \overfullrule0pt
     \def\pagewidth{468pt}
     \def\pageheight{9truein}
     \hsize\pagewidth
     \vsize\pageheight
     \nologo
     \def\copyrightdate##1{}
     \def\folio{\ifnum\pageno<0{}\else\number\pageno\fi}
     \def\mtfootline{}
     \stylenumber=10
     \pageno=-1000}

%With-proofs, results-only versions
\newif\ifwithproofs
\withproofstrue
\newif\ifresultsonly
\resultsonlyfalse
\def\noproofs{\withproofsfalse}
\def\resultsonly{\resultsonlytrue\withproofsfalse}
\long\def\proof#1{\ifwithproofs{\medskip

     \noindent{\bf proof} #1}\else\fi}
\long\def\prooflet#1{\ifwithproofs{#1}\else\fi}
%\long\def\Prooflet#1{\prooflet{\Prf\ #1 \Qed\ }}
\long\def\endnotes#1{\ifresultsonly\else{#1}\fi}
\long\def\exercises#1{\ifresultsonly\else{#1}\fi}
\long\def\references#1{\ifresultsonly\else{#1}\fi}
\long\def\dvro#1#2{\ifresultsonly{#1}\else{#2}\fi}

%Headline assignments

\def\topparagraph{}
\def\bottomparagraph{}
\def\newparagraph{}
\newif\ifpageturned

\newdimen\rhoffset
\newdimen\lhoffset
\lhoffset=0cm
\rhoffset=0cm

\newcount\volumeno
\volumeno=5
\newcount\luluvolumeno
\luluvolumeno=0

\let\plainpagebody=\pagebody
\def\pagebody{\ifodd\pageno\global\hoffset=\rhoffset
         \else\global\hoffset=\lhoffset\fi
     \plainpagebody
     \turnpage}
\def\turnpage{
     \iffirstpage\else{\baselineskip=24pt\line{\mtfootline}}\fi
%     \immediate\write0{turning page \folio,
%           top paragraph \topparagraph,   bottom \bottomparagraph,
%           new \newparagraph, pagetotal \the\pagetotal}
     \global\pageturnedtrue
     \global\firstpagefalse
     \global\let\topparagraph=\bottomparagraph
     \global\let\bottomparagraph=\newparagraph
     }

\newdimen\pagetotalplus
\newdimen\pagetotalminus

%wheader parameters:  #1 new paragraph name
 % #2#3#4  \vskip parameters if page break not forced (no. of points)
 % #5  spare height demanded if page break not to be forced
\long\def\wheader#1#2#3#4#5{\gdef\newparagraph{#1}
  \pagetotalplus=\the\pagetotal
  \advance\pagetotalplus by #5
  \pagetotalminus=\the\pagetotal
  \advance\pagetotalminus by -\pageshrink
  %\showthe\pagetotal \showthe\pageshrink \showthe\pagegoal
  \global\pageturnedfalse
  \ifdim\pagetotalminus<\pagegoal\ifdim\pagetotalplus>\pagegoal
%     \immediate\write0{forcing page break}
     \eject
     \else\vskip #2pt plus #3pt minus #4pt\fi
     \else\vskip #2pt plus #3pt minus #4pt\fi
  \ifpageturned\global\let\topparagraph=\newparagraph
%     \immediate\write0{paragraph change on turning page}
     \fi
%  \immediate\write0{top paragraph \topparagraph, bottom
%       \bottomparagraph, pagetotal \the\pagetotal}
  \ifdim\pagetotal>\pagegoal\else\gdef\bottomparagraph{#1}\fi
%   \immediate\write0{top paragraph \topparagraph, bottom
%       \bottomparagraph, pagetotal \the\pagetotal}
  }
\def\header#1{
     \wheader{#1}{10}{4}{4}{24pt}}
\def\leader#1#2{\immediate\write0{#1 #2}
     \wheader{#1}{14}{7}{5}{24pt}
     {\bf #1 #2}}
\def\vleader#1#2#3{\immediate\write0{#2 #3}
     \wheader{#2}{14}{7}{5}{#1}
     {\bf #2 #3}}
\def\Notesheader#1{\immediate\write0{#1 Notes and comments}
     \wheader{#1 Notes}{16}{8}{5}{48pt}
     \noindent{\bf #1 Notes and comments}}
\def\vNotesheader#1#2{\immediate\write0{#2 Notes and comments}
     \wheader{#2 Notes}{16}{8}{5}{#1}
     \noindent{\bf #2 Notes and comments}}
\def\spheader#1#2#3#4#5{\header{#1#2#3#4#5}{\bf (#5)}}
\def\spstheader#1#2#3#4#5{\header{*#1#2#3#4#5}{\bf (#5)}}
\def\vspheader#1#2#3#4#5#6{
     \wheader{#2#3#4#5#6}{10}{4}{4}{#1}{\bf (#6)}}
\def\sqheader#1#2#3#4#5{\header{#1#2#3#4#5}{$\pmb{>}${\bf (#5)}}}
\def\errheader#1#2#3{
     \wheader{#3}{6}{4}{2}{24pt}
     {\bf p #1 l #2}}
\long\def\woddheader#1#2#3#4#5{
   \ifodd\pageno\wheader{#1}{#2}{#3}{#4}{#5}
   \else\vskip #2pt plus #3pt minus #4pt\fi}

\def\indexmedskip{
  \pagetotalplus=\the\pagetotal
  \advance\pagetotalplus by 18 pt
  \pagetotalminus=\the\pagetotal
  \advance\pagetotalminus by -\pageshrink
  \ifdim\pagetotalminus<\pagegoal\ifdim\pagetotalplus>\pagegoal
      \else\medskip\fi\else\medskip\fi}

\def\indexheader#1{\immediate\write0{#1}\vindexheader{#1}{24}}
\def\vindexheader#1#2{
  \gdef\newparagraph{#1}
  \pagetotalplus=\the\pagetotal
  \advance\pagetotalplus by #2 pt
  \pagetotalminus=\the\pagetotal
  \advance\pagetotalminus by -\pageshrink
  %\showthe\pagetotal \showthe\pageshrink \showthe\pagegoal
  \global\pageturnedfalse
%  \ifdim\pagetotal>\pagegoal
%      \immediate\write0{warning: pagetotal greater than pagegoal}
%      \immediate\write0{...   last paragraph reached is #1}
%      \immediate\write0{...   check for underfull vbox here}
%      \else\fi
  \ifdim\pagetotalminus<\pagegoal\ifdim\pagetotalplus>\pagegoal{\eject}
      \fi\fi
  \ifdim\pagetotal>\pagegoal\else\gdef\bottomparagraph{#1}\fi
  \ifpageturned{\global\let\topparagraph=\newparagraph
%     \immediate\write0{paragraph change on turning page}
      }\else\fi
%  \immediate\write0{top paragraph \topparagraph, bottom
%    \bottomparagraph}
  }

\def\leftheadline{\rlap{\folio}\hfill{\eightit\chaptername}\hfill
       \llap{\eightbf\topparagraph}}
\def\rightheadline{\rlap{\eightbf\bottomparagraph}\hfill
       {\eightit\headlinesectionname}\hfill\llap{\folio}}

%Footline assignments
\def\mtfootline{\ifodd\pageno\rightfootline\else\leftfootline\fi}
\def\rightfootline{\hfill\eightsmc D.H.Fremlin}
\def\leftfootline{\ifwithproofs{\eightsmc Measure Theory\hfill}
    \else{\eightsmc Measure Theory \eightit (abridged version)\hfill}
    \fi}

%Style assignments
\def\newchapter#1{\gdef\topparagraph{}
   \gdef\bottomparagraph{Chap.\ #1 {\it intro.}}
   \gdef\newparagraph{Chap.\ #1 {\it intro.}}
   \footmarkcount=0
   \gdef\sectionname{Introduction}
   \gdef\headlinesectionname{Introduction}
   \centerline{\bf Chapter #1}
   \medskip
   \centerline{\bf \chaptername}
   \medskip
   }
\def\newsection#1{\gdef\bottomparagraph{\S#1 {\it intro.}}
   \noindent{\bf #1 \sectionname}
   \smallskip
   \let\headlinesectionname=\sectionname}
\def\newvolume#1{\gdef\topparagraph{}
   \gdef\bottomparagraph{Vol.\ #1 {\it intro.}}
   \Loadfont{\twelvebf=cmbx12}
   \def\chaptername{Volume #1}
   \def\sectionname{\volumename}
   \def\headlinesectionname{\volumename}
   \centerline{\twelvebf Volume #1:  \volumename}
   \bigskip
   }

%font variations
\newif\ifvolwp
\volwpfalse
\def\tenrmstretch#1{\fontdimen3\tenrm=#1}
\font\elevenex=cmex10 at 11pt

%contents and introductions
\newif\iflargelogo
\def\vtmpb#1{}
\def\newvolume#1{}
\def\pagereference#1#2{\hfill\hbox{\ifresultsonly#2\qquad \hskip 1truein
                                                 \else#1\qquad \fi}}
\long\def\section#1#2#3#4#5#6{\ifresultsonly
    \qquad #1 #2\pagereference{#4}{#5}
  \else\vskip 2pt plus 1pt minus 0pt
    \vbox{\qquad #1 #2\pagereference{#4}{#5}\par
    \abstr{#6}
    }\vskip -1pt plus 1pt minus 0pt
  \fi}
\long\def\chapintrosection#1#2#3{\ifresultsonly
    \qquad Introduction\pagereference{#2}{#3}
  \else\vskip 1pt plus 1pt minus 0pt
    \qquad Introduction\pagereference{#2}{#3}
    \vskip 1pt plus 1pt minus 0pt
  \fi}
\def\abstr#1{\ifresultsonly{}
   \else{\advance\leftskip by\parindent
     \advance\rightskip by\parindent
     \advance\leftskip by\parindent
     \advance\rightskip by\parindent
     \advance\leftskip by\parindent
     \advance\rightskip by\parindent
     \advance\leftskip by\parindent
     \vskip1pt plus 0pt minus0pt
     \parindent=0pt
     \smallerfonts #1%
     \hbox{\vrule height0pt depth2pt width0pt}
     \vskip1pt plus 0pt minus0pt}\fi}
     
%language variations
\newif\ifUSEnglish
\USEnglishfalse
%\USEnglishtrue

%"Later editions only"
\newcount\latereditionpage
\newcount\latereditionmark
\latereditionpage=-1
\latereditionmark=0
\newcount\newpageno

%removing the \write0 instruction altogether disorganizes it!
\def\Latereditions{\newpageno=\pageno%
  \ifdim\pagetotal>\pagegoal\advance\newpageno by 1\fi%
     \divide\newpageno by 2%
  \immediate\write0{%
%     "Later editions only";
%      pagetotal \the\pagetotal, pagegoal \the\pagegoal,
%      pageno \number\pageno,\par newpageno \number\newpageno,
%      latereditionpage \number\latereditionpage%
  }\ifnum\latereditionpage=\newpageno%
     \ifDHFamspptloaded$^{\number\latereditionmark}$%
     \else\footnote{Later editions only.}\fi%
  \else\footnote{Later editions only.}%
  \global\latereditionpage=\newpageno%
  \global\latereditionmark=\footmarkcount\fi}%
%  \immediate\write0{newpageno \number\newpageno,
%      latereditionpage \number\latereditionpage}

   
%general definitions
\def\action{\mathchoice{\lower-.3ex\hbox{$\ssbullet$}}
                       {\lower-.3ex\hbox{$\ssbullet$}}
                       {\ssbullet}
                       {\ssbullet}}  %see also \varaction
\def\add{\mathop{\text{add}}\nolimits}
\long\def\allowmorestretch#1#2{
  \ifdim\pagewidth<#1pt\else
    \fontdimen3\tenrm=2pt\fontdimen4\tenrm=1.67pt\fi
  #2
    \fontdimen3\tenrm=1.67pt\fontdimen4\tenrm=1.11pt}
\def\am{atom\discretionary{-}{}{}lessly-measurable}
\def\AmuA{\Aut_{\bar\mu}\mskip-2mu\frak A}
\def\ASint{\mathchoice{\Bbb A\mskip-5mu}
                  {\Bbb A\mskip-3mu}
                  {\Bbb A\mskip-2mu}
                  {\Bbb A\mskip-2mu}\Sint}
\def\Asisfour{$(\frak A,\bar\mu,T,\family{t}{T}{\frak A_t})$}
\def\Asissix{$(\frak A,\bar\mu,T,
   \penalty-100\family{t}{T}{\frak A_t},
   \penalty-100\Cal T,\family{\tau}{\Cal T}{\frak A_{\tau}})$}
\def\Asisfourt{$(\frak A,\bar\mu,\coint{0,\infty},
   \langle\frak A_t\rangle_{t\ge 0})$}
\def\Asissixt{$(\frak A,\bar\mu,\coint{0,\infty},
   \penalty-100\langle\frak A_t\rangle_{t\ge 0},
   \penalty-100\Cal T,\family{\tau}{\Cal T}{\frak A_{\tau}})$}
\def\Atop#1#2{{\scriptstyle #1}\atop{\scriptstyle #2}}
\def\Aut{\mathop{\text{Aut}}}

\def\Bcap{\Booleanop{\cap}}
\def\Bcup{\Booleanop{\cup}}
\def\Booleanop#1{
   \ifamstexsupported{\mathchoice
    {\hbox{\hskip.2em\scriptfont2=\eightsy$\scriptstyle#1$\hskip.2em}}
    {\hbox{\hskip.2em\scriptfont2=\eightsy$\scriptstyle#1$\hskip.2em}}
 %   {{\mskip2mu\scriptstyle#1\mskip2mu}}  %until 15.10.17
    {{\scriptscriptstyle#1}}  %from 15.10.17
    {{\scriptscriptstyle#1}}}
   \else{\mathchoice{{\scriptstyle#1}}{{\scriptstyle#1}}
    {{\scriptstyle#1}}{{\scriptscriptstyle#1}}}
    \fi}
\def\Bsetminus{\Booleanop{\setminus}}
\def\Bsubseteq{\hskip.1em\Booleanop{\subseteq}\hskip.1em}
\def\Bsubset{\Booleanop{\subset}}
\def\Bsupseteq{\Booleanop{\supseteq}}
\def\Bsupset{\Booleanop{\supset}}
\def\Bsymmdiff{\Booleanop{\symmdiff}}
\def\Bcapshort{\Booleanopshort{\cap}}
\def\Bcupshort{\Booleanopshort{\cup}}
\def\Booleanopshort#1{
   \ifamstexsupported{\mathchoice
    {\hbox{\scriptfont2=\eightsy$\scriptstyle#1$}}
    {\hskip-1pt\hbox{\scriptfont2=\eightsy$\scriptstyle#1$}}
    {{\scriptstyle#1}}
    {{\scriptscriptstyle#1}}}
   \else{\mathchoice{{\scriptstyle#1}}{{\scriptstyle#1}}
    {{\scriptstyle#1}}{{\scriptscriptstyle#1}}}
    \fi}
\def\Bsetminusshort{\Booleanopshort{\setminus}}
\def\BSint{\mathchoice{\Bbb B\mskip-5mu}
                  {\Bbb B\mskip-3mu}
                  {\Bbb B\mskip-2mu}
                  {\Bbb B\mskip-2mu}\Sint}
%\def\Bsubseteqshort{\hskip.1em\Booleanopshort{\subseteq}\hskip.1em}
\def\Bsubseteqshort{\hskip.1em\Booleanopshort{\subseteq}}
\def\Bsubsetshort{\Booleanopshort{\subset}}
\def\Bsupseteqshort{\Booleanopshort{\supseteq}}
\def\Bsymmdiffshort{\Booleanopshort{\symmdiff}}

\def\closeBooleanop#1{
   \ifamstexsupported{\mathchoice
    {\hbox{\scriptfont2=\eightsy$\scriptstyle#1$}}
    {\hbox{\scriptfont2=\eightsy$\scriptstyle#1$}}
    {{\scriptstyle#1}}
    {{\scriptscriptstyle#1}}}
   \else{\mathchoice{{\scriptstyle#1}}{{\scriptstyle#1}}
    {{\scriptstyle#1}}{{\scriptscriptstyle#1}}}
    \fi}
\def\closeBcup{\closeBooleanop{\cup}}
\def\closeBcap{\closeBooleanop{\cap}}
\def\closeBsetminus{\closeBooleanop{\setminus}}
\def\closeBsymmdiff{\closeBooleanop{\symmdiff}}

\def\biggerHeint{\textfont3=\twelveex
   \raise 0.6pt\hbox to 0pt{\hskip 2.6pt{\sevenrm H}\hss}
   \intop\nolimits}
\def\biggerPfint{\textfont3=\twelveex
   \raise 0.6pt\hbox to 0pt{\hskip 1.3pt{\sevenrm Pf}\hss}
   \intop\nolimits\hskip 1.2pt}
\def\biggerSint{\textfont3=\twelveex
   \raise 0.6pt\hbox to 0pt{\hskip 3.6pt{\sevenrm S}\hss}
   \intop\nolimits}
\def\bu{\mathop{\text{bu}}}
\def\Bvalue#1{\mathchoice
   {\hbox{$[\![#1]\!]$}}
   {\hbox{$[\![#1]\!]$}}
   {\hbox{$\scriptstyle[\![#1]\!]$}}
   {\hbox{$\scriptscriptstyle[\![#1]\!]$}}}
\def\cadlag{c\`adl\`ag}
\def\CalBa{\Cal{B}\eurm{a}}
   %note: neither  \cal  nor  \eusm  has lower case
\def\CalFr{\Cal F_{\text{Fr}}}
\def\callal{c\`all\`al}
\def\CalNwd{\Cal{N}\hskip-1.2pt\eurm{w}\hskip-.8pt\eurm{d}}

\def\CalRbg{\Cal{R}\mskip-1mu\eurm{b}\mskip-1mu\eurm{g}}
\def\CalSmz{\Cal{S}\mskip-2mu\eurm{mz}}
\def\CalUn{\Cal{U}\eurm{n}}
\def\capacity{\mathop{\text{cap}}\nolimits}
\def\Caratheodory{Cara\-th\'eo\-dory}
\def\Cdlg{C_{\text{dlg}}}
\def\cff{\cf\mskip1mu}
\def\checkpmb#1{\check{\pmb #1}}
\def\ci{\mathop{\text{ci}}}
\def\cl{\mathop{\text{cl}}}
\def\Clll{\hbox{$C$
  \hskip-4.5pt\raise4pt\hbox{\char'23
  \hskip-1.6pt\vrule height6.5pt width.5pt depth0pt
  \hskip.7pt\vrule height6.5pt width.5pt depth0pt
  \hskip-2.5pt\lower5.4pt\hbox{\char'23}}}}
\def\clMsimp{\overline{M}_{\text{simp}}}
\def\closeuparrow{\mathchar"0222}
\def\closedownarrow{\mathchar"0223}
\def\clstar{\text{cl*}}
\def\cov{\mathop{\text{cov}}}
\def\covar#1#2{\mathchoice
  {[#1{\lower4pt\hbox{${\scriptstyle *}\atop{\vrule height 18pt depth -10pt}$}}#2]}
  {[#1{*\atop{\vrule height 8pt depth 0pt}}#2]}
  {[#1{*\atop{\vrule height 6pt depth 0pt}}#2]}
  {[#1{*\atop{\vrule height 6pt depth 0pt}}#2]}}
\def\covSh{\mathop{\text{cov}}\nolimits_{\text{Sh}}}
\def\cdownarrow{c\mskip2mu^{\downarrow}}
\def\cycle#1{(\overleftarrow{#1})}
\def\ddaggerloc{$(\ddagger)_{\text{loc}}$}
\def\ddownarrow{d\mskip2mu^{\downarrow}}
%\def\DiniD{\setover{
%  \hbox{\hskip.1em\vrule height-1pt width.5em depth1.4pt\hfil}}D}
%\def\Dinid{\hskip.1em
%  \hbox{\hskip.05em\vrule height-1.3pt width.5em depth1.7pt}\hskip-.65em D}
\def\Dinid{\hbox{$D$\hskip-8.3pt\vrule height-1pt width7pt depth1.85pt}
   \hskip1pt}
\def\DiniD{\hbox{$D$\hskip-8.3pt\vrule height9.2pt width7ptdepth-8.35pt}
   \hskip1pt}

\def\discrcenter#1#2{\ifdim\pagewidth>#1{#2}\else{

  \Centerline{#2}

  \noindent}\fi}
    %note:  in \discrcenter, make sure that #2 ends in a space
    % (if you want one to appear) and that the continuation
    % follows without
    % a space after the final } of #2, so that it will be
    % truly unindented
\def\discrpage{\ifnum\stylenumber=1{\frnewpage}  %redefined in mt.tex
        \else\ifnum\stylenumber=3{\frnewpage}
        \else\ifnum\stylenumber=5{\frnewpage}
        \else\wheader{}{18}{14}{2}{72pt}\fi\fi\fi}
\def\displaycause#1{\noalign{\noindent (#1)}}
\def\Displaycause#1{\cr\displaycause{#1}&}
\def\dotproduct{\hskip.1em\tbf{.}\hskip.1em}
\def\doubleheadrightarrow{\rightarrow\mskip-12mu\rightarrow}
\long\def\doubleinset#1{\inset{\inset{\parindent=-20pt #1}}}
\def\duparrow{d\mskip2mu^{\uparrow}}
\def\dvAformerly#1{\discrversionA{\footnote{Formerly #1.}}{}}
\def\dvAnew#1{\discrversionA{\footnote{New #1.}}{}}
\def\dvArevised#1{\discrversionA{\footnote{Revised #1.}}{}}
\def\dvrocolon#1{\dvro{#1: }{#1}}
\def\eightVVdash{\mskip5mu\vrule height 6pt depth 2pt width 0.4pt
  \mskip2mu\vrule height 6pt depth 2pt width 0.4pt
  \vrule height 2.25pt depth -1.8pt width 3.2pt\mskip2mu}
\def\embedsinto{\mskip5mu\raise1.5pt\hbox{$\subset$}\mskip-15mu
   \lower3.5pt\hbox{$\rightarrow$}\mskip5mu}
\def\energy{\mathop{\text{energy}}\nolimits}
%\eqalign:  see \S478 for breaks during \eqalign
\def\equivGT{\equiv_{\text{GT}}}
\def\equivT{\equiv_{\text{T}}}
\def\Eta{\text{H}}
\def\Expn{\Bbb E}
\long\def\Extract{\oldfootnote{}{
   \noindent{\eightit
       \ifresultsonly{Extract from {\eightsmc Measure Theory,}
          results-only version,
          by {\eightsmc D.H.Fremlin,}  University of Essex,
Colchester.}
       \else\ifwithproofs{Extract from {\eightsmc Measure Theory,}
          by {\eightsmc D.H.Fremlin,}  University of Essex,
Colchester.}
       \else{Extract from {\eightsmc Measure Theory,} abridged version,
          by {\eightsmc D.H.Fremlin,}  University of Essex,
Colchester.}
       \fi\fi
   This material is copyright.   It is issued under the terms of the
   Design Science License as published in
   {\eighttt http://dsl.org/copyleft/dsl.txt}.
   This is a development version and the
   source files are not permanently archived, but current versions are
normally
   accessible through
   {\eighttt https://www1.essex.ac.uk/maths/people/fremlin/mt.htm}.
   For further information contact {\eighttt david\@fremlin.org}.}
   }}
\def\Fn{\mathop{\text{Fn}}\nolimits}
\def\FN{\mathop{\text{FN}}\nolimits}
\def\formerly#1{\footnote{Formerly #1.}}
  %use only once in a chapter, never in same volume as reference
\def\fraction#1{\mathchoice{\hbox{$<$}#1\hbox{$>$}}
     {\hbox{$<$}#1\hbox{$>$}}
     {\hbox{$\scriptstyle<$}#1\hbox{$\scriptstyle>$}}
     {\hbox{$\scriptscriptstyle<$}#1\hbox{$\scriptscriptstyle>$}}}
\def\frakmctbl{\frak m_{\text{countable}}}
\def\frmedcirc{{\raise 2.5pt\hbox{$_{\bigcirc}$}}}
\def\grad{\mathop{\text{grad}}\nolimits}
\def\hatpmb#1{\hat{\pmb{#1}}}
\def\hattilde#1{\skew2\hat{\tilde#1}}
\def\hattildepmb#1{\skew1\hat{\tildepmb#1}}
\def\hcov{\mathop{\text{hcov}}\nolimits}
\def\Heint{\mathchoice{\hbox to 0pt{\hskip 3.1pt H\hss}}
     {\raise 0.6pt\hbox to 0pt{\hskip 1.9pt{\sevenrm H}\hss}}
     {\raise 0.4pt\hbox to 0pt{\hskip 1.5pt{\fiverm H}\hss}}
     {\raise 0.4pt\hbox to 0pt{\hskip 1.5pt{\fiverm H}\hss}}
     \intop\nolimits}
\def\high#1{\lceil#1\rceil}
\def\highoverline#1{\overline{\vrule height6pt width0pt depth 0pt#1}}
\def\Hint#1{({\it Hint\/}: #1)}
\def\hL{\mathop{\text{hL}}}
\def\Hmp{\text{Hom}_{\bar\mu,\pi}}
\def\hp{\mathop{\text{hp}}\nolimits}
\def\hwtr{hereditarily weakly $\theta$-{\vthsp}refinable}
\def\iiv{ii_{\pmb{v}}}
\def\iivu{\iiv(\pmb{u})}
\def\imp{inverse-{\vthsp}measure-{\vthsp}preserving}
\def\innerprod#1#2{(#1|#2)}
\def\intstar{\text{int*}}
\def\CalIS{\Cal I(\Cal S)}
\def\jump{\mathop{\text{jump}}\nolimits}
\def\Jump{\mathop{\text{Jump}}\nolimits}
\def\Krein{Kre\v{\i}n}
\def\Lambdapv{\Lambda_{\text{pv}}}
\def\leRB{\le_{\text{RB}}}
\def\leRK{\le_{\text{RK}}}
\def\link{\mathord{\text{link}}}
\def\locmafp{\widehat{\otimes}_{\text{loc}}}
\def\Locmafp{\widehat{\bigotimes}^{\text{loc}}}
\def\low#1{\lfloor#1\rfloor}
\def\lti{left-\vthsp translation-\vthsp invariant}
\def\magnitude{\mathop{\text{mag}}}
\def\Mah{\mathop{\text{Mah}}}
\def\Mahcr{\Mah_{\text{cr}}}
\def\MahcrR{\Mah_{\text{crR}}}
\def\MahqR{\Mah_{\text{qR}}}
\def\MahR{\Mah_{\text{R}}}
\def\MahspcrR{\Mah_{\text{spcrR}}}
\def\Malm{M_{\text{alm}}}
\def\Marik{Ma\v r\'\i k} 
\def\Mbv{M_{\text{bv}}}
\def\Mcdlg{M_{\text{cdlg}}}
\def\Mdm{M_{\text{D-M}}}
\def\med{\mathop{\text{med}}\nolimits}
\def\Mfa{M_{\text{fa}}}
\def\Mhsr{\mathop{\text{Mhsr}}\nolimits}
\def\Mjf{\mathop{M_{\text{j-f}}}\nolimits}
\def\Mlbv{M_{\text{lbv}}}
\def\Mlob{M_{\text{lo-b}}}
\def\Mlns{M_{\text{ln-s}}}
\def\Mmo{M_{\text{mo}}}
\def\Mlmo{M_{\text{lmo}}}
\def\Mns{M_{\text{n-s}}}
\def\Mnsu{M_{\text{n-s}}^{\uparrow}}
%\def\Mnsu{M_{\text{n-s}\uparrow}}
\def\MMah{M_{\text{Mah}}}
\def\Mob{M_{\text{o-b}}}
\def\Mpob{M_{\text{po-b}}}
\def\Mpv{M_{\text{pv}}}
\def\MSi{M_{\text{S-i}}}
\def\Msimp{M_{\text{simp}}}
\def\Mth{Ma\-ha\-ram-{\vthsp}type-{\vthsp}homogeneous}
\def\Mvlm{M_{\text{vlm}}}
\def\negquad{\hskip-1em\relax}
\def\NN{\BbbN^{\Bbb N}}
\def\non{\mathop{\text{non}}}
\def\notBooleanop#1{
   \ifamstexsupported{\mathchoice
{\hbox{\hskip.2em$\not\scriptfont2=\eightsy\scriptstyle#1$\hskip.2em}}
{\hbox{\hskip.2em$\not\scriptfont2=\eightsy\scriptstyle#1$\hskip.2em}}
    {{\not\scriptstyle#1}}
    {{\not\scriptscriptstyle#1}}}
   \else{\mathchoice{{\not\scriptstyle#1}}{{\not\scriptstyle#1}}
    {{\not\scriptstyle#1}}{{\not\scriptscriptstyle#1}}}
    \fi}
\def\notBsubseteq{\notBooleanop{\subseteq}}
\def\notBsupseteq{\notBooleanop{\supseteq}}
\def\notVVdash{\not\mskip4mu\vrule height 7.5pt depth 2.5pt width 0.5pt
  \mskip2.5mu\vrule height 7.5pt depth 2.5pt width 0.5pt
  \vrule height 2.75pt depth -2.25pt width 4pt\mskip-2mu}
\def\nuprime{\nu^{\mskip0.7mu\prime}} %better  \nu'  in superscripts
\def\nw{\mathop{\text{nw}}}
\def\Osclln{\mathop{\text{Osclln}}\nolimits}
\def\otp{\mathop{\text{otp}}}
\def\partstar{\mathchoice{\partial\hbox{*}}
     {\partial\hbox{*}}
     {\partial\hbox{\sevenrm *}}
     {\partial\hbox{\fiverm *}}}
\def\per{\mathop{\text{per}}}
\def\Pfint{\mathchoice{\hbox to 0pt{\hskip 3.1pt Pf\hss}}
     {\raise 0.6pt\hbox to 0pt{\hskip 1.3pt{\sevenrm Pf}\hss}}
     {\raise 0.4pt\hbox to 0pt{\hskip 1.0pt{\fiverm Pf}\hss}}
     {\raise 0.4pt\hbox to 0pt{\hskip 1.0pt{\fiverm Pf}\hss}}
     \intop\nolimits
     \mathchoice{\hskip 2pt}{\hskip 1.2pt}{\hskip 1pt}{\hskip 1pt}}
\def\Pou{\mathop{\text{Pou}}\nolimits}
\def\prforc{\preccurlyeq_{\text{forcing}}}
\def\prGT{\preccurlyeq_{\text{GT}}}
\def\prT{\preccurlyeq_{\text{T}}}
\def\pssqa{power set $\sigma$-quotient algebra}
\def\qm{quasi-{\vthsp}measurable}
\def\Reverse#1{\setover
   {\lower0.4ex\hbox{$\scriptscriptstyle\leftrightarrow$}}{#1}}
\def\rhoKR{\rho_{\text{KR}}} 
\def\rhotv{\rho_{\text{tv}}} 
\def\Rint{\mathchoice{\hbox to 0pt{\hskip 3.1pt R\hss}}
     {\raise 0.6pt\hbox to 0pt{\hskip 1.9pt{\sevenrm R}\hss}}
     {\raise 0.4pt\hbox to 0pt{\hskip 1.5pt{\fiverm R}\hss}}
     {\raise 0.4pt\hbox to 0pt{\hskip 1.5pt{\fiverm R}\hss}}
     \intop\nolimits}
\def\RO{\text{RO}}
\def\rti{right-{\vthsp}translation-{\vthsp}invariant}
\def\rvm{real-{\vthsp}valued-{\vthsp}measurable}
\def\sat{\mathop{\text{sat}}}
\def\sfamilysS#1{\family{\sigma}{\Cal S}{#1_{\sigma}}}
\def\sfcc{$\sigma$-finite-cc}
\def\sgn{\mathop{\text{sgn}}}
\def\shr{\mathop{\text{shr}}\nolimits}
\def\Sii{\text{S}ii}
\def\Siiv{\Sii_{\pmb v}}
\def\Sint{\mathchoice{\hbox to 0pt{\hskip 4.1pt\ninerm S\hss}} 
     {\raise 0.6pt\hbox to 0pt{\hskip 3.0pt{\sevenrm S}\hss}}
     {\raise 0.4pt\hbox to 0pt{\hskip 1.5pt{\fiverm S}\hss}} %unchecked
     {\raise 0.4pt\hbox to 0pt{\hskip 1.5pt{\fiverm S}\hss}} %unchecked
     \intop\nolimits}
\def\Sinteightpoint{\raise 0.6pt\hbox to 0pt{\hskip 2.3pt{\fiverm S}\hss}
     \intop\nolimits}
\def\SL{\text{SL}}
\def\sm{semi-martingale} %hyphen to remind me that it's the "virtual" kind
\def\Sous{\Cal S\text{ous}}  %used in vol 6
\def\spread{\mathop{\text{spr}}}
\def\ssplrarrow{^{\scriptscriptstyle\leftrightarrow}}
\def\ssptilde{{\raise1.2ex\hbox{$\scriptscriptstyle\sim$}}}
%\def\sspvec{{\raise1.2ex\hbox{$\scriptscriptstyle\rightarrow$}}}
\def\sspvec{\mskip3mu\vec{\null}\mskip5mu}
\def\Sti{\mathop{\text{Sti}}\nolimits}
\def\strbar{^{\scriptscriptstyle\pmb{-}}}
\def\strplus{^{\scriptscriptstyle\pmb{+}}}
\def\strprime{^{\mathchoice{\mskip1.5mu\pmb{\prime}}
                  {\mskip1.5mu\pmb{\prime}}
                  {\mskip1.5mu\pmb{\prime}}
                  {\mskip1mu\pmb{\prime}}}}
%\def\strprime{\,\pmb{\prime}}
\def\supp{\mathop{\text{supp}}}
\def\tausm{\tau_{\sigma\text{-m}}}
\def\tbfoneminus#1{\rlap{$\tbf 1^{(#1)}$}\rlap{\hskip7pt$_<$}
   \phantom{\tbf 1^{(#1)}}}
%\def\tbfoneminus#1{\rlap{$\tbf 1^{(#1)}$}\rlap{\hskip7pt$_-$}
%   \phantom{\tbf 1^{(#1)}}}
\def\tensorhat{\widehat{\otimes}}
\def\Tensorhat{\widehat{\bigotimes}}
\def\tensorhatsigma{\widehat{\otimes}_{\sigma}}
\def\tildeClll{\hbox{$\tilde C$
  \hskip-4.5pt\raise4pt\hbox{\char'23
  \hskip-1.6pt\vrule height6.5pt width.5pt depth0pt
  \hskip.7pt\vrule height6.5pt width.5pt depth0pt
  \hskip-2.5pt\lower5.4pt\hbox{\char'23}}}}
\def\tildehat#1{\skew2\tilde{\hat#1}}
\def\tildehatpmb#1{\tilde{\hatpmb#1}}
\def\tildepmb#1{\tilde{\pmb{#1}}}
\def\tinyplus{{\vrule height 1.6pt depth -1.3pt width 2.5pt}
    \hskip-1.3pt{\vrule height 2.6pt depth -0.1pt width 0.3pt}}
\def\Tr{\mathop{\text{Tr}}}
\def\trs{^{\top}\hskip-1pt}
\def\triplepc#1#2#3{(#1,#2,\hbox{$<$}#3)}
\def\TSi{\frak T_{\text{S-i}}}
\def\undphi{\underline{\phi}{}}
\def\undpsi{\underline{\psi}{}}
\def\undtheta{\underline{\theta}{}}
\def\updown{\mathchoice
  {\hbox{\hskip-1.5pt\char'23
    \hskip-1.7pt\vrule height6.5pt width.7pt depth1.3pt
    \hskip.9pt\vrule height6.5pt width.7pt depth1.3pt
    \hskip-3.3pt\lower6.7pt\hbox{\char'23}\hskip-1.4pt}}
  {\hbox{\hskip-2pt\char'23
    \hskip-1.7pt\vrule height6.5pt width.7pt depth1.3pt
    \hskip.9pt\vrule height6.5pt width.7pt depth1.3pt
    \hskip-2.5pt\lower6.7pt\hbox{\char'23}\hskip-1.4pt}}
  {\hbox{\hskip-2pt\char'23  %OK for subscript, fails for superscript
    \hskip-1.7pt\vrule height6.5pt width.7pt depth0pt
    \hskip.9pt\vrule height6.5pt width.7pt depth0pt
    \hskip-2.5pt\lower5.4pt\hbox{\char'23}\hskip-1.4pt}}
  {\hbox{\hskip-2pt\char'23  %not tested
    \hskip-1.7pt\vrule height6.5pt width.7pt depth1.3pt
    \hskip.9pt\vrule height6.5pt width.7pt depth1.3pt
    \hskip-2.5pt\lower6.7pt\hbox{\char'23}\hskip-1.4pt}}}
\def\updownarrows{\uparrow\mskip-4mu\downarrow}
\def\updowncgt{$\updown$-convergent}
\def\upr{\mathop{\text{upr}}\nolimits}
\def\ureg{\mathop{\text{ureg}}}

\def\Var{\mathop{\text{Var}}\nolimits}

\def\varaction{\mathchoice{\lower-.2ex\hbox{$\smallcirc$}}
                       {\lower-.2ex\hbox{$\smallcirc$}}
                       {\ssbullet}
                       {\ssbullet}}
\def\varcheck#1{{\setover
       {\lower 0.4ex\hbox{$\scriptscriptstyle\vee$}}{#1}}}
\def\varcheckf{{\setover{\,\lower 0.4ex
              \hbox{$\scriptscriptstyle\vee$}}{f}}}
\def\varchecki{\mskip-3mu\text{\it{\accent"14{\hbox{\hskip-3.5pt\i}}}}\mskip2mu}
\def\varspcheck{^{\scriptscriptstyle\vee}}
%\def\varspcheck{{\setover{\scriptscriptstyle\vee}{\phantom{g}}}}
%\def\varspcheckf{{\setover{\lower 0.4ex
%              \hbox{$\scriptscriptstyle\vee$}}{\phantom{f}}}}
\def\varhat#1{{\setover{\lower 0.4ex
              \hbox{$\scriptscriptstyle\wedge$}}{#1}}}
\def\varhatf{{\setover{\,\lower 0.4ex
              \hbox{$\scriptscriptstyle\wedge$}}{f}}}
\def\varinnerprod#1#2{#1\dotproduct#2}
\def\varsphat{^{\scriptscriptstyle\wedge}}
% or use  \hbox{\^{\null}}
%\def\varsphat{{\setover{\scriptscriptstyle\wedge}{\phantom{g}}}}
%\def\varsphatf{{\setover{\lower 0.4ex
%              \hbox{$\scriptscriptstyle\wedge$}}{\phantom{f}}}}
\def\varBspcheck{\mathchoice
  {\hskip-2pt
  {\fourteenrm\hbox{\lower2pt\hbox{\v{\null}}}}\hskip-1pt}
  {\hskip-2pt
  {\fourteenrm\hbox{\lower2pt\hbox{\v{\null}}}}\hskip-1pt}
  {\hskip-2.5pt
  {\fourteenrm\hbox{\lower4pt\hbox{\v{\null}}}}\hskip-1pt}
  {\hskip-2.5pt
  {\fourteenrm\hbox{\lower4pt\hbox{\v{\null}}}}\hskip-1pt}}
\def\varCspcheck{\mathchoice
  {\hskip-1pt\check{\phantom x}}   %not tested
  {\hskip-1pt\check{\phantom x}}
  {\hskip-1.2pt\check{\phantom x}}
  {\hskip-1.5pt\check{\phantom x}}}  %not tested
\def\varCsphat{\mathchoice
  {\hskip-1pt\hat{\phantom x}}   %not tested
  {\hskip-1pt\hat{\phantom x}}
  {\hskip-1.2pt\hat{\phantom x}}  %not tested
  {\hskip-1.5pt\hat{\phantom x}}}  %not tested
\def\VeqL{$V\hbox{$=$}L$}
\def\VVdash{\mskip5mu\vrule height 7.5pt depth 2.5pt width 0.5pt
  \mskip2.5mu\vrule height 7.5pt depth 2.5pt width 0.5pt
  \vrule height 2.75pt depth -2.25pt width 4pt\mskip2mu}
\def\VVdP{\VVdash_{\Bbb P}}
\def\VVdPk{\VVdash_{\Bbb P_{\kappa}}}
\def\wdistr{\mathop{\text{wdistr}}}
\def\weight{\mathop{\text{wt}}}
\def\w1lim{\mathop{\text{w1lim}}}
\def\wsid{weakly $(\sigma,\infty)$-{\vthsp}distributive}
\def\1lim{\mathop{1\mskip-1.5mu\text{lim}}}
\def\2vm{two-{\vthsp}valued-{\vthsp}measurable}

%Cara\-th\'e\-o\-dory

\def\chaptername{}
\def\sectionname{}
\def\headlinesectionname{}

\loadeusm\loadeurm
\Loadfourteens
\Loadfont{\ninerm=cmr9}

%note:  $..\cmmnt{=..}..$  is better as  \cmmnt{\mskip5mu
 
  
%----------------------------------------------------------%  
  
\smallprint  
%\USsmallprint  
%\noproofs  
%\resultsonly  
%\atUEssex  
\usegraphicx
  
\Extract  
  
%----------------------------------------------------------%  

\frfilename{mt55.tex}
\versiondate{7.1.15}
\copyrightdate{2007}

\def\chaptername{Possible worlds}

\newchapter{55}

In my original plans for this volume, I hoped to cover the most important
consistency proofs relating to undecidable questions in measure theory.
Unhappily my ignorance of forcing means that for the majority of results I
have nothing useful to offer.   I have therefore restricted my account to
the very narrow range of ideas in which I feel I have achieved
some understanding beyond what I have read in the standard texts.

For a measure theorist, by far the most important forcings are those
of `adding random reals'.   I give three sections (\S\S552-553 and 555)
to these.
Without great difficulty, we can determine the behaviour of the cardinals
in Cicho\'n's diagram (552B, 552C, 552F-552I), %552F 552G 552H 552I
at least if many random reals are
added.   Going deeper, there are things to be said about outer
measure and Sierpi\'nski sets (552D, 552E),
and extensions of Radon measures
(552N).   In the same section I give a version of the fundamental
result that simple iteration of random real forcings gives random
real forcings (552P).   In \S553 I collect results which are connected with
other topics dealt with above (Rothberger's property, precalibers,
ultrafilters, cellularity,
trees, medial limits, universally measurable sets) and in
which the arguments seem to me to develop properties of measure algebras
which may be of independent interest.
In preparation for this work, and also for \S554, I start with a section
(\S551) devoted to a rather technical
general account of forcings with quotients of
$\sigma$-algebras of sets, aiming to find effective representations of
names for points, sets, functions, measure algebras and filters.

Very similar ideas can also take us a long way with Cohen real forcing
(\S554).   Here I give little more than obvious parallels to the first part
of \S552, with an account of Freese-Nation numbers sufficient to support
Carlson's theorem that a Borel lifting for Lebesgue measure can exist when
the continuum hypothesis is false (554I).

One of the most remarkable applications of random reals is in Solovay's
proof that if it is consistent to suppose that there is a
two-valued-measurable cardinal, then it is consistent to suppose that there
is an atomlessly-measurable cardinal (555D).   By taking a bit of trouble
over the lemmas, we can get a good deal more, including the corresponding
theorem relating supercompact cardinals to the normal measure axiom (555N);
and similar techniques show the possibility of interesting \pssqa s
(555G, 555K).

I end the chapter with something quite different (\S556).
A familiar phenomenon
in ergodic theory is that once one has proved a theorem for ergodic
transformations one can expect, possibly at the cost of substantial effort,
but without having to find any really new idea, a corresponding result
for general measure-preserving transformations.
There is more than one way to look at this, but here I present a method in
which the key step, in each application,
is an appeal to the main theorem of forcing.   A similar approach gives a
description of the completion of the asymptotic density algebra.   The
technical details take up a good deal of space, but are based on the same
principles as those in \S551, and are essentially straightforward.

\discrpage

\frfilename{mt551.tex}
\versiondate{2.12.13}
\copyrightdate{2007}

\def\chaptername{Possible worlds}
\def\sectionname{Forcing with quotient algebras}

\def\BbbPk{\Bbb P_{\kappa}}
\def\VVdPk{\VVdash_{\Bbb P_{\kappa}}}


\newsection{551}

In preparation for the discussion of random real forcing in the next two
sections, I
introduce some techniques which can be applied whenever a forcing
notion is described in terms of a Loomis-Sikorski representation of
its regular
open algebra.   The first step is just a translation of the correspondence
between names for real numbers in the forcing language and members of
$L^0(\RO(\Bbb P))$, as described in 5A3L, when $L^0(\RO(\Bbb P))$ can be
identified with a quotient of a space $L^0(\Sigma)$ of measurable
functions.   More care is needed,
but we can find a similar formulation of names for members
of $\{0,1\}^I$ for any set $I$ (551C).   Going a step farther,
it turns out that there are very useful descriptions
of Baire subsets of $\{0,1\}^I$ (551D-551F), Baire
measurable functions (551N),
the usual measure on $\{0,1\}^I$ (551I-551J) and its measure
algebra (551P).   In some special cases, these methods can be used to
represent iterated forcing notions (551Q).
I end with a construction for a forcing
extension of a filter on a countable set (551R).

\leader{551A}{Definition (a)} A {\bf measurable space with negligibles} is
a triple $(\Omega,\Sigma,\Cal I)$ where $\Omega$ is a set, $\Sigma$ is a
$\sigma$-algebra of subsets of $\Omega$ and $\Cal I$ is a $\sigma$-ideal of
subsets of $\Omega$ generated by $\Sigma\cap\Cal I$.
In this case $\frak A=\Sigma/\Sigma\cap\Cal I$ is a
Dedekind $\sigma$-complete Boolean algebra\cmmnt{ (314C)}.

\cmmnt{In this context I will use the phrase
`$\Cal I$-almost everywhere' to mean `except on a
set belonging to $\Cal I$'.}

\spheader 551Ab\cmmnt{ I will say that} $(\Omega,\Sigma,\Cal I)$ is
{\bf non-trivial} if $\Omega\notin\Cal I$, so that $\frak A\ne\{0\}$.
In this case, the forcing notion $\Bbb P$
{\bf associated} with $(\Omega,\Sigma,\Cal I)$ is
$(\frak A^+,\Bsubseteqshort,\Omega^{\ssbullet},\downarrow)$\cmmnt{
(5A3M)}.   If $\frak A$ is Dedekind complete we can identify
$\frak A$ with the regular open algebra
$\RO(\Bbb P)$\cmmnt{ (514Sb, 5A3M)}.

\spheader 551Ac\cmmnt{ I will say that} $(\Omega,\Sigma,\Cal I)$ is {\bf
$\omega_1$-saturated} if $\Sigma\cap\Cal I$ is $\omega_1$-saturated in
$\Sigma$\cmmnt{ in the sense of 541A, that is, if there is no
uncountable disjoint family in
$\Sigma\setminus\Cal I$, that is, if $\frak A$ and $\Bbb P$ are ccc}.
In this case, $\frak A$ is Dedekind complete\cmmnt{ (316Fa, 541B)}.

\spheader 551Ad\cmmnt{ I will say that} $(\Omega,\Sigma,\Cal I)$ is
{\bf complete} if $\Cal I\subseteq\Sigma$\cmmnt{ (cf.\ 211A)}.

\cmmnt{\medskip

\noindent{\bf Remark} For an account of the general theory of measurable
spaces with negligibles, see {\smc Fremlin 87}.
}

\leader{551B}{Definition} Let $(\Omega,\Sigma,\Cal I)$ be a
non-trivial measurable space
with negligibles with a Dedekind complete quotient algebra $\frak A$,
and $\Bbb P$ its associated forcing notion.
\cmmnt{Recall from 364Ib
that }$L^0(\frak A)$ can be regarded as a quotient of
the space of $\Sigma$-measurable functions from $\Omega$ to
$\Bbb R$.   If $h:\Omega\to\Bbb R$ is $\Sigma$-measurable, write
$\vec h=(h^{\ssbullet})\sspvec$ where $h^{\ssbullet}$ is the equivalence
class of $h$ in $L^0(\frak A)$, identified with $L^0(\RO(\Bbb P))$, and
$(h^{\ssbullet})\sspvec$ is the $\Bbb P$-name for a real number as
defined in 5A3L.   Then

\Centerline{$\VVdP\,\vec h$ is a real number,}

\noindent and for any $\alpha\in\Bbb Q$

\Centerline{$\Bvalue{\vec h>\check\alpha}
=\Bvalue{(h^{\ssbullet})\sspvec>\check\alpha}
=\Bvalue{h^{\ssbullet}>\alpha}
=\{\omega:h(\omega)>\alpha\}^{\ssbullet}$.}

\noindent From 5A3Lc, we see that if $h_0$, $h_1$ are
$\Sigma$-measurable real-valued functions on $\Omega$, then

\Centerline{$\VVdP\,(h_0+h_1)\sspvec=\vec h_0+\vec h_1$,
$(h_0\times h_1)\sspvec=\vec h_0\times\vec h_1$,}

\noindent and that if $\sequencen{h_n}$ is a sequence of measurable
functions with limit $h$,

\Centerline{$\VVdP\,\vec h=\lim_{n\to\infty}\vec h_n$ in
$\Bbb R$.}

\leader{551C}{Definition} Let $(\Omega,\Sigma,\Cal I)$ be a
non-trivial measurable space
with negligibles with a Dedekind complete quotient algebra $\frak A$,
and $\Bbb P$ its associated forcing notion.

\spheader 551Ca If $f:\Omega\to\{0,1\}$ is $\Sigma$-measurable, let
$\vec f$ be the $\Bbb P$-name

\Centerline{$\{(\varchecki,f^{-1}[\{i\}]^{\ssbullet}):i\in\{0,1\}$,
$f^{-1}[\{i\}]\notin\Cal I\}$.}

\noindent\cmmnt{Then }$\VVdP\,\vec f\in\{0,1\}$ and
$\Bvalue{\vec f=\varchecki}=f^{-1}[\{i\}]^{\ssbullet}$ for both $i$.
\cmmnt{(I will try always
to make it clear when this definition of $\vec f$ is intended to overrule
the definition in 551B;  but we see from 551Xf that any confusion is
unlikely to matter.)}

Observe that if a $\Bbb P$-name $\dot x$ and $p\in\frak A^+$ are
such that $p\VVdP\,\dot x\in\{0,1\}$, then there is a measurable
$f:\Omega\to\{0,1\}$ such that $p\VVdP\,\dot x=\vec f$\prooflet{;
take $f=\chi E$ where $E\in\Sigma$ is such that
$E^{\ssbullet}=\Bvalue{\dot x=1}$ in $\frak A$}.

\spheader 551Cb Now let $I$ be any set, and $f:\Sigma\to\{0,1\}^I$ a
$(\Sigma,\CalBa_I)$-measurable function, where
$\CalBa_I=\CalBa(\{0,1\}^I)$ is
the Baire $\sigma$-algebra of $\{0,1\}^I$\cmmnt{,
that is, the $\sigma$-algebra of
subsets of $\{0,1\}^I$ generated by sets of the form
$\{x:x\in\{0,1\}^I$, $x(i)=1\}$ for $i\in I$ (4A3Na)}.   For each
$i\in I$, set $f_i(\omega)=f(\omega)(i)$ for $\omega\in\Omega$;
then\cmmnt{ $f_i:\Omega\to\{0,1\}$ is measurable, so} we have a
$\Bbb P$-name $\vec f_i$\cmmnt{ as in (a)}.   Let $\vec f$ be
the $\Bbb P$-name
$\{(\family{i}{\check I}{\vec f_i},\Bbbone)\}$\cmmnt{ (interpreting
the subformula $\family{i}{\check I}{\ldots}$ in the forcing
language, of course, by the convention of 5A3Fb)}.   \cmmnt{Then}

\Centerline{$\VVdP\,\vec f\in\{0,1\}^{\check I}$,}

\noindent and for every $i\in I$

\Centerline{$\VVdP\,\vec f(\varchecki)=\vec f_i$.}

\spheader 551Cc In the other direction, if a $\Bbb P$-name $\dot x$
and $p\in\frak A^+$ are such that
$p\VVdP\,\dot x\in\{0,1\}^{\check I}$, then for each $i\in I$ we have a
$\Bbb P$-name $\dot x(\varchecki)$ and a measurable
$f_i:\Omega\to\{0,1\}$ such that
$p\VVdP\,\dot x(\varchecki)=\vec f_i$;  setting
$f(\omega)=\familyiI{f_i(\omega)}$ for $\omega\in\Omega$, $f$ is
$(\Sigma,\CalBa_I)$-measurable and $p\VVdP\,\vec f=\dot x$.

\cmmnt{\spheader 551Cd I ought to remark that there is a problem with
equality for the $\Bbb P$-names $\vec f$.
If, in the context of (b)-(c) above, we have two
$(\Sigma,\CalBa_I)$-measurable functions $f$ and $g$, and if
$p\in\frak A^+$, then

$$\eqalign{p\VVdP\,\vec f=\vec g
&\iff\text{ for every }i\in I,\,p\VVdP\,\vec f_i=\vec g_i\cr
&\iff\text{ for every }i\in I,\,
p\Bsubseteq\{\omega:f_i(\omega)=g_i(\omega)\}^{\ssbullet}
  \text{ in }\frak A.\cr}$$

\noindent In particular, $\VVdP\,\vec f=\vec g$ iff $f_i=g_i$
$\Cal I$-a.e.\ for
every $i\in I$.   If $I$ is uncountable we can easily have
$\VVdP\,\vec f=\vec g$
while $f(\omega)\ne g(\omega)$ for every $\omega\in\Omega$.   But if $I$ is
countable then we shall have

\Centerline{$p\VVdP\,\vec f=\vec g
\iff p\Bsubseteq\{\omega:f(\omega)=g(\omega)\}^{\ssbullet}$.}

\noindent For a context in which these considerations are vital, see (a-ii) of
the proof of 551E.
}%end of comment

\spheader 551Ce Suppose that $x$ is any point of $\{0,1\}^I$.
Then we have a
corresponding $\Bbb P$-name $\check x$, and
$\VVdP\,\check x\in\{0,1\}^{\check I}$.   For each $i\in I$,
$\VVdP\,\check x(\varchecki)=x(i)\varBspcheck\in\{0,1\}$.
If we set $e_x(\omega)=x$ for
every $\omega\in\Omega$, then\cmmnt{ $e_x(\omega)(i)=x(i)$
for every $i\in I$ and
$\omega\in\Omega$, so $\VVdP\,\vec e_x(\varchecki)=x(i)\varBspcheck$
for every $i\in I$, and} $\VVdP\,\vec e_x=\check x$.

\vleader{60pt}{551D}{Definition} Let $(\Omega,\Sigma,\Cal I)$ be a
non-trivial measurable space
with negligibles with a Dedekind complete quotient algebra,
and $\Bbb P$ its associated forcing notion.   Let $I$ be any set.
If $W\subseteq\Omega\times\{0,1\}^I$, let $\vec W$ be the
$\Bbb P$-name

$$\eqalign{\{(\vec f,E^{\ssbullet}):E&\in\Sigma\setminus\Cal I,\,
f:\Omega\to\{0,1\}^I\text{ is }(\Sigma,\CalBa_I)\text{-measurable},\cr
&\mskip170mu(\omega,f(\omega))\in W\text{ for every }\omega\in E\},\cr}$$

\noindent interpreting $\vec f$ as in 551C.
\cmmnt{I give the definition for arbitrary sets $W$, but
it is useful primarily when $W\in\Sigma\tensorhat\CalBa_I$,
as in most of the
next proposition.   Perhaps I can note straight away that}

\Centerline{$\VVdP\,\vec W\subseteq\{0,1\}^{\check I}$}

\noindent and\cmmnt{ that} if $W=\Omega\times\{0,1\}^I$ then

\Centerline{$\VVdP\,\vec W=\{0,1\}^{\check I}$\ifresultsonly{.}\fi}

\cmmnt{\noindent (using 551Cb-551Cc).}

\leader{551E}{Proposition} Let $(\Omega,\Sigma,\Cal I)$ be a
non-trivial measurable space
with negligibles with a Dedekind complete quotient
algebra\cmmnt{ $\frak A$},
$\Bbb P$ its associated forcing notion, and $I$ a set.

(a) If $W\in\Sigma\tensorhat\CalBa_I$ and $f:\Omega\to\{0,1\}^I$ is
$(\Sigma,\CalBa_I)$-measurable, then
$\{\omega:(\omega,f(\omega))\in W\}$ belongs to $\Sigma$, and
$\Bvalue{\vec f\in\vec W}=\{\omega:(\omega,f(\omega))\in W\}^{\ssbullet}$.

(b) If $V$, $W\in\Sigma\tensorhat\CalBa_I$ then

\doubleinset{$\VVdP\,\vec V\cap\vec W=(V\cap W)\sspvec$,
$\vec V\cup\vec W=(V\cup W)\sspvec$,
$\vec V\setminus\vec W=(V\setminus W)\sspvec$ and
$\vec V\symmdiff\vec W=(V\symmdiff W)\sspvec$.}

(c) If $V$, $W\subseteq\Omega\times\{0,1\}^I$ and $V\subseteq W$ then

\Centerline{$\VVdP\,\vec V\subseteq\vec W$.}

(d) If $\sequencen{W_n}$ is a sequence in $\Sigma\tensorhat\CalBa_I$
with union $W$ and intersection $V$, then

\Centerline{$\VVdP\,\bigcup_{n\in\Bbb N}\vec W_n=\vec W$
and $\bigcap_{n\in\Bbb N}\vec W_n=\vec V$.}

(e) Suppose that $J\subseteq I$ is countable, $z\in\{0,1\}^J$, $E\in\Sigma$
and

\Centerline{$W
=\{(\omega,x):\omega\in E$, $x\in\{0,1\}^I$, $x\restr J=z\}$.}

\noindent Then

\Centerline{$E^{\ssbullet}
=\Bvalue{\vec W=\{x:x\in\{0,1\}^{\check I}$, $\check z\subseteq x\}}$,}

\Centerline{$1\Bsetminus E^{\ssbullet}
=\Bvalue{\vec W=\emptyset}$.}

\proof{{\bf (a)(i)} Let $\Cal W$ be the family of subsets of
$\Omega\times\{0,1\}^I$ such that
$F_W=\{\omega:(\omega,f(\omega))\in W\}\in\Sigma$.
Then $\Cal W$ is a Dynkin class of subsets of $\Omega\times\{0,1\}^I$, just
because $\Sigma$ is a $\sigma$-algebra.
If $H\in\Sigma$, $J\subseteq I$ is finite, $z\in\{0,1\}^J$ and
$W=\{(\omega,x):\omega\in H$, $z\subseteq x\in\{0,1\}^I\}$ then
$F_W=H\cap\{\omega:f(\omega)(i)=z(i)$ for every $i\in J\}$ belongs to
$\Sigma$ because $f$ is $(\Sigma,\CalBa_I)$-measurable, so $W\in\Cal W$.
By the Monotone Class Theorem (136B),
$\Cal W$ includes the $\sigma$-algebra generated by sets of this
form, which is just $\Sigma\tensorhat\CalBa_I$.

\medskip

\quad{\bf (ii)} Now suppose that $W\in\Sigma\tensorhat\CalBa_I$.
If $F_W\in\Cal I$ then
surely $F_W^{\ssbullet}=0\Bsubseteq\Bvalue{\vec f\in\vec W}$.   If
$F_W\notin\Cal I$
then $(\vec f,F_W^{\ssbullet})\in\vec W$,
$F_W^{\ssbullet}\VVdP\,\vec f\in\vec W$
and again $F_W^{\ssbullet}\Bsubseteq\Bvalue{\vec f\in\vec W}$.

\Quer\ I wish to apply 5A3E.
If $F_W^{\ssbullet}\ne\Bvalue{\vec f\in\vec W}$, set
$p=\Bvalue{\vec f\in\vec W}\Bsetminus F_W^{\ssbullet}$.   Since
$p\VVdP\,\vec f\in\vec W$ there must be a $q\in\frak A^+$ and a
$\Bbb P$-name
$\dot x$ and an $r$ stronger than both $p$ and $q$ such that

\Centerline{$r\VVdP\,\dot x=\vec f$ and $(\dot x,q)\in\vec W$.}

\noindent Now there must be a
$(\Sigma,\CalBa_I)$-measurable function $g$ and an
$E\in\Sigma\setminus\Cal I$
such that $\dot x=\vec g$, $q=E^{\ssbullet}$
and $(\omega,g(\omega))\in W$ for
every $\omega\in E$.   In this case, $r=G^{\ssbullet}$ for some
$G\subseteq E\setminus F_W$, and $r\VVdP\,\vec f=\vec g$.

Because $W\in\Sigma\tensorhat\CalBa_I$,
there is a countable set $J\subseteq I$ such that
$W$ factors through $\Omega\times\{0,1\}^J$.   For each $i\in J$, we have
$r\VVdP\,\vec f(\varchecki)=\vec g(\varchecki)$, that is,

\Centerline{$r\Bsubseteq\Bvalue{\vec f(\varchecki)=\vec g(\varchecki)}
=\{\omega:f(\omega)(i)=g(\omega)(i)\}^{\ssbullet}$.}

\noindent So $f(\omega)(i)=g(\omega)(i)$ for
$\Cal I$-almost every $\omega\in G$.   This
is true for every $i\in J$, so $f(\omega)\restr J=g(\omega)\restr J$ for
$\Cal I$-almost
every $\omega\in G$.   But this means that, for
$\Cal I$-almost every $\omega\in G$,
$(\omega,f(\omega))\in W$ iff $(\omega,g(\omega))\in W$.   However,
$G\subseteq E\setminus F_W$, so $(\omega,g(\omega))\in W$ and
$(\omega,f(\omega))\notin W$ for every $\omega\in G$.\ \Bang

So we must have $F_W^{\ssbullet}=\Bvalue{\vec f\in\vec W}$, as claimed.

\medskip

{\bf (b)} These are now elementary.   The point is that if a $\Bbb P$-name
$\dot x$ and $p\in\frak A^+$ are such that $p\VVdP\,\dot x\in\vec V\cap\vec W$,
then $p\VVdP\,\dot x\in\{0,1\}^{\check I}$, so there is a
$(\Sigma,\CalBa_I)$-measurable $f:\Omega\to\{0,1\}^I$ such that
$p\VVdP\,\dot x=\vec f$, and $p\VVdP\,\vec f\in\vec V\cap\vec W$.
Now (a) shows that

$$\eqalign{\Bvalue{\vec f\in(V\cap W)\sspvec}
&=\{\omega:(\omega,f(\omega))\in V\cap W\}^{\ssbullet}\cr
&=\{\omega:(\omega,f(\omega))\in V\}^{\ssbullet}
  \Bcap\{\omega:(\omega,f(\omega))\in W\}^{\ssbullet}\cr
&=\Bvalue{\vec f\in\vec V}\Bcap\Bvalue{\vec f\in\vec W}
\Bsupseteq p\cr}$$

\noindent and

\Centerline{$p\VVdP\,\dot x=\vec f\in(V\cap W)\sspvec$.}

\noindent As $p$ and $\dot x$ are arbitrary,

\Centerline{$\VVdP\,\vec V\cap\vec W\subseteq(V\cap W)\sspvec$.}

\noindent The other seven inequalities are equally straightforward.

\medskip

{\bf (c)} This is immediate from the definition in 551D, since we actually
have $\vec V\subseteq\vec W$.

\medskip

{\bf (d)} We can repeat the method of (b).   If a $\Bbb P$-name
$\dot x$ and $p\in\frak A^+$ are such that
$p\VVdP\,\dot x\in\bigcap_{n\in\Bbb N}\vec W_n$,
then there is a
$(\Sigma,\CalBa_I)$-measurable $f:\Omega\to\{0,1\}^I$ such that
$p\VVdP\,\dot x=\vec f$, and $p\VVdP\,\vec f\in\vec W_n$ for every $n$.
Now

$$\eqalignno{\Bvalue{\vec f\in\vec V}
&=\{\omega:(\omega,f(\omega))\in\bigcap_{n\in\Bbb N}W_n\}^{\ssbullet}\cr
&=(\bigcap_{n\in\Bbb N}\{\omega:(\omega,f(\omega))\in W_n\})^{\ssbullet}
=\inf_{n\in\Bbb N}\{\omega:(\omega,f(\omega))\in W_n\}^{\ssbullet}\cr
\displaycause{because $\Sigma\cap\Cal I$ is a $\sigma$-ideal of $\Sigma$,
so $E\mapsto E^{\ssbullet}$ is sequentially order-continuous, by 313Qb}
&\Bsupseteq p\cr}$$

\noindent and

\Centerline{$p\VVdP\,\dot x=\vec f\in\vec V$.}

\noindent As $p$ and $\dot x$ are arbitrary,

\Centerline{$\VVdP\,\bigcap_{n\in\Bbb N}\vec W_n\subseteq\vec V$.}

\noindent On the other hand, (c) tells us that

\Centerline{$\VVdP\,\vec V\subseteq\bigcap_{n\in\Bbb N}\vec W_n$,
so we have equality.}

Putting this together with (b) (and recalling that
$\VVdP\,(\Omega\times\{0,1\}^I)\sspvec=\{0,1\}^{\check I}$), we get

\Centerline{$\VVdP\,\bigcup_{n\in\Bbb N}\vec W_n=\vec W$.}

\medskip

{\bf (e)(i)} Suppose that $p\in\frak A^+$ and that $\dot x$ is a
$\Bbb P$-name such that $p\VVdP\,\dot x\in\vec W$.
Let $f:\Omega\to\{0,1\}^I$ be
a $(\Sigma,\CalBa_I)$-measurable function such that
$p\VVdP\,\dot x=\vec f$ (551Cc).   Then

\Centerline{$p\Bsubseteq\Bvalue{\vec f\in\vec W}
=\{\omega:\omega\in E$, $z\subseteq f(\omega)\}^{\ssbullet}$}

\noindent by (a) above;  that is, $p\Bsubseteq E^{\ssbullet}$ and

\Centerline{$p\VVdP\,\vec f(\varchecki)=z(i)\varBspcheck
=\check z(\varchecki)$}

\noindent for every $i\in J$, so

\Centerline{$p\VVdP\,\check z\subseteq\vec f$.}

\noindent As $p$ and $\dot x$ are arbitrary,

\Centerline{$\Bvalue{\vec W\ne\emptyset}\subseteq E^{\ssbullet}$}

\noindent and

\Centerline{$\VVdP\,\vec W
\subseteq\{x:\check z\subseteq x\in\{0,1\}^{\check I}\}$.}

\wheader{551E}{6}{2}{2}{48pt}

\quad{\bf (ii)} If $E\in\Cal I$ then
$\VVdP\,\vec W=\emptyset$ and we can stop.
Otherwise, suppose that $p\in\frak A^+$ is stronger than
$E^{\ssbullet}$ and that $\dot x$ is a $\Bbb P$-name such that

\Centerline{$p\VVdP\,\check z\subseteq\dot x\in\{0,1\}^{\check I}$.}

\noindent Let $f$ be a $(\Sigma,\CalBa_I)$-measurable function such that
$p\VVdP\,\dot x=\vec f$.   Then $p\VVdP\,\vec f_i=z(i)\varBspcheck$
for each $i\in J$, where $f_i(\omega)=f(\omega)(i)$ for every $\omega$, so
$p\Bsubseteq\{\omega:z\subseteq f(\omega)\}^{\ssbullet}$.   But also
$p\Bsubseteq E^{\ssbullet}$, so

\Centerline{$p
\Bsubseteq\{\omega:\omega\in E$, $z\subseteq f(\omega)\}^{\ssbullet}
=\Bvalue{\vec f\in\vec W}$,}

\noindent and $p\VVdP\,\dot x\in\vec W$.   As $p$ and
$\dot x$ are arbitrary,

\Centerline{$\Bvalue{\{x:\check z\subseteq x\}\subseteq\vec W}
\Bsupseteq E^{\ssbullet}$}

\noindent and we have

\Centerline{$\Bvalue{\{x:\check z\subseteq x\}=\vec W}
=E^{\ssbullet}$,
\quad$\Bvalue{\emptyset=\vec W}
=1\Bsetminus E^{\ssbullet}$.}
}%end of proof of 551E

\leader{551F}{Proposition} Let $(\Omega,\Sigma,\Cal I)$ be a
non-trivial measurable space
with negligibles with a Dedekind complete quotient
algebra\cmmnt{ $\frak A$},
$\Bbb P$ its associated forcing notion, and $I$ a set.

(a) If $W\in\Sigma\tensorhat\CalBa_I$ then

\Centerline{$\VVdP\,\vec W\in\CalBa_{\check I}$.}

(b) Suppose that $(\Omega,\Sigma,\Cal I)$ is $\omega_1$-saturated,
$p\in\frak A^+$, and that
$\dot W$ is a $\Bbb P$-name such that

\Centerline{$p\VVdP\,\dot W\in\CalBa_{\check I}$.}

\noindent Then there is a $W\in\Sigma\tensorhat\CalBa_I$ such that

\Centerline{$p\VVdP\,\dot W=\vec W$.}

\proof{{\bf (a)} Let $\Cal W$ be the family of those
$W\in\Sigma\tensorhat\CalBa_I$
such that $\VVdP\,\vec W\in\CalBa_{\check I}$.
551Eb and 551Ed tell us that $\Cal W$ is a $\sigma$-subalgebra of
$\Sigma\tensorhat\CalBa_I$, and 551Ee tells us that $E\times H\in\Cal W$
whenever $E\in\Sigma$ and $H$ is a basic cylinder set in $\{0,1\}^I$.
So $\Cal W$ must be the whole of $\Sigma\tensorhat\CalBa_I$.

\medskip

{\bf (b)(i)} Suppose that $p\in\frak A^+$ and that $\dot W$ is a
$\Bbb P$-name such that

\Centerline{$p\VVdP\,\dot W$ is a basic cylinder set in
$\{0,1\}^{\check I}$.}

\noindent Then there is a $W\in\Sigma\tensorhat\CalBa_I$ such that
$p\VVdP\,\dot W=\vec W$.   \Prf\ We know that

\Centerline{$p\VVdP$ there is a $z\in\Fn_{<\omega}(\check I;\{0,1\})$
such that $\dot W=\{x:z\subseteq x\in\{0,1\}^{\check I}\}$.}

\noindent So there is a $\Bbb P$-name $\dot z$ such that

\Centerline{$p\VVdP\,\dot z\in\Fn_{<\omega}(\check I;\{0,1\})$ and
$\dot W=\{x:\dot z\subseteq x\}$;}

\noindent adjusting $\dot z$ if necessary, we can suppose that

\Centerline{$\VVdP\,\dot z\in\Fn_{<\omega}(\check I;\{0,1\})$.}

\noindent But this means that there is a maximal antichain (that is, a
partition of unity) $C\subseteq\frak A^+$
and a family $\family{c}{C}{z_c}$ in
$\Fn_{<\omega}(I;\{0,1\})$ such that

\Centerline{$c\VVdP\,\dot z=\check z_c$}

\noindent for every $c\in C$.   Because $\Cal I$ is $\omega_1$-saturated,
$\frak A$ is ccc and $C$ is countable.   We can therefore find a partition
$\family{c}{C}{E_c}$ of $\Omega$ into members of $\Sigma$ such that
$E_c^{\ssbullet}=c$ for every $c\in C$.   Consider

\Centerline{$W_c=E_c\times\{x:z_c\subseteq x\in\{0,1\}^I\}$ for $c\in C$,
\quad$W=\bigcup_{c\in C}W_c$.}

\noindent Of course $W\in\Sigma\tensorhat\CalBa_I$.   By 551Ee,

\Centerline{$c\VVdP\,\vec W_c=\{x:\check z_c\subseteq x\}
=\{x:\dot z\subseteq x\}$,
\quad$c\VVdP\,\vec W_d=\emptyset$}

\noindent whenever $c$, $d\in C$ are distinct.
Because $C$ is countable, 551Ed tells us that

\Centerline{$c\VVdP\,\vec W
=\bigcup_{d\in\check C}\vec W_d=\{x:\dot z\subseteq x\}$}

\noindent for every $c\in C$;  because $C$ is a maximal antichain,

\Centerline{$\VVdP\,\vec W=\{x:\dot z\subseteq x\}$}

\noindent and

\Centerline{$p\VVdP\,\vec W=\dot W$.   \Qed}

\medskip

\quad{\bf (ii)} Suppose that $p\in\frak A^+$ and that $\dot W$ is a
$\Bbb P$-name such that

\Centerline{$p\VVdP\,\dot W$ is a cozero set in $\{0,1\}^{\check I}$.}

\noindent Then there is a $W\in\Sigma\tensorhat\CalBa_I$ such that
$p\VVdP\,\dot W=\vec W$.   \Prf\ Set $p'=\Bvalue{\dot W=\emptyset}$.
If $p\Bsubseteq p'$ we can take $W=\emptyset$ and stop.   Otherwise,
let $E\in\Sigma$ be such that $E^{\ssbullet}=1\Bsetminus p'$.   We have

\Centerline{$p\Bsetminus p'\VVdP\,
\dot W$ is the union of a sequence of basic cylinder sets,}

\noindent so there is a sequence $\sequencen{\dot W_n}$ of $\Bbb P$-names
such that

\Centerline{$p\Bsetminus p'\VVdP\,\dot W_n$
is a basic cylinder set for every $n$ and
$\dot W=\bigcup_{n\in\Bbb N}\dot W_n$.}

\noindent By (i), we have for each $n\in\Bbb N$ a
$W_n\in\Sigma\tensorhat\CalBa_I$ such that
$p\Bsetminus p'\VVdP\,\dot W_n=\vec W_n$;
now $V=\bigcup_{n\in\Bbb N}W_n$ belongs to
$\Sigma\tensorhat\CalBa_I$ and
$\VVdP\,\vec V=\bigcup_{n\in\Bbb N}\vec W_n$, so
$p\Bsetminus p'\VVdP\,\vec V=\dot W$.   Finally, setting
$W=(E\times\{0,1\}^I)\cap V$, $p\VVdP\,\vec W=\dot W$.\ \Qed

\medskip

\quad{\bf (iii)} Suppose that $p\in\frak A^+$, $\alpha<\omega_1$ and that
$\dot W$ is a $\Bbb P$-name such that

\Centerline{$p\VVdP\,\dot W\in\CalBa_{\check\alpha}(\{0,1\}^{\check I})$,}

\noindent defining $\CalBa_{\alpha}$ as in 5A4Ga.
Then there is a $W\in\Sigma\tensorhat\CalBa_I$ such that
$p\VVdP\,\dot W=\vec W$.   \Prf\ Induce on $\alpha$.   The case $\alpha=0$
is (ii) above.   For the inductive step to $\alpha>0$, we have

\Centerline{$p\VVdP\,\dot W\in\CalBa_{\check\alpha}(\{0,1\}^{\check I})$,}

\noindent so

\doubleinset{$p\VVdP$ there is a sequence $\sequencen{W_n}$ in
$\bigcup_{\beta<\check\alpha}\CalBa_{\beta}(\{0,1\}^{\check I})$
such that $\dot W=\bigcup_{n\in\Bbb N}\{0,1\}^{\check I}\setminus W_n$;}

\noindent let $\sequencen{\dot W_n}$ be a sequence of $\Bbb P$-names
such that

\Centerline{$p\VVdP\,
\dot W_n\in\bigcup_{\beta<\check\alpha}\CalBa_{\beta}(\{0,1\}^{\check I})$
for every $n\in\Bbb N$ and
$\dot W=\bigcup_{n\in\Bbb N}\{0,1\}^{\check I}\setminus\dot W_n$.}

\noindent For $n\in\Bbb N$, $\beta<\alpha$ set

\Centerline{$b_{n\beta}
=\Bvalue{\dot W_n\in\CalBa_{\check\beta}(\{0,1\}^{\check I})
\setminus\bigcup_{\gamma<\check\beta}\CalBa_{\gamma}(\{0,1\}^{\check I})}$,}

\noindent and choose $E_{n\beta}\in\Sigma$ such that
$E_{n\beta}^{\ssbullet}=b_{n\beta}$.
Writing $A_n=\{\beta:\beta<\alpha$, $b_{n\beta}\ne 0\}$,
$p\Bsubseteq\sup_{\beta\in A_n}b_{n\beta}$.  If $\beta\in A_n$, then

\Centerline{$b_{n\beta}
 \VVdP\,\dot W_n\in\CalBa_{\check\beta}(\{0,1\}^{\check I})$,}

\noindent so by the inductive hypothesis there is a
$W_{n\beta}\in\Sigma\tensorhat\CalBa_I$ such that
$b_{n\beta}\VVdP\,\dot W_n=\vec W_{n\beta}$.   For
$\beta\in\alpha\setminus A_n$ set $W_{n\beta}=\emptyset$.

Set $W_n=\bigcup_{\beta<\alpha}(E_{n\beta}\times\{0,1\}^I)\cap W_{n\beta}$.
Then

\Centerline{$\VVdP\,\vec W_n
=\bigcup_{\beta<\check\alpha}(E_{n\beta}\times\{0,1\}^I)\sspvec
  \cap\vec W_{n\beta}$,}

\noindent so if $\beta\in A_n$

\Centerline{$b_{n\beta}\VVdP\,\vec W_n=\vec W_{n\beta}=\dot W_n$}

\noindent because

\Centerline{$b_{n\beta}\VVdP\,(E_{n\gamma}\times\{0,1\}^I)\sspvec
 =\emptyset$}

\noindent if $\gamma<\alpha$ and $\gamma\ne\beta$, and

\Centerline{$b_{n\beta}\VVdP\,
(E_{n\beta}\times\{0,1\}^I)\sspvec=\{0,1\}^{\check I}$.}

\noindent As $p\Bsubseteq\sup_{\beta\in A_n}b_{n\beta}$,

\Centerline{$p\VVdP\,\vec W_n=\dot W_n$.}

This is true for every $n\in\Bbb N$.   So if we set
$W=\bigcup_{n\in\Bbb N}(\Omega\times\{0,1\}^I)\setminus W_n$,
we shall have $W\in\Sigma\tensorhat\CalBa_I$ and

\Centerline{$p\VVdP\,\vec W
=\bigcup_{n\in\Bbb N}\{0,1\}^{\check I}\setminus\vec W_n
=\bigcup_{n\in\Bbb N}\{0,1\}^{\check I}\setminus\dot W_n
=\dot W$.   \Qed}

\medskip

\quad{\bf (iv)} Finally, suppose that $p\in\frak A^+$ and that $\dot W$
is a $\Bbb P$-name such that

\Centerline{$p\VVdP\,\dot W\in\CalBa_{\check I}$.}

\noindent Then there is a $W\in\Sigma\tensorhat\CalBa_I$ such that
$p\VVdP\,\dot W=\vec W$.   \Prf\ Because $\Bbb P$ is ccc,

\Centerline{$\VVdP\,\check\omega_1$ is the first uncountable ordinal}

\noindent (5A3Nb), so

\Centerline{$\VVdP\,\CalBa(\{0,1\}^{\check I})
=\bigcup_{\alpha<\check\omega_1}\CalBa_{\alpha}(\{0,1\}^{\check I})$.}

\noindent For $\alpha<\omega_1$ set

\Centerline{$b_{\alpha}
=\Bvalue{\dot W\in\CalBa_{\check\alpha}(\{0,1\}^{\check I})}$.}

\noindent Then
$p\Bsubseteq\sup_{\alpha<\omega_1}b_{\alpha}$.   Again because $\frak A$ is
ccc, there is a $\gamma<\omega_1$ such that
$p\Bsubseteq\sup_{\alpha<\gamma}b_{\alpha}$.   If $\alpha<\gamma$ and
$c_{\alpha}=b_{\alpha}\Bsetminus\sup_{\beta<\alpha}b_{\beta}$ is non-zero,
choose $W_{\alpha}\in\Sigma\tensorhat\CalBa_I$ such that
$c_{\alpha}\VVdP\,\vec W_{\alpha}=\dot W$;  for other $\alpha<\gamma$
set $W_{\alpha}=\emptyset$.   Choose $F_{\alpha}\in\Sigma$ such that
$F_{\alpha}^{\ssbullet}=c_{\alpha}$ for each $\alpha$.
Set $W=\bigcup_{\alpha<\gamma}(F_{\alpha}\times\{0,1\}^I)\cap W_{\alpha}
\in\Sigma\tensorhat\CalBa_I$.   As in (iii) just above,

\Centerline{$c_{\alpha}\VVdP\,\vec W=\vec W_{\alpha}=\dot W$}

\noindent whenever $c_{\alpha}\ne 0$, so

\Centerline{$p\VVdP\,\vec W=\dot W$.  \Qed}
}%end of proof of 551F

\vleader{48pt}{551G}{}\cmmnt{ I noted above that there are difficulties in
computing
$\Bvalue{\vec f=\vec g\,}$ for functions $f$, $g:\Sigma\to\{0,1\}^I$.
For $W$, $V\in\Sigma\tensorhat\CalBa_I$ the corresponding question about
$\Bvalue{\vec W=\vec V}$ turns out to be simpler, at least in some
important cases.

\medskip

\wheader{551G}{0}{0}{0}{72pt}
\noindent}{\bf Proposition} Let $(\Omega,\Sigma,\Cal I)$ be a non-trivial
measurable space with negligibles with a Dedekind complete quotient
algebra $\frak A$,
$\Bbb P$ the associated forcing notion and $I$ a set.   Suppose
that $\Sigma$ is closed under Souslin's operation.

(a) If $W\in\Sigma\tensorhat\CalBa_I$ then
$F=\{\omega:W[\{\omega\}]\ne\emptyset\}$ belongs to $\Sigma$ and
$\Bvalue{\vec W\ne\emptyset}=F^{\ssbullet}$ in $\frak A\cong\RO(\Bbb P)$.

(b) If $W$, $V\in\Sigma\tensorhat\CalBa_I$ then
$\Bvalue{\vec W=\vec V}=\{\omega:W[\{\omega\}]=V[\{\omega\}]\}^{\ssbullet}$.

\proof{{\bf (a)(i)} The point is that there is a $\Sigma$-measurable
function
$f:\Omega\to\{0,1\}^I$ such that $(\omega,f(\omega))\in W$ for every
$\omega\in F$.

\medskip

\Prf\grheada\ Suppose first that $I$ is countable.   Let
$\Cal V$ be the family of subsets of $\Omega\times\{0,1\}^I$ obtainable by
Souslin's operation $\Cal S$ from
$\{E\times H:E\in\Sigma$, $H\subseteq\{0,1\}^I$ is closed$\}$.   The family
$\Cal W=\{V:V\in\Cal V$, $(\Omega\times\{0,1\}^I)\setminus V\in\Cal V\}$ is a
$\sigma$-algebra and contains $E\times H$ whenever $E\in\Sigma$ and
$H\subseteq\{0,1\}^I$ is open-and-closed, so
$\Cal W\supseteq\Sigma\tensorhat\CalBa_I$ and $W\in\Cal W\subseteq\Cal V$.
By 423N, there is a selector $g$ for $W$ which is measurable for the
$\sigma$-algebra $\Tau$ of subsets of $\Omega$ generated by
$\Cal S(\Sigma)$;
but we are supposing that this is just $\Sigma$.   Also
$F=\dom g$ belongs to $\Tau=\Sigma$.   If $f$ is any extension of $g$ to a
$\Sigma$-measurable function from $\Omega$ to $\{0,1\}^I$, then $f$ has the
required property.

\medskip

\qquad\grheadb\ For the general case, note that $W\in\Sigma\tensorhat\CalBa_I$
factors through $\Omega\times\{0,1\}^J$ for some countable $J\subseteq I$, that
is, there is a $W_1\in\Sigma\tensorhat\CalBa_J$ such that

\Centerline{$W
=\{(\omega,x):\omega\in\Omega$, $x\in\{0,1\}^I$, $(\omega,x\restr J)\in W_1\}$.}

\noindent Now ($\alpha$) tells us that
$F_1=\{\omega:W_1[\{\omega\}]\ne\emptyset\}$ belongs to $\Sigma$ and that
there is a $\Sigma$-measurable $f_1:\Omega\to\{0,1\}^J$ such that
$(\omega,f_1(\omega))\in W_1$ for every $\omega\in F_1$.   Of course $F_1=F$,
and if we set

$$\eqalign{f(\omega)(i)
&=f_1(\omega)(i)\text{ for }\omega\in\Omega,\,i\in J,\cr
&=0\text{ for }\omega\in\Omega,\,i\in I\setminus J,\cr}$$

\noindent then $f$ is $(\Sigma,\CalBa_I)$-measurable and
$(\omega,f(\omega))\in W$ for every $\omega\in F$.\ \Qed

\medskip

\quad{\bf (ii)} Using 551Ea, it follows that

\Centerline{$\Bvalue{\vec W\ne\emptyset}\Bsupseteq\Bvalue{\vec f\in\vec W}
=\{\omega:(\omega,f(\omega))\in W\}^{\ssbullet}=F^{\ssbullet}$.}

\noindent On the other hand, if $a=\Bvalue{\vec W\ne\emptyset}$ is non-zero,
then there is a $\Bbb P$-name $\dot x$ such that $a\VVdP\,\dot x\in\vec W$.   By
551Cc, there is a $(\Sigma,\CalBa_I)$-measurable $g$ such that
$a\VVdP\,\dot x=\vec g$, in which case

\Centerline{$a\Bsubseteq\Bvalue{\vec g\in\vec W}
=\{\omega:(\omega,g(\omega))\in W\}^{\ssbullet}\Bsubseteq F^{\ssbullet}$.}

\noindent So $\Bvalue{\vec W\ne\emptyset}=F^{\ssbullet}$ exactly.

\medskip

{\bf (b)} Apply (a) to $W\symmdiff V$ (using 551Eb, as usual).
}%\end of proof of 551G

\leader{551H}{Examples} \cmmnt{Cases in which a $\sigma$-algebra is
closed under Souslin's operation, so that the conditions of 551G can be
satisfied, include the following.

\medskip

}{\bf (a)} If $(X,\Sigma,\mu)$ is a complete locally
determined measure space, then $\Sigma$ is closed under Souslin's
operation\prooflet{ (431A)}.

\spheader 551Hb If $(\Omega,\Sigma,\Cal I)$ is a complete
$\omega_1$-saturated
measurable space with negligibles, then $\Sigma$ is closed under Souslin's
operation\prooflet{ (431G)}.

\spheader 551Hc If $X$ is any topological space, then its Baire-property
algebra $\widehat{\Cal B}(X)$ is closed under Souslin's
operation\cmmnt{ (431Fb)}.

\leader{551I}{Theorem} Let $(\Omega,\Sigma,\Cal I)$ be a
non-trivial measurable space
with negligibles with a Dedekind complete quotient algebra,
$\Bbb P$ its associated forcing notion, and $I$ a set.
Let $W$ be any member of $\Sigma\tensorhat\CalBa_I$.  Then

(i) $h(\omega)=\nu_IW[\{\omega\}]$ is defined for every
$\omega\in\Omega$, where $\nu_I$ is the usual measure of $\{0,1\}^I$;

(ii) $h:\Omega\to[0,1]$ is $\Sigma$-measurable;

(iii) $\VVdP\,\nu_{\check I}\vec W=\vec h$,

\noindent where\cmmnt{ in this formula} $\vec h$ is the
$\Bbb P$-name for a real number defined from $h$ as in 551B,
and $\nu_{\check I}$ is an
abbreviation for `the usual measure on $\{0,1\}^{\check I}$'.

\proof{ I follow the method of 551Ea and 551Fa.

\medskip

{\bf (a)} Suppose that $W$ is of the form
$E\times\{x:z\subseteq x\in\{0,1\}^I\}$, where $z\in\{0,1\}^J$ for some finite
$J\subseteq I$.   Then (using 551Ee)

\Centerline{$E^{\ssbullet}
=\Bvalue{\vec W=\{x:\check z\subseteq x\in\{0,1\}^{\check I}\}}
\subseteq\Bvalue{\nu_{\check I}\vec W=2^{-\#(\check J)}}$,}

\Centerline{$1\setminus E^{\ssbullet}
=\Bvalue{\vec W=\emptyset}
\subseteq\Bvalue{\nu_{\check I}\vec W=0}$;}

\noindent while also $h=2^{-\#(J)}\chi E$ so

\Centerline{$E^{\ssbullet}=\Bvalue{\vec h=2^{-\#(\check J)}}$,
\quad$1\Bsetminus E^{\ssbullet}=\Bvalue{\vec h=0}$.}

\noindent So in this case

\Centerline{$\VVdP\,\nu_{\check I}\vec W=\vec h$.}

\medskip

{\bf (b)} Now 551E shows that the set of those $W\in\CalBa_I$ for which
(i)-(iii) are true is a Dynkin class, so by the Monotone Class Theorem once more
we have the result.
}%\end of proof of 551I

\vleader{72pt}{551J}{Corollary} Let $(\Omega,\Sigma,\Cal I)$ be a
non-trivial $\omega_1$-saturated measurable space with negligibles,
$\Bbb P$ its associated forcing notion,
$P\cmmnt{\mskip5mu=(\Sigma/\Sigma\cap\Cal I)^+}$
the partially ordered set underlying $\Bbb P$, and $I$ a set.
If $p\in P$ and $\dot W$ is a $\Bbb P$-name such that

\Centerline{$p\VVdP\,\dot W\subseteq\{0,1\}^{\check I}$ is
$\nu_{\check I}$-negligible,}

\noindent then there is a $W\in\Sigma\tensorhat\CalBa_I$
such that $\nu_IW[\{\omega\}]=0$ for every $\omega\in\Omega$ and

\Centerline{$p\VVdP\,\dot W\subseteq\vec W$.}

\proof{ Because

\Centerline{$\VVdP$ the usual measure on $\{0,1\}^{\check I}$
is a completion regular Radon measure,}

\noindent we know that

\Centerline{$p\VVdP$ there is a $\nu_{\check I}$-negligible
member of $\CalBa_{\check I}$ including $\dot W$.}

\noindent Let $\dot V$ be a $\Bbb P$-name such that

\Centerline{$p\VVdP\,\dot W\subseteq\dot V\in\CalBa_{\check I}$ and
$\nu_{\check I}\dot V=0$.}

\noindent By 551Fb, there is a $V\in\Sigma\tensorhat\CalBa_I$ such that
$p\VVdP\,\dot V=\vec V$.   Set $h(\omega)=\nu_IV[\{\omega\}]$ for
$\omega\in\Omega$;  then

\Centerline{$p\VVdP\,\vec h=\nu_{\check I}\vec V=0$}

\noindent (551I), so $p\Bsubseteq E^{\ssbullet}$, where $E=h^{-1}[\{0\}]$.
Set $W=(E\times\{0,1\}^I)\cap V$.   Then
$W\in\Sigma\tensorhat\CalBa_I$, $\nu_IW[\{\omega\}]=0$
for every $\omega$, and (using 551Gb)

\Centerline{$p\VVdP\,\vec W=\vec V=\dot V\supseteq\dot W$,}

\noindent as required.
}%end of proof of 551J

\leader{551K}{}\cmmnt{ We have been looking here at general sets
$W\in\Sigma\tensorhat\CalBa_I$.
A special case of obvious importance is
when $W$ is of the form $\Omega\times H$ where $H\in\CalBa_I$.   For
these it is worth refining the results slightly.

\medskip

\noindent}{\bf Proposition} Let $(\Omega,\Sigma,\Cal I)$ be a
non-trivial measurable space with negligibles with a Dedekind complete
quotient algebra, $\Bbb P$ the associated forcing notion, and $I$ a set.
For $H\subseteq\{0,1\}^I$ set
$\tilde H=(\Omega\times H)\sspvec$\cmmnt{ as defined in 551D}.

(a) If $H=\{x:z\subseteq x\in\{0,1\}^I\}$, where
$z\in\Fn_{<\omega}(I;\{0,1\})$, then

\Centerline{$\VVdP\,\tilde H=\{x:\check z\subseteq x\in\{0,1\}^{\check I}\}$.}

(b)(i) If $G$, $H\in\CalBa_I$ then

$$\eqalign{\VVdP\,\tilde G\cup\tilde H&=(G\cup H)\ssptilde,\,
\tilde G\cap\tilde H=(G\cap H)\ssptilde,\cr
&\tilde G\setminus\tilde H=(G\setminus H)\ssptilde,\,
\tilde G\symmdiff\tilde H=(G\symmdiff H)\ssptilde.\cr}$$

\quad(ii) If $\sequencen{H_n}$ is any sequence in $\CalBa_I$ then

\Centerline{$\VVdP\,
\bigcup_{n\in\Bbb N}\tilde H_n=(\bigcup_{n\in\Bbb N}H_n)\ssptilde$,
$\bigcap_{n\in\Bbb N}\tilde H_n=(\bigcap_{n\in\Bbb N}H_n)\ssptilde$.}

(c) If $\alpha<\omega_1$ and
$H\in\CalBa_{\alpha}(\{0,1\}^I)$,\cmmnt{ once again defining
$\CalBa_{\alpha}$ as in 5A4Ga,} then

\Centerline{$\VVdP\,\tilde H\in\CalBa_{\check\alpha}(\{0,1\}^{\check I})$.}

(d) If $H$ is measured by the usual measure $\nu_I$ of $\{0,1\}^I$, then

\Centerline{$\VVdP\,\nu_{\check I}\tilde H=(\nu_IH)\varBspcheck$.}

\proof{{\bf (a)} This is covered by 551Ee.

\medskip

{\bf (b)} This is a special case of parts (b) and (d) of 551E.

\medskip

{\bf (c)} A subset of $\{0,1\}^I$ is a cozero set iff it is
empty or expressible as
the union of a sequence of basic cylinder sets, so if $H$ is a cozero set then
(a) and (b-ii) tell us that

\Centerline{$\VVdP\,\tilde H$ is a cozero set in $\{0,1\}^{\check I}$.}

\noindent Now an induction on $\alpha$ shows that if
$H\in\CalBa_{\alpha}(\{0,1\}^I)$ then

\Centerline{$\VVdP\,\tilde H\in\CalBa_{\check\alpha}(\{0,1\}^{\check I})$.}

\medskip

{\bf (d)} We have $H_0$, $H_1\in\CalBa_I$ such that
$H_0\subseteq H\subseteq H_1$ and $\nu_IH_0=\nu_IH=\nu_IH_1$.
Applying 551I(iii) to $\Omega\times H_0$ and $\Omega\times H_1$,

\Centerline{$\VVdP\,\nu_{\check I}\tilde H_0=\nu_{\check I}\tilde H_1
=(\nu_IH)\varBspcheck$,}

\noindent while of course
$\VVdP\,\tilde H_0\subseteq\tilde H\subseteq\tilde H_1$ (551Ec), so

\Centerline{$\VVdP\,\nu_{\check I}\tilde H=(\nu_IH)\varBspcheck$.}
}%end of proof of 551K

\cmmnt{
\leader{551L}{Remark} If I ask you to think of your favourite Baire set in
$\{0,1\}^I$, it is likely to come with a definition;  for instance, the set $H$
of those $x\in\{0,1\}^{\Bbb N}$ such that
$\lim_{n\to\infty}\Bover1{n+1}\sum_{i=1}^nx(i)=\Bover12$.
The point of 551K is that we shall automatically get

\Centerline{$\VVdP\,\tilde H
=\{x:x\in\{0,1\}^{\Bbb N}$,
$\lim_{n\to\infty}\Bover1{n+1}\sum_{i=1}^nx(i)=\Bover12\}$.}

\noindent\Prf\

\Centerline{$H
=\bigcap_{n\in\Bbb N}\bigcup_{m\in\Bbb N}\bigcap_{k\ge m}
  \bigcup_{z\in L_{nk}}\{x:z\subseteq x\in\{0,1\}^{\Bbb N}\}$,}

\noindent where

\Centerline{$L_{nk}=\{z:z\in\{0,1\}^{k+1}$,
$|\Bover1{k+1}\sum_{i=0}^kz_i-\Bover12|\le\Bover1{n+1}\}$.}

\noindent So 551K tells us that

\Centerline{$\VVdP\,\tilde H
=\bigcap_{n\in\Bbb N}\bigcup_{m\in\Bbb N}\bigcap_{k\ge m}
\bigcup_{z\in\check L_{nk}}
\{x:z\subseteq x\in\{0,1\}^{\Bbb N}\}$,}

\noindent and of course

\Centerline{$\VVdP\,\check L_{nk}=\{z:z\in\{0,1\}^{\check k+1}$,
$|\Bover1{\check k+1}\sum_{i=0}^{\check k}z_i-\Bover12|
  \le\Bover1{\check n+1}\}$.\ \Qed}

What I am trying to say here is that the process
$H\mapsto(\Omega\times H)\sspvec=\tilde H$ builds a $\Bbb P$-name for the
`right' subset of $\{0,1\}^I$, in the sense that any adequately concrete
definition of $H$ will also, when interpreted in $V^{\Bbb P}$,
be a definition of $\tilde H$.
}%end of comment

\leader{551M}{}\cmmnt{ We can go still farther.

\medskip

\noindent}{\bf Definition} Let $(\Omega,\Sigma,\Cal I)$ be a non-trivial
measurable space with negligibles, and $\Bbb P$ its associated forcing
notion.   Let $I$ be any set.   If $\psi:\Omega\times\{0,1\}^I\to\Bbb R$ is
$(\Sigma\tensorhat\CalBa_I)$-measurable, let $\vec\psi$ be the
$\Bbb P$-name

$$\eqalign{\{((\vec f,\vec h),\Bbbone):
f&\text{ is a }(\Sigma,\CalBa_I)\text{-measurable function from }
  \Omega\text{ to }\{0,1\}^I,\cr
&h:\Omega\to\Bbb R\text{ is }\Sigma\text{-measurable},\,
h(\omega)=\psi(\omega,f(\omega))\text{ for every }\omega\in\Omega\},\cr}$$

\noindent where in this formula $\vec f$ is to be interpreted as a
$\Bbb P$-name for a member of $\{0,1\}^{\check I}$, as in 551C, and
$\vec h$ as a $\Bbb P$-name for a real number, as in 551B.

\leader{551N}{Proposition} Let $(\Omega,\Sigma,\Cal I)$ be a non-trivial
measurable space with negligibles with a Dedekind complete quotient
algebra $\frak A$, $\Bbb P$ its associated forcing
notion, and $I$ a set.   Suppose that
$\psi:\Omega\times\{0,1\}^I\to\Bbb R$ is
$(\Sigma\tensorhat\CalBa_I)$-measurable, and define $\vec\psi$ as in 551M.

(a) $\VVdP\,\vec\psi$ is a real-valued
function on $\{0,1\}^{\check I}$.

(b) If $\phi:\Omega\times\{0,1\}^I\to\Bbb R$ is another
$(\Sigma\tensorhat\CalBa_I)$-measurable function, and $\alpha\in\Bbb R$,
then

\Centerline{$\VVdP\,(\phi+\psi)\sspvec=\vec\phi+\vec\psi$,
$(\alpha\phi)\sspvec=\check\alpha\vec\phi$.}

(c) If $\sequencen{\psi_n}$ is a sequence of
$(\Sigma\tensorhat\CalBa_I)$-measurable real-valued functions on
$\Omega\times\{0,1\}^I$ and
\ifnum\stylenumber=12 we set \fi
$\psi(\omega,x)=\penalty-100\lim_{n\to\infty}\psi_n(\omega,x)$ for every
$\omega\in\Omega$ and $x\in\{0,1\}^I$, then

\Centerline{$\VVdP\,\vec\psi(x)=\lim_{n\to\infty}\vec\psi_n(x)$ for every
$x\in\{0,1\}^{\check I}$.}

\wheader{551N}{0}{0}{0}{30pt}
(d) If $W\in\Sigma\tensorhat\CalBa_I$, then

\Centerline{$\VVdP\,(\chi W)\sspvec=\chi\vec W$.}

(e) $\VVdP\,\vec\psi$ is $\CalBa_{\check I}$-measurable.

(f) If $h(\omega)=\int\psi(\omega,x)\nu_I(dx)$ is defined for every
$\omega\in\Omega$, then

\Centerline{$\VVdP\,\int\vec\psi\,d\nu_{\check I}$ is defined and equal to
$\vec h$.}

\proof{{\bf (a)(i)} Suppose that we have two members
$((\vec f_0,\vec h_0),\Bbbone)$
and $((\vec f_1,\vec h_1),\Bbbone)$ of $\vec\psi$, and that
$E\in\Sigma\setminus\Cal I$ is such that
$E^{\ssbullet}\VVdP\,\vec f_0=\vec f_1$.   Then
$E^{\ssbullet}\VVdP\,\vec h_0=\vec h_1$.   \Prf\ Let $J\subseteq I$ be a
countable set such that $\psi$ factors through $\Omega\times\{0,1\}^J$, in
the sense that $\psi(\omega,x)=\psi(\omega,y)$ whenever $\omega\in\Omega$
and $x$, $y\in\{0,1\}^I$ are such that $x\restr J=y\restr J$.   For each
$i\in J$,

\Centerline{$E^{\ssbullet}\VVdP\,
\vec f_0(\varchecki)=\vec f_1(\varchecki)$,}

\noindent so that $f_0(\omega)(i)=f_1(\omega)(i)$ for
$\Cal I$-almost every
$\omega\in E$.   Consequently $f_0(\omega)\restr J=f_1(\omega)\restr J$ and

\Centerline{$h_0(\omega)=\psi(\omega,f_0(\omega))=\psi(\omega,f_1(\omega))
=h_1(\omega)$}

\noindent for $\Cal I$-almost every $\omega\in E$;  that is,
$E^{\ssbullet}\VVdP\,\vec h_0=\vec h_1$.\ \Qed

It follows that

\Centerline{$\VVdP\,\vec\psi$ is a function}

\noindent (5A3Ea).

\medskip

\quad{\bf (ii)} By the constructions in 551Cb and 551B,

\Centerline{$\VVdP\,\vec\psi\subseteq\{0,1\}^{\check I}\times\Bbb R$.}

\medskip

\quad{\bf (iii)} If $\dot x$ is a $\Bbb P$-name and $p\in\frak A^+$ is such
that $p\VVdP\,\dot x\in\{0,1\}^{\check I}$, then there
is a $(\Sigma,\CalBa_I)$-measurable $f:\Omega\to\{0,1\}^I$ such that
$p\VVdP\,\dot x=\vec f$ (551Cc again).
Setting $h(\omega)=\psi(\omega,f(\omega))$ for $\omega\in\Omega$,
$((\vec f,\vec h),\Bbbone)\in\vec\psi$, so

\Centerline{$p\VVdP\,\dot x=\vec f$ and $(\vec f,\vec h)\in\vec\psi$,
so $\dot x\in\dom(\vec\psi)$.}

\noindent As $p$ and $\dot x$ are arbitrary,

\Centerline{$\VVdP\,\dom(\vec\psi)=\{0,1\}^{\check I}$.}

\medskip

{\bf (b)} This is easy.   If $p\in\frak A^+$ and $\dot x$ is a
$\Bbb P$-name such that $p\VVdP\,\dot x\in\{0,1\}^{\check I}$, take a
$(\Sigma,\CalBa_I)$-measurable $f:\Omega\to\{0,1\}^I$ such that
$p\VVdP\,\dot x=\vec f$;  set

\Centerline{$h_0(\omega)=\phi(\omega,f(\omega))$,
\quad$h_1(\omega)=\psi(\omega,f(\omega))$}

\noindent for $\omega\in\Omega$;  then

$$\eqalign{p\VVdP\,
(\phi+\psi)\sspvec(\dot x)
&=(\phi+\psi)\sspvec(\vec f\,)
=(h_0+h_1)\sspvec\cr
&=\vec h_0+\vec h_1
=\vec\phi(\vec f\,)+\vec\psi(\vec f\,)
=\vec\phi(\dot x)+\vec\psi(\dot x),\cr
(\alpha\phi)\sspvec(\dot x)
&=(\alpha h_0)\sspvec
=\check\alpha\vec h_0
=\check\alpha\vec\phi(\dot x)\cr}$$

\noindent by 5A3Lc.   As $p$ and $\dot x$ are arbitrary,

\Centerline{$\VVdP\,(\phi+\psi)\sspvec=\vec\phi+\vec\psi$,
$(\alpha\phi)\sspvec=\check\alpha\vec\phi$.}

\medskip

{\bf (c)} In the same way,
if $p\in\frak A^+$ and $\dot x$ is a
$\Bbb P$-name such that $p\VVdP\,\dot x\in\{0,1\}^{\check I}$, take a
$(\Sigma,\CalBa_I)$-measurable $f:\Omega\to\{0,1\}^I$ such that
$p\VVdP\,\dot x=\vec f$.  Set

\Centerline{$h_n(\omega)=\psi_n(\omega,f(\omega))$,
\quad$h(\omega)=\psi(\omega,f(\omega))$}

\noindent for $\omega\in\Omega$ and $n\in\Bbb N$;  then
$h=\lim_{n\to\infty}h_n$, so

\Centerline{$p\VVdP\,\vec\psi(\dot x)=\vec h
=\lim_{n\to\infty}\vec h_n=\lim_{n\to\infty}\vec\psi_n(\dot x)$.}

\medskip

{\bf (d)} Take $p\in\frak A^+$  and a $\Bbb P$-name $\dot x$ such that
$p\VVdP\,\dot x\in\{0,1\}^{\check I}$.   Let $f:\Omega\to\{0,1\}^I$ be a
$(\Sigma,\CalBa_I)$-measurable function such that
$p\VVdP\,\dot x=\vec f$;  set $h(\omega)=\chi W(\omega,f(\omega))$
for $\omega\in\Omega$, so that

\Centerline{$\VVdP\,\vec h=(\chi W)\sspvec(\vec f\,)$,
\quad$p\VVdP\,\vec h=(\chi W)\sspvec(\dot x)$.}

\noindent If $p=E^{\ssbullet}$ where $E\in\Sigma\setminus\Cal I$,

$$\eqalignno{p\VVdP\,&(\chi W)\sspvec(\dot x)=1\cr
&\iff p\VVdP\,\vec h=1\cr
&\iff h(\omega)=1\text{ for }\Cal I\text{-almost every }\omega\in E\cr
&\iff (\omega,f(\omega))\in W\text{ for }\Cal I\text{-almost every }
  \omega\in E\cr
&\iff p\Bsubseteq\Bvalue{\vec f\in\vec W}\cr
\displaycause{551Ea}
&\iff p\VVdP\,\dot x\in\vec W;\cr}$$

\noindent similarly,

$$\eqalignno{p\VVdP\,&(\chi W)\sspvec(\dot x)=0\cr
&\iff h(\omega)=0\text{ for }\Cal I\text{-almost every }\omega\in E\cr
&\iff (\omega,f(\omega))\notin W\text{ for }\Cal I\text{-almost every }
   \omega\in E\cr
&\iff p\VVdP\,\dot x\notin\vec W.\cr}$$

\wheader{551N}{0}{0}{0}{36pt}
\noindent As $p$ and $\dot x$ are arbitrary,

\Centerline{$\VVdP\,
(\chi W)\sspvec=\chi\vec W$.}

\medskip

{\bf (e)} Assembling (a)-(d), and recalling 551Fa,
we see that the result is true when $\psi$ is
a linear multiple of the indicator function
of a set in $\Sigma\tensorhat\CalBa_I$, whenever $\psi$ is a sum
of such functions, and whenever $\psi$ is the limit of a sequence of such
sums;  that is, whenever $\psi$ is $(\Sigma\tensorhat\CalBa_I)$-measurable.

\medskip

{\bf (f)} Similarly, (d) and 551I tell us that the result is true for the
indicator function of a member of $\Sigma\tensorhat\CalBa_I$.   Once
again, we can move to a linear combination of such functions,
using (b), and thence to a non-negative
$(\Sigma\tensorhat\CalBa_I)$-measurable function,
using (c);  finally, with (b) again, we get the general case.
}%end of proof of 551N

\leader{551O}{Measure algebras}\cmmnt{ With a little more effort we can
get a representation of the standard measure algebras in the same style.}
Let $I$ be a set, $\nu_I$ the usual measure on $\{0,1\}^I$ and
$(\frak B_I,\bar\nu_I)$ its measure algebra.   It will be important to
appreciate that these are abbreviations for formulae in set theory with a
single parameter $I$;   so that if we have a forcing notion $\Bbb P$ and a
$\Bbb P$-name $\tau$, we shall have $\Bbb P$-names $\frak B_{\tau}$ and
$\bar\nu_{\tau}$, uniquely defined as soon as we have settled on the
exact formulations we wish to apply when interpreting the basic
constructions $\{\ldots\}$, $\Cal P$ in the forcing language.
Similarly, if we write
$\Bbb P_I=(\frak B_I^+,\Bsubseteqshort,1,\downarrow)$
for the forcing notion based on the Boolean algebra $\frak B_I$, this also
is a formula which can be interpreted in forcing languages.

\leader{551P}{Theorem} Let $(\Omega,\Sigma,\Cal I)$ be a non-trivial
$\omega_1$-saturated measurable space with negligibles
such that $\Sigma$ is closed under Souslin's operation.
Let $\Bbb P$ be the associated forcing notion,
$P\cmmnt{\mskip5mu=(\Sigma/\Sigma\cap\Cal I)^+}$
its underlying partially ordered set, and $I$ a set.   Set

\Centerline{$\Lambda=\Sigma\tensorhat\CalBa_I$,
\quad$\Cal J=\{W:W\in\Lambda$, $\nu_IW[\{\omega\}]=0$ for
$\Cal I$-almost every $\omega\in\Omega\}$;}

\noindent then $\Cal J$ is a $\sigma$-ideal of $\Lambda$\cmmnt{ (cf.\
527B)};  let $\frak C$ be the quotient algebra $\Lambda/\Cal J$.   For
$W\in\Lambda$ and $\omega\in\Omega$
set $h_W(\omega)=\nu_IW[\{\omega\}]$.
For $a\in\frak C$ let $\vec a$ be the $\Bbb P$-name

\Centerline{$\{(\vec W,\Bbbone):W\in\Lambda$, $W^{\ssbullet}=a\}$}

\noindent where the $\Bbb P$-names $\vec W$ are defined as in 551D.
Consider the $\Bbb P$-names

\Centerline{$\dot{\frak D}=\{(\vec a,\Bbbone):a\in\frak C\}$,
\quad$\dot\pi
=\{(((W^{\ssbullet})\sspvec,(\vec W)^{\ssbullet}),\Bbbone):
W\in\Lambda\}$.}

(a) $\VVdP\,\dot\pi$ is a bijection between $\dot{\frak D}$ and
$\frak B_{\check I}$.

\wheader{551P}{0}{0}{0}{48pt}

(b) If $a$, $b\in\frak C$, $V\in\Lambda$ and
$V^{\ssbullet}=a$, then

\Centerline{$\VVdP\,
\dot\pi(a\Bsymmdiff b)\sspvec=\dot\pi\vec a\Bsymmdiff\dot\pi\vec b$,
\quad$\dot\pi(a\Bcap b)\sspvec=\dot\pi\vec a\Bcap\dot\pi\vec b$,
\quad$\bar\nu_{\check I}(\dot\pi\vec a)=\vec h_V$,}

\noindent defining $h_V$ and $\vec h_V$ as in 551I.

(c) Let $\varepsilon:\Sigma/\Sigma\cap\Cal I\to\frak C$ be the canonical
map defined by the formula

\Centerline{$\varepsilon(E^{\ssbullet})=(E\times\{0,1\}^I)^{\ssbullet}$
for $E\in\Sigma$.}

\noindent If $p\in(\Sigma/\Sigma\cap\Cal I)^+$ and $a$,
$b\in\frak C$, then

\Centerline{$p\VVdP\,\dot\pi\vec a=\dot\pi\vec b$}

\noindent iff $a\Bcap\varepsilon(p)=b\Bcap\varepsilon(p)$.

\cmmnt{\medskip

\noindent{\bf Remarks} Note that in the formula

\Centerline{$\{(((W^{\ssbullet})\sspvec,(\vec W)^{\ssbullet}),\Bbbone):
W\in\Lambda\}$}

\noindent the first $^{\ssbullet}$ is
interpreted in the ordinary universe as the canonical map from
$\Lambda$ to $\frak C$, and the second is interpreted in the forcing
language as the canonical map from $\CalBa_{\check I}$ to
$\frak B_{\check I}$;  while among the brackets $(\ldots)$, some are just
separators, some are to be interpreted as an ordered-pair construction in
the ordinary universe, and some are to be interpreted as and ordered-pair
construction in the forcing language.    Similarly, in the formula

\Centerline{$\VVdP\,
\dot\pi(a\Bsymmdiff b)\sspvec=\dot\pi\vec a\Bsymmdiff\dot\pi\vec b$}

\noindent the first $\Bsymmdiff$ is to be interpreted in the ordinary
universe as symmetric difference in the algebra $\frak C$,
while the second is to be interpreted in the forcing language as symmetric
difference in $\frak B_{\check I}$.
}

\proof{{\bf (a)(i)}
$\VVdP\,\dot\pi$ is a function with domain $\dot{\frak D}$ and
$\dot\pi[\dot{\frak D}]=\dot B$.

\Prf\Quer\ Suppose, if possible, that $V$, $W\in\Lambda$ and
$E\in\Sigma\setminus\Cal I$ are such that

\Centerline{$E^{\ssbullet}\VVdP\,
(V^{\ssbullet})\sspvec=(W^{\ssbullet})\sspvec$,
$\vec V^{\ssbullet}\ne\vec W^{\ssbullet}$.}

\noindent By 551I(iii) and 551Eb,

\Centerline{$E^{\ssbullet}\VVdP\,\vec h_{V\symmdiff W}
=\nu_{\check I}(V\symmdiff W)\sspvec\ne 0$.}

\noindent On the other hand,

\Centerline{$E^{\ssbullet}\VVdP\,\vec V\in(V^{\ssbullet})\sspvec
=(W^{\ssbullet})\sspvec$,}

\noindent so there must be a $W_1\in\Lambda$ and an
$F\in\Sigma\setminus\Cal I$ such that $F^{\ssbullet}$ is stronger than
$E^{\ssbullet}$,
$W_1\symmdiff W\in\Cal J$ and $F^{\ssbullet}\VVdP\,\vec V=\vec W_1$.
Now, calculating in $\Sigma/\Sigma\cap\Cal I$,

$$\eqalignno{F^{\ssbullet}
&\Bsubseteq\{\omega:V[\{\omega\}]=W_1[\{\omega\}]\}^{\ssbullet}\cr
\displaycause{551Gb}
&\Bsubseteq
\{\omega:\nu_I(V[\{\omega\}]\symmdiff W_1[\{\omega\}])=0\}^{\ssbullet}
=\{\omega:\nu_I(V[\{\omega\}]\symmdiff W[\{\omega\}])=0\}^{\ssbullet}\cr
\displaycause{because $W\symmdiff W_1\in\Cal J$, so
$\nu_I(W[\{\omega\}]\symmdiff W_1[\{\omega\}])=0$ for
$\Cal I$-almost every $\omega$}
&=\Bvalue{\vec h_{V\symmdiff W}=0}\cr}$$

\noindent (551B);  which is impossible, because
$E^{\ssbullet}\Bsubseteq\Bvalue{\vec h_{V\symmdiff W}\ne 0}$.\ \BanG\
So 5A3E tells us that

\Centerline{$\VVdP\,\dot\pi$ is a function with domain $\dot{\frak D}$
and $\dot\pi[\dot{\frak D}]=\dot B$,}

\noindent where $\dot B$ is the $\Bbb P$-name
$\{(\vec W^{\ssbullet},\Bbbone):W\in\Lambda\}$.

\medskip

\quad{\bf (ii)} Now $\VVdP\,\dot\pi$ is injective.
\Prf\ I aim to use the condition (ii) in 5A3Eb.  I
take the argument in two bites.

\medskip

\qquad\grheada\Quer\
Suppose, if possible, that $V$, $W\in\Lambda$ and
$p\in P$ are such that $p\VVdP\,\vec V^{\ssbullet}=\vec W^{\ssbullet}$ but
$p\notVVdash_{\Bbb P}\,
(V^{\ssbullet})\sspvec\subseteq(W^{\ssbullet})\sspvec$.   Then there are
a $q\in P$, stronger than $p$, and a $\Bbb P$-name $\dot x$ such that

\Centerline{$q\VVdP\,
\dot x\in(V^{\ssbullet})\sspvec\setminus(W^{\ssbullet})\sspvec$.}

\noindent By the definition in 5A3Cb, there are an $r\in P$, stronger than
$q$, and a $V_1\in\Lambda$ such that
$V_1^{\ssbullet}=V^{\ssbullet}$ and $r\VVdP\,\dot x=\vec V_1$.
Let $E\in\Sigma\setminus\Cal I$ be such that $E^{\ssbullet}=r$, and set

\Centerline{$W_1=(V_1\cap(E\times\{0,1\}^I))
  \cup(W\cap((\Omega\setminus E)\times\{0,1\}^I))\in\Lambda$.}

\noindent Then $E^{\ssbullet}\Bsubseteq\Bvalue{\vec W_1=\vec V_1}$
(551Gb again).   At the same time,

\Centerline{$E^{\ssbullet}
\Bsubseteq\Bvalue{\vec V^{\ssbullet}=\vec W^{\ssbullet}}
=\Bvalue{\vec h_{V\symmdiff W}=0}$}

\noindent as in (i) just above.   Now

$$\eqalign{&\{\omega:\omega\in\Omega,\,
  \nu_I(W_1[\{\omega\}]\symmdiff W[\{\omega\}])>0\}
\cr&\mskip80mu
=\{\omega:\omega\in E,\,
  \nu_I(V_1[\{\omega\}]\symmdiff W[\{\omega\}])>0\}
\cr&\mskip80mu
\subseteq\{\omega:\omega\in\Omega,\,
   \nu_I(V_1[\{\omega\}]\symmdiff V[\{\omega\}])>0\}
 \cup\{\omega:\omega\in E,\,h_{V\symmdiff W}(\omega)>0\}
\cr&\mskip80mu
\in\Cal I.\cr}$$

\noindent But this means that $W_1^{\ssbullet}=W^{\ssbullet}$ and
$(\vec W_1,\Bbbone)\in(W^{\ssbullet})\sspvec$, so that

\Centerline{$\VVdP\,\vec W_1\in(W^{\ssbullet})\sspvec$,
\quad$r\VVdP\,\dot x=\vec V_1=\vec W_1\in(W^{\ssbullet})\sspvec$;}

\noindent but $r$ is stronger than $q$ and

\Centerline{$q\VVdP\,\dot x\notin(W^{\ssbullet})\sspvec$,}

\noindent so we have a contradiction.\ \Bang

\medskip

\qquad\grheadb\ Thus if $V$, $W\in\Lambda$ and
$p\in P$ are such that $p\VVdP\,\vec V^{\ssbullet}=\vec W^{\ssbullet}$,
we must have
$p\VVdP\,(V^{\ssbullet})\sspvec\subseteq(W^{\ssbullet})\sspvec$.
Similarly,
$p\VVdP\,(W^{\ssbullet})\sspvec\subseteq(V^{\ssbullet})\sspvec$ and
$p\VVdP\,(W^{\ssbullet})\sspvec=(V^{\ssbullet})\sspvec$.
Accordingly the condition 5A3Eb(ii) is satisfied and
$\VVdP\,\dot\pi$ is injective.\ \Qed

\medskip

\quad{\bf (iii)} We need to check that

\Centerline{$\VVdP\,\dot B=\frak B_{\check I}$.}

\medskip

\Prf\grheada\ Suppose that
$E\in\Sigma\setminus\Cal I$ and a $\Bbb P$-name $\dot x$ are such that
$E^{\ssbullet}\VVdP\,\dot x\in\dot B$.   Then there must be an
$F\in\Sigma\setminus\Cal I$ and a $W\in\Lambda$
such that $F^{\ssbullet}$ is stronger than $E^{\ssbullet}$ and
$F^{\ssbullet}\VVdP\,\dot x=\vec W^{\ssbullet}\in\frak B_{\check I}$;
as $E^{\ssbullet}$ and $\dot x$ are arbitrary,

\Centerline{$\VVdP\,\dot B\subseteq\frak B_{\check I}$.}

\medskip

\qquad\grheadb\ Suppose that
$E\in\Sigma\setminus\Cal I$ and a $\Bbb P$-name $\dot x$ are such that
$E^{\ssbullet}\VVdP\,\dot x\in\frak B_{\check I}$.   Then there must be a
$\Bbb P$-name $\dot W$ such that

\Centerline{$E^{\ssbullet}\VVdP\,\dot W\in\CalBa_{\check I}$ and
$\dot x=\dot W^{\ssbullet}$.}

\noindent By 551Fb, there is a $W\in\Lambda$ such that

\Centerline{$E^{\ssbullet}\VVdP\,\vec W=\dot W$,
so $\dot x=\vec W^{\ssbullet}\in\dot B$;}

\noindent as $E$ and $\dot x$ are arbitrary,

\Centerline{$\VVdP\,\frak B_{\check I}\subseteq\dot B$
and $\dot B=\frak B_{\check I}$.\ \Qed}

\medskip

\quad{\bf (iv)} Putting these together,

\Centerline{$\VVdP\,\dot\pi$ is a bijection between $\dot{\frak D}$ and
$\frak B_{\check I}$.}

\medskip

{\bf (b)} This is now easy.   If $V$, $W\in\Lambda$, $a=V^{\ssbullet}$ and
$b=W^{\ssbullet}$, then


$$\eqalignno{\VVdP\,\dot\pi(a\Bsymmdiff b)\sspvec
&=\dot\pi((V\symmdiff W)^{\ssbullet})\sspvec
=((V\symmdiff W)\sspvec)^{\ssbullet}
=(\vec V\symmdiff\vec W)^{\ssbullet}\cr
\displaycause{551Eb}
&=\vec V^{\ssbullet}\Bsymmdiff\vec W^{\ssbullet}
=\dot\pi\vec a\Bsymmdiff\dot\pi\vec b,\cr}$$

\noindent and similarly

\Centerline{$\VVdP\,\dot\pi(a\Bcap b)\sspvec
=\dot\pi\vec a\Bcap\dot\pi\vec b$.}

\noindent Finally,

\Centerline{$\VVdP\,\bar\nu_{\check I}(\dot\pi\vec a)
=\bar\nu_{\check I}\vec V^{\ssbullet}
=\nu_{\check I}\vec V
=\vec h_V$}

\noindent by 551I(iii) again.

\medskip

{\bf (c)} Let $E\in\Sigma\setminus\Cal I$ and $V$, $W\in\Lambda$ be such
that $E^{\ssbullet}=p$, $V^{\ssbullet}=a$ and $W^{\ssbullet}=b$.   Then
(b) tells us that

\Centerline{$\VVdP\,
\bar\nu_{\check I}(\dot\pi\vec a\Bsymmdiff\dot\pi\vec b)
=\bar\nu_{\check I}\dot\pi(a\Bsymmdiff b)\sspvec
=\bar\nu_{\check I}\dot\pi((V\Bsymmdiff W)^{\ssbullet})\sspvec
=\vec h_{V\symmdiff W}$.}

\noindent So

$$\eqalignno{p\VVdP\,\dot\pi\vec a=\dot\pi\vec b
&\iff p\VVdP\,\bar\nu_{\check I}(\dot\pi\vec a\Bsymmdiff\dot\pi\vec b)=0\cr
&\iff p\VVdP\,\vec h_{V\symmdiff W}=0\cr
&\iff h_{V\symmdiff W}(\omega)=0\text{ for }\Cal I\text{-almost every }
   \omega\in E\cr
\displaycause{551B}
&\iff \nu_I(V[\{\omega\}]\symmdiff W[\{\omega\}])=0
   \text{ for }\Cal I\text{-almost every }\omega\in E\cr
&\iff (E\times\{0,1\}^I)\cap(V\symmdiff W)\in\Cal J\cr
&\iff\varepsilon(p)\Bcap(a\Bsymmdiff b)=0
\iff a\Bcap\varepsilon(p)=b\Bcap\varepsilon(p).\cr}$$
}%end of proof of 551P

\leader{551Q}{Iterated \dvrocolon{forcing}}\cmmnt{ The machinery just
developed can
be used to establish one of the most important properties of
random real forcing.

\medskip

\noindent}{\bf Theorem} Let $(\Omega,\Sigma,\Cal I)$ be a non-trivial
$\omega_1$-saturated measurable space with negligibles
such that $\Sigma$ is closed under Souslin's operation,
$\Bbb P$ its associated forcing notion, and $I$ a set.    As in
551P, set $\Lambda=\Sigma\tensorhat\CalBa_I$,

\Centerline{$\Cal J=\{W:W\in\Lambda$, $\nu_IW[\{\omega\}]=0$ for
$\Cal I$-almost every $\omega\in\Omega\}$}

\noindent and $\frak C=\Lambda/\Cal J$.   Then

\Centerline{$\frak C\cong\RO(\Bbb P*\Bbb P_{\check I})$,}

\noindent where the $\Bbb P$-name $\Bbb P_{\check I}$ is defined as in
551O\cmmnt{, and $\Bbb P*\Bbb P_{\check I}$ is the iterated forcing notion
defined in 5A3O}.

\proof{{\bf (a)} Since I wish to follow {\smc Kunen 80} as closely as
possible, I should perhaps start with a remark on the interpretation of
names for forcing notions.   There is, strictly speaking, a distinction to
be made between a name for a forcing notion, which is a name for a
quadruplet of the form $(P,\le,\Bbbone,\updownarrows)$, and
a quadruplet of names, the first for a set, the second for a pre-order on
that set, and so on;  and the latter is easier to work with
({\smc Kunen 80}, \S VIII.5, and 5A3O).
In the present case, we do not need
any new manoeuvres, since the construction of
the name $\Bbb P_{\check I}$ is based
on $\Bbb P$-names for $\frak B_{\check I}$,
$\Bsubseteqshort_{\frak B_{\check I}}$ and
$1_{\frak B_{\check I}}$.

As usual in this section, I will write $\frak A$ for
$\Sigma/\Sigma\cap\Cal I$.

\medskip

{\bf (b)} Now $\Bbb P*\Bbb P_{\check I}$ is based on the set $P$ of pairs
$(p,\dot b)$ where $p\in\frak A^+$, $\dot b\in B$ and
$p\VVdP\,\dot b\in\frak B_{\check I}^+$;  here $B$ is the
domain of the $\Bbb P$-name $\frak B_{\check I}^+$ (5A3Ba).
If we say that
$(p,\dot b)\le(p',\dot b')$ if $p\Bsubseteq p'$ and
$p\VVdP\,\dot b\Bsubseteq\dot b'$, then $P$ is pre-ordered by $\le$ and
$\Bbb P*\Bbb P_{\check I}$ is active downwards.

We have a unique function $\theta:P\to\frak C^+$ such that

\Centerline{$\theta(p,\dot b)\Bsubseteq\varepsilon(p)$,
\quad$p\VVdP\,\dot\pi\theta(p,\dot b)\sspvec=\dot b$,}

\noindent whenever $(p,\dot b)\in P$, where $\varepsilon$, $\dot\pi$ and
$\vec a$, for $a\in\frak C$, are defined as
in 551P.    \Prf\ If $(p,\dot b)\in P$, so that
$p\VVdP\,\dot b\in\frak B_{\check I}$,
there is a $\Bbb P$-name $\dot b_1$ such that

\Centerline{$\VVdP\,\dot b_1\in\frak B_{\check I}$,}

\Centerline{$p\VVdP\,\dot b_1=\dot b$.}

\noindent Next, there is an
$a_0\in\frak C$ such that $\VVdP\,\dot\pi\vec a_0=\dot b_1$
(551Pa).   Set $a=a_0\Bcap\varepsilon(p)$.   Then 551Pc tells us that

\Centerline{$p\VVdP\,\dot\pi\vec a=\dot\pi\vec a_0
=\dot b_1=\dot b\ne 0$,}

\noindent and $a\ne 0$.
To see that $a$ is unique, observe that if $c\in\frak C$ is
such that $p\VVdP\,\dot\pi\vec c=\dot b$, then
$c\Bcap\varepsilon(p)=a\Bcap\varepsilon(p)$, by 551Pc again;  so if
$c\Bsubseteq\varepsilon(p)$, $c=a$.   We therefore can, and must,
take $a$ for $\theta(p,\dot b)$.\ \Qed

\wheader{551Q}{4}{2}{2}{42pt}

{\bf (c)(i)} If $(p,\dot b)$, $(p',\dot b')\in P$ and $(p,\dot b)$ is
stronger than $(p',\dot b')$, then $p\Bsubseteq p'$ and
$p\VVdP\,\dot b\Bsubseteq\dot b'$.   In this case,
\ifnum\stylenumber=12

\Centerline{$p\VVdP\,\dot\pi\theta(p',\dot b')\sspvec=\dot b'$ and
$\dot\pi(\theta(p,\dot b)\Bcap\theta(p',\dot b'))\sspvec
=\dot\pi\theta(p,\dot b)\sspvec\Bcap\dot\pi\theta(p',\dot b')\sspvec
=\dot b\Bcap\dot b'=\dot b$}
\else

\doubleinset{$p\VVdP\,\dot\pi\theta(p',\dot b')\sspvec=\dot b'$ and
$\dot\pi(\theta(p,\dot b)\Bcap\theta(p',\dot b'))\sspvec
=\dot\pi\theta(p,\dot b)\sspvec\Bcap\dot\pi\theta(p',\dot b')\sspvec
=\dot b\Bcap\dot b'=\dot b$,}
\fi

\noindent while
$\theta(p,\dot b)\Bcap\theta(p',\dot b')\Bsubseteq\varepsilon(p)$, so
$\theta(p,\dot b)\Bcap\theta(p',\dot b')=\theta(p,\dot b)$ and
$\theta(p,\dot b)\Bsubseteq\theta(p',\dot b')$.

\medskip

\allowmorestretch{468}{
\quad{\bf (ii)}
If $(p,\dot b)$ and $(p',\dot b')\in P$ are incompatible, then
$\theta(p,\dot b)\Bcap\theta(p',\dot b')=0$.   \Prf\Quer\ Otherwise,
writing $a$ for $\theta(p,\dot b)\Bcap\theta(p',\dot b')$,
}

\Centerline{$\varepsilon(p\Bcap p')=\varepsilon(p)\Bcap\varepsilon(p')
\Bsupseteq a\ne 0$}

\noindent so $p\Bcap p'\ne 0$ and

\Centerline{$p\Bcap p'\VVdP\,\dot\pi\vec a
\Bsubseteq\dot\pi\theta(p,\dot b)\sspvec
  \Bcap\dot\pi\theta(p',\dot b')\sspvec
=\dot b\Bcap\dot b'
=0$;}

\noindent as $a\Bsubseteq\varepsilon(p\Bcap p')$, $a$ must be $0$;  which
is absurd.\ \Bang\Qed

\medskip

\quad{\bf (iii)} If $a\in\frak C^+$, there is a
$(p,\dot b)\in P$ such that $\theta(p,\dot b)\Bsubseteq a$.   \Prf\
By 551Pc, $\notVVdash_{\Bbb P}\dot\pi\vec a=0$, that is, there is a
$p_0\in\frak C^+$ such that $p_0\VVdP\,\dot\pi\vec a\ne 0$.
Now there must be a
$p\in\frak C^+$, stronger than $p_0$,
and a $\dot b\in B$ such that
$p\VVdP\,\dot b=\dot\pi\vec a$, in which case $(p,\dot b)\in P$ and
$p\VVdP\,\dot\pi\theta(p,\dot b)\sspvec=\dot\pi\vec a$.   Accordingly

\Centerline{$\theta(p,\dot b)=\theta(p,\dot b)\Bcap\varepsilon(p)
=a\Bcap\varepsilon(p)\Bsubseteq a$.   \Qed}

\medskip

{\bf (d)} Observe now that $\frak C$ is ccc (527L), therefore Dedekind
complete, and (c) tells us that
$\theta:P\to\frak C^+$ satisfies the conditions
of 514Sa.     So
$\RO(\Bbb P*\Bbb P_{\check I})=\RO^{\downarrow}(P)$ is
isomorphic to $\frak C$.
}%end of proof of 551Q

\leader{551R}{Extending \dvrocolon{filters}}\cmmnt{ The following
device will be useful in \S553.
%553N

\medskip

\noindent}{\bf Proposition} Let $(\Omega,\Sigma,\Cal I)$ be a non-trivial
$\omega_1$-saturated
measurable space with negligibles, $\frak A$ its quotient algebra,
$\Bbb P$ the associated forcing notion,
$I$ a countable set and $\Cal F$ a filter on $I$.

(a) For $H\in\Sigma\tensorhat\Cal PI$,
write $\vec H$ for the $\Bbb P$-name
$\{(\varchecki,H^{-1}[\{i\}]^{\ssbullet}):
i\in I$, $H^{-1}[\{i\}]\notin\Cal I\}$.

\quad(i) $\VVdP\,\vec H\subseteq\check{I}$.

\quad(ii) If $\dot F$ is a $\Bbb P$-name and $p\in\frak A^+$ is such that
$p\VVdP\,\dot F\subseteq\check{I}$, then there is an
$H\in\Sigma\tensorhat\Cal P I$ such that
$p\VVdP\,\dot F=\vec H$.

(b) Write $\vec{\Cal F}$ for the $\Bbb P$-name

\Centerline{$\{(\vec H,E^{\ssbullet}):H\in\Sigma\tensorhat\Cal P I$,
$E\in\Sigma\setminus\Cal I$, $H[\{\omega\}]\in\Cal F$ for every
$\omega\in E\}$.}

\noindent Then

\Centerline{$\VVdP\,\vec{\Cal F}$ is a filter on $\check I$.}

\proof{{\bf (a)(i)} is elementary, just because

\Centerline{$\VVdP\,\varchecki\in\check{I}$}

\noindent for every $i\in I$.

\medskip

\quad{\bf (ii)} Because $(\Omega,\Sigma,\Cal I)$ is $\omega_1$-saturated,
$\frak A$ is Dedekind complete and can be identified with
$\RO(\Bbb P)$.   We therefore have, for each $i\in I$, an
$E_i\in\Sigma$ such that $E_i^{\ssbullet}$ can be identified with
$\Bvalue{\varchecki\in\dot F}$.   Set $H=\bigcup_{i\in I}E_i\times\{i\}$.
Then

\Centerline{$\Bvalue{\varchecki\in\dot F}
=H^{-1}[\{i\}]^{\ssbullet}=\Bvalue{\varchecki\in\vec H}$}

\noindent for every $i\in I$, so

\Centerline{$p\VVdP\,\dot F=\dot F\cap\check{I}=\vec H\cap\check{I}=\vec H$.}

\medskip

{\bf (b)(i)} By (a-i), $\VVdP\,\vec{\Cal F}\subseteq\check{I}$.

\medskip

\quad{\bf (ii)}
Since $((\Omega\times I)\sspvec,\Bbbone)\in\vec{\Cal F}$ and

\Centerline{$\VVdP\,(\Omega\times I)\sspvec=\check{I}$}

\noindent (551D), we have

\Centerline{$\VVdP\,\check{I}\in\vec{\Cal F}$.}

\medskip

\quad{\bf (iii)} If $(\vec H,p)\in\vec{\Cal F}$ then
$p\VVdP\,\vec H\ne\emptyset$.   \Prf\ Express $p$ as $E^{\ssbullet}$ where
$E\in\Sigma$ and $H[\{\omega\}]\in\Cal F$ for every $\omega\in E$.
Then $E\subseteq\bigcup_{i\in I}H^{-1}[\{i\}]$.   So if $q\in\frak A^+$
is stronger than $p$, there must be an $i\in I$ such that
$r=q\Bcap H^{-1}[\{i\}]^{\ssbullet}$ is non-zero;  in which case $r$ is
stronger than $q$ and

\Centerline{$r\VVdP\,\varchecki\in\vec H$, so $\vec H\ne\emptyset$.}

\noindent As $q$ is arbitrary, $p\VVdP\,\vec H\ne\emptyset$.\ \Qed

It follows at once that $\VVdP\,\emptyset\notin\vec{\Cal F}$.

\medskip

\quad{\bf (iv)} If $\dot F_0$, $\dot F_1$ are $\Bbb P$-names and
$p\in\frak A^+$ is such that

\Centerline{$p\VVdP\,\dot F_0$, $\dot F_1\in\vec{\Cal F}$,}

\noindent then

\Centerline{$p\VVdP\,\dot F_0\cap\dot F_1\in\vec{\Cal F}$.}

\noindent\Prf\ If $q\in\frak A^+$ is stronger than $p$ there must be
$(\vec H_0,E_0^{\ssbullet})$, $(\vec H_1,E_1^{\ssbullet})\in\vec{\Cal F}$
and an $r$ stronger than $q$, $E_0^{\ssbullet}$ and
$E_1^{\ssbullet}$ such that

\Centerline{$r\VVdP\,\dot F_0=\vec H_0$ and $\dot F_1=\vec H_1$.}

\noindent Now
$((H_0\cap H_1)\sspvec,(E_0\cap E_1)^{\ssbullet})\in\vec{\Cal F}$ and

\Centerline{$r\VVdP\,\dot F_0\cap\dot F_1=\vec H_0\cap\vec H_1
=(H_0\cap H_1)\sspvec\in\vec{\Cal F}$.}

\noindent As $q$ is arbitrary,

\Centerline{$p\VVdP\,\dot F_0\cap\dot F_1\in\vec{\Cal F}$.\ \Qed}

Accordingly

\Centerline{$\VVdP\,\vec{\Cal F}$ is closed under $\cap$.}

\medskip

\quad{\bf (v)} Suppose that $\dot F_0$, $\dot F_1$ are $\Bbb P$-names and
$p\in\frak A^+$ is such that

\Centerline{$p\VVdP\,\dot F_0\subseteq\dot F_1\subseteq\check{I}$,
$\dot F_0\in\vec{\Cal F}$.}

\noindent By (a-ii), there is an $H_1\in\Sigma\tensorhat\Cal P I$ such
that $p\VVdP\,\dot F_1=\vec H_1$.   If $q$ is stronger than $p$, there are
an $(\vec H_0,E_0^{\ssbullet})\in\vec{\Cal F}$ and an $r$ stronger than
both $q$ and $E_0^{\ssbullet}$ such that

\Centerline{$r\VVdP\,\vec H_0=\dot F_0\subseteq\dot F_1=\vec H_1$.}

\noindent Expressing $r$ as $E^{\ssbullet}$ where
$E\in\Sigma\setminus\Cal I$, we have

\Centerline{$E\cap H_0^{-1}[\{i\}]\setminus H_1^{-1}[\{i\}]\in\Cal I$}

\noindent for every $i\in I$.   Set

\Centerline{$E_1
=E\setminus\bigcup_{i\in I}(H_0^{-1}[\{i\}]\setminus H_1^{-1}[\{i\}])$;}

\noindent then $(\vec H_1,E_1^{\ssbullet})\in\vec{\Cal F}$, so

\Centerline{$r=E_1^{\ssbullet}\VVdP\,\dot F_1=\vec H_1\in\vec{\Cal F}$.}

\noindent As $q$ is arbitrary,

\Centerline{$p\VVdP\,\dot F_1\in\vec{\Cal F}$.}

\noindent As $p$, $\dot F_0$ and $\dot F_1$ are arbitrary,

\Centerline{$\VVdP\,\vec{\Cal F}$ is a filter on $\check{I}$.}
}%end of proof of 551R

\exercises{\leader{551X}{Basic exercises (a)}
%\spheader 551Xa
Let $(\Omega,\Sigma,\Cal I)$ be any measurable space with
negligibles.   Set $\hat\Sigma=\{E\symmdiff F:E\in\Sigma$, $F\in\Cal I\}$.
(i) Show that $(\Omega,\hat\Sigma,\Cal I)$ is a complete measurable space
with negligibles;  we may call it the {\bf completion} of
$(\Omega,\Sigma,\Cal I)$.
(ii) Show that the algebras $\Sigma/\Sigma\cap\Cal I$ and
$\hat\Sigma/\Cal I$ are canonically isomorphic (cf.\ 322Da).
(iii) Show that $(\Omega,\hat\Sigma,\Cal I)$ is $\omega_1$-saturated iff
$(\Omega,\Sigma,\Cal I)$ is.
%551A

\spheader 551Xb
Let $(\Omega,\Sigma,\mu)$ be a measure space
and $\Cal N(\mu)$ the null ideal of $\mu$.  (i) Show that
$(\Omega,\Sigma,\Cal N(\mu))$ is a measurable space with negligibles.
(ii) Show that if the completion of $(\Omega,\Sigma,\mu)$ (212C) is
$(\Omega,\hat\Sigma,\hat\mu)$,
then $(\Omega,\hat\Sigma,\Cal N(\hat\mu))$
is the completion of $(\Omega,\Sigma,\Cal N(\mu))$.
%551A

\spheader 551Xc Let $X$ be a topological space, $\Cal B(X)$ the Borel
$\sigma$-algebra of $X$, $\widehat{\Cal B}(X)$ the Baire-property algebra of $X$
 and $\Cal M$ the ideal of meager subsets of $X$.
(i) Show that $(X,\Cal B(X),\Cal M)$ is a measurable space with negligibles.
(ii) Show that its completion is $(X,\widehat{\Cal B}(X),\Cal M)$.
%551A

\spheader 551Xd Let $(\Omega,\Sigma,\Cal I)$ be a non-trivial measurable
space with negligibles, and $\Bbb P$ the associated forcing notion.
(i) Show that
the regular open algebra of $\Bbb P$ can be identified with the Dedekind
completion of $\Sigma/\Sigma\cap\Cal I$.   (ii) Show that if
$\Cal E$ is any coinitial subset of $\Sigma\setminus\Cal I$ containing
$\Omega$, then the forcing notion $\Cal E$, active downwards,
has regular open algebra isomorphic to $\RO(\Bbb P)$.
%551A

\spheader 551Xe Let $(\Omega,\Sigma,\Cal I)$ and $(\Upsilon,\Tau,\Cal J)$
be measurable spaces with negligibles.   Show that
$(\Omega\times\Upsilon,\Sigma\tensorhat\Tau,
\Cal I\ltimes_{\Sigma\tensorhat\Tau}\Cal J)$,
as defined in 527Bc, is a measurable space with negligibles.
%551A

\spheader 551Xf Let $(\Omega,\Sigma,\Cal I)$ be a non-trivial measurable
space with negligibles, and $\Bbb P$ the associated forcing notion.   Let
$f:\Omega\to\{0,1\}$ be a measurable function, and define $\Bbb P$-names
$\dot x$, $\dot y$ by saying that $\dot x$ is the $\Bbb P$-name for a
real number as defined from $f$ in 551B, while
$\dot y$ is the $\Bbb P$-name for a member of $\{0,1\}$
as defined in 551Ca.   Show that

\Centerline{$\VVdP$ regarding $0$ and $1$ as real numbers,
$\dot x=\dot y$.}
%551C

\spheader 551Xg Let $(\Omega,\Sigma,\Cal I)$ be a non-trivial measurable
space
with negligibles, $\Bbb P$ the associated forcing notion, and $I$ a set.
Suppose that $f:\Omega\to\{0,1\}^I$ is a $(\Sigma,\CalBa_I)$-measurable
function, and that $\Gamma_f\subseteq\Omega\times\{0,1\}^I$ is its graph
(for once, I distinguish between $f$ and $\Gamma_f$).   Let $\vec f$ and
$\vec\Gamma_f$ be the $\Bbb P$-names for a point of $\{0,1\}^{\check I}$
and a subset of $\{0,1\}^{\check I}$ defined by the formulae in
551C and 551D respectively.   Show that
$\VVdP\,\vec\Gamma_f=\{\vec f\,\}$.
%551D

\spheader 551Xh Let $(\Omega,\Sigma,\Cal I)$ be a non-trivial measurable
space with negligibles, and $(\Omega,\hat\Sigma,\Cal I)$ its completion;
write $\Bbb P$, $\hat{\Bbb P}$ for the associated forcing notions, so that
$\Bbb P$ and $\hat{\Bbb P}$ are canonically isomorphic.
Let $I$ be a set, $W$ a member of
$\Sigma\tensorhat\{0,1\}^I\subseteq\hat\Sigma\tensorhat\{0,1\}^I$ and
$\vec W$ the $\Bbb P$-name, $\hat{\vec W}$ the $\hat{\Bbb P}$-name defined
by the formula in 551D.   Explain what it ought to mean to say that
$\VVdash_{\hat{\Bbb P}}\,\hat{\vec W}=\vec W$, and why this is true.
%551D 551Xa

\spheader 551Xi Let $(\Omega,\Sigma,\Cal I)$ be a non-trivial
$\omega_1$-saturated measurable space with negligibles with a Dedekind
complete quotient algebra,
$\Bbb P$ the associated forcing notion, and $I$ a set.   Suppose that
$W\in\Sigma\tensorhat\CalBa(\{0,1\}^I)$ is such that
$\VVdP\,\vec W\in\CalBa_{\check\alpha}(\{0,1\}^{\check I})$,
where $\alpha<\omega_1$.
Show that $W[\{\omega\}]\in\CalBa_{\alpha}(\{0,1\}^I)$ for
$\Cal I$-almost every $\omega\in\Omega$.
%551K

\leader{551Y}{Further exercises (a)}
%\spheader 551Ya
Investigate the difficulties which arise if we try to represent
names for Borel subsets of $\{0,1\}^{\check I}$ as members of
$\Sigma\tensorhat\{0,1\}^I$, when $I$ is uncountable.
Show that some of these are
resolvable if $\Omega$ is actually the Stone space of $\RO(\Bbb P)$.
%551F

\spheader 551Yb Let $(\Omega,\Sigma,\Cal I)$ be a non-trivial
measurable space
with negligibles, $\Bbb P$ the associated forcing notion and
$P$ the partially ordered set underlying $\Bbb P$.
(i) Let $p\in P$
and a $\Bbb P$-name $\dot G$ be such that

\Centerline{$p\VVdP\,\dot G$ is a dense open subset of $\{0,1\}^{\omega}$.}

\noindent Show that there is a $W\in\Sigma\tensorhat\CalBa_{\omega}$ such that
every vertical section of $W$ is a dense open set and $p\VVdP\,\dot G=\vec W$.
(ii) Let $p\in P$
and a $\Bbb P$-name $\dot A$ be such that

\Centerline{$p\VVdP\,\dot A$ is a meager subset of $\{0,1\}^{\omega}$.}

\noindent Show that there is a $W\in\Sigma\tensorhat\CalBa_{\omega}$ such that
every vertical section of $W$ is a meager set and
$p\VVdP\,\dot A\subseteq\vec W$.
%551F

}%end of exercises

\endnotes{
\Notesheader{551}
There are real metamathematical difficulties in forcing, and we need to
find new compromises between formal rigour and intuitive accessibility.
In the formulae
of this section I am taking a path with rather more explicit
declarations than is customary.   The definitions
of $\vec u$ in 5A3L, $\vec f$ in 551Ca and $\vec W$ in 551D are
supposed to be $\Bbb P$-names in the exact sense used in {\smc Kunen 80}.
This leads to rather odd sentences of the form

\Centerline{$(\vec f,p)\in\vec W$ so $p\VVdP\,\vec f\in\vec W$}

\noindent (as in (a-ii) of the proof of 551E, for example),
in which the symbol $\in$ is
being used in different ways in the two halves;
but it has the advantage that
we can move from $W$ to $\vec W$ without further explanation, as in the
statements of 551E-551J.  %551E 551F 551I 551J
But you will observe that elsewhere I allow such terms as
$\CalBa$ and $\nu_{\ldots}$ to enter sentences in the forcing language,
since these correspond to definitions which can be expanded there.
Note that I am being less strict than usual in requiring formulae to
be unambiguous (see 551Xf and 551Xg).

There is always room for variation in the matter of which terms
should be decorated with $\varBspcheck$s when they appear in expressions of the
forcing language, and I have tried to be reasonably consistent;  but there
are particular difficulties with transferring names for families (5A3Fb),
which appear here in such formulae as
`$\VVdP\,\lim_{n\to\infty}\vec h_n=\lim_{n\to\infty}\vec\psi_n(\dot x)$'
(part (c) of the proof of 551N).

I hope that it is not too confusing
to have the formula $\Bvalue{\ldots}$ used in two different ways, not
infrequently in the same sentence:  sometimes as a
`Boolean value' in the forcing sense, and sometimes in the sense of
Chapter 36.   If you look back at the definitions in \S364 you will see
that the expression $f^{\ssbullet}$
also shifts in interpretation as we move between the
formally distinct algebras $\frak A$ and $\RO(\Bbb P)$.   There are some
particularly difficult formulae to parse in 551P;  following the statement
of the theorem I offer a remark on
the expression $(((W^{\ssbullet})\sspvec,(\vec W)^{\ssbullet}),\Bbbone)$,
and some of the same difficulties arise in the line

\Centerline{$\VVdP\,\dot\pi(a\Bsymmdiff b)\sspvec
=\dot\pi((V\symmdiff W)^{\ssbullet})\sspvec
=((V\symmdiff W)\sspvec)^{\ssbullet}
=(\vec V\symmdiff\vec W)^{\ssbullet}$}

\noindent in part (b) of the proof, where as well as the ambiguities in
$^{\ssbullet}$ and $(\ldots)$ we have the symbols $\Bsymmdiff$ and
$\symmdiff$ being used first for an operation in the ground-model
Boolean algebra $\frak C$, then for symmetric difference in the ordinary
universe, and finally for symmetric
difference in the forcing language.
}%end of notes

\discrpage



\frfilename{mt552.tex}
\versiondate{29.1.14}
\copyrightdate{2007}

\Loadfourteens

\def\chaptername{Possible worlds}
\def\sectionname{Random reals I}

\def\BbbPk{\Bbb P_{\kappa}}
\def\eightVVdash{\mskip5mu\vrule height 6pt depth 2pt width 0.4pt
  \mskip2mu\vrule height 6pt depth 2pt width 0.4pt
  \vrule height 2.25pt depth -1.8pt width 3.2pt\mskip2mu}
\def\VVdPk{\VVdash_{\Bbb P_{\kappa}}}


\newsection{552}

From the point of view of a measure theorist, `random real forcing' has a
particular significance.   Because the forcing notions are defined directly
from the central structures of measure theory (552A), they can be
expected to provide worlds in which measure-theoretic questions are
answered.   Thus we find ourselves with many Sierpi\'nski sets (552E),
information on cardinal functions
(552C, 552F-552J), %552F 552G 552H 552I 552J
and theorems on extension of measures (552N).   But there is a
second reason why any measure theorist or probabilist should pay attention
to random real forcing.   Natural questions in the forcing language, when
translated into propositions about the ground model, are likely to hinge on
properties of measure algebras, giving us a new source of challenging
problems.   Perhaps the deepest intuitions are those associated with
iterated random real forcing (552P).

\leader{552A}{Notation (a)} As usual, if $\mu$ is a measure then
$\Cal N(\mu)$ will be its null ideal.   It
will be convenient to have a special notation for certain sets of
finite functions:  if $I$ is a set,
$\Fn_{<\omega}(I;\{0,1\})$ will be
$\bigcup_{K\in[I]^{<\omega}}\{0,1\}^K$.

For any set $I$ I\cmmnt{ will} write $\nu_I$
for the usual completion regular Radon probability
measure on $\{0,1\}^I$, $\Tau_I$ for its domain and
$(\frak B_I,\bar\nu_I)$ for its measure algebra;
$\CalBa_I\cmmnt{\mskip5mu=\CalBa(\{0,1\}^I)}$
will be the Baire $\sigma$-algebra of $\{0,1\}^I$.
\cmmnt{(It will sometimes be convenient, when
applying the results of \S551, to regard $\frak B_I$ as the quotient
$\CalBa_I/\CalBa_I\cap\Cal N(\nu_I)$.)}   \cmmnt{In this context,} I\cmmnt{ will}
write $\familyiI{e_i}$ for the standard generating family in
$\frak B_I$\cmmnt{ (525A)}.   $\Bbb P_I$ will be the forcing notion
$\frak B_I^+=\frak B_I\setminus\{0\}$, active downwards.
For a formula $\phi$
in the corresponding forcing language I\cmmnt{ will} write $\Bvalue{\phi}$ for the
truth value of $\phi$, interpreted as a member of
$\frak B_I$\cmmnt{ (5A3M)}.
\cmmnt{Note that as }$\Bbb P_I$\cmmnt{ is ccc (cf.\ 511Db),
it} preserves cardinals\cmmnt{ (5A3Nb)}.

As in \S551, the formulae $\nu_I$, $\frak B_I$ etc.\ are to be regarded as
formulae of set theory with one free variable into which the parameter $I$
has been substituted, so that we have corresponding names $\nu_{\dot I}$,
$\frak B_{\dot I}$ in any forcing language, and in particular (once the
context has established a forcing notion $\Bbb P$) we have $\Bbb P$-names
$\nu_{\check I}$, $\frak B_{\check I}$ for any ground-model set $I$.

\spheader 552Ab A great deal of the work of this chapter will involve
interpretations of names for standard objects\cmmnt{ (in particular, for
cardinals)} in forcing languages.
\cmmnt{Reflecting suggestions in 5A3H and 5A3N,}
I\cmmnt{ will} try to signal intended interpretations by using the superscript
$\varBspcheck$.
\cmmnt{Thus $\frak c$ will always be an abbreviation for `the
initial ordinal equipollent with the set of subsets of the natural
numbers', whether I am using the ordinary
language of set theory or speaking in a forcing language;  and
$\check{\frak c}$, in a forcing language, will refer to the name
$\{(\check\xi,\Bbbone):\xi<\frak c\}$, where it is to be understood that
the symbol $\frak c$ must now
be interpreted in the ordinary universe.   As I
shall avoid arguments involving more than one forcing notion (and, in
particular, iterated forcing), there will I hope be little scope for
confusion, even in such sentences as

\Centerline{$\VVdPk\,\frak b=\check{\frak b}$}

\noindent (552C).   The leading $\VVdPk$ declares that the rest of the
sentence is in the language of $\Bbb P_{\kappa}$-forcing;
the first $\frak b$, and the $\varBspcheck$, are therefore to be
interpreted in that language;  but the second $\frak b$, being subject to
the $\mskip5mu\varBspcheck$, is to be interpreted in the ground model.
(Many authors would write $\frak b^V$ at this point.)   Similarly, in

\Centerline{$\VVdPk\,2^{\check\lambda}=(\kappa^{\lambda})\varBspcheck$}

\noindent (552B), the subformula $\kappa^{\lambda}$ is to be interpreted in
the ordinary universe, but $2^{\check\lambda}=\#(\Cal P\check\lambda)$ is
to be interpreted in the forcing language.   I hope that
the resulting gains in directness and conciseness
will not be at the expense of leaving you uncertain of the meaning.}

\leader{552B}{Theorem} Suppose that $\lambda$ and $\kappa$ are infinite
cardinals.   Then

\Centerline{$\VVdPk\,2^{\check\lambda}=(\kappa^{\lambda})\varBspcheck$,}

\noindent where $\kappa^{\lambda}$ is the cardinal power\cmmnt{
(interpreted in the ordinary universe, of course)}.

\proof{{\bf (a)} Recall that $\#(\frak B_{\kappa})=\kappa^{\omega}$
(524Ma), so that

\Centerline{$\#(\frak B_{\kappa}^{\lambda})
=\#(\kappa^{\omega\times\lambda})$.}

If $\dot A$ is a $\BbbPk$-name for a subset of
$\check\lambda$, then we have a corresponding family
$\ofamily{\eta}{\lambda}{\Bvalue{\check\eta\in\dot A}}$ of truth values;
and if $\dot A$, $\dot B$ are two such names, and
$\Bvalue{\check\eta\in\dot A}=\Bvalue{\check\eta\in\dot B}$
for every $\eta<\lambda$, then

\Centerline{$\VVdPk\,\check\eta\in\dot A\iff\check\eta\in\dot B$}

\noindent for every $\eta<\lambda$, so

\Centerline{$\VVdPk\,\dot A=\dot B$.}

\noindent So

\Centerline{$\VVdPk\,2^{\check\lambda}=\#(\Cal P\check\lambda)
\le\#((\frak B_{\kappa}^{\lambda})\varBspcheck)=(\kappa^{\lambda})\varBspcheck$.}

\medskip

{\bf (b)} In the other direction,
consider first the case in which $\lambda\le\kappa$.
Let $F$ be the set of all functions from $\lambda$ to $\kappa$,
so that $\#(F)=\kappa^{\lambda}$.
Then there is a set $G\subseteq F$ such that
$\#(G)=\kappa^{\lambda}$ and $\{\eta:\eta<\lambda$, $f(\eta)\ne g(\eta)\}$
is infinite
whenever $f$, $g\in G$ are distinct.   \Prf\ If $\kappa=\kappa^{\lambda}$ we can take
$G$ to be the set of constant functions.   Otherwise, for $f$, $g\in F$,
say that $f=^*g$ if
$\{\eta:f(\eta)\ne g(\eta)\}$ is finite;  this is an equivalence relation.
Let $G\subseteq F$ be a set meeting
each equivalence class in just one element.   Then we have
$\#(\{g:g=^*f\})=\kappa<\kappa^{\lambda}$ for every $f\in F$, so $\#(G)=\kappa^{\lambda}$, as
required.\ \Qed

Let $\langle e_{\xi\eta}\rangle_{\xi<\kappa,\eta<\lambda}$ be a
stochastically independent family in $\frak B_{\kappa}$ of elements of
measure $\bover12$.   For $f\in G$ let $\dot A_f$ be a $\BbbPk$-name for
a subset of $\check\lambda$ such that

\Centerline{$\Bvalue{\check\eta\in\dot A_f}=e_{f(\eta),\eta}$}

\noindent for every $\eta<\lambda$.   If $f$, $g\in G$ are distinct, set
$I=\{\eta:f(\eta)\ne g(\eta)\}$;  then

\Centerline{$\Bvalue{\dot A_f\ne\dot A_g}
=\sup_{\eta<\lambda}e_{f(\eta),\eta}\Bsymmdiff e_{g(\eta),\eta}
=\sup_{\eta\in I}e_{f(\eta),\eta}\Bsymmdiff e_{g(\eta),\eta}
=1$}

\noindent because
$\family{\eta}{I}{e_{f(\eta),\eta}\Bsymmdiff e_{g(\eta),\eta}}$
is an infinite
stochastically independent family of elements of measure $\bover12$.

Thus in the forcing language we have a name for an injective function from
$\check G$ to $\Cal P\check\lambda$,
corresponding to the map $f\mapsto\dot A_f$
from $G$ to names of subsets of $\lambda$.   So

\Centerline{$\VVdPk\,2^{\check\lambda}\ge\#(\check G)
=(\kappa^{\lambda})\varBspcheck$.}

\noindent Putting this together with (a), we have

\Centerline{$\VVdPk\,2^{\check\lambda}=(\kappa^{\lambda})\varBspcheck$.}

\medskip

{\bf (c)} If $\lambda>\kappa$, then (in the ordinary universe)
$2^{\lambda}=\kappa^{\lambda}$.   Now

\Centerline{$\VVdPk\,(\Cal P\lambda)\varBspcheck
\subseteq\Cal P\check\lambda$,}

\noindent so

\Centerline{$\VVdPk\,(\kappa^{\lambda})\varBspcheck=\#((\Cal P\lambda)\varBspcheck)
\le\#(\Cal P\check\lambda)=2^{\check\lambda}$,}

\noindent and again we have

\Centerline{$\VVdPk\,2^{\check\lambda}=(\kappa^{\lambda})\varBspcheck$.}
}%end of proof of 552B

\leader{552C}{Theorem} Let $\kappa$ be any cardinal.   Then

\Centerline{$\VVdPk\,\frak b=\check{\frak b}$ and
$\frak d=\check{\frak d}$.}

\proof{{\bf (a)} The point is that if $\dot f$ is any $\BbbPk$-name for a
member of $\BbbN^{\Bbb N}$, then there is an $h\in\BbbN^{\Bbb N}$ such that

\Centerline{$\VVdPk\,\dot f\le^*\check h$,}

\noindent where I write $f\le^*g$ to mean that $\{n:g(n)<f(n)\}$ is finite,
as in 522C.   \Prf\ For $n$, $i\in\Bbb N$ set
$a_{ni}=\Bvalue{\dot f(\check n)=\varchecki}$.
Then $D_n=\{a_{ni}:i\in\Bbb N\}$ is a partition of unity in
$\frak B_{\kappa}$ for each $n\in\Bbb N$.   Because $\frak B_{\kappa}$ is
\wsid\ (322F), there is a partition of unity $D$ such that
$\{i:a_{ni}\Bcap d\ne 0\}$ is finite for each $n$ and each $d\in D$.
Let $\sequence{k}{d_k}$ be a sequence running over $D$ and take $h(n)$ such
that $a_{mi}\Bcap d_n=0$ whenever $m\le n$ and $i>h(n)$.   Now

\Centerline{$\Bvalue{\check h(\check m)<\dot f(\check m)}
=\Bvalue{h(m)\varBspcheck<\dot f(\check m)}
=\sup\{a_{mi}:i>h(m)\}\Bsubseteq 1\Bsetminus d_n$}

\noindent whenever $n\le m$.   So

$$\eqalign{\Bvalue{\check h(n)<\dot f(n)\text{ for infinitely many }n}
&=\inf_{n\in\Bbb N}\sup_{m\ge n}
  \Bvalue{h(m)\varBspcheck<\dot f(\check m)}\cr
&\Bsubseteq\inf_{n\in\Bbb N}1\Bsetminus d_n
=0,\cr}$$

\noindent that is,

\Centerline{$\VVdPk\,\dot f\le^*\check h$.  \Qed}

\medskip

{\bf (b)(i)} Let $\ofamily{\xi}{\lambda}{\dot f_{\xi}}$ be a family of
$\BbbPk$-names for members of $\BbbN^{\Bbb N}$, where $\lambda<{\frak b}$.
Then for each $\xi<\lambda$ we can find an $h_{\xi}\in\BbbN^{\Bbb N}$ such
that $\VVdPk\,\dot f_{\xi}\le^*\check h_{\xi}$.   As $\lambda<{\frak b}$, there
is an $h\in\BbbN^{\Bbb N}$ such that $h_{\xi}\le^*h$ for every
$\xi<\lambda$.   Now $\VVdPk\,\check h_{\xi}\le^*\check h$ for every $\xi$, so
$\VVdPk\,\dot f_{\xi}\le^*\check h$ for every $\xi$.   As $\lambda$ and
$\ofamily{\xi}{\lambda}{\dot f_{\xi}}$ are arbitrary,

\Centerline{$\VVdPk\,\frak b\ge\check{\frak b}$.}

\medskip

\quad{\bf (ii)} Let $\ofamily{\xi}{{\frak b}}{h_{\xi}}$ be a family in
$\BbbN^{\Bbb N}$ which has no $\le^*$-upper bound in $\BbbN^{\Bbb N}$.
Then

\Centerline{$\VVdPk\,\{\check h_{\xi}:\xi<\check{\frak b}\}$
has no $\le^*$-upper bound.}

\noindent\Prf\Quer\ Otherwise, there are a $\BbbPk$-name $\dot f$ for a
member of $\BbbN^{\Bbb N}$ and an $a\in\frak B_{\kappa}^+$ such that

\Centerline{$a\VVdPk\,\check h_{\xi}\le^*\dot f$ for every
$\xi<\check{\frak b}$.}

\noindent Now there is an $h\in\BbbN^{\Bbb N}$ such that
$\VVdPk\,\dot f\le^*\check h$.
There must be a $\xi<{\frak b}$ such that $h_{\xi}\not\le^*h$.   We have
$a\VVdPk\,\check h_{\xi}\le^*\dot f\le^*\check h$, so there are an $a'$,
stronger than $a$, and an $n\in\Bbb N$ such that

\Centerline{$a'\VVdPk\,\check h_{\xi}(i)\le\check h(i)$ for every
$i\ge\check n$.}

\noindent However, there is an $i\ge n$ such that $h(i)<h_{\xi}(i)$, in
which case

\Centerline{$\VVdPk\,\varchecki\ge\check n$ and
$\check h(\varchecki)<\check h_{\xi}(\varchecki)$;}

\noindent  which is impossible.\ \Bang\Qed

So

\Centerline{$\VVdPk\,\frak b\le\check{\frak b}$, therefore
$\frak b=\check{\frak b}$.}

\medskip

{\bf (c)(i)} Let $\ofamily{\xi}{\lambda}{\dot f_{\xi}}$ be a family of
$\BbbPk$-names for members of $\BbbN^{\Bbb N}$ where
$\lambda<\check{\frak d}$.
Then for each $\xi<\lambda$ we can find an $h_{\xi}\in\BbbN^{\Bbb N}$ such
that $\VVdPk\,\dot f_{\xi}\le^*\check h_{\xi}$.   As $\lambda<{\frak d}$, there
is an $h\in\BbbN^{\Bbb N}$ such that $h\not\le^*h_{\xi}$ for every
$\xi<\lambda$.   Now

\Centerline{$\VVdPk\,\check h\not\le^*\check h_{\xi}$ for every
$\xi<\check\lambda$,}

\noindent so

\Centerline{$\VVdPk\,\check h\not\le^*\dot f_{\xi}$ for every
$\xi<\check\lambda$.}

\noindent As $\lambda$ and
$\ofamily{\xi}{\lambda}{\dot f_{\xi}}$ are arbitrary,

\Centerline{$\VVdPk\,\frak d\ge\check{\frak d}$.}

\medskip

\quad{\bf (ii)} Let $\ofamily{\xi}{{\frak d}}{h_{\xi}}$ be a family in
$\BbbN^{\Bbb N}$ which is $\le^*$-cofinal with $\BbbN^{\Bbb N}$.
Then

\Centerline{$\VVdPk\,\{\check h_{\xi}:\xi<\check{\frak d}\}$ is
$\le^*$-cofinal with $\BbbN^{\Bbb N}$.}

\noindent\Prf\ Let $\dot f$ be a $\BbbPk$-name for a
member of $\BbbN^{\Bbb N}$.  There are an $h\in\BbbN^{\Bbb N}$ such that
$\VVdPk\,\dot f\le^*\check h$, and a $\xi<{\frak d}$ such that
$h\le^*h_{\xi}$.   In this case,

\Centerline{$\VVdPk\,\dot f\le^*\check h\le^*\check h_{\xi}$.  \Qed}

\noindent So

\Centerline{$\VVdPk\,\frak d\le\check{\frak d}$.}
}%end of proof of 552C

\vleader{36pt}{552D}{Lemma} Let $\lambda$ and $\kappa$ be infinite
cardinals, and $A$ any subset of $\{0,1\}^{\lambda}$.   Then

\Centerline{$\VVdPk\,\nu_{\check\lambda}^*(\check A)
=(\nu_{\lambda}^*A)\varBspcheck$.}

\proof{{\bf (a)} \Quer\ Suppose, if possible, that

\Centerline{$\neg\VVdPk\,(\nu_{\lambda}^*A)\varBspcheck
\le\nu_{\check\lambda}^*(\check A)$.}

\noindent Then there are an $a\in\frak B_{\kappa}^+$ and a $q\in\Bbb Q$ such that
$q<\nu_{\lambda}^*A$ and

\Centerline{$a\VVdPk\,\nu_{\check\lambda}^*(\check A)<\check q$.}

\noindent Let $\dot E$ be a $\BbbPk$-name such that

\Centerline{$a\VVdPk\,\check A\subseteq\dot E$,
$\dot E\in\CalBa_{\check\lambda}$ and
$\nu_{\check\lambda}\dot E<\check q$.}

\noindent Of course we can arrange that
$1\Bsetminus a\VVdPk\,\dot E=\emptyset$,
so that $\VVdPk\,\dot E\in\CalBa_{\check\lambda}$ and there is a
$W\in\Tau_{\kappa}\tensorhat\CalBa_{\lambda}$ such that
$\VVdPk\,\dot E=\vec W$ (551Fb).   Setting $h(x)=\nu_{\lambda}W[\{x\}]$ for
$x\in\{0,1\}^{\kappa}$,

\Centerline{$\VVdPk\,\vec h=\nu_{\check\lambda}\vec W$}

\noindent (551I(iii)), so

\Centerline{$a\VVdPk\,\vec h=\nu_{\check\lambda}\dot E<\check q$}

\noindent and $a\Bsubseteq\{x:h(x)<q\}^{\ssbullet}$.
Take $F\in\Tau_{\kappa}$
such that $F^{\ssbullet}=a$;  then $h(x)<q$ for almost every $x\in F$ and
$(\nu_{\kappa}\times\nu_{\lambda})(W\cap(F\times\{0,1\}^{\lambda}))
<q\nu_{\kappa}F$.

For each $y\in A$, let $e_y:\{0,1\}^{\kappa}\to\{0,1\}^{\lambda}$ be the
constant function with value $y$.   Then $\VVdPk\,\vec e_y=\check y$
(551Ce), so

\Centerline{$a\VVdPk\,\vec e_y\in\check A\subseteq\vec W$}

\noindent and $(x,y)\in W$ for almost every $x\in F$.   But if we set

\Centerline{$H=\{y:(x,y)\in W$ for $\nu_{\kappa}$-almost every $x\in F\}$,}

\noindent $H\in\Tau_{\lambda}$, $A\subseteq H$ and

\Centerline{$\nu_{\kappa}F\cdot\nu_{\lambda}H
\le(\nu_{\kappa}\times\nu_{\lambda})(W\cap(F\times\{0,1\}^{\lambda}))
<q\nu_{\kappa}F$.}

\noindent It follows that

\Centerline{$\nu_{\lambda}^*A\le\nu_{\lambda}H<q$,}

\noindent contrary to hypothesis.\ \BanG\  So

\Centerline{$\VVdPk\,(\nu_{\lambda}^*A)\varBspcheck
\le\nu_{\check\lambda}^*(\check A)$.}


\medskip

{\bf (b)} In the other direction, let $E\in\CalBa_{\lambda}$ be such that
$A\subseteq E$ and $\nu_{\lambda}E=\nu_{\lambda}^*A$, and consider
$W=\{0,1\}^{\kappa}\times E$.   Then

\Centerline{$\VVdPk\,\check A\subseteq\vec W$ and
$\nu_{\check\lambda}\vec W=(\nu_{\lambda}E)\varBspcheck$,}

\noindent so

\Centerline{$\VVdPk\,\dot\nu_{\lambda}^*\check A
\le(\nu_{\lambda}^*A)\varBspcheck$.}
}%end of proof of 552D

\leader{552E}{Theorem} Let $\kappa$ and $\lambda$ be infinite
cardinals, with $\kappa\ge\max(\omega_1,\lambda)$.
Then

\Centerline{$\VVdPk$ there is a strongly Sierpi\'nski set for
$\nu_{\check\lambda}$ with cardinal $\check\kappa$.}

\proof{{\bf (a)} As $\kappa\ge\lambda$, $\Bbb P_{\kappa}$ is isomorphic to
$\Bbb P=\Bbb P_{\kappa\times\lambda}$.   For each $\xi<\kappa$, let
$f_{\xi}:\{0,1\}^{\kappa\times\lambda}\to\{0,1\}^{\lambda}$ be given by
setting $f_{\xi}(x)(\eta)=x(\xi,\eta)$ for every
$x\in\{0,1\}^{\kappa\times\lambda}$ and $\eta<\lambda$;  then, taking
$\vec f_{\xi}$ to be the $\Bbb P$-name defined by the
process of 551Cb,

\Centerline{$\VVdP\,\vec f_{\xi}\in\{0,1\}^{\check\lambda}$.}

\noindent If $\xi$, $\xi'<\kappa$ are distinct, then for any finite set
$I\subseteq\lambda$

$$\eqalign{\Bvalue{\vec f_{\xi}(\eta)
=\vec f_{\xi'}(\eta)\text{ for every }\eta\in\check I}
&=\{x:f_{\xi}(x)(\eta)=f_{\xi'}(x)(\eta)
  \text{ for every }\eta\in I\}^{\ssbullet}\cr
&=\{x:x(\xi,\eta)=x(\xi',\eta)\text{ for every }\eta\in I\}^{\ssbullet}\cr}$$

\noindent has measure $2^{-\#(I)}$, so, because $\lambda$ is infinite,
$\bar\nu\Bvalue{\vec f_{\xi}=\vec f_{\xi'}}=0$ and

\Centerline{$\VVdP\,\vec f_{\xi}\ne\vec f_{\xi'}$.}

\noindent So, taking $\dot A$ to be the $\Bbb P$-name
$\{(\vec f_{\xi},\Bbbone):\xi<\kappa\}$, we have

\Centerline{$\VVdP\,\dot A\subseteq\{0,1\}^{\check\lambda}$ has
cardinal $\check\kappa\ge\omega_1$, so is uncountable.}

\noindent(As remarked in 5A3Nb, we do not need to distinguish between
$\omega_1$ and $\check\omega_1$ in the last formula.)

\medskip

{\bf (b)} Let $r\ge 1$ be an integer and $\dot W$ a
$\Bbb P$-name such that

\Centerline{$\VVdP\,\dot W$ is a subset of
$(\{0,1\}^{\check\lambda})^{\check r}$
which is negligible for the usual measure.}

\noindent Then there is a Baire subset $W$ of
$\{0,1\}^{\kappa\times\lambda}\times(\{0,1\}^{\lambda})^r$,
negligible for the usual measure on this space, such that

\Centerline{$\VVdP\,\dot W\subseteq\vec W$}

\noindent (551J, applied to
$\{0,1\}^{\lambda\times r}\cong(\{0,1\}^{\lambda})^r$).
Let $J\subseteq\kappa$ be a countable set such that $W$ factors
through $\{0,1\}^{J\times\lambda}\times(\{0,1\}^{\lambda})^r$, that is,
there is a negligible Baire set
$W_1\subseteq\{0,1\}^{J\times\lambda}\times(\{0,1\}^{\lambda})^r$ such that
$W=\{(x,y):(x\restr J\times\lambda,y)\in W_1\}$.
If $\xi_0,\ldots,\xi_{r-1}$
are distinct elements of $\kappa\setminus J$, then

\Centerline{$\VVdP\,(\vec f_{\xi_0},\ldots,\vec f_{\xi_{r-1}})\notin\vec W$.}

\noindent\Prf\ Applying 551Ea to the function
$x\mapsto(f_{\xi_0}(x),\ldots,f_{\xi_{r-1}}(x))$, we have

\Centerline{$\Bvalue{(\vec f_{\xi_0},\ldots,\vec f_{\xi_{r-1}})\in\vec W}
=\{x:(x,(f_{\xi_0}(x),\ldots,f_{\xi_{r-1}}(x)))\in W\}^{\ssbullet}$.}

\noindent Set $K=J\cup\{\xi_0,\ldots,\xi_{r-1}\}$ and for
$w\in\{0,1\}^{K\times\lambda}$, $i<r$, $\eta<\lambda$
set $g_i(w)(\eta)=w(\xi_i,\eta)$.   Then
$w\mapsto(w\restr J\times\lambda,(g_0(w),\ldots,g_{r-1}(w))$ is a
measure space isomorphism
between $\{0,1\}^{K\times\lambda}$ and
$\{0,1\}^{J\times\lambda}\times(\{0,1\}^{\lambda})^r$, so

\Centerline{$W_2=\{w:w\in\{0,1\}^{K\times\lambda}$,
$(w\restr J\times\lambda,(g_0(w),\ldots,g_{r-1}(w))\in W_1\}$}

\noindent is negligible.   Consequently


$$\eqalign{\{x:(x,(f_{\xi_0}(x),\ldots,f_{\xi_{r-1}}(x)))\in W\}
&=\{x:(x\restr J\times\lambda,(f_{\xi_0}(x),\ldots,f_{\xi_{r-1}}(x)))
  \in W_1\}\cr
&=\{x:x\restr K\times\lambda\in W_2\}\cr}$$

\noindent is negligible and
$\Bvalue{(\vec f_{\xi_0},\ldots,\vec f_{\xi_{r-1}})\in\vec W}=0$, that is,

\Centerline{$\VVdP\,(\vec f_{\xi_0},\ldots,\vec f_{\xi_{r-1}})\notin\vec W$.
\Qed}

Now if we set $\dot B=\{(\vec f_{\xi},\Bbbone):\xi\in J\}$, we have

\doubleinset{$\VVdP\,\dot B$ is a countable subset of $\dot A$ and
$(x_0,\ldots,x_{r-1})\notin\dot W$ whenever $x_0,\ldots,x_{r-1}$ are distinct
members of $\dot A\setminus\dot B$.}

\noindent As $\dot W$ is arbitrary,

\Centerline{$\VVdP\,\dot A$ is a
strongly Sierpi\'nski set with cardinal $\check\kappa$.}

\noindent As $\Bbb P$ is isomorphic to $\Bbb P_{\kappa}$,

\Centerline{$\VVdPk$ there is a
strongly Sierpi\'nski set for $\nu_{\check\lambda}$
with cardinal $\check\kappa$.}
}%end of proof of 552E

\leader{552F}{Theorem} Let $\kappa$ and $\lambda$ be infinite cardinals.

(a) If either $\kappa$ or $\lambda$ is uncountable,

\Centerline{$\VVdPk\,\add\Cal N(\nu_{\check\lambda})=\omega_1$.}

(b) $\VVdash_{\Bbb P_{\omega}}\,\add\Cal N(\nu_{\omega})
=(\add\Cal N(\nu_{\omega}))\varBspcheck$.

\proof{{\bf (a)(i)} If $\lambda$ is uncountable, then

\Centerline{$\VVdPk\,\check\lambda$ is
uncountable, so $\add\Cal N(\nu_{\check\lambda})=\omega_1$}

\noindent (5A3Nb, 521Jb/523E).

\medskip

\quad{\bf (ii)} If $\kappa$ is uncountable, then

\Centerline{$\VVdPk$ there is a Sierpi\'nski set for $\nu_{\omega}$, so
$\omega_1\le\add\Cal N(\nu_{\check\lambda})
\le\add\Cal N(\nu_{\omega})=\omega_1$}

\noindent (552E, 523B, 537B(a-i)).
%523B Cichon

\medskip

{\bf (b)(i)}
Let $\ofamily{\xi}{\add\Cal N(\nu_{\omega})}{H_{\xi}}$ be a family of
negligible Borel sets in $\{0,1\}^{\omega}$ such that
$A=\bigcup_{\xi<\add\Cal N(\nu_{\omega})}H_{\xi}$ is not negligible.
Then 552D tells us that

\doubleinset{$\VVdash_{\Bbb P_{\omega}}\,
\check H_{\xi}$ is negligible for every
$\xi<(\add\Cal N(\nu_{\omega}))\varBspcheck$,
but $\check A=\bigcup_{\xi<(\add\Cal N(\nu_{\omega}))\varBspcheck}\check H_{\xi}$ is not,
so $\add\Cal N(\nu_{\omega})\le(\add\Cal N(\nu_{\omega}))\varBspcheck$.}

\medskip

\quad{\bf (ii)} \Quer\ If

\Centerline{$\neg\VVdash_{\Bbb P_{\omega}}\,
\add\Cal N(\nu_{\omega})\ge(\add\Cal N(\nu_{\omega}))\varBspcheck$,}

\noindent then there are an $a\in\frak B_{\kappa}^+$ and a
$\theta<\add\Cal N(\nu_{\omega})$ such that

\Centerline{$a\VVdash_{\Bbb P_{\omega}}\,
\add\Cal N(\nu_{\omega})=\check\theta$.}

\noindent Now there is a family $\ofamily{\xi}{\theta}{\dot W_{\xi}}$ of
$\BbbPk$-names such that

\Centerline{$a\VVdash_{\Bbb P_{\omega}}\,
\dot W_{\xi}\in\Cal N(\nu_{\omega})\Forall\xi<\check\theta$,
$\bigcup_{\xi<\check\theta}\dot W_{\xi}\notin\Cal N(\nu_{\omega})$.}

\noindent By 551J, there is for each $\xi<\theta$ a
$W_{\xi}\in\Tau_{\omega}\tensorhat\CalBa_{\omega}$ such that

\Centerline{$a\VVdash_{\Bbb P_{\omega}}\,
\dot W_{\xi}\subseteq\vec W_{\xi}$}

\noindent and all the vertical sections of every $W_{\xi}$ are negligible.   But
this means that $W_{\xi}$ is negligible for the product measure
$\nu_{\omega}\times\nu_{\omega}$.   Because

\Centerline{$\theta<\add\Cal N(\nu_{\omega})
=\add\Cal N(\nu_{\omega}\times\nu_{\omega})$,}

\noindent
$\bigcup_{\xi<\theta}W_{\xi}$ also is negligible, and there is a negligible
$W\in\Tau_{\omega}\tensorhat\CalBa_{\omega}$ including every $W_{\xi}$.
In this case, 551I(iii) tells us that

\Centerline{$\VVdash_{\Bbb P_{\omega}}\,\nu_{\omega}\vec W=0$,}

\noindent so

\Centerline{$a\VVdash_{\Bbb P_{\omega}}\,
\bigcup_{\xi<\check\theta}\dot W_{\xi}
\subseteq\bigcup_{\xi<\check\theta}\vec W_{\xi}
\subseteq\vec W$ is negligible.   \Bang}

Putting this together with (i),

\Centerline{$\VVdash_{\Bbb P_{\omega}}\,
  \add\Cal N(\nu_{\omega})=(\add\Cal N(\nu_{\omega}))\varBspcheck$.}
}%end of proof of 552F

\leader{552G}{Theorem} Let $\kappa$ and $\lambda$ be infinite cardinals.

(a) $\VVdPk\,\cov\Cal N(\nu_{\check\lambda})
\ge\max(\kappa,\cov\Cal N(\nu_{\lambda}))\varBspcheck$.

(b)\cmmnt{ ({\smc Pawlikowski 86})}
$\VVdPk\,\cov\Cal N(\nu_{\omega})\ge\frak b$.

(c)\cmmnt{ ({\smc Miller 82})}
If $\kappa\ge\frak c$ then
$\VVdPk\,\cov\Cal N(\nu_{\omega})=\frak c$.\cmmnt{\footnote{\smallerfonts
Remember that the final
$\frak c$ here is to be interpreted in the forcing language.}}

(d)\cmmnt{ ({\smc Miller 82})} Suppose that $\kappa$ and $\lambda$
are uncountable.   Then

\Centerline{$\VVdPk\,\cov\Cal N(\nu_{\check\lambda})
\le(\sup_{\delta<\kappa}\delta^{\omega})\varBspcheck$,}

\noindent where each $\delta^{\omega}$ is the cardinal power.

\proof{{\bf (a)(i)} If $\kappa=\omega$ then of course
$\VVdPk\,\cov\Cal N(\nu_{\check\lambda})\ge\check\kappa$.
If $\kappa$ is uncountable, then

\Centerline{$\VVdPk\,\cov\Cal N(\nu_{\check\kappa})\ge\check\kappa$}

\noindent by 552E and 537B(a-i), so by 523F we have

\Centerline{$\VVdPk\,\cov\Cal N(\nu_{\check\lambda})\ge\check\kappa$.}

\medskip

\quad{\bf (ii)} \Quer\ If

\Centerline{$\neg\VVdPk\,\cov\Cal N(\nu_{\check\lambda})
  \ge(\cov\Cal N(\nu_{\lambda}))\varBspcheck$}

\noindent then we have an $a\in\frak B_{\kappa}^+$, a cardinal
$\theta<\cov\Cal N(\nu_{\lambda})$ and a
family $\ofamily{\xi}{\theta}{\dot W_{\xi}}$ of $\BbbPk$-names such that

\Centerline{$a\VVdPk\,\{\dot W_{\xi}:\xi<\check\theta\}$ is a cover of
$\{0,1\}^{\check\lambda}$ by negligible sets.}

\noindent By 551J again, we have for each $\xi<\theta$ a
$(\nu_{\kappa}\times\nu_{\lambda})$-negligible set $W_{\xi}$ such that
$a\VVdPk\,\dot W_{\xi}\subseteq\vec W_{\xi}$.   Set

\Centerline{$V_{\xi}=\{y:y\in\{0,1\}^{\lambda}$, $W_{\xi}^{-1}[\{y\}]$ is not
$\nu_{\kappa}$-negligible$\}$;}

\noindent then $\nu_{\lambda}V_{\xi}=0$ for every $\xi<\theta$,
so there is a
$y\in\{0,1\}^{\lambda}\setminus\bigcup_{\xi<\theta}V_{\xi}$.
In this case, let
$e_y:\{0,1\}^{\kappa}\to\{0,1\}^{\lambda}$ be the constant function with
value $y$.   Then we have a $\BbbPk$-name $\vec e_y$ for a member of
$\{0,1\}^{\lambda}$, and for each $\xi<\theta$

\Centerline{$\Bvalue{\vec e_y\in\vec W_{\xi}}
=\{x:(x,e_y(x))\in W_{\xi}\}^{\ssbullet}
=W_{\xi}^{-1}[\{y\}]^{\ssbullet}=0$}

\noindent (551Ea).   So

\Centerline{$\VVdPk\,\vec e_y\notin\vec W_{\xi}$ for every
$\xi<\check\theta$,}

\noindent and

\Centerline{$a\VVdPk\,
\vec e_y\in\{0,1\}^{\check\lambda}
  \setminus\bigcup_{\xi<\check\theta}\dot W_{\xi}$.  \Bang}

We conclude that

\Centerline{$\VVdPk\,\cov\Cal N(\nu_{\check\lambda})
  \ge(\cov\Cal N(\nu_{\lambda}))\varBspcheck$.}

\medskip

{\bf (b)(i)} Set $S_2=\bigcup_{n\in\Bbb N}\{0,1\}^n$ and let
$\sequencen{(\sigma_n,\tau_n,k_n)}$ enumerate
$S_2\times S_2\times\Bbb N$ with cofinal repetitions.
Let $D$ be the set of those $\alpha\in\BbbN^{\Bbb N}$ such
that

\inset{$\sequencen{k_{\alpha(n)}}$ is strictly increasing,

$\#(\sigma_{\alpha(m)})\le\alpha(n)$ whenever $n\in\Bbb N$ and
$m<k_{\alpha(n+1)}$,

$\sum_{i=k_{\alpha(n)}}^{k_{\alpha(n+1)}-1}
2^{-\#(\sigma_{\alpha(i)})-\#(\tau_{\alpha(i)})}\le 4^{-n}$ for every
$n\in\Bbb N$.}

\noindent For $\alpha\in D$ set

\Centerline{$G_{\alpha}
=\bigcap_{n\in\Bbb N}\bigcup_{m\ge n}\{(u,y):u$, $y\in\{0,1\}^{\omega}$,
$u\supseteq\sigma_{\alpha(m)}$, $y\supseteq\tau_{\alpha(m)}\}$.}

\medskip

\quad{\bf (ii)} For every $\alpha\in D$, $G_{\alpha}$ is negligible for the
product measure on $\{0,1\}^{\omega}\times\{0,1\}^{\omega}$.   \Prf\
For any $n\in\Bbb N$, the measure of $G_{\alpha}$ is at most

$$\eqalign{\sum_{m=k_{\alpha(n)}}^{\infty}
  2^{-\#(\sigma_{\alpha(m)})-\#(\tau_{\alpha(m)})}
&\le\sum_{j=n}^{\infty}\sum_{m=k_{\alpha(j)}}^{k_{\alpha(j+1)}-1}
2^{-\#(\sigma_{\alpha(m)})-\#(\tau_{\alpha(m)})}\cr
&\le\sum_{j=n}^{\infty}4^{-j}
=\Bover43\cdot 4^{-n}.  \text{ \Qed}\cr}$$

\medskip

\quad{\bf (iii)} If $G\subseteq\{0,1\}^{\omega}\times\{0,1\}^{\omega}$ is
negligible, there is an $\alpha\in D$ such that $G\subseteq G_{\alpha}$.
\Prf\ For each $i\in\Bbb N$, let $H_i\supseteq G$ be an open set such that
$(\nu_{\omega}\times\nu_{\omega})(H_i)\le 2^{-i}$;  we can suppose that
$H_i$ is not open-and-closed.   $H_i$ can be expressed as
the union of a sequence of open-and-closed sets;  it can therefore be expressed
as the union of a disjoint sequence of open-and-closed sets;  each of these is
expressible as the union of a disjoint family of sets of the form
$\{u:\sigma\subseteq x\}\times\{y:\tau\subseteq y\}$ where
$\sigma$, $\tau\in S_2$;  so $H_i$ is expressible as
$\bigcup_{j\in\Bbb N}\{u:\sigma'_{ij}\subseteq u\}
\times\{y:\tau'_{ij}\subseteq y\}$, with

\Centerline{$\sum_{j\in\Bbb N}2^{-\#(\sigma'_{ij})-\#(\tau'_{ij})}\le 2^{-i}$.}

\noindent Re-indexing
$\langle (\sigma'_{ij},\tau'_{ij})\rangle_{i,j\in\Bbb N}$ as
$\sequence{m}{(\sigma''_m,\tau''_m)}$, we have

\Centerline{$G\subseteq\bigcap_{n\in\Bbb N}\bigcup_{m\ge n}
\{(u,y):\sigma''_m\subseteq u$, $\tau''_m\subseteq y\}$,}

\noindent and

\Centerline{$\sum_{m\in\Bbb N}2^{-\#(\sigma''_m)-\#(\tau''_m)}
\le\sum_{i=0}^{\infty}2^{-i}<\infty$.}

Let $\gamma:\Bbb N\to\Bbb N$ be a strictly increasing function such
that

\Centerline{$\sum_{m=\gamma(n)}^{\gamma(n+1)-1}
2^{-\#(\sigma''_m)-\#(\tau''_m)}
\le 4^{-n}$}

\noindent for every $n\in\Bbb N$.   Now choose $\sequencen{\alpha(n)}$ so that

\Centerline{$k_{\alpha(n)}=\gamma(n)$,
\quad$\sigma_{\alpha(n)}=\sigma''_n$,
\quad$\tau_{\alpha(n)}=\tau''_n$,
\quad$\alpha(n)\ge\#(\sigma''_m)$ whenever $m<\gamma(n+1)$}

\noindent for each $n\in\Bbb N$.   Then $\alpha\in D$ and
$G\subseteq G_{\alpha}$.\ \Qed

\medskip

\quad{\bf (iv)} Define $h:\BbbN^{\Bbb N}\to\BbbN^{\Bbb N}$ by setting
$h(\beta)(n)=n+\sum_{i=0}^n\beta(i)$ for $\beta\in\BbbN^{\Bbb N}$
and $n\in\Bbb N$.   For $\beta\in\BbbN^{\Bbb N}$ define
$f_{\beta}:\{0,1\}^{\omega}\to\{0,1\}^{\omega}$ by setting
$f_{\beta}(u)(n)=u(h(\beta)(n))$ for $\beta\in\BbbN^{\Bbb N}$ and $n\in\Bbb N$;
note that $f_{\beta}$ is continuous.
For $\alpha$, $\beta\in\BbbN^{\Bbb N}$ say that $\alpha\le^*\beta$ if
$\{n:\beta(n)<\alpha(n)\}$ is finite.

\medskip

\quad{\bf (v)} If $\alpha\in D$, $\beta\in\BbbN^{\Bbb N}$ and
$\alpha\le^*\beta$, then
$C=\{u:u\in\{0,1\}^{\omega}$, $(u,f_{\beta}(u))\in G_{\alpha}\}$ is
$\nu_{\omega}$-negligible.   \Prf\ Let $n_0$ be such that $\alpha(n)\le\beta(n)$
for $n\ge n_0$.   $C$ is just $\bigcap_{n\in\Bbb N}\bigcup_{m\ge n}C_m$
where

\Centerline{$C_m
=\{u:\sigma_{\alpha(m)}\subseteq u,\,
\tau_{\alpha(m)}\subseteq f_{\beta}(u)\}$}

\noindent for each $m$.
We know that $\sequence{j}{k_{\alpha(j)}}$ is strictly increasing;
if $m\ge k_{\alpha(n_0)}$, let $j\ge n_0$ be such that
$k_{\alpha(j)}\le m<k_{\alpha(j+1)}$, and set

$$\eqalign{C'_m
&=\{u:\sigma_{\alpha(m)}\subseteq u,\,
\tau_{\alpha(m)}(i)=f_{\beta}(u)(i)\text{ for }
j\le i<\#(\tau_{\alpha(m)})\}\cr
&=\{u:u(i)=\sigma_{\alpha(m)}(i)\text{ for }i<\#(\sigma_{\alpha(m)}),\cr
&\mskip100mu
u(h(\beta)(i))=\tau_{\alpha(m)}(i)\text{ for }
j\le i<\#(\tau_{\alpha(m)})\}\cr
&\supseteq C_m.\cr}$$

\noindent We know that

\Centerline{$\#(\sigma_{\alpha(m)})\le\alpha(j)\le\beta(j)
\le h(\beta)(i)$}

\noindent whenever $i\ge j$ and that $h(\beta)$ is a
strictly increasing function, so

\Centerline{$\nu_{\omega}C'_m
\le 2^{-\#(\sigma_{\alpha(m)})-\#(\tau_{\alpha(m)})+j}$.}

\noindent But this means that

$$\eqalign{\sum_{m=k_{\alpha(n_0)}}^{\infty}\nu_{\omega}C_m
&=\sum_{j=n_0}^{\infty}
\sum_{m=k_{\alpha(j)}}^{k_{\alpha(j+1)}-1}\nu_{\omega}C_m\cr
&\le\sum_{j=n_0}^{\infty}
2^{j}\sum_{m=k_{\alpha(j)}}^{k_{\alpha(j+1)}-1}
  2^{-\#(\sigma_{\alpha(m)})-\#(\tau_{\alpha(m)})}
\le\sum_{j=n_0}^{\infty}2^{j}\cdot 4^{-j}\cr}$$

\noindent is finite, and $C$ is negligible.\ \Qed

\medskip

\quad{\bf (vi)} Let $\Phi$ be the set of all continuous
functions from $\{0,1\}^{\kappa}$ to $\{0,1\}^{\omega}$, and
$\Cal E$ the set of $(\nu_{\kappa}\times\nu_{\omega})$-negligible sets in
$\Tau_{\kappa}\tensorhat\CalBa_{\omega}$;  let $R$ be the relation

\Centerline{$\{(W,g):W\in\Cal E$, $g\in\Phi$,
$\{x:(x,g(x))\in W\}\in\Cal N(\nu_{\kappa})\}$.}

\noindent Then $(\Cal E,R,\Phi)\prGT(\BbbN^{\Bbb N},\le^*,\BbbN^{\Bbb N})$.
\Prf\ For $W\in\Cal E$ set

$$\eqalign{V_W
&=\{(u,y):u,\,y\in\{0,1\}^{\omega},\cr
&\mskip150mu\{v:v\in\{0,1\}^{\kappa\setminus\omega},\,(u\cup v,y)\in W\}
\text{ is not }\nu_{\kappa\setminus\omega}\text{-negligible}\}.\cr}$$

\noindent Then $V_W$ is $(\nu_{\omega}\times\nu_{\omega})$-negligible;  by
(iii), we can find $\phi(W)\in D$ such that $V_W\subseteq G_{\phi(W)}$.
In the other direction, given $\beta\in\BbbN^{\Bbb N}$, define
$\psi(\beta)\in\Phi$ by saying that $\psi(\beta)(x)=f_{\beta}(x\restr\omega)$
for $x\in\{0,1\}^{\kappa}$.

If $W\in\Cal E$ and $\beta\in\BbbN^{\Bbb N}$
are such that $\phi(W)\le^*\beta$, we have $\nu_{\omega}C=0$ where
$C=\{u:(u,f_{\beta}(u))\in G_{\phi(W)}\}$, by (v).   But if
$C'=\{x:(x,\psi(\beta)(x))\in W\}$, and $u\in\{0,1\}^{\omega}\setminus C$, then

\Centerline{$\{v:v\in\{0,1\}^{\kappa\setminus\omega},\,u\cup v\in C'\}
=\{v:v\in\{0,1\}^{\kappa\setminus\omega},\,(u\cup v,f_{\beta}(u))\in W\}$}

\noindent must be $\nu_{\kappa\setminus\omega}$-negligible, since
$(u,f_{\beta}(u))\notin V_W$.   So $C'$ is negligible and
$(W,\psi(\beta))\in R$.   As $W$ and $\beta$ are arbitrary, $(\phi,\psi)$
is a Galois-Tukey connection and
$(\Cal E,R,\Phi)\prGT(\BbbN^{\Bbb N},\le^*,\penalty-100\BbbN^{\Bbb N})$.\
\Qed

Consequently $\add(\Cal E,R,\Phi)\ge\frak b$ (522C(i), 512Ea, 512Db).
%512Ea add P = add(P,\le,P)
%512Db add and \prGT
%522C(i)  add(\NN,\le^*)=\frak b

\medskip

\quad{\bf (vii)} By 552C, we do not need to distinguish between the
interpretations of $\frak b$ in the ordinary universe and in the
forcing language.   Suppose that $a\in\frak B_{\kappa}^+$ and that $\dot{\Cal A}$ is a
$\BbbPk$-name such that

\Centerline{$a\VVdPk\,\dot{\Cal A}\subseteq\Cal N(\nu_{\omega})$ and
$\#(\dot{\Cal A})<\frak b$.}

\noindent Then there are a $b\in\frak B_{\kappa}^+$, stronger than $a$, a cardinal
$\theta<\frak b$ and a family $\ofamily{\xi}{\theta}{\dot W_{\xi}}$ of
$\BbbPk$-names such that

\Centerline{$b\VVdPk\,\dot{\Cal A}=\{\dot W_{\xi}:\xi<\check\theta\}$.}

\noindent For each $\xi<\theta$,
we have a $(\nu_{\kappa}\times\nu_{\omega})$-negligible
$W_{\xi}\in\Tau_{\kappa}\tensorhat\CalBa_{\omega}$ such that
$b\VVdPk\,\dot W_{\xi}\subseteq\vec W_{\xi}$
(551J, as usual).   Each $W_{\xi}$ belongs to
$\Cal E$.   Since $\theta<\frak b\le\add(\Cal E,R,\Phi)$,
there is a $g\in\Phi$ such that
$(W_{\xi},g)\in R$ for every $\xi<\theta$, that is,
$\{x:(x,g(x))\in W_{\xi}\}$ is negligible for every $\xi<\theta$.
But this means that

\Centerline{$\VVdPk\,\vec g\in\{0,1\}^{\omega}\setminus\vec W_{\xi}$}

\noindent for every $\xi<\theta$.   So

\Centerline{$b\VVdPk\,\vec g\notin\bigcup\dot{\Cal A}$ and $\dot{\Cal A}$
does not cover $\{0,1\}^{\omega}$.}

\noindent As $a$ and $\dot{\Cal A}$ are arbitrary,

\Centerline{$\VVdPk\,\cov\Cal N(\nu_{\omega})\ge\frak b$.}

\medskip

{\bf (c)} Write $\theta$ for the cardinal power
$\kappa^{\omega}$, so that $\VVdPk\,\frak c=\check\theta$ (552B).
\Quer\ If

\Centerline{$\neg\VVdPk\,\cov\Cal N(\nu_{\omega})=\frak c$,}

\noindent then there must be an $a\in\frak B_{\kappa}^+$, a cardinal $\delta<\theta$
and a family $\ofamily{\xi}{\delta}{\dot W_{\xi}}$ of $\BbbPk$-names such
that

\Centerline{$a\VVdPk\,\{\dot W_{\xi}:\xi<\check\delta\}$ is a cover of
$\{0,1\}^{\omega}$ by negligible sets.}

\noindent For each $\xi<\delta$, let
$W_{\xi}\in\Tau_{\kappa}\tensorhat\CalBa_{\omega}$ be a
$(\nu_{\kappa}\times\nu_{\omega})$-negligible set such that
$a\VVdPk\,\dot W_{\xi}\subseteq\vec W_{\xi}$;  expanding it if
necessary, we can suppose that $W_{\xi}$ is a Baire set.   Let
$I_{\xi}\subseteq\kappa$ be a countable set
such that $(u,y)\in W_{\xi}$ whenever
$(x,y)\in W_{\xi}$, $u\in\{0,1\}^{\kappa}$ and
$u\restr I_{\xi}=x\restr I_{\xi}$.   Set

\Centerline{$W'_{\xi}=\{(v,y):(u,y)\in W_{\xi}$, $v\in\{0,1\}^{\kappa}$,
$\{\eta:\eta<\kappa$, $u(\eta)\ne v(\eta)\}\in[I_{\xi}]^{<\omega}\}$.}

\noindent Then $W'_{\xi}$ is still
$(\nu_{\kappa}\times\nu_{\omega})$-negligible.

Because $\kappa\ge\frak c$ and $\delta<\kappa^{\omega}$,
there is a countably infinite $K\subseteq\kappa$ such that $K\cap I_{\xi}$
is finite for every $\xi<\delta$ (5A1Gc).   Enumerate $K$ as
$\sequencen{\eta_n}$ and define $f:\{0,1\}^{\kappa}\to\{0,1\}^{\omega}$ by
setting $f(u)=\sequencen{u(\eta_n)}$ for $u\in\{0,1\}^{\kappa}$.

For each $\xi<\kappa$, $\{u:(u,f(u))\in W_{\xi}\}$ is
$\nu_{\kappa}$-negligible.   \Prf\ Set $J=\kappa\setminus K$,
so that $\{0,1\}^{\kappa}$ can be identified with
$\{0,1\}^J\times\{0,1\}^K$.   Because $I_{\xi}\setminus J$ is finite,
$W'_{\xi}$ is equal to

\Centerline{$\{(v,y):(u,y)\in W_{\xi}$, $v\in\{0,1\}^{\kappa}$,
$\{\eta:\eta<\kappa$, $u(\eta)\ne v(\eta)\}\in[J\cap I_{\xi}]^{<\omega}\}$}

\noindent and can be expressed as
$\{(u,y):(u\restr J,y)\in V\}$ where
$V\subseteq\{0,1\}^J\times\{0,1\}^{\omega}$ must be
negligible.   Now the map $u\mapsto(u\restr J,f(u)):
\{0,1\}^{\kappa}\to\{0,1\}^J\times\{0,1\}^{\omega}$ is just a copy of the
map $u\mapsto(u\restr J,u\restr K)$, so is a measure space isomorphism
between $\{0,1\}^{\kappa}$ and $\{0,1\}^J\times\{0,1\}^{\omega}$, and
$V'=\{u:u\in\{0,1\}^{\kappa}$, $(u\restr J,f(u))\in V\}$ is negligible.
But observe now that

\Centerline{$\{u:(u,f(u))\in W_{\xi}\}
\subseteq\{u:(u,f(u))\in W'_{\xi}\}
=\{u:(u\restr J,f(u))\in V\}=V'$}

\noindent is negligible.\ \Qed

Turn now to 551E.   In the language there, we have
$\Bvalue{\vec f\in\vec W_{\xi}}=0$, that is,
$\VVdPk\,\vec f\notin\vec W_{\xi}$ and $a\VVdPk\,\vec f\notin\dot W_{\xi}$.
So

\Centerline{$a\VVdPk\,
\bigcup_{\xi<\check\delta}\dot W_{\xi}\ne\{0,1\}^{\check\lambda}$,}

\noindent which is impossible.\ \Bang

So we have the result claimed.

\medskip

{\bf (d)(i)}
If $\cf\kappa>\omega$ then
$\sup_{\delta<\kappa}\delta^{\omega}=\kappa^{\omega}$;  but
this means that

\Centerline{$\VVdPk\,\cov\Cal N_{\check\lambda}
\le\frak c
=(\kappa^{\omega})\varBspcheck
=(\sup_{\delta<\kappa}\delta^{\omega})\varBspcheck$.}

\noindent
So henceforth suppose that $\cf\kappa=\omega$.   By 523B, we may also
assume that $\lambda=\omega_1$.
%523B Cichon

\medskip

\quad{\bf (ii)} Let $D$ be the set of all pairs $(\pmb{\xi},y)$ where
$\pmb{\xi}\in\omega_1^{\Bbb N}$ is one-to-one
and $y$ is a Baire measurable function from
$\{0,1\}^{\delta}$ to $\{0,1\}^{\omega}$ for some cardinal $\delta<\kappa$.
Then $\#(D)=\sup_{\delta<\kappa}\delta^{\omega}$ (use 5A4G(b-ii)).
For $(\pmb{\xi},y)\in D$, let
$W_{\pmb{\xi}y}\subseteq\{0,1\}^{\kappa}\times\{0,1\}^{\omega_1}$
be the set

\Centerline{$\{(u,v):\lim_{n\to\infty}
  \Bover1n\#(\{i:i<n,\,v(\xi_i)=y(u\restr\alpha)(i)\})
=\Bover12\}$,}

\noindent where $\pmb{\xi}=\sequence{i}{\xi_i}$ and
$\dom y=\{0,1\}^{\alpha}$.   Then $W_{\pmb{\xi}y}$ is a Baire set;  also
the vertical section
$W_{\pmb{\xi}y}[\{u\}]$ is $\nu_{\omega_1}$-conegligible for almost every
$u\in\{0,1\}^{\kappa}$.   \Prf\ The set

\Centerline{$V=\{x:x\in\{0,1\}^{\Bbb N}$,
$\lim_{n\to\infty}\Bover1n\#(\{i:i<n$, $x(i)=y(u\restr\alpha)(i)\})
=\Bover12\}$}

\noindent is conegligible in $\{0,1\}^{\Bbb N}$, by the strong law of large
numbers (273F).   But $W_{\pmb{\xi}y}[\{u\}]$ is the inverse image of $V$
under the \imp\ map $v\mapsto\sequence{i}{v(\xi_i)}$, so is
$\nu_{\omega_1}$-conegligible.\ \Qed

\medskip

\quad{\bf (iii)} Consequently

\Centerline{$\VVdPk\,\vec W_{\pmb{\xi}y}$ is conegligible in
$\{0,1\}^{\omega_1}$}

\noindent whenever $(\pmb{\xi},y)\in D$ (551I(iii)).   Now

\Centerline{$\VVdPk\,
\bigcap_{(\pmb{\xi},y)\in\check D}\vec W_{\pmb{\xi}y}$ is empty.}

\noindent\Prf\Quer\ Otherwise, there are an $a\in\frak B_{\kappa}^+$ and a
$\BbbPk$-name $\dot x$ such that

\Centerline{$a\VVdPk\,\dot x\in
\bigcap_{(\pmb{\xi},y)\in\check D}\vec W_{\pmb{\xi}y}$.}

\noindent Let $f:\{0,1\}^{\kappa}\to\{0,1\}^{\omega_1}$ be a
$(\Tau_{\kappa},\CalBa_{\omega_1})$-measurable function such
that $a\VVdPk\,\vec f=\dot x$ (551Cc).   Set
$\epsilon=\bover14\bar\nu_{\kappa}a$ and for each $\xi<\omega_1$ let
$E_{\xi}$ be an open-and-closed subset of $\{0,1\}^{\kappa}$ such that
$\bar\nu_{\kappa}(E_{\xi}\symmdiff\{u:f(u)(\xi)=1\})\le\epsilon$.
Let $\alpha_{\xi}<\kappa$ be such that $E_{\xi}$ is determined by
coordinates less than $\alpha_{\xi}$.   Because $\cf\kappa=\omega$,
there is a cardinal $\delta<\kappa$ such that
$A=\{\xi:\alpha_{\xi}\le\delta\}$ is infinite;  let
$\pmb{\xi}=\sequence{i}{\xi_i}$ enumerate a subset of $A$.   For each
$i\in\Bbb N$ let $F_i\subseteq\{0,1\}^{\delta}$ be an open-and-closed set
such that $E_{\xi_i}=\{u:u\restr\delta\in F_i\}$.   Define
$y:\{0,1\}^{\delta}\to\{0,1\}^{\Bbb N}$ by saying that
$y(v)(i)=\chi F_i(v)$ for $v\in\{0,1\}^{\delta}$ and $i\in\Bbb N$;  then
$y$ is Baire measurable, so $W_{\pmb{\xi}y}$ is defined and
$a\VVdPk\,\vec f\in\vec W_{\pmb{\xi}y}$.

Set $H=\{u:(u,f(u))\in W_{\pmb{\xi}y}\}$.   Then

\Centerline{$a\Bsubseteq\Bvalue{\vec f\in\vec W_{\pmb{\xi}y}}
=H^{\ssbullet}$}

\noindent so $\nu_{\kappa}H\ge\bar\nu_{\kappa}a=4\epsilon$.

But consider the sets

$$\eqalign{H_i
&=\{u:f(u)(\xi_i)=y(u\restr\delta)(i)\}
\cr&=\{0,1\}^{\kappa}
  \setminus(\{u:f(u)(\xi_i)=1\}\symmdiff\{u:y(u\restr\delta)(i)=1\})
\cr&=\{0,1\}^{\kappa}\setminus(E_{\xi_i}\symmdiff\{u:f(u)(\xi_i)=1\})\cr}$$

\noindent for $i\in\Bbb N$.   These all have measure at least $1-\epsilon$.
For $n\ge 1$ set

\Centerline{$\gamma_n
=\nu_{\kappa}\{u:\#(\{i:i<n,\,u\in H_i\})\le\Bover{2n}3\}$;}

\noindent  then

\Centerline{$n(1-\epsilon)
\le\sum_{i<n}\nu_{\kappa}H_i
=\int\#(\{i:i<n,\,u\in H_i\})\nu_{\kappa}(du)
\le\Bover{2n}3\gamma_n+n(1-\gamma_n)$}

\noindent and $\gamma_n\le 3\epsilon$.   So

$$\eqalign{H
&=\{u:\lim_{n\to\infty}\Bover1n\#(\{i:i<n,\,u\in H_i\})=\Bover12\}\cr
&\subseteq\bigcup_{m\in\Bbb N}\bigcap_{n\ge m}
  \{u:\#(\{i:i<n,\,u\in H_i\})\le\Bover23\}\cr}$$

\noindent has measure at most $3\epsilon$;  which is impossible.\
\Bang\Qed

\medskip

\quad{\bf (iv)} Thus

\Centerline{$\VVdPk\,
\bigcap_{(\pmb{\xi},y)\in\check D}\vec W_{\pmb{\xi}y}$ is empty and
$\{0,1\}^{\omega_1}$ can be covered by $(\sup_{\delta<\kappa}\delta^{\omega})\varBspcheck$ negligible sets,}

\noindent which is what we needed to know.
}%end of proof of 552G

\vleader{60pt}{552H}{Theorem}
Let $\kappa$ and $\lambda$ be infinite cardinals.

(a) $\VVdPk\,\non\Cal N(\nu_{\check\lambda})
\le(\non\Cal N(\nu_{\lambda}))\varBspcheck$.

\wheader{552H}{0}{0}{0}{36pt}
(b) If $\kappa\ge\max(\lambda,\omega_1)$ then

\Centerline{$\VVdPk\,\non\Cal N(\nu_{\check\lambda})=\omega_1$.}

(c)\cmmnt{ ({\smc Pawlikowski 86})}

\Centerline{$\VVdPk\,\non\Cal N(\nu_{\omega})\le\frak d$.}

\proof{{\bf (a)} Let $A\subseteq\{0,1\}^{\lambda}$ be a non-negligible set of
size $\non\Cal N(\nu_{\lambda})$.   Then 552D tells us that

\Centerline{$\VVdPk\,\check A$ is a non-negligible set with cardinal
$(\non\Cal N(\nu_{\lambda}))\varBspcheck$, so $\non\Cal N(\nu_{\check\lambda})\le(\non\Cal N(\nu_{\lambda}))\varBspcheck$.}

\medskip

{\bf (b)} Put 552E and 537Ba together again, as in part (a) of the proof of
552F.

\medskip

{\bf (c)} Continue the argument from the end of (b-vi) of the proof of
552G above.    We have
$(\Cal E,R,\Phi)\prGT(\BbbN^{\Bbb N},\le^*,\BbbN^{\Bbb N})$, so
$\cov(\Cal E,R,\Phi)\le\frak d$ (522C(i), 512Ea, 512Da).
%522C(i)  cf(\NN,\le^*)=\frak d
%512Ea  cov(P,\le,P)=\cf P
%512Da  cov and \prGT
So there is a
family $\ofamily{\xi}{\frak d}{g_{\xi}}$ in $\Phi$ such that for
every $W\in\Cal E$ there is a $\xi<\frak d$ such that $(W,g_{\xi})\in R$,
that is, $\{x:(x,g_{\xi}(x))\in W\}$ is negligible, that is,
$\VVdPk\,\vec g_{\xi}\notin\vec W$.   Now

\Centerline{$\VVdPk\,\{\vec g_{\xi}:\xi<\frak d\}$ is not negligible.}

\noindent\Prf\Quer\ Otherwise, there are an $a\in\frak B_{\kappa}^+$ and a $\BbbPk$-name
$\dot W$ such that

\Centerline{$a\VVdPk\,\dot W$ is a negligible set containing $\vec g_{\xi}$ for
every $\xi<\frak d$.}

\noindent By 551J once again, there is a $W\in\Cal E$ such that
$a\VVdPk\,\dot W\subseteq\vec W$.   But now we have a $\xi<\frak d$ such that
$\VVdPk\,\vec g_{\xi}\notin\vec W$, which is impossible.\ \Bang\QeD\
So $\VVdPk\,\non\Cal N(\nu_{\omega})\le\frak d$.
}%end of proof of 552H

\leader{552I}{Theorem} Let $\kappa$ and $\lambda$ be infinite cardinals.
Set $\theta_0=\max(\cf\Cal N(\nu_{\omega}),\cff[\kappa]^{\le\omega},
\cff[\lambda]^{\le\omega})$.   Then

\Centerline{$\VVdPk\,\cf\Cal N(\nu_{\check\lambda})=\check\theta_0$.}

\proof{{\bf (a)} $\VVdPk\,\cf\Cal N(\nu_{\omega})
\ge\cff[\check\kappa]^{\le\omega}=(\cff[\kappa]^{\le\omega})\varBspcheck$.
\Prf\ If $\kappa=\omega$ this is trivial.   Otherwise it follows from
552E, % Sierpinski set with cardinal \check\kappa
537B(a-ii) % \VVdPk\,\cf\Cal N(\nu_{\omega})
  %\ge\cff[\check\kappa]^{\le\omega}
and 5A3Nd.\ \Qed

\medskip

{\bf (b)} Set

\Centerline{$\theta_1=\cf\Cal N(\nu_{\lambda})
=\max(\cf\Cal N(\nu_{\omega}),\cff[\lambda]^{\le\omega})$}

\noindent (523N).   Then
$\VVdPk\,\cf\Cal N(\nu_{\check\lambda})\ge\check\theta_1$.
\Prf\Quer\ Otherwise, there are $a\in\frak B_{\kappa}$,
$\theta<\theta_1$ and a family $\ofamily{\xi}{\theta}{\dot W_{\xi}}$ of
$\Bbb P_{\kappa}$-names such that

\Centerline{$a\VVdPk\,\{\dot W_{\xi}:\xi<\check\theta\}$ is a cofinal
family in $\Cal N(\nu_{\check\lambda})$.}

\noindent For each $\xi$ choose a $(\nu_{\kappa}\times\nu_{\lambda})$-negligible
$W_{\xi}\in\Tau_{\kappa}\tensorhat\CalBa_{\lambda}$ such that
$a\VVdPk\,\dot W_{\xi}\subseteq\vec W_{\xi}$.   Then

\Centerline{$V_{\xi}
=\{y:y\in\{0,1\}^{\lambda}$, $W_{\xi}^{-1}[\{y\}]$ is not
$\nu_{\kappa}$-negligible$\}$}

\noindent is $\nu_{\lambda}$-negligible.   Because
$\theta<\cf\Cal N(\nu_{\lambda})$, there is a $V\in\Cal N(\nu_{\lambda})$ such
that $V\not\subseteq V_{\xi}$ for every $\xi<\theta$, and (enlarging $V$
slightly if necessary) we can arrange that
$V\in\CalBa_{\lambda}$.

Set $W=\{0,1\}^{\kappa}\times V$.   Then
$W\in\Tau_{\kappa}\tensorhat\CalBa_{\lambda}$ and every vertical
section of $W$ is negligible, so 551I tells us that

\Centerline{$\VVdPk\,\vec W$ is negligible in $\{0,1\}^{\check\lambda}$.}

\noindent Accordingly

\Centerline{$a\VVdPk$ there is a $\xi<\check\theta$ such that
$\vec W\subseteq\dot W_{\xi}\subseteq\vec W_{\xi}$,}

\noindent and there must be a $b\in\frak B_{\kappa}$,
stronger than $a$, and a
$\xi<\theta$ such that $b\VVdPk\,\vec W\subseteq\vec W_{\xi}$.   But now take
any point $y$ of $V\setminus V_{\xi}$ and consider the constant function
$e_y$ on $\{0,1\}^{\kappa}$ with value $y$.   Then
$\{x:(x,e_y(x))\in W\setminus W_{\xi}\}$
is conegligible, so 551E tells us that

\Centerline{$\VVdPk\,\vec e_y\in\vec W\setminus\vec W_{\xi}$, so
$\vec W\not\subseteq\vec W_{\xi}$,}

\noindent contrary to the choice of $\xi$.\ \Bang\Qed

\medskip

{\bf (c)} Now

\Centerline{$\VVdPk\,\cf\Cal N(\nu_{\check\lambda})
\ge\max(\cf\Cal N(\nu_{\omega}),\check\theta_1)
\ge\max(\cff[\check\kappa]^{\le\omega},\check\theta_1)
=\check\theta_0$.}

\medskip

{\bf (d)} In the other direction, let
$\mu=\nu_{\kappa}\times\nu_{\lambda}$ be the product measure on
$\{0,1\}^{\kappa}\times\{0,1\}^{\lambda}$.   Again by 523N,
$\cf\Cal N(\mu)=\theta_0$;  let $\ofamily{\xi}{\theta_0}{W_{\xi}}$ be a
cofinal family in $\Cal N(\mu)$ consisting of sets in
$\Tau_{\kappa}\tensorhat\CalBa_{\lambda}$.   By 551J,

\Centerline{$\VVdPk\,\{\vec W_{\xi}:\xi<\check\theta_1\}$ is cofinal with
$\Cal N(\nu_{\check\lambda})$,
so $\cf\Cal N(\nu_{\check\lambda})\le\check\theta_0$.}

\noindent Putting this together with (c),

\Centerline{$\VVdPk\,\cf\Cal N(\nu_{\check\lambda})=\check\theta_0$,}

\noindent and the proof is complete.
}%end of proof of 552I

\leaveitout{
\leader{552Q}{Proposition} If $\frakmctbl=\theta_1$ and $\non\Cal M=\theta_2$
then

\Centerline{$\VVdash_{\Bbb P_{\omega}}\frakmctbl=\check\theta_1$ and
$\non\Cal M=\check\theta_2$.}

\proof{{\smc Bartoszy\'nski \& Judah 95}, 3.2.43.   Can use idea of 5x46O for
$\non\le\check\non$.
}
}%end of leaveitout

\leader{552J}{Theorem} Let $\kappa$ and $\lambda$ be infinite cardinals;
set $\theta_0=\shr\Cal N(\nu_{\lambda})$ and let
$\theta_1$ be the cardinal power $\lambda^{\omega}$.
Then

\Centerline{$\VVdPk\,\check\theta_0
\le\shr\Cal N(\nu_{\check\lambda})\le\check\theta_1$.}

\proof{{\bf (a)} \Quer\ Suppose, if possible, that

\Centerline{$\neg\VVdPk\,
\check\theta_0\le\shr\Cal N(\nu_{\check\lambda})$.}

\noindent Then there are an
$a\in\frak B_{\kappa}^+$ and a cardinal $\theta'<\theta_0$ such that

\Centerline{$a\VVdPk\,\shr\Cal N(\nu_{\check\lambda})=\check\theta'$.}

\noindent Of course $\theta'$ is infinite.
Let $A\subseteq\{0,1\}^{\lambda}$ be such that
$\nu_{\lambda}^*A>0$ but $B\in\Cal N(\nu_{\lambda})$ for every
$B\in[A]^{\le\theta'}$.   By 552D,

\Centerline{$\VVdPk\,\nu_{\check\lambda}^*(\check A)>0$.}

\noindent
There must therefore be a $\BbbPk$-name $\dot B$ for a
subset of $\check A$ with cardinal at most $\check\theta'$ such that

\Centerline{$a\VVdPk\,\nu_{\check\lambda}^*(\dot B)>0$.}

\noindent By 5A3Nc there is a $B\subseteq A$ such that
$\#(B)\le\max(\omega,\theta')=\theta'$ and

\Centerline{$a\VVdPk\,\dot B\subseteq\check B$.}

\noindent By 552D, in the other direction,

\Centerline{$a\VVdPk\,0<\nu_{\check\lambda}^*(\dot B)
\le\nu_{\check\lambda}^*(\check B)
=(\nu_{\lambda}^*B)\varBspcheck$}

\noindent and $\nu_{\lambda}^*B>0$, contrary to the choice of $A$.\ \Bang

\medskip

{\bf (b)} (In this part of the
proof it will be convenient to regard $\frak B_{\kappa}$ as the measure
algebra of $\nu_{\kappa}\restr\CalBa_{\kappa}$.)

\medskip

\quad{\bf (i)} \Quer\
Suppose, if possible, that

\Centerline{$\neg\VVdPk\,
\shr\Cal N(\nu_{\check\lambda})\le\check\theta_1$.}

\noindent Then there are an $a\in\frak B_{\kappa}^+$
and a $\BbbPk$-name $\dot A$ such that

\doubleinset{$a
\VVdPk\,\dot A$ is a non-negligible subset of $\{0,1\}^{\check\lambda}$
and every subset of $\dot A$ with cardinal at most
$\check\theta_1$ is negligible.}

\medskip

\quad{\bf (ii)} Let $\ofamily{\xi}{\kappa}{e_{\xi}}$
be the standard generating family in $\frak B_{\kappa}$.
Choose $\ofamily{\xi}{\theta_1^+}{f_{\xi}}$,
$\ofamily{\xi}{\theta_1^+}{J_{\xi}}$, $\ofamily{\xi}{\theta_1^+}{K_{\xi}}$,
$\ofamily{\xi}{\theta_1^+}{W_{\xi}}$ and $\ofamily{\xi}{\theta_1^+}{V_{\xi}}$
inductively, as follows.   $K_{\xi}=\bigcup_{\eta<\xi}J_{\eta}$.
Given that $\xi<\theta_1^+$ and that, for each $\eta<\xi$,
$f_{\eta}:\{0,1\}^{\kappa}\to\{0,1\}^{\lambda}$ is a
$(\CalBa_{\kappa},\CalBa_{\lambda})$-measurable function such that
$a\VVdPk\,\vec f_{\eta}\in\dot A$ (where $\vec f_{\eta}$ is
the $\Bbb P_{\kappa}$-name for a member of $\{0,1\}^{\check\lambda}$
as defined in 551Cb), then

\Centerline{$a\VVdPk\,\{\vec f_{\eta}:\eta<\check\xi\}$ is negligible,}

\noindent so by 551J there is a set
$W_{\xi}\in\CalBa_{\kappa}\tensorhat\CalBa_{\lambda}$,
negligible for the product measure on
$\{0,1\}^{\kappa}\times\{0,1\}^{\lambda}$,
such that $a\VVdPk\,\vec f_{\eta}\in\vec W_{\xi}$ for every $\eta<\xi$.

Set

$$\eqalign{V_{\xi}
&=\{(x,y):x\in\{0,1\}^{\kappa},\,
y\in\{0,1\}^{\lambda},\cr
&\mskip100mu\{t:t\in\{0,1\}^{\kappa\setminus K_{\xi}},\,
  ((x\restr K_{\xi})\cup t,y)\in W_{\xi}\}\text{ is
not }\nu_{\kappa\setminus K_{\xi}}\text{-negligible}\}.\cr}$$

\noindent Then
$V_{\xi}\in\CalBa_{\kappa}\tensorhat\CalBa_{\lambda}$ is
negligible, so $\VVdPk\,\vec V_{\xi}\in\Cal N(\nu_{\check\lambda})$ and
$a\VVdPk\,\dot A\not\subseteq\vec V_{\xi}$;  let
$f_{\xi}:\{0,1\}^{\kappa}\to\{0,1\}^{\lambda}$ be a
$(\CalBa_{\kappa},\CalBa_{\lambda})$-measurable function such that
$a\VVdPk\,\vec f_{\xi}\in\dot A\setminus\vec V_{\xi}$ (551Cc).   Let
$J_{\xi}\subseteq\kappa$ be a set with cardinal at most $\lambda$ such that
$\{x:f_{\xi}(x)(\zeta)=1\}$ is determined by coordinates in $J_{\xi}$ for every
$\zeta<\lambda$, and continue.

\medskip

\quad{\bf (iii)} If $\eta<\xi<\theta_1^+$ then
$a\VVdPk\,\vec f_{\eta}\in\vec V_{\xi}$.
\Prf\ As $J_{\eta}\subseteq K_{\xi}$,
we have a function $g:\{0,1\}^{K_{\xi}}\to\{0,1\}^{\lambda}$ such that
$f_{\eta}(x)=g(x\restr K_{\xi})$ for every $x\in\{0,1\}^{\lambda}$.   Now, for
any $s\in\{0,1\}^{K_{\xi}}$ and $y\in\{0,1\}^{\lambda}$, the set

\Centerline{$\{t:t\in\{0,1\}^{\kappa\setminus K_{\xi}}$,
$(s\cup t,y)\in W_{\xi}\setminus V_{\xi}\}$}

\noindent is $\nu_{\kappa\setminus K_{\xi}}$-negligible, so

\Centerline{$E=\{(s,t):(s\cup t,g(s))\in W_{\xi}\setminus V_{\xi}\}$}

\noindent is
$(\nu_{K_{\xi}}\times\nu_{\kappa\setminus K_{\xi}})$-negligible.
($E\in\CalBa_{K_{\xi}}\tensorhat\CalBa_{\kappa\setminus K_{\xi}}$ because
$W_{\xi}$ and $V_{\xi}$ belong to
$\CalBa_{\kappa}\tensorhat\CalBa_{\lambda}$
and $g$ is $(\CalBa_{K_{\xi}},\penalty-100\CalBa_{\lambda})$-measurable.)
But if we
identify $\{0,1\}^{K_{\xi}}\times\{0,1\}^{\kappa\setminus K_{\xi}}$
with $\{0,1\}^{\kappa}$, then
$E$ becomes $\{x:(x,f_{\eta}(x))\in W_{\xi}\setminus V_{\xi}\}$.   Now

$$\eqalignno{a
&\Bsubseteq\Bvalue{\vec f_{\eta}\in\vec W_{\xi}}
=\{x:(x,f_{\eta}(x))\in W_{\xi}\}^{\ssbullet}\cr
\displaycause{551Ea}
&\Bsubseteq\{x:(x,f_{\eta}(x))\in V_{\xi}\}^{\ssbullet}
=\Bvalue{\vec f_{\eta}\in\vec V_{\xi}},\cr}$$

\noindent and $a\VVdPk\,\vec f_{\eta}\in\vec V_{\xi}$.\ \Qed

\medskip

\quad{\bf (iv)} For each $\xi<\theta_1^+$, $V_{\xi}$ factors through
$\{0,1\}^{K_{\xi}}\times\{0,1\}^{\lambda}$ and belongs to
$\CalBa_{\kappa}\tensorhat\CalBa_{\lambda}$.
There is therefore a countable set $L_{\xi}\subseteq K_{\xi}$
such that $V_{\xi}$ factors through
$\{0,1\}^{L_{\xi}}\times\{0,1\}^{\lambda}$.   Let $S$ be the set
$\{\xi:\xi<\theta_1^+$, $\cf\xi>\omega\}$.
Because $\theta_1\ge\omega_1$, $S$ is stationary in $\theta_1^+$
(5A1Ac).   For each $\xi\in S$, let $g(\xi)<\xi$ be such that
$L_{\xi}\subseteq K_{g(\xi)}$.   By the Pressing-Down Lemma there is a
$\gamma<\theta_1^+$ such that $S'=\{\xi:\xi\in S$, $g(\xi)=\gamma\}$ is
stationary.

For $\xi\in S'$, we have a
$V'_{\xi}\in\CalBa_{K_{\gamma}}\tensorhat\CalBa_{\lambda}$ such that

\Centerline{$V_{\xi}
=\{(x,y):x\in\{0,1\}^{\kappa}$, $y\in\{0,1\}^{\lambda}$,
$(x\restr K_{\gamma},y)\in V'_{\xi}\}$.}

\noindent But $\#(K_{\gamma})\le\lambda$, so

\Centerline{$\#(\CalBa_{K_{\gamma}}\tensorhat\CalBa_{\lambda})
\le\lambda^{\omega}=\theta_1<\#(S')$,}

\noindent and there are $\xi$, $\eta\in S'$ such that $\eta<\xi$ and
$V'_{\eta}=V'_{\xi}$ and $V_{\eta}=V_{\xi}$.   But also

\Centerline{$a\VVdPk\,\vec f_{\eta}\in\vec V_{\xi}\setminus\vec V_{\eta}$,}

\noindent so this is impossible.\ \Bang
}%\end of proof of 552J

\leader{552K}{Lemma} Let $I$ be a set.   Let
$q:\Fn_{<\omega}(I;\{0,1\})\to\coint{0,\infty}$ be a
function such that $q(\emptyset)=1$ and

\Centerline{$q(z)=q(z\cup\{(i,0)\})+q(z\cup\{(i,1)\})$}

\noindent whenever $z\in\Fn_{<\omega}(I;\{0,1\})$ and
$i\in I\setminus\dom z$.   Then there is a
unique Radon measure $\mu$ on $\{0,1\}^I$ such that

\Centerline{$\mu\{x:z\subseteq x\in\{0,1\}^I\}=q(z)$}

\noindent for every $z\in\Fn_{<\omega}(I;\{0,1\})$.

\proof{{\bf (a)} For each $K\in[I]^{<\omega}$,
let $\mu_K$ be the measure on the
finite set $\{0,1\}^K$ defined by saying that
$\mu_KA=\sum_{z\in A}q(z)$  for every $A\subseteq\{0,1\}^K$.   For
$K\subseteq L\in[I]^{<\omega}$ set $f_{KL}(z)=z\restr K$ for
$z\in\{0,1\}^L$;  then $f_{KL}$ is \imp\ for $\mu_K$ and $\mu_L$.
\Prf\ It is enough to consider the case $L=K\cup\{i\}$ where
$i\in I\setminus K$.   In this case, for $A\subseteq\{0,1\}^K$,

$$\eqalign{\mu_Lf^{-1}[A]
&=\sum_{w\in f^{-1}[A]}q(w)
=\sum_{z\in A}q(z\cup\{(i,0)\})+q(z\cup\{(i,1)\})\cr
&=\sum_{z\in A}q(z)
=\mu_KA.  \text{ \Qed}\cr}$$

\medskip

{\bf (b)}
Let $\Cal E$ be the algebra of open-and-closed sets in $\{0,1\}^I$, that
is, the family $\{f_K^{-1}[A]:K\in[I]^{<\omega}$, $A\subseteq\{0,1\}^K\}$,
where $f_K(x)=x\restr K$ for $x\in\{0,1\}^I$.   Then we can define a
functional $\nu:\Cal E\to[0,1]$ by setting

\Centerline{$\nu f_K^{-1}[A]=\mu_KA$ whenever $K\in[I]^{<\omega}$,
$A\subseteq\{0,1\}^K$;}

\noindent by (a), this is well-defined.
By 416Qa, there is a unique Radon measure $\mu$ on
$\{0,1\}^I$ extending $\nu$, so that

\Centerline{$\mu\{x:z\subseteq x\}=\nu\{x:z\subseteq x\}=\mu_K\{z\}=q(z)$}

\noindent whenever $K\subseteq I$ is finite and $z\in\{0,1\}^K$.
}%end of proof of 552K

\leader{552L}{Lemma} Let $\theta$ be a regular infinite cardinal such
that the cardinal power $\delta^{\omega}$ is less than $\theta$ for every
$\delta<\theta$, and
$S\subseteq\theta$ a stationary set such that $\cf\xi>\omega$ for every
$\xi\in S$.   Let $\ofamily{\xi}{\theta}{M_{\xi}}$
be a family of sets with cardinal less than $\theta$, and $I$ a set with cardinal less
than $\theta$;  suppose that for each $i\in I$ we are given a function
$f_i$ with domain $S$ such that $f_i(\xi)\in\bigcup_{\eta<\xi}M_{\eta}$ for
every $\xi\in S$.   Then there is an $\omega_1$-complete filter $\Cal F$
on $\theta$, containing every closed cofinal subset of $\theta$, such that
for every $i\in I$ there is a $D\in\Cal F$ such that $D\subseteq S$ and
$f_i$ is constant on $D$.

\proof{{\bf (a)}
Set $M=\bigcup_{\xi<\theta}M_{\xi}$, so that $\#(M)\le\theta$;
let $\ofamily{\xi}{\theta}{x_{\xi}}$ run over $M$.   Set

\Centerline{$F^*=\{\xi:\xi<\theta$,
$\bigcup_{\eta<\xi}M_{\eta}=\{x_{\eta}:\eta<\xi\}\}$;}

\noindent then $F^*$ is a closed cofinal subset of $\theta$, because
$\theta$ is regular and uncountable.   Set $S_1=S\cap F^*$, so that $S_1$
is stationary.   For $\xi\in S_1$ and $i\in I$
let $h_{\xi}(i)<\xi$ be such that $f_i(\xi)=x_{h_{\xi}(i)}$.
For $J\in[I]^{\le\omega}$, $\xi\in S_1$ set

\Centerline{$D_{\xi J}
=\{\eta:\eta\in S\cap F^*$, $h_{\eta}\restr J=h_{\xi}\restr J\}$.}

\medskip

{\bf (b)} There is a $\xi\in S_1$ such that
$D_{\xi J}\cap F\ne\emptyset$ for every closed cofinal set
$F\subseteq\theta$ and every $J\in[I]^{\le\omega}$.   \Prf\Quer\ Otherwise,
for each $\xi\in S_1$ choose $J_{\xi}\in[I]^{\le\omega}$ and
a closed cofinal set $F_{\xi}$ not meeting $D_{\xi J_{\xi}}$.
Let $F$ be the diagonal intersection
$\{\xi:\xi<\theta$, $\xi\in F_{\eta}$ whenever $\eta\in S_1\cap\xi\}$,
so that $F$ is a closed cofinal set (4A1B(c-ii))
and $S_2=S_1\cap F$ is stationary.
For $\xi\in S_2$ let $g(\xi)<\xi$ be such that
$h_{\xi}[J_{\xi}]\subseteq g(\xi)$.   Then there is
a $\gamma<\theta$ such that $S_3=\{\xi:\xi\in S_2$, $g(\xi)=\gamma\}$ is
stationary, by the Pressing-Down Lemma (4A1Cc).   Now
$h_{\xi}\restr J_{\xi}\in[I\times\gamma]^{\le\omega}$ for every
$\xi\in S_1$, and $\#([I\times\gamma]^{\le\omega})
\le\max(\#(I),\gamma,\omega)^{\omega}<\theta$,
so there are $\xi$, $\eta\in S_3$ such that
$h_{\xi}\restr J_{\xi}=h_{\eta}\restr J_{\eta}$ and $\eta<\xi$.
But in this case we have $\xi\in F_{\eta}\cap D_{\eta J_{\eta}}$, which is
supposed to be impossible.\ \Bang\Qed

\medskip

{\bf (c)} If now $\sequencen{F_n}$ is any sequence of closed cofinal sets
in $\theta$, and $\sequencen{J_n}$ is any sequence in $[I]^{\le\omega}$,

\Centerline{$\bigcap_{n\in\Bbb N}D_{\xi J_n}\cap F_n=D_{\xi J}\cap F$}

\noindent is non-empty, where $J=\bigcup_{n\in\Bbb N}J_n$ and
$F=\bigcap_{n\in\Bbb N}F_n$.   So we have an $\omega_1$-complete filter
$\Cal F$ on $\theta$ generated by

\Centerline{$\{D_{\xi J}:J\in[I]^{\le\omega}\}
\cup\{F:F\subseteq\theta$ is closed and cofinal$\}$.}

\noindent If $i\in I$ then $f_i$ is constant on $D_{\xi,\{i\}}\in\Cal F$,
so we're done.
}%end of proof of 552L

\vleader{72pt}{552M}{Proposition} Let $\kappa$ and $\lambda$ be infinite
cardinals.   Then the following are equiveridical:

(i) if $\Cal A\subseteq\Cal P(\{0,1\}^{\kappa})$ and
$\#(\Cal A)\le\lambda$ then there is an extension of
$\nu_{\kappa}$ to a measure measuring every member of $\Cal A$;

(ii) for every function
$f:\{0,1\}^{\kappa}\to\{0,1\}^{(\kappa+\lambda)\setminus\kappa}$,
there is a Baire measure $\mu$ on $\{0,1\}^{\kappa+\lambda}$ such that
$\mu\{y:y\in\{0,1\}^{\kappa+\lambda}$, $z\subseteq y\}=2^{-\#(K)}$
whenever $K\in[\kappa]^{<\omega}$ and $z\in\{0,1\}^K$, and
$\mu^*\{x\cup f(x):x\in\{0,1\}^{\kappa}\}=1$;

(iii) if $(X,\Sigma,\mu)$ is a locally compact\cmmnt{ (definition:
342Ad)} semi-finite measure space with Maharam type at most $\kappa$,
$\Cal A\subseteq\Cal PX$ and $\#(\Cal A)\le\lambda$, then there is an
extension of $\mu$ to a measure measuring every member of $\Cal A$.

\proof{{\bf (i)$\Rightarrow$(ii)} Assume (i).   If
$f:\{0,1\}^{\kappa}\to\{0,1\}^{(\kappa+\lambda)\setminus\kappa}$
is a function, set
$\Cal A=\{\{x:f(x)(\xi)=1\}:\kappa\le\xi<\kappa+\lambda\}$, so that
$\Cal A$ is a family of subsets of $\{0,1\}^{\kappa}$ and
$\#(\Cal A)\le\lambda$.   Let $\nu$ be a measure on $\{0,1\}^{\kappa}$,
extending $\nu_{\kappa}$ and measuring every member of $\Cal A$.
Then $\{x:(x\cup f(x))(\xi)=1\}\in\dom\nu$ for every
$\xi\in\kappa+\lambda$, so we have a Baire measure $\mu$ on
$\{0,1\}^{\kappa+\lambda}$ defined by saying that
$\mu E=\nu\{x:x\cup f(x)\in E\}$ for Baire sets
$E\subseteq\{0,1\}^{\kappa+\lambda}$.   If $K\in[\kappa]^{<\omega}$ and
$z\in\{0,1\}^{\kappa}$, then

\Centerline{$\mu\{y:z\subseteq y\}=\nu\{x:z\subseteq x\cup f(x)\}
=\nu\{x:z\subseteq x\}=\nu_{\kappa}\{x:z\subseteq x\}
=2^{-\#(K)}$;}

\noindent while if $E\in\CalBa_{\kappa+\lambda}$ and
$x\cup f(x)\in E$ for every
$x\in\{0,1\}^{\kappa}$, then $\mu E=\nu\{0,1\}^{\kappa}=1$, so
$\mu^*\{x\cup f(x):x\in\{0,1\}^{\kappa}\}=1$.   Thus (ii) is true.

\medskip

{\bf (ii)$\Rightarrow$(i)} Assume (ii).   Let $\Cal A$ be a family of
subsets of $\{0,1\}^{\kappa}$ with $\#(\Cal A)\le\lambda$.   Let
$\ofamily{\eta}{\lambda}{A_{\eta}}$ run over $\Cal A\cup\{\emptyset\}$.
Define $f:\{0,1\}^{\kappa}\to\{0,1\}^{(\kappa+\lambda)\setminus\kappa}$
by saying that
$f(x)(\kappa+\eta)=(\chi A_{\eta})(x)$ whenever $\eta<\lambda$ and
$x\in\{0,1\}^{\kappa}$.   Let $\mu$ be a Baire measure on
$\{0,1\}^{\kappa+\lambda}$ satisfying the conditions of (ii).   Set
$g(x)=x\cup f(x)$ for $x\in\{0,1\}^{\kappa}$.
Because $g[\{0,1\}^{\kappa}]$ has full outer measure for
$\mu$, we have a measure $\nu$ on $\{0,1\}^{\kappa}$ such that
$\nu g^{-1}[E]=\mu E$ for every Baire set
$E\subseteq\{0,1\}^{\kappa+\lambda}$ (234F);
let $\hat\nu$ be the completion of $\nu$.   Now
$A_{\eta}=g^{-1}[\{y:y(\kappa+\eta)=1\}]$ is measured by $\nu$ and
$\hat\nu$.   Also

\Centerline{$\hat\nu\{x:z\subseteq x\}
=\nu\{x:z\subseteq x\}=\nu\{x:z\subseteq g(x)\}
=\mu\{y:z\subseteq y\}=2^{-\#(K)}$}

\noindent whenever
$K\in[\kappa]^{<\omega}$ and $z\in\{0,1\}^{\kappa}$, so $\hat\nu$ extends
$\nu_{\kappa}$ (254G) and is an extension of $\nu_{\kappa}$ measuring every
member of $\Cal A$.

\medskip

{\bf (i)$\Rightarrow$(iii)} Suppose that (i) is true.

\medskip

\quad\grheada\ Let $(X,\Sigma,\mu)$ be a compact probability space of
Maharam
type at most $\kappa$, and $\Cal A$ a family of subsets of $X$ with cardinal at
most $\lambda$.   Then there is a function $h:\{0,1\}^{\kappa}\to X$ which
is \imp\ for $\nu_{\kappa}$ and $\mu$.   \Prf\ By 332P, the measure algebra
of $\mu$ can be embedded into $\frak B_{\kappa}$;  by 343B, this embedding
can be realized by an \imp\ function from $\{0,1\}^{\kappa}$ to $X$.\ \QeD\
Now $\Cal C=\{h^{-1}[A]:A\in\Cal A\}$ has cardinal at most $\lambda$,
so there is
an extension $\nu$ of $\nu_{\kappa}$ measuring every member of $\Cal C$;
and the image measure $\nu h^{-1}$ extends $\mu$ and measures every member
of $\Cal A$.

\medskip

\quad\grheadb\ It follows at once that if $(X,\Sigma,\mu)$ is a compact
totally finite measure space with Maharam
type at most $\kappa$, and $\Cal A$ a family of subsets of $X$ with cardinal at
most $\lambda$, then $\mu$ can be extended to every member of $\Cal A$.
(If $\mu X=0$ this is trivial, and otherwise we can apply ($\alpha$) to a
scalar multiple of $\mu$.)

\medskip

\quad\grheadc\ Now suppose that $(X,\Sigma,\mu)$ is a locally compact
semi-finite measure space with Maharam
type at most $\kappa$, and $\Cal A$ a family of subsets of $X$ with cardinal at
most $\lambda$.   In the measure algebra $(\frak A,\bar\mu)$ of $\mu$,
let $D$ be a
partition of unity consisting of elements of finite measure;  for $d\in D$
choose $E_d\in\Sigma$ such that $E_d^{\ssbullet}=d$.   If $G\in\Sigma$ then

\Centerline{$\mu G=\bar\mu G^{\ssbullet}
=\sum_{d\in D}\bar\mu(d\Bcap G^{\ssbullet})=\sum_{d\in D}\mu(E_d\cap G)$.}

\noindent For each $d\in D$, the subspace measure $\mu_{E_d}$ on $E_d$ is
compact and totally finite and has Maharam type at most $\kappa$ (put
331Hc and 322Ja together),
so by ($\beta$) can be extended to a measure
$\mu'_{E_d}$ measuring $A\cap E_d$ for every $A\in\Cal A$.   Set
$\mu'F=\sum_{d\in D}\mu'_{E_d}(F\cap E_d)$ whenever $F\subseteq X$ is such
that the sum is defined;  then $\mu'$ is a measure on $X$, extending $\mu$
and measuring every set in $\Cal A$, as required.

\medskip

{\bf (iii)$\Rightarrow$(i)} is trivial.
}%end of proof of 552M

\leader{552N}{Theorem}\cmmnt{ ({\smc Carlson 84})} Let $\kappa$ and
$\lambda$ be infinite cardinals such that $\kappa$ is greater than the
cardinal power $\lambda^{\omega}$.   Then

\doubleinset{$\VVdPk$ if $\Cal A\subseteq\Cal P(\{0,1\}^{\check\kappa})$
and $\#(\Cal A)\le\check\lambda$, there is an extension of
$\nu_{\check\kappa}$ to a measure measuring every member of $\Cal A$.}

%what if we have a family $\Cal A$ expressible as the union of
%at most  \check\lambda  well-ordered sets?  see 214P

\proof{{\bf (a)}
Let $\langle e_{\xi\zeta}\rangle_{\xi,\zeta<\kappa}$
be a re-indexing of the standard generating family in $\frak B_{\kappa}$.
For $J\subseteq\kappa\times\kappa$ let $\frak C_J$ be the closed subalgebra of
$\frak B_{\kappa}$ generated by $\{e_{\xi\zeta}:(\xi,\zeta)\in J\}$.
Recall that $\#(L^{\infty}(\frak C_J))\le\max(\omega,\#(J)^{\omega})$
for every $J$ (524Ma, 515Mb);
%515Mb  \#(L^0)=\/(\frak A)
we shall also need to know that
every element of $\frak B_{\kappa}$ belongs to $\frak C_J$ for
a smallest $J\subseteq\kappa\times\kappa$, and this
$J$ is countable (254Rc, 531Jb).

Set
$I=(\kappa+\lambda)\setminus\kappa$, where $\kappa+\lambda$ is the ordinal
sum, so that $I$ is disjoint from $\kappa$ and $\#(I)=\lambda$.
Let $\dot f$ be a $\BbbPk$-name for a function
from $\{0,1\}^{\check\kappa}$ to $\{0,1\}^{\check I}$.

For each $\xi<\kappa$ let $\dot x_{\xi}$ be a $\BbbPk$-name for a member of
$\{0,1\}^{\check\kappa}$ such that
$\Bvalue{\dot x_{\xi}(\check\zeta)=1}=e_{\xi\zeta}$
for every $\zeta<\kappa$.   For
$z\in\Fn_{<\omega}(\kappa+\lambda;\{0,1\})$ and $\xi<\kappa$, set

\Centerline{$a_{\xi z}
=\Bvalue{\check z\subseteq\dot x_{\xi}\cup\dot f(\dot x_{\xi})}$}

\noindent and let
$J_{\xi z}\subseteq\kappa\times\kappa$ be the smallest set such that
$a_{\xi z}\in\frak C_{J_{\xi z}}$.   Note that

\Centerline{$a_{\xi z}
=a_{\xi,z\restr I}
  \Bcap\inf_{\zeta\in\kappa\cap z^{-1}[\{1\}]}e_{\xi\zeta}
  \Bsetminus\sup_{\zeta\in\kappa\cap z^{-1}[\{0\}]}e_{\xi\zeta}$,}

\noindent so that

\Centerline{$J_{\xi z}
\subseteq J_{\xi,z\restr I}\cup(\{\xi\}\times\dom z)$.}

Set $\theta=(\lambda^{\omega})^+\le\kappa$.
For $\xi\le\theta$ let $L_0(\xi)\subseteq\kappa$ be the smallest set
such that $\xi\subseteq L_0(\xi)$ and
$J_{\eta w}\subseteq L_0(\xi)\times L_0(\xi)$
for every $\eta<\xi$ and $w\in\Fn_{<\omega}(I;\{0,1\})$;  set
$L(\xi)=L_0(\xi)\times L_0(\xi)$.   Then
$\#(L(\xi))\le\max(\omega,\lambda,\#(\xi))<\theta$ for every
$\xi<\theta$, and $L(\xi)=\bigcup_{\eta<\xi}L(\eta)$ for limit
$\xi\le\theta$.   Set

\Centerline{$D^*=\{\xi:\xi<\theta$ is a limit ordinal,
$\xi>\sup(\theta\cap L_0(\eta))$ for every $\eta<\xi\}$;}

\noindent then $D^*$ is a closed cofinal subset of $\theta$,
and $\xi\notin L_0(\xi)$ for every $\xi\in D^*$.   So

\Centerline{$S=\{\xi:\xi\in D^*$, $\cf(\xi\cap D^*)\ge\omega_1\}$}

\noindent is stationary in $\theta$.

\medskip

{\bf (b)} For $J\subseteq\kappa\times\kappa$, let
$P_J:L^1(\frak B_{\kappa})
\to L^1(\frak C_J)\subseteq L^1(\frak B_{\kappa})$ be the
conditional expectation operator defined by saying that
$P_Ju\in L^1(\frak C_J)$ and
$\int_cP_Ju=\int_cu$ whenever $c\in\frak C_J$ and
$u\in L^1(\frak B_{\kappa})$ (254R, 365Q\formerly{3{}65R}).
We need to know that $P_{J\cap J'}=P_JP_{J'}$ for all $J$,
$J'\subseteq\kappa$ (254Ra), and that $P_J(u\times v)=u\times P_Jv$
whenever $u\in L^1(\frak C_J)$,
$v\in L^1(\frak B_{\kappa})$ and $u\times v\in L^1(\frak B_{\kappa})$
(242L).   It follows that if $J$, $J'$,
$J''\subseteq\kappa\times\kappa$, $u\in L^1(\frak C_J)$ and
$J\cap J'=J\cap J''$, then

\Centerline{$P_{J'}(u)=P_{J'}P_J(u)=P_{J\cap J'}(u)=P_{J\cap J''}(u)
=P_{J''}(u)$.}

If $h:\kappa\times\kappa\to\kappa\times\kappa$ is any permutation, then we
have a corresponding measure-preserving automorphism
$\pi:\frak B_{\kappa}\to\frak B_{\kappa}$ defined by saying that
$\pi e_{\xi\zeta}=e_{\xi'\zeta'}$ if $(\xi',\zeta')=h(\xi,\zeta)$, and
a Banach lattice automorphism
$T:L^1(\frak B_{\kappa})\to L^1(\frak B_{\kappa})$ defined by
saying that $T(\chi a)=\chi\pi a$ for $a\in\frak B_{\kappa}$ (see
365N\formerly{3{}65O});
note that $T\restr L^{\infty}(\frak B_{\kappa})$ is multiplicative
(compare the formulae in 365Nb and 364Pa).
%Now, for any $J\subseteq\kappa\times\kappa$,
%$P_{h[J]}=TP_JT^{-1}$.   \Prf\ Write $\phi:\frak C_J\to\frak B_{\kappa}$
%and $\psi:\frak C_{h[J]}\to\frak B_{\kappa}$ for the identity maps.
%Then $\psi\pi=\pi\phi:\frak C_J\to\frak B_{\kappa}$, because
%$\pi[\frak C_J]=\frak C_{h[J]}$.   By 365O,
%
%\Centerline{$P_{\pi}P_{\psi}=P_{\psi\pi}=P_{\pi\phi}=P_{\phi}P_{\pi}$,}
%
%\noindent where $P_{\pi}$, $P_{\psi}$, $P_{\phi}$ are the conditional
%expectation operators associated with the measure-preserving Boolean
%homomorphisms $\pi$, $\psi$ and $\phi$.   But $\pi$ is an automorphism, so
%$P_{\pi}$ is just $T^{-1}$ (365P), while in the language here we have
%$P_{\phi}=P_J$ and $P_{\psi}=P_{h[J]}$;  so
%$T^{-1}P_{h[J]}=P_JT^{-1}$, that is, $P_{h[J]}=TP_JT^{-1}$.\ \Qed

If $J\subseteq\kappa\times\kappa$ and $u\in L^1(\frak C_J)$, then
$Tu\in L^1(\frak C_{h[J]})$.    \Prf\ By 314H,
$\frak C_{h[J]}=\pi[\frak C_J]$;  now use the fact that
$\Bvalue{Tu>\alpha}=\pi\Bvalue{u>\alpha}$ for every $\alpha\in\Bbb R$.\
\QeD\   If
$h\restr J$ is the identity, then $\pi a=a$ for every $a\in\frak C_J$ and
$Tv=v$ for every $v\in L^{\infty}(\frak C_J)$.   Consequently $P_JT=P_J$.
\Prf\ If $u\in L^1(\frak B_{\kappa})$ and $c\in\frak C_J$ then

$$\eqalign{\int_cP_JTu
&=\int_cTu
=\int Tu\times\chi c
=\int T^{-1}(Tu\times\chi c)
\cr&=\int u\times T^{-1}\chi c
=\int u\times\chi c
=\int_cu;\cr}$$

\noindent as $P_JTu$ certainly belongs to $L^1(\frak C_J)$, it must be
equal to $P_Ju$.\ \Qed

\medskip

{\bf (c)} Fix $\xi\in S$ and $w\in\Fn_{<\omega}(I;\{0,1\})$ for the moment.
Because $\xi\cap D^*$ has
uncountable cofinality and $J_{\xi w}$ is countable, there is a
$g_w(\xi)\in\xi\cap D^*$ such that
$J_{\xi w}\cap L(\xi)\subseteq L(g_w(\xi))$ and
$\{\eta:\eta\in L_0(\xi)$, $(\xi,\eta)\in J_{\xi w}\}
\subseteq L_0(g_w(\xi))$.
Let $h_{\xi w}:\kappa\times\kappa\to\kappa\times\kappa$ be the involution
defined by saying that

$$\eqalign{h_{\xi w}(\eta,\zeta)
&=(g_w(\xi),\zeta)\text{ if }\eta=\xi,\cr
&=(\xi,\zeta)\text{ if }\eta=g_w(\xi),\cr
&=(\eta,\zeta)\text{ otherwise;}\cr}$$

\noindent note that

\Centerline{$h_{\xi w}[J_{\xi w}]\cap L(\xi)
\subseteq L(g_w(\xi))\cup(\{g_w(\xi)\}\times L_0(g_w(\xi))
\subseteq L(g_w(\xi)+1)$,}

\noindent while $h_{\xi w}$ is the identity on $L(g_w(\xi))$, since
neither $\xi$ nor $g_w(\xi)$ belongs to $L_0(g_w(\xi))$.   Let
$T_{\xi w}:L^1(\frak B_{\kappa})\to L^1(\frak B_{\kappa})$ be
the corresponding Banach lattice isomorphism defined as in (b) above.
Then (b) tells us that

\Centerline{$P_{L(g_w(\xi))}T_{\xi w}=P_{L(g_w(\xi))}$.}

Set

\Centerline{$u_{\xi w}=P_{L(g_w(\xi)+1)}T_{\xi w}(\chi a_{\xi w})
\in L^{\infty}(\frak C_{L(g_w(\xi)+1)})$.}

\medskip

{\bf (d)} Setting $M(\eta)
=\eta\times L^{\infty}(\frak C_{L(\eta)})$ for $\eta<\theta$, we
see that

\Centerline{$\#(M(\eta))\le\#(\eta)^{\omega}\le\lambda^{\omega}<\theta$}

\noindent whenever $\eta\ge 2$ (see (a) above), while
$(g_w(\xi),u_{\xi w})\in\bigcup_{\eta<\xi}M(\eta)$ whenever $\xi\in S$ and
$w\in\Fn_{<\omega}(I;\{0,1\})$.   Since
$\#(\Fn_{<\omega}(I;\{0,1\}))=\lambda<\theta$, 552L tells us that there is
an $\omega_1$-complete filter $\Cal F$ on $\theta$,
containing $\theta\setminus\zeta$
for every $\zeta<\theta$, such that for every
$w\in\Fn_{<\omega}(I;\{0,1\})$ there is a $D\in\Cal F$ such that
$D\subseteq S$ and $g_w$ and
$\xi\mapsto u_{\xi w}$ are constant on $D$.

\medskip

{\bf (e)} For $z\in\Fn_{<\omega}(\kappa+\lambda;\{0,1\})$ and
$\xi<\theta$, set $v_{\xi z}=P_{L(\xi)}(\chi a_{\xi z})$.
Then there is a
$D\in\Cal F$ such that $D\subseteq S$
and $\xi\mapsto v_{\xi z}$ is constant on $D$.
\Prf\ Set $z'=z\restr L_0(\theta)$,
$z''=z\restr\kappa\setminus L_0(\theta)$
and $w=z\restr I$, so that
$a_{\xi z}=a_{\xi z'}\Bcap a_{\xi z''}\Bcap a_{\xi w}$.   Set
$m=\#(z'')$, so that
$a_{\xi z''}=\Bvalue{\check z''\subseteq\dot x_{\xi}}$
has measure $2^{-m}$ for every $\xi$.   Also
$a_{\xi z''}\in\frak C_{\kappa\times(\kappa\setminus L_0(\theta))}$ is
stochastically independent of $\frak C_{L(\theta)}$, so
$P_{L(\theta)}(\chi a_{\xi z''})=2^{-m}\chi 1$;
while $a_{\xi z'}$ and $a_{\xi w}$ belong to
$\frak C_{L(\theta)}$, so, using the formulae in (b),

$$\eqalign{v_{\xi z}
&=P_{L(\xi)}(\chi a_{\xi z})
=P_{L(\xi)}P_{L(\theta)}(\chi a_{\xi z})\cr
&=P_{L(\xi)}P_{L(\theta)}
  (\chi a_{\xi z'}\times\chi a_{\xi z''}\times\chi a_{\xi w})\cr
&=P_{L(\xi)}(\chi a_{\xi z'}\times\chi a_{\xi w}
  \times P_{L(\theta)}(\chi a_{\xi z''}))\cr
&=2^{-m}P_{L(\xi)}(\chi a_{\xi z'}\times\chi a_{\xi w}).\cr}$$

Let $\xi_0<\theta$ be such that $\dom z'\subseteq L_0(\xi_0)$.   By (d),
there are $D_0\in\Cal F$, $\zeta<\theta$ and
$u\in L^{\infty}(\frak B_{\kappa})$ such that
$D_0\subseteq S$ and $g_w(\xi)=\zeta$ and
$u_{\xi w}=u$ for every $\xi\in D_0$.   For
$\xi\in D_0\setminus\xi_0$ take $h_{\xi w}$ and $T_{\xi w}$ as in (c).
Then, writing $\pi_{\xi w}$ for the measure-preserving
automorphism defined from $h_{\xi w}$ as in (b),

$$\eqalign{\pi_{\xi w}a_{\xi z'}
&=\pi_{\xi w}(\inf_{z'(\eta)=1}e_{\xi\eta}
  \Bsetminus\sup_{z'(\eta)=0}e_{\xi\eta})
=\inf_{z'(\eta)=1}\pi_{\xi w}e_{\xi\eta}
  \Bsetminus\sup_{z'(\eta)=0}\pi_{\xi w}e_{\xi\eta}\cr
&=\inf_{z'(\eta)=1}e_{\zeta\eta}
 \Bsetminus\sup_{z'(\eta)=0}e_{\zeta\eta}
=a_{\zeta z'};\cr}$$

\noindent consequently $T_{\xi w}(\chi a_{\xi z'})=\chi a_{\zeta z'}$.
Now

$$\eqalignno{P_{L(\xi)}(\chi a_{\xi z'}\times\chi a_{\xi w})
&=P_{L(\zeta)}(\chi a_{\xi z'}\times\chi a_{\xi w})\cr
\displaycause{because
$\chi a_{\xi z'}\times\chi a_{\xi w}
\in L^{\infty}(\frak C_{(\{\xi\}\times L_0(\xi))\cup J_{\xi w}})$, and
$(\{\xi\}\times L_0(\xi))\cup J_{\xi w})\cap L(\xi)
=J_{\xi w}\cap L(\xi)\subseteq L(\zeta)\subseteq L(\xi)$}
&=P_{L(\zeta)}T_{\xi w}(\chi a_{\xi z'}\times\chi a_{\xi w})\cr
\displaycause{see (c) above}
&=P_{L(\zeta)}
  (T_{\xi w}\chi a_{\xi z'}\times T_{\xi w}\chi a_{\xi w})\cr
\displaycause{because $T_{\xi w}$ is multiplicative on
$L^{\infty}(\frak B_{\kappa})$}
&=P_{L(\zeta)}
  (\chi a_{\zeta z'}\times T_{\xi w}\chi a_{\xi w})\cr
&=P_{L(\zeta)}P_{L(\xi)}
  (\chi a_{\zeta z'}\times T_{\xi w}\chi a_{\xi w})\cr
\displaycause{because $L(\zeta)\subseteq L(\xi)$}
&=P_{L(\zeta)}
  (\chi a_{\zeta z'}\times P_{L(\xi)}T_{\xi w}\chi a_{\xi w})\cr
\displaycause{because
$a_{\zeta z'}\in\frak C_{\{\zeta\}\times L_0(\xi)}
\subseteq\frak C_{L(\xi)}$}
&=P_{L(\zeta)}
  (\chi a_{\zeta z'}\times P_{L(\zeta+1)}T_{\xi w}\chi a_{\xi w})\cr
\displaycause{because
$T_{\xi w}\chi a_{\xi w}\in L^{\infty}(\frak C_{h_{\xi w}[J_{\xi w}]})$ and
$h_{\xi w}[J_{\xi w}]\cap L(\xi)\subseteq L(\zeta+1)$}
&=P_{L(\zeta)}(\chi a_{\zeta z'}\times u_{\xi w})
=P_{L(\zeta)}(\chi a_{\zeta z'}\times u).\cr}$$

Finally, we get

\Centerline{$v_{\xi z}
=2^{-m}P_{L(\xi)}(\chi a_{\xi z'}\times\chi a_{\xi w})
=2^{-m}P_{L(\zeta)}(\chi a_{\zeta z'}\times u)$}

\noindent for every $\xi\in D=D_0\setminus\xi_0$,
so we have the required constant value.\ \Qed

\medskip

{\bf (f)} For each $z\in\Fn_{<\omega}(\kappa+\lambda;\{0,1\})$ set
$v_z=\lim_{\xi\to\Cal F}v_{\xi z}$, the limit being defined in the strong
sense that $\{\xi:\xi\in S$, $v_{\xi z}=v_z\}$ belongs to $\Cal F$.
Observe that
$0\le v_z\le\chi 1$ and that $v_{\emptyset}=\chi 1$, because
$a_{\xi\emptyset}=1$ for every $\xi\in S$.   If
$z\in\Fn_{<\omega}(\kappa+\lambda;\{0,1\})$ and
$\eta\in(\kappa+\lambda)\setminus\dom z$,
then $a_{\xi,z\cup\{(\eta,0)\}}$ and $a_{\xi,z\cup\{(\eta,1)\}}$ are
disjoint and have union $a_{\xi z}$, so

\Centerline{$v_{\xi z}=P_{L(\xi)}(\chi a_{\xi z})
=P_{L(\xi)}
  (\chi a_{\xi,z\cup\{(\eta,0)\}}+\chi a_{\xi,z\cup\{(\eta,1)\}})
=v_{\xi,z\cup\{(\eta,0)\}}+v_{\xi,z\cup\{(\eta,1)\}}$}

\noindent for every $\xi\in S$, and
$v_z=v_{z\cup\{(\eta,0)\}}+v_{z\cup\{(\eta,1)\}}$.
Let $\vec v_z$ be the $\BbbPk$-name for a real number corresponding
to $v_z$ as defined in 5A3L.

\medskip

{\bf (g)} We have a $\BbbPk$-name $\dot\mu$ for a
Baire probability measure on $\{0,1\}^{\kappa+\lambda}$ such that

\Centerline{$\VVdPk\,
\dot\mu\{x:\check z\subseteq x\in\{0,1\}^{\check\kappa+\check\lambda}\}
=\vec v_z$}

\noindent for every $z\in\Fn_{<\omega}(\kappa+\lambda;\{0,1\})$.
\Prf\ Start by setting

\Centerline{$\dot\mu_0
=\{((\check z,\vec v_z),1_{\frak B_{\kappa}}):
z\in\Fn_{<\omega}(\kappa+\lambda;\{0,1\})\}$,}

\noindent so that $\dot\mu_0$ is a $\Bbb P_{\kappa}$-name for a function
from $\Fn_{<\omega}(\check\kappa+\check\lambda;\{0,1\})$ to $[0,1]$ and

\Centerline{$\VVdPk\,
\dot\mu_0\check z=\vec v_z$}

\noindent for every $z\in\Fn_{<\omega}(\kappa+\lambda;\{0,1\})$;  note that

\Centerline{$\VVdPk\,
  \check\kappa+\check\lambda=(\kappa+\lambda)\varBspcheck$
and $\Fn_{<\omega}(\check\kappa+\check\lambda;\{0,1\})
=(\Fn_{<\omega}(\kappa+\lambda;\{0,1\}))\varBspcheck$.}

\noindent Now (f) tells us that

\Centerline{$\VVdPk\,\dot\mu_0\emptyset=1$,}

$$\eqalign{\VVdPk\,\dot\mu_0(z)
&=\dot\mu_0(z\cup\{(\eta,0)\})+\dot\mu_0(z\cup\{(\eta,1)\})\cr
&\mskip100mu\text{whenever }
  z\in\Fn_{<\omega}(\check\kappa+\check\lambda;\{0,1\})
\text{ and }\eta\in(\check\kappa+\check\lambda)\setminus\dom z.\cr}$$

\noindent By 552K, copied into $V^{\BbbPk}$,

$$\eqalign{\VVdPk\,
&\text{ there is a Radon measure }
  \mu\text{ on }\{0,1\}^{\check\kappa+\check\lambda}\text{ such that }\cr
&\mskip150mu\mu\{x:z\subseteq x\}=\dot\mu_0(z)
  \text{ for every }
  z\in\Fn_{<\omega}(\check\kappa+\check\lambda;\{0,1\}).\cr}$$

\noindent In fact we don't really want the Radon measure here, but its
restriction to the Baire $\sigma$-algebra.
Let $\dot\mu$ be a $\BbbPk$-name for a Baire measure on
$\{0,1\}^{\check\kappa+\check\lambda}$ such that

\Centerline{$\VVdPk\,
\dot\mu\{x:\check z\subseteq x\in\{0,1\}^{\check\kappa+\check\lambda}\}
=\dot\mu_0(\check z)=\vec v_z$}

\noindent for every $z\in\Fn_{<\omega}(\kappa+\lambda;\{0,1\})$;
this is what we have been looking for.\ \Qed

\medskip

{\bf (h)} I had better check that

\Centerline{$\VVdPk\,\dot\mu\{x:\check z\subseteq x\}
=(2^{-\#(z)})\varBspcheck$}

\noindent whenever $z\in\Fn_{<\omega}(\kappa;\{0,1\})$.
\Prf\ If $z\in\Fn_{<\omega}(\kappa;\{0,1\})$ and
$\xi\in S$,
then $a_{\xi z}$ belongs to $\frak C_{\{\xi\}\times\kappa}$, so is
stochastically independent of $\frak C_{L(\xi)}$, and

\Centerline{$v_{\xi z}
=P_{L(\xi)}(\chi a_{\xi z})
=(\bar\nu_{\kappa}a_{\xi z})\chi 1=2^{-\#(z)}\chi 1$.}

\noindent Accordingly
$v_z=2^{-\#(z)}\chi 1$ and

\Centerline{$\VVdPk\,\dot\mu\{x:\check z\subseteq x\}=\vec v_z
=(2^{-\#(z)})\varBspcheck$.   \Qed}

\medskip

{\bf (i)} Finally, we come to the key fact:

\Centerline{$\VVdPk\,
\{\dot x_{\xi}\cup\dot f(\dot x_{\xi}):\xi<\check\theta\}$
has full outer measure for $\dot\mu$.}

\noindent\Prf\Quer\ Otherwise, there are a non-zero $b\in\frak B_{\kappa}$,
a rational number $q<1$ and a sequence $\sequencen{\dot C_n}$ of
$\BbbPk$-names for basic cylinder sets in
$\{0,1\}^{\check\kappa+\check\lambda}$ such that

\Centerline{$b\VVdPk\,\dot x_{\xi}\cup\dot f(\dot x_{\xi})
\in\bigcup_{n\in\Bbb N}\dot C_n$}

\noindent for every $\xi<\theta$, while also

\Centerline{$b\VVdPk\,\sum_{n=0}^{\infty}\dot\mu\dot C_n\le\check q$.}

\noindent Because $\frak B_{\kappa}$ is ccc, we can find for each
$n\in\Bbb N$ a partition $\sequence{i}{b_{ni}}$ of unity in
$\frak B_{\kappa}$, and a sequence
$\sequence{i}{z_{ni}}$ in $\Fn_{<\omega}(\kappa+\lambda;\{0,1\})$, such
that $b_{ni}\VVdPk\,\dot C_n=\{x:\check z_{ni}\subseteq x\}$ for each $i$.
Now

\Centerline{$\Bvalue{\dot x_{\xi}\cup\dot f(\dot x_{\xi})\in\dot C_n}
=\sup_{i\in\Bbb N}b_{ni}
  \Bcap\Bvalue{\check z_{ni}\subseteq\dot x_{ni}\cup\dot f(\dot x_{\xi}}
=\sup_{i\in\Bbb N}b_{ni}\Bcap a_{\xi,z_{ni}}$}

\noindent so we must have

\Centerline{$b\Bsubseteq\sup_{i,n\in\Bbb N}b_{ni}\Bcap a_{\xi,z_{ni}}$}

\noindent for every $\xi$.   At the same time,

\Centerline{$b_{ni}\VVdPk\,\dot\mu\dot C_n=\vec v_{z_{ni}}$}

\noindent for each $n$ and $i$, so
$\dot\mu\dot C_n$ is represented by
$\sum_{i\in\Bbb N}\chi b_{ni}\times v_{z_{ni}}$ in
$L^{\infty}(\frak B_{\kappa})$ and
$\min(1,\sum_{n=0}^{\infty}\dot\mu\dot C_n)$ is
represented by
$\chi 1\wedge\sum_{n,i\in\Bbb N}\chi b_{ni}\times v_{z_{ni}}$.
Since
$\Bvalue{\sum_{n=0}^{\infty}\dot\mu\dot C_n\le\check q}$ includes $b$,

\Centerline{$\sum_{n,i\in\Bbb N}\chi b\times\chi b_{ni}\times v_{z_{ni}}
\le q\chi b$.}

Let $J\subseteq\kappa\times\kappa$ be a countable set such that
$b\Bcap b_{ni}\in\frak C_J$ for every $n$, $i\in\Bbb N$.   Then,
because $\Cal F$ is $\omega_1$-complete and contains $S\setminus\zeta$ for
every $\zeta<\theta$, there is a $\xi\in S$ such that

\Centerline{$v_{\xi z_{ni}}=v_{z_{ni}}$ whenever $n$, $i\in\Bbb N$,}

\Centerline{$J\cap L(\theta)=J\cap L(\xi)$,
\quad$J\cap(\{\xi\}\times\kappa)=\emptyset$.}

\noindent Set $L=L(\theta)\cup(\{\xi\}\times\kappa)$.   If $n$, $i\in\Bbb N$, then

$$\eqalignno{\int\chi b\times\chi b_{ni}\times v_{z_{ni}}
&=\int\chi b\times\chi b_{ni}\times P_{L(\xi)}\chi a_{\xi,z_{ni}}\cr
&=\int P_{L(\xi)}(\chi b\times\chi b_{ni}
  \times P_{L(\xi)}\chi a_{\xi,z_{ni}})\cr
&=\int P_{L(\xi)}(\chi b\times\chi b_{ni})
  \times P_{L(\xi)}\chi a_{\xi,z_{ni}}\cr
&=\int P_{L(\xi)}(\chi b\times\chi b_{ni})
  \times\chi a_{\xi,z_{ni}}\cr
&=\int P_L(\chi b\times\chi b_{ni})
  \times\chi a_{\xi,z_{ni}}\cr
\displaycause{because $J\cap L=J\cap L(\xi)$}
&=\int P_L(\chi b\times\chi b_{ni}
  \times\chi a_{\xi,z_{ni}})\cr
\displaycause{because $J_{\xi,z_{ni}}
\subseteq J_{\xi,z_{ni}\restr I}\cup(\{\xi\}\times\kappa)
\subseteq L$, so $a_{\xi,z_{ni}}\in\frak C_L$}
&=\int\chi b\times\chi b_{ni}\times\chi a_{\xi,z_{ni}}.\cr}$$

\noindent Summing over $n$ and $i$, we have

$$\eqalignno{\bar\nu_{\kappa}b
&\le\sum_{n=0}^{\infty}\sum_{i=0}^{\infty}
  \bar\nu_{\kappa}(b\Bcap b_{ni}\Bcap a_{\xi,z_{ni}})
=\sum_{n=0}^{\infty}\sum_{i=0}^{\infty}\int\chi b\times\chi b_{ni}
  \times\chi a_{\xi,z_{ni}}\cr
&=\sum_{n=0}^{\infty}\sum_{i=0}^{\infty}\int\chi b\times\chi b_{ni}
  \times v_{z_{ni}}
\le\int q\chi b
=q\bar\nu_{\kappa}b,\cr}$$

\noindent which is impossible.\ \Bang\Qed

\medskip

{\bf (j)} What all this shows is that

\doubleinset{$\VVdPk$ for every
$f:\{0,1\}^{\check\kappa}
  \to\{0,1\}^{(\check\kappa+\check\lambda)\setminus\check\kappa}$
there is a Baire measure $\mu$
on $\{0,1\}^{\check\kappa+\check\lambda}$ such that
$\mu\{y:y\in\{0,1\}^{\check\kappa+\check\lambda},\,z\subseteq y\}
=2^{-\#(K)}$
whenever $K\in[\check\kappa]^{<\omega}$
and $z\in\{0,1\}^K$,
and $\mu^*\{x\cup f(x):x\in\{0,1\}^{\check\kappa}\}=1$.}

\noindent By 552M, copied into $V^{\BbbPk}$,

\doubleinset{$\VVdPk$ if $\Cal A\subseteq\Cal P(\{0,1\}^{\check\kappa})$
and $\#(\Cal A)\le\check\lambda$, there is an extension of
$\nu_{\check\kappa}$ to a measure measuring every member of $\Cal A$,}

\noindent as required.
}%end of proof of 552N

\leader{552O}{Proposition} Suppose that $(X,\Sigma,\mu)$ is a probability
space such that for every countable family $\Cal A$
of subsets of $X$ there is a measure on $X$
extending $\mu$ and measuring every member of $\Cal A$.

(a) If $Y$ is a universally negligible\cmmnt{ (definition:  439B)}
second-countable T$_0$ space, then $\#(Y)<\cov\Cal N(\mu)$.

(b) $\cov\Cal N(\mu)>\non\Cal N(\nu_{\omega})$.

\proof{{\bf (a)} \Quer\ Otherwise,
let $\family{y}{Y}{E_y}$ be a cover of $X$ by
$\mu$-negligible sets, and $f:X\to Y$ a function such that $x\in E_{f(x)}$
for every $x\in X$.   Let $\Cal U$ be a countable base for the topology of
$Y$ and $\Cal A=\{f^{-1}[U]:U\in\Cal U\}$;  let $\tilde\mu$
be a measure on $X$
extending $\mu$ and measuring every member of $\Cal A$.   Consider the
image measure $\tilde\mu f^{-1}$ on $Y$.
This measures every member of $\Cal U$ so measures every Borel set in $Y$;
let $\nu$ be its restriction to the Borel $\sigma$-algebra of $Y$.   Then
$\nu$ is a Borel probability measure on $Y$.   Take any $y\in Y$.
Because $Y$ has a T$_0$ topology, $\Cal U$ must separate the points of $Y$
and $\{y\}$ is a Borel set;  now

\Centerline{$\nu\{y\}=\tilde\mu f^{-1}[\{y\}]\le\tilde\mu^*E_y
\le\mu^*E_y=0$.}

\noindent So $\nu$ is zero on singletons and witnesses that $Y$ is not
universally negligible.\ \Bang

\medskip

{\bf (b)} By Grzegorek's theorem (439Fc), there is a universally negligible
set $Y\subseteq[0,1]$ with cardinal $\non\Cal N(\nu_{\omega})$.   (Recall
that the Lebesgue null ideal is isomorphic to $\Cal N(\nu_{\omega})$, as
noted in 522W(a-i).)
}%end of proof of 552O

%question:  do we really need "second-countable"?

\leader{552P}{Theorem} Let $\kappa$ and $\lambda$ be infinite
cardinals.   Then the iterated forcing notion
$\BbbPk*\Bbb P_{\check\lambda}$ has regular open algebra
isomorphic to $\frak B_{\max(\kappa,\lambda)}$.

\cmmnt{\medskip

\noindent{\bf Remark} Here $\Bbb P_{\check\lambda}$ represents a standard
$\BbbPk$-name for random real forcing;  see 551O.
}

\proof{ In Theorem 551Q, take $\Omega=\{0,1\}^{\kappa}$,
$\Sigma=\Tau_{\kappa}$, $\Cal I=\Cal N(\nu_{\kappa})$ and $I=\lambda$.
If we identify $\{0,1\}^{\kappa}\times\{0,1\}^{\lambda}$ with
$\{0,1\}^{\kappa+\lambda}$, where $\kappa+\lambda$ is the ordinal sum,
then $\Lambda=\Sigma\tensorhat\CalBa_{\lambda}$
becomes a $\sigma$-algebra intermediate between $\CalBa_{\kappa+\lambda}$
and $\Tau_{\kappa+\lambda}$, while

\Centerline{$\Cal J=\{W:W\in\Lambda$, $\nu_{\lambda}W[\{x\}]=0$ for
$\nu_{\kappa}$-almost every $x\in\{0,1\}^{\kappa}\}$}

\noindent is just $\Lambda\cap\Cal N(\nu_{\kappa+\lambda})$.   It follows
at once that the algebra $\frak A=\Lambda/\Cal J$ is isomorphic to
$\frak B_{\kappa+\lambda}$;  and 551Q tells us that
$\RO(\BbbPk*\Bbb P_{\check\lambda})$ is isomorphic to $\frak A$.   Since we
are supposing that $\kappa$ and $\lambda$ are infinite,
$\frak B_{\kappa+\lambda}\cong\frak B_{\max(\kappa,\lambda)}$ and we're
done.
}%end of proof of 552P

%\leader{552R}
%\leader{552S}
%\leader{552T}
%\leader{552U}
%\leader{552V}
%\leader{552W}

\exercises{\leader{552X}{Basic exercises (a)}
%\spheader 552Xa
Let $\kappa$ be an infinite cardinal.
Show that $\VVdPk\,\check\kappa^{\omega}=(\kappa^{\omega})\varBspcheck$,
where these are all cardinal powers.
%552B

\spheader 552Xb ({\smc Miller 82})
Suppose that $\frak c<\omega_{\omega}$.   Show that

\Centerline{$\VVdash_{\Bbb P_{\omega_{\omega}}}
\,\cov\Cal N(\nu_{\omega_1})=\omega_{\omega}
<\cov\Cal N(\nu_{\omega})$.}
%552G

\spheader 552Xc\dvAnew{2009}
Suppose that the continuum hypothesis is true.   Show that

\Centerline{$\VVdash_{\Bbb P_{\omega_2}}\,\,\frak c$ is a
precaliber of every measurable algebra.}

\noindent\Hint{525K.}
%552H out of order query

\spheader 552Xd\dvAnew{2010}
Describe Cicho\'n's diagram in the forcing universe
$V^{\Bbb P_{\omega_2}}$ (i) if we start with $\frak c=\omega_1$
(ii) if we start with $\frak m=\frak c=\omega_2$.   Locate the shrinking
number of Lebesgue measure in each case.
%552J

\spheader 552Xe\dvAnew{2010}
Suppose that the continuum hypothesis is true.   Show that

\doubleinset{$\VVdash_{\Bbb P_{\omega_2}}\,
\ci(\Cal P(\{0,1\}^{\omega})\setminus\Cal N(\nu_{\omega}))=\frak c$,
so there is a
family $\ofamily{\xi}{\frak c}{\nu_{\xi}}$ of additive functionals on
$\Cal P([0,1])$ such that $\sup_{\xi<\frak c}\nu_{\xi}A$ is the Lebesgue
outer measure of $A$ for every $A\subseteq[0,1]$.}

\noindent(See {\smc Lipecki 09}.)
%552J

\spheader 552Xf Suppose that the continuum hypothesis is true.
Show that there
is a sequence $\sequencen{A_n}$ of subsets of $[0,1]$ such that there is no
measure extending Lebesgue measure which measures every $A_n$.
\Hint{there is a sequence $\sequencen{f_n}$ of functions from $\omega_1$
to itself such that
$\{f_n(\xi):n\in\Bbb N\}=\{\eta:\eta\le\xi\}$ for every $\xi<\omega_1$.}
%552N

\leader{552Y}{Further exercises (a)}
%\spheader 552Ya
Let $\kappa$ and $\lambda$ be infinite cardinals, and $\mu$
a Baire measure on $\{0,1\}^{\lambda}$.   (i) Show that there is a
$\BbbPk$-name $\dot\mu$ for a Baire measure on $\{0,1\}^{\check\lambda}$
such that $\VVdPk\,\dot\mu\{x:\check z\subseteq x\}
=(\mu\{x:z\subseteq x\})\varBspcheck$
for every $z\in\Fn_{<\omega}(\lambda;\{0,1\})$.   (ii) Show that if
$A\subseteq\{0,1\}^{\lambda}$,
then $\VVdPk\,\dot\mu^*(\check A)=(\mu^*A)\varBspcheck$.
%552D, 552O

\spheader 552Yb(i) Show that for any non-zero
cardinals $\kappa$, $\lambda$ there are
cardinals $\theta^{\text{cov}}_{\kappa\lambda}$ and
$\theta^{\text{non}}_{\kappa\lambda}$ such that

\Centerline{$\VVdPk\,
\cov\Cal N(\nu_{\check\lambda})
=\check\theta^{\text{cov}}_{\kappa\lambda}$,
\quad$\non\Cal N(\nu_{\check\lambda})
=\check\theta^{\text{non}}_{\kappa\lambda}$.}

\noindent\Hint{if $\kappa$ is infinite, $\frak B_{\kappa}$ is homogeneous.}
(ii) Show that $\theta^{\text{cov}}_{\kappa\lambda}$ increases with
$\kappa$ and decreases with $\lambda$, while
$\theta^{\text{non}}_{\kappa\lambda}$ decreases with $\kappa$ and
increases with $\lambda$.   \Hint{552P.}
%552P 552G 552H
}%end of exercises


\endnotes{
\Notesheader{552} In any forcing model, all the open questions of ZFC
re-present themselves for our attention.   The first and most important
question concerns the continuum hypothesis, and in most cases we can say
something useful.   So I start with 552B:  `if you add $\kappa$ random
reals, then the continuum rises to $\kappa^{\omega}$'.   Any mnemonic of
this kind has to come with footnotes concerning the interpretation of the
terms, because we cannot rely on the formula
`$\kappa^{\omega}$' meaning the
same thing in the universe we start from and the forcing model we move to.
Indeed, in general forcing models, the symbol `$\kappa$' has to be watched,
since I normally reserve it for cardinals, and cardinals sometimes
collapse;  but here, at least, we have a ccc forcing notion, and cardinals
are preserved (5A3Nb).   Actually, `$\kappa^{\omega}$' also is safe in the
present context (552Xa);  but we find this out afterwards.

One of the central properties of random real
forcing concerns iteration:
if you do it twice, you still have random real forcing.   Of
course `iterated forcing', in a vast variety of forms, is an indispensable
technique, and two-stage forcing, as in 552P, is the easiest kind.   I do
not expect to quote this result very often in this book, but that is
because (for random reals) I am interested as much in the forcing notions
themselves, and the measure algebras which are their regular open algebras,
as in the propositions which are true in the forcing models.   So when I
see a proof which depends on repeated random real forcing my first impulse
is to examine the relevant properties of measure algebras, and this
generally leads to a direct proof in terms of single-stage forcing.
Note the form of Theorem 552P:  as in 551Q,
it does not claim that the iteration
$\BbbPk*\Bbb P_{\check\lambda}$ is isomorphic to
$\Bbb P_{\max(\kappa,\lambda)}$, but only that they have isomorphic regular
open algebras, and therefore lead to the same mathematical worlds (5A3I).

A typical example is in 552J.    Random real forcing does not change outer
measures (552D).   If we think of $\Bbb P_{\kappa}$ as an iteration
$\Bbb P_{\kappa\setminus J}*\Bbb P_{\check J}$, and we have a
$\Bbb P_{\kappa}$-name $\dot E$ for a `new' negligible set, then, thinking
in $V^{\Bbb P_{\check J}}$,
the set of members of $\{0,1\}^{\check\lambda}$ contained in
$\dot E$ will have to be negligible.
Back in the ordinary universe, we shall
have a $\Bbb P_J$-name for a negligible set containing every member of
$\{0,1\}^{\check\lambda}$ with a $\Bbb P_J$-name which belongs to $\dot E$.
In 552J, the idea is that if $\dot A$ is a set in
$V^{\Bbb P_{\kappa}}$ and every small subset of $\dot A$ is negligible
in $V^{\Bbb P_{\kappa}}$,
then at every stage the set of members of $\dot A$ which have
been named so far must be negligible in $V^{\Bbb P_{\kappa}}$,
just because there are not very
many names yet available, and therefore is also negligible
in the intermediate universe of the forcing notion $\Bbb P_{K_{\xi}}$.
This must be witnessed by a countable structure
in the intermediate universe, and the Pressing-Down Lemma tells us that
there is a stationary set of levels for which the same countable structure
will serve;  it follows easily that we have a name in $V^{\Bbb P_{\kappa}}$
for a negligible set including $\dot A$.   I invite you to seek out the
elements of the formal exposition in 552J which correspond to this
sketch.

552E can also be approached as a result about iterated random real forcing.
Here, $\dot A$ is just the set of `random reals' $\dot x_{\xi}$ built
directly from the regular open algebra $\frak B_{\kappa}$.   To see that
this is a Sierpi\'nski set, we need to look at a negligible set.   A
negligible set in
$V^{\Bbb P_{\kappa}}$ is included in one which has a name $\dot C$ in
$V^{\Bbb P_J}$ for some countable $J\subseteq\kappa$.   Thinking in
$V^{\Bbb P_J}$, all but countably many of the $\dot x_{\xi}$ are still
random, because they are the `random reals' of
$V^{\Bbb P_{\kappa\setminus J}}$, and therefore do not belong to $\dot C$.
The proof of 552E is no more than a formal elaboration of this idea, with
the extra technical device necessary to reach `strongly Sierpi\'nski'.

In 552C all we need to know is that $\BbbPk$ is weakly
$\sigma$-distributive, and the key fact is that for every name $\dot f$ for
a member of $\BbbN^{\Bbb N}$ there is an $h$ in the ordinary universe such
that $\VVdPk\,\dot f\le^*\check h$;
this is why such partial orders are sometimes
called `$\omega^{\omega}$-bounding'.    The rest of the argument is
based on the same ideas as part (d) of the proof of 5A3N.

In 552F-552J %552F 552G 552H 552I 552J
I list the results known to me about the
additivity, covering number, uniformity, cofinality and shrinking number
of the ideals
$\Cal N(\nu_{\lambda})$ after random real forcing.   Covering number,
uniformity and shrinking number are the
difficult ones, and even the most basic case, when $\lambda=\omega$ and
we are forcing with $\Bbb P_{\omega}$, seems not to have been completely
sorted out.   552Gb and 552Hc show that there is room for surprises.
My method throughout is to use the results of \S551 to relate
$\Cal N(\nu_{\lambda})$ in $V^{\Bbb P_{\kappa}}$ to
$\Cal N(\nu_{\kappa}\times\nu_{\lambda})$ in the original universe.  Given a
$\Bbb P_{\kappa}$-name $\dot W$ for a negligible set in
$\{0,1\}^{\check\lambda}$, we
have a negligible $W\subseteq\{0,1\}^{\kappa}\times\{0,1\}^{\lambda}$ such that
$\VVdPk\,\dot W\subseteq\vec W$, and then a negligible
$V\subseteq\{0,1\}^{\lambda}$, corresponding to the non-negligible horizontal
sections of $W$, such that $(\{0,1\}^{\lambda}\setminus V)\varBspcheck$ is
disjoint from $\vec W$ and $\dot W$ in $V^{\Bbb P_{\kappa}}$.

In 552K-552M %552K 552L 552M
I give some lemmas which apparently have nothing to do with forcing.   The
intention is to express as much as possible of the argument of Carlson's
theorem 552N as results in ZFC.   In this section I am taking forcing
arguments particularly laboriously;  but even when you have got to the
point where they seem elementary to you, I believe that it is still worth
while minimising the regions in which one has to deal with more than one
model of set theory at a time.
In 552M the parts (ii) and (iii) contrast oddly.   Part
(ii) is there to serve
as a combinatorial form of (i) which will be
accessible for the purposes of 552N.    Part (iii) is there to give
a notion of the scope of 552N, and in particular to show that in
random real models we have extension theorems for many measures not
obviously similar to the basic measures $\nu_{\kappa}$.   I have already
noted a similar result in 543G.

In \S439 I described a number of examples of probability spaces
$(X,\Sigma,\mu)$ with a countable family $\Cal A\subseteq\Cal PX$ such that
$\mu$ has no extension to a measure measuring every member of $\Cal A$.
In particular, as observed in 439Xk, Grzegorek's theorem 439Fc
gives us an example of a subspace of $[0,1]$ for which the subspace measure
fails to be extendable to some countably-generated $\sigma$-algebra.
These are ZFC examples;  we really do need something like `compactness'
in 552M(iii).

Note that {\smc Carlson 84} gives a rather sharper form of Theorem 552N,
carrying information about the covering numbers
of the measures constructed in $V^{\BbbPk}$.

}%end of notes

\discrpage

\frfilename{mt553.tex}
\versiondate{3.5.14}
\copyrightdate{2005}

\def\chaptername{Possible worlds}
\def\sectionname{Random reals II}

\Loadfourteens

\def\BbbPk{\Bbb P_{\kappa}}
\long\def\doubleinset#1{\inset{\inset{\parindent=-20pt #1}}}
\def\rank{\mathop{\text{rank}}}

\newsection{553}

In this section I collect some further properties of random real models
which seem less directly connected with the main topics of this book than
those treated in \S552.   The first concerns strong measure zero or
`Rothberger's property'\cmmnt{ (534C\formerly{5{}34C, 5{}34E})} and
gives a bound for the sizes of sets with this property.   The second relates
perfect sets in $V^{\Bbb P_{\kappa}}$ to negligible F$_{\sigma}$ sets in
the original universe;  it shows that a random real model can have
properties relevant to a question in \S531 (553F).
Following these, I discuss properties of ultrafilters and partially
ordered sets which are not obviously connected with measure theory,
but where the arguments needed to establish the truth of sentences in
$V^{\Bbb P_{\kappa}}$ involve interesting properties of measure algebras
(553G-553M).   I conclude with notes on medial limits (553N) and
universally measurable sets (553O).

%553B 553C strong measure 0
%553E 553F perfect sets meet ground-model negl sets
%553G 553H rapid p-point ufilters
%553I 553J ccc posets
%553K 553L 553M special Aronszajn trees
%553N medial limits

\leader{553A}{Notation}\cmmnt{ I repeat some formulae from 552A.}
For any set $I$, $\nu_I$ will be
the usual measure on $\{0,1\}^I$, $\Tau_I$ its domain,
$\Cal N(\nu_I)$ its null ideal and
$(\frak B_I,\bar\nu_I)$
its measure algebra.   $\CalBa_I$ will be the
Baire $\sigma$-algebra of $\{0,1\}^I$.   For a cardinal $\kappa$,
$\BbbPk$ will be the forcing notion
$\frak B_{\kappa}^+$, active downwards.

\leader{553B}{Lemma} If $A\in\CalRbg(\{0,1\}^{\Bbb N})$,
then for any $f:\Bbb N\to\Bbb N$ there is a sequence $\sequencen{z_n}$ such
that $z_n\in\{0,1\}^{f(n)}$ for each $n$ and
$A\subseteq\bigcap_{n\in\Bbb N}
\bigcup_{m\ge n}\{x:z_m\subseteq x\in\{0,1\}^{\Bbb N}\}$.

\proof{ By 534Eb\formerly{5{}34Fd},
$A$ has strong measure zero with respect to the metric
$\rho$ on $\{0,1\}^{\Bbb N}$ defined by saying that

\Centerline{$\rho(x,y)=2^{-n}$ if $x\restr n=y\restr n$
and $x(n)\ne y(n)$.}

\noindent So
for each $n\in\Bbb N$ we can find a sequence $\sequence{i}{A_{ni}}$ of
subsets of $\{0,1\}^{\Bbb N}$ such that
$A\subseteq\bigcup_{i\in\Bbb N}A_{ni}$ and
$\diam A_{ni}\le 2^{-f(2^n(2i+1))}$ for each $i$.   Take $x_{ni}\in A_{ni}$
if $A_{ni}$ is non-empty, any member of $\{0,1\}^{\Bbb N}$ otherwise, and
set $z_m=x_{ni}\restr f(2^n(2i+1))$ if $m=2^n(2i+1)$;  take $z_0$ to be any
member of $\{0,1\}^{f(0)}$.   Then $z_m\in\{0,1\}^{f(m)}$ for every $m$,
and if $n\in\Bbb N$ then

$$\eqalign{A
&\subseteq\bigcup_{i\in\Bbb N}A_{ni}
\subseteq\bigcup_{i\in\Bbb N}\{x:\rho(x,x_{ni})\le 2^{-f(2^n(2i+1))}\}\cr
&=\bigcup_{i\in\Bbb N}\{x:x\supseteq z_{2^n(2i+1)}\}
\subseteq\bigcup_{m\ge n}\{x:x\supseteq z_m\},\cr}$$

\noindent as required.
}%end of proof of 553B

\leader{553C}{Proposition} Let $\kappa$ be any cardinal.
Then

\Centerline{$\VVdPk$ $\#(A)\le\check{\frak c}$ for every
$A\in\CalRbg(\{0,1\}^{\Bbb N})$.}

\proof{ (See {\smc Bartoszy\'nski \& Judah 95}, 8.2.11.)

\medskip

{\bf (a)}
Let $\dot A$ be a $\BbbPk$-name for a member of $\CalRbg(\{0,1\}^{\Bbb N})$.
Take any $f\in\BbbN^{\Bbb N}$.
Applying 553B in the forcing language, we must have

\doubleinset{$\VVdPk$ there is a sequence $\sequencen{z_n}$
such that $z_n\in\{0,1\}^{\check f(n)}$ for every $n\in\Bbb N$ and
$\dot A\subseteq\bigcap_{n\in\Bbb N}\bigcup_{m\ge n}\{x:x\supseteq z_n\}$.}

\noindent Let $\sequencen{\dot z_f(n)}$ be a sequence of $\BbbPk$-names
such that

\Centerline{$\VVdPk\dot z_f(n)\in\{0,1\}^{\check f(n)}$ for every
$n\in\Bbb N$,}

\Centerline{$\VVdPk\dot A\subseteq\bigcap_{n\in\Bbb N}\bigcup_{m\ge n}
\{x:x\supseteq\dot z_f(n)\}$.}

\medskip

{\bf (b)} Let $\ofamily{\xi}{\kappa}{e_{\xi}}$ be the standard generating
family in $\frak B_{\kappa}$ (525A).
Let $J\subseteq\kappa$ be a set with cardinal at
most $\frak c$ such that $\Bvalue{\dot z_f(n)=\check z}$ belongs to the
closed subalgebra $\frak C_J$ generated by $\{e_{\xi}:\xi\in J\}$ for every
$f\in\BbbN^{\Bbb N}$ and $z\in\bigcup_{n\in\Bbb N}\{0,1\}^{f(n)}$.
Let $P_J:L^{\infty}(\frak B_{\kappa})\to L^{\infty}(\frak C_J)$ be the
conditional expectation operator (365Q\formerly{3{}65R}).

Observe that $\frak C_J$ and $\frak C_J^{\Bbb N}$ have
cardinal at most $\frak c^{\omega}=\frak c$.   So we have a family
$\ofamily{\eta}{\frak c}{\dot y_{\eta}}$ of $\BbbPk$-names for members of
$\{0,1\}^{\Bbb N}$ such that
whenever $\sequencen{d_n}$ is a sequence in $\frak C_J$ there is an
$\eta<\frak c$ such that $\Bvalue{\dot y_{\eta}(\check n)=1}=d_n$ for every
$n\in\Bbb N$.

\medskip

{\bf (c)}
Let $\dot x$ be a $\BbbPk$-name for a member of $\{0,1\}^{\Bbb N}$, and
suppose that $a=\Bvalue{\dot x\in\dot A}$ is non-zero.
For $z\in\bigcup_{n\in\Bbb N}\{0,1\}^n$ set
$b_z=\Bvalue{\dot x\supseteq\check z}$.   For $m$, $n\in\Bbb N$, set

\Centerline{$c_{nm}=\sup_{z\in\{0,1\}^m}
   \Bvalue{P_J(\chi(a\Bcap b_z))>2^{-n}}\in\frak C_J$.}

\noindent Note that if $k\le m$ and $z\in\{0,1\}^m$ then
$b_z\Bsubseteq b_{z\restr k}$, so

\Centerline{$\Bvalue{P_J(\chi(a\Bcap b_z))>2^{-n}}
\Bsubseteq\Bvalue{P_J(\chi(a\Bcap b_{z\restr k}))>2^{-n}}$,}

\noindent and $c_{nm}\Bsubseteq c_{nk}$;  set $c_n=\inf_{m\in\Bbb N}c_{nm}$.
\Quer\ Suppose, if possible, that $c_n=0$ for every $n$.
Let $f\in\BbbN^{\Bbb N}$ be such
that $\sum_{n=0}^{\infty}\bar\nu_{\kappa}c_{n,f(n)}<\bar\nu_{\kappa}a$, and
set

\Centerline{$d=\sup_{n\in\Bbb N}(\sup_{z\in\{0,1\}^{f(n)}}
   \Bvalue{P_J(\chi(a\Bcap b_z))>2^{-n}})
\in\frak C_J$.}

\noindent Then $\bar\nu_{\kappa}d<\bar\nu_{\kappa}a$, so
$a\Bsetminus d\ne 0$;  while

$$\eqalignno{
\bar\nu_{\kappa}((a\Bsetminus d)\Bcap\Bvalue{\dot x\supseteq\dot z_f(n)})
&=\sum_{z\in\{0,1\}^{f(n)}}
  \bar\nu_{\kappa}(a\Bcap b_z
    \Bcap\Bvalue{\dot z_f(n)=\check z}\Bsetminus d)\cr
\displaycause{because $\VVdPk\dot z_f(n)\in\{0,1\}^{f(n)\varBspcheck}$}
&=\sum_{z\in\{0,1\}^{f(n)}}
  \int_{\Bvalue{\dot z_f(n)=\check z}\Bsetminus d}P_J(\chi(a\Bcap b_z))\cr
&\le 2^{-n}\sum_{z\in\{0,1\}^{f(n)}}
  \bar\nu_{\kappa}\Bvalue{\dot z_f(n)=\check z}\cr
\displaycause{because $d$ includes
$\Bvalue{P_J(\chi(a\Bcap b_z))>2^{-n}}$ for every $z\in\{0,1\}^{f(n)}$}
&=2^{-n}\cr}$$

\noindent for every $n$.   Consequently

\Centerline{$(a\Bsetminus d)\Bcap\inf_{n\in\Bbb N}\sup_{m\ge n}
  \Bvalue{\dot x\supseteq\dot z_f(m)}=0$,}

\noindent that is,

\Centerline{$a\Bsetminus d\VVdPk
  \dot x\supseteq\dot z_f(n)$ for only finitely many $n$.}

\noindent But this implies that

\Centerline{$a\Bsetminus d\VVdPk
  \dot x\notin\dot A$,}

\noindent contrary to hypothesis.\ \Bang

\medskip

{\bf (d)} Continuing from (c), we find that there are an
$a'\in\frak B_{\kappa}^+$, stronger than $a$, and an $\eta<\frak c$ such that
$a'\VVdPk\dot x=\dot y_{\eta}$.

\Prf\ Let $n\in\Bbb N$ be such that $c_n$, as defined in (c),
is non-zero.   For $m\in\Bbb N$ and $w\in\{0,1\}^m$, set

\Centerline{$d_w=\inf_{k\ge m}\sup_{z\in\{0,1\}^k,z\supseteq w}
  \Bvalue{P_J(\chi(a\Bcap b_z))>2^{-n}}\in\frak C_J$.}

\noindent Then

\Centerline{$\sup_{w\in\{0,1\}^m}d_w
  =\inf_{k\ge m}\sup_{z\in\{0,1\}^k}\Bvalue{P_J(\chi(a\Bcap b_z))>2^{-n}}
=c_n$}

\noindent because

\Centerline{$\langle\sup_{z\in\{0,1\}^k,z\supseteq w}
  \Bvalue{P_J(\chi(a\Bcap b_z))>2^{-n}}\rangle_{k\ge m}$}

\noindent is non-increasing for each $w$.    In particular,
$d_{\emptyset}=c_n$.   Also
$d_w=d_{w\cup\{(m,0)\}}\Bcup d_{w\cup\{(m,1)\}}$ for every $w\in\{0,1\}^m$.
So if we set

\Centerline{$d'_{\emptyset}=1$,}

\Centerline{$d'_{w\cup\{(m,0)\}}=d'_w\Bcap d_{w\cup\{(m,0)\}}$,
\quad$d'_{w\cup\{(m,1)\}}=d'_w\Bsetminus d_{w\cup\{(m,0)\}}$}

\noindent for every $m\in\Bbb N$ and $w\in\{0,1\}^m$, $d'_w\in\frak C_J$
and $d_w=d'_w\Bcap c_n$ for every $w$,
and there must be an $\eta<\frak c$ such that

\Centerline{$\Bvalue{\dot y_{\eta}(\check n)=1}
=\sup_{w\in\{0,1\}^{n+1},w(n)=1}d'_w$}

\noindent for every $n\in\Bbb N$, in which case

\Centerline{$\Bvalue{\check w\subseteq\dot y_{\eta}}=d'_w$ for every
$w\in\bigcup_{n\in\Bbb N}\{0,1\}^n$.}

If $m\in\Bbb N$, then

$$\eqalignno{\bar\nu_{\kappa}
  (a\Bcap\Bvalue{\dot x\restr\check m=\dot y_{\eta}\restr\check m})
&=\sum_{w\in\{0,1\}^m}\bar\nu_{\kappa}(a\Bcap b_w\Bcap d'_w)\cr
&=\sum_{w\in\{0,1\}^m}\int_{d'_w}P_J(\chi(a\Bcap b_w))
\ge\sum_{w\in\{0,1\}^m}2^{-n}\bar\nu_{\kappa}(c_n\Bcap d'_w)\cr
\displaycause{because $c_n\Bcap d'_w=d_w
\Bsubseteq\Bvalue{P_J(\chi(a\Bcap b_w))>2^{-n}}$ for every $w$}
&=2^{-n}\bar\nu_{\kappa}c_n.\cr}$$

\noindent So if we set $a'=a\Bcap\Bvalue{\dot x=\dot y_{\eta}}$, then

$$\eqalign{\bar\nu_{\kappa}a'
&=\bar\nu_{\kappa}
  (\inf_{m\in\Bbb N}a
    \Bcap\Bvalue{\dot x\restr\check m=\dot y_{\eta}\restr\check m})\cr
&=\inf_{m\in\Bbb N}\bar\nu_{\kappa}
  (a\Bcap\Bvalue{\dot x\restr\check m=\dot y_{\eta}\restr\check m})
\ge 2^{-n}\bar\nu_{\kappa}c_n
>0,\cr}$$

\noindent and $a'\ne 0$, while $a'\Bsubseteq a$ and
$a'\VVdPk\dot x=\dot y_{\eta}$.   So we have a suitable pair $a'$,
$\eta$.\ \Qed

\medskip

\wheader{553C}{0}{0}{0}{36pt}
{\bf (e)} Putting (c) and (d) together, we see that for any
name $\dot x$ for a member of $\{0,1\}^{\Bbb N}$,

\Centerline{$\Bvalue{\dot x\in\dot A}
\Bsubseteq\sup_{\eta<\frak c}\Bvalue{\dot x=\dot y_{\eta}}$.}

\noindent But this means that

\Centerline{$\VVdPk\dot A\subseteq\{\dot y_{\eta}:\eta<\check{\frak c}\}$}

\noindent and

\Centerline{$\VVdPk\#(\dot A)\le\check{\frak c}$.}
}%end of proof of 553C

\medskip

\cmmnt{
\leader{553D}{Remark} If $\kappa>\frak c$ then

\doubleinset{$\VVdPk$ for any countable family of subsets of
$\{0,1\}^{\omega}$
there is an extension of $\nu_{\omega}$ measuring every member of
the family}

\noindent (552N).   By 552Oa and 552Gc we see that in this case

\doubleinset{$\VVdPk$ any universally negligible subset of
$\{0,1\}^{\omega}$ has cardinal less than $\check{\frak c}$.}

\noindent The proposition here tells us that

\doubleinset{$\VVdPk$ any subset of
$\{0,1\}^{\omega}$ with strong measure zero
has cardinal at most $\check{\frak c}$}

\noindent without restriction on $\kappa$.
}%end of comment

\vleader{72pt}{553E}{Proposition}
Let $\kappa$ and $\lambda$ be infinite cardinals,
and $\dot K$ a $\BbbPk$-name such that

\Centerline{$\VVdPk\,\dot K$ is a compact
subset of $\{0,1\}^{\check\lambda}$ which is not scattered.}

\noindent Then there is a
negligible F$_{\sigma}$ set $G\subseteq\{0,1\}^{\lambda}$ such that

\Centerline{$\VVdPk\,\dot K\cap\tilde G\ne\emptyset$}

\noindent where
$\tilde G\cmmnt{\mskip5mu=(\{0,1\}^{\kappa}\times G)^{\sspvec}}$
is the $\BbbPk$-name for an F$_{\sigma}$ set in
$\{0,1\}^{\check\lambda}$ corresponding to $G$\cmmnt{ as described in
551K}.

\woddheader{553E}{0}{0}{0}{30pt}

\proof{{\bf (a)} If $a\in\frak B_{\kappa}^+$,
$\dot A$ is a $\BbbPk$-name such that

\Centerline{$a\VVdPk\,\dot A$ is an infinite subset of
$\{0,1\}^{\check\lambda}$}

\noindent and $\epsilon>0$, then there is an open-and-closed subset $H$ of
$\{0,1\}^{\lambda}$ such that $\nu_{\lambda}H\le\epsilon$ and
$\bar\nu_{\kappa}(a\Bcap\Bvalue{\dot A\cap\tilde H=\emptyset})\le\epsilon$.
\Prf\ We may suppose that $\epsilon=2^{-k}$ for some $k\in\Bbb N$.
Let $\sequence{i}{\dot y_i}$ be a sequence of $\BbbPk$-names such that

\Centerline{$a\VVdPk\,\dot y_i\in\dot A$ and $\dot y_i\ne\dot y_j$}

\noindent whenever $i$, $j\in\Bbb N$ are distinct.
Let $N\in\Bbb N$ be so large that $e^{-\epsilon N}<\bover12\epsilon$.
Then

\Centerline{$a\VVdPk$ there is a finite $J\subseteq\check\lambda$ such that
$\dot y_i\restr J\ne\dot y_j\restr J$ whenever $i<j<\check N$,}

\noindent that is,

\Centerline{$\sup_{J\in[\lambda]^{<\omega}}
\Bvalue{\dot y_i\restr\check J\ne\dot y_j\restr\check J
  \text{ whenever }i<j<\check N}
\Bsupseteq a$,}

\noindent and there is a finite set $J\subseteq\lambda$ such that

\Centerline{$\bar\nu_{\kappa}(a\Bsetminus
\Bvalue{\dot y_i\restr\check J\ne\dot y_j\restr\check J
  \text{ whenever }i<j<\check N})
\le\Bover12\epsilon$;}

\noindent enlarging $J$ if necessary,
we can suppose that $m=\#(J)$ is such that $m\ge k$ and
\penalty-100$(1-\Bover{N}{2^m})^{2^m\epsilon}\le\Bover12\epsilon$.

For each $i\in\Bbb N$, let $f_i:\{0,1\}^{\kappa}\to\{0,1\}^{\lambda}$ be a
$(\Tau_{\kappa},\CalBa_{\lambda})$-measurable function such that
(in the language of 551Cc) $a\VVdPk\,\dot y_i=\vec f_i$ .   Set

\Centerline{$E=\{x:f_i(x)\restr J\ne f_j(x)\restr J$ whenever
$i<j<N\}$;}

\noindent then

\Centerline{$E^{\ssbullet}
=\Bvalue{\dot y_i\restr\check J\ne\dot y_j\restr\check J
\text{ whenever }i<j<\check N}$,}

\noindent and
$\bar\nu_{\kappa}(a\Bsetminus E^{\ssbullet})\le\bover12\epsilon$.

Let $L$ be a subset of $\{0,1\}^J$ obtained by a
stochastic process in which
we pick $2^{m-k}=2^m\epsilon$ points independently with the uniform
distribution, and take $L$ to be the set of these points.
For any $x\in E$,

\Centerline{$\Pr(f_i(x)\restr J\notin L\Forall i<N)
=\Pr((L\cap\{f_i(x)\restr J:i<N\})=\emptyset)
=(1-\Bover{N}{2^m})^{2^m\epsilon}
\le\Bover12\epsilon$.}

\noindent By Fubini's theorem, there must be an $L\subseteq\{0,1\}^J$ such that
$\#(L)\le 2^m\epsilon$ and

\Centerline{$\nu_{\kappa}\{x:x\in E$,
$f_i(x)\restr J\notin L\Forall i<N\}
\le\Bover12\epsilon\nu_{\kappa}E\le\Bover12\epsilon$.}

Set $H=\{y:y\in\{0,1\}^{\lambda}$, $y\restr J\in L\}$ and
$b=\Bvalue{\dot A\cap\tilde H=\emptyset}$.   Then $H$ is open-and-closed,
$\nu_{\lambda}H=2^{-m}\#(L)\le\epsilon$ and

$$\eqalign{a\Bcap b
&\Bsubseteq a\Bcap\Bvalue{\dot y_i\notin\tilde H\Forall i<\check N}
=a\Bcap\{x:f_i(x)\notin H\Forall i<N\}^{\ssbullet}\cr
&=a\Bcap\{x:f_i(x)\restr J\notin L\text{ for every }i<N\}^{\ssbullet}\cr
&\Bsubseteq(a\Bsetminus E^{\ssbullet})
\Bcup\{x:x\in E,\,
f_i(x)\restr J\notin L\text{ for every }i<N\}^{\ssbullet}\cr}$$

\noindent has measure at most $\epsilon$, as required.\ \Qed

\medskip

{\bf (b)} Because every non-scattered space has a non-empty closed
subset with no isolated points, we may suppose that

\Centerline{$\VVdPk\,\dot K$ has no isolated points.}

\noindent For any $\epsilon\in\ooint{0,1}$
there is a compact negligible set $F\subseteq\{0,1\}^{\lambda}$ such that

\Centerline{$\bar\nu_{\kappa}\Bvalue{\dot K\cap\tilde F\ne\emptyset}
\ge 1-\epsilon$.}

\noindent\Prf\ Choose $\sequencen{a_n}$, $\sequencen{H_n}$
and $\sequencen{\dot K_n}$ inductively, as
follows.   $a_0=1$ and $\dot K_0=\dot K$.   Given that

\Centerline{$a_n\VVdPk\,\dot K_n$ is a non-empty compact set in
$\{0,1\}^{\check\lambda}$ without isolated points}

\allowmorestretch{468}{
\noindent and $\bar\nu_{\kappa}a_n>1-\epsilon$,
let $H_n\subseteq\{0,1\}^{\lambda}$ be an open-and-closed set of
measure at most $2^{-n}$ such that
$a_{n+1}=\penalty-100a_n\Bcap\Bvalue{\dot K_n\cap\tilde H_n\ne\emptyset}$
has measure greater than $1-\epsilon$.
Now let $\dot K_{n+1}$ be a $\BbbPk$-name such that
$\VVdPk\,\dot K_{n+1}=\dot K_n\cap\tilde H_n$.   Because
}

\doubleinset{$a_{n+1}\VVdPk\,\dot K_n$ is a compact set without
isolated points, $\tilde H_n$ is
open-and-closed and $\dot K_n\cap\tilde H_n\ne\emptyset$, so
$\dot K_{n+1}$ is a non-empty compact set without isolated points,}

\noindent the induction continues.

At the end of the induction, set $F=\bigcap_{n\in\Bbb N}H_n$ and
$a=\inf_{n\in\Bbb N}a_n$.   Then $\bar\nu_{\kappa}a\ge 1-\epsilon$ and
$\VVdPk\,\tilde F=\bigcap_{n\in\Bbb N}\tilde H_n$, so

\doubleinset{$a\VVdPk\,\tilde F\cap\dot K$ is the intersection of the
non-increasing sequence $\sequencen{\dot K_n}$ of non-empty compact sets,
so is not empty.}

\noindent So

\Centerline{$\bar\nu_{\kappa}\Bvalue{\tilde F\cap\dot K\ne\emptyset}
\ge\bar\nu_{\kappa}a\ge 1-\epsilon$.}

\noindent Also, of course, $\nu_{\kappa}F=0$, as required.\ \Qed

\medskip

\wheader{553E}{0}{0}{0}{48pt}
{\bf (c)} Finally, let $\sequencen{F_n}$ be a sequence of compact negligible
sets such that

\Centerline{$\bar\nu_{\kappa}\Bvalue{\dot K\cap\tilde F_n\ne\emptyset}
\ge 1-2^{-n}$}

\noindent for every $n$, and set $G=\bigcup_{n\in\Bbb N}F_n$;  this works.
}%end of proof of 553E

\leader{553F}{Corollary} Suppose that $\cf\Cal N(\nu_{\omega})=\omega_1$
and that $\kappa\ge\omega_2$ is a cardinal.   Then

\doubleinset{$\VVdPk\,\omega_1$ is a precaliber of every measurable
algebra but does not have Haydon's property.}

\proof{ By 523N,

\Centerline{$\cf\Cal N(\nu_{\omega_1})
=\max(\cf\Cal N(\nu_{\omega}),\cff[\omega_1]^{\le\omega})=\omega_1$;}

\noindent let
$\ofamily{\xi}{\omega_1}{H_{\xi}}$ be a cofinal family in
$\Cal N(\nu_{\omega_1})$.   Now 552Ga and 525J tell us that

\doubleinset{$\VVdPk\,\cov\Cal N(\nu_{\lambda})>\omega_1$ for every
infinite cardinal $\lambda$, so $\omega_1$ is a precaliber of every
measurable algebra.}

\noindent Next, defining $\tilde H_{\xi}$ from $H_{\xi}$ as in 551K and
553E,

\Centerline{$\VVdPk\,\tilde H_{\xi}\in\Cal N(\nu_{\omega_1})$
for every $\xi<\omega_1$}

\noindent (551Kd;  remember that in this context we do not need to
distinguish between $\omega_1$ and $\check\omega_1$, by 5A3Nb), while

\Centerline{$\VVdPk$ if $K\subseteq\{0,1\}^{\omega_1}$ is a non-scattered
compact set then $K$ meets $\bigcup_{\xi<\omega_1}\tilde H_{\xi}$.}

\noindent\Prf\ Suppose that $a\in\frak B_{\kappa}^+$ and $\dot K$ is a
$\BbbPk$-name such that

\Centerline{$a\VVdPk\,\dot K\subseteq\{0,1\}^{\omega_1}$ is a non-scattered
compact set.}

\noindent If $a=1$ set $\dot K'=\dot K$;  otherwise let
$\dot K'$ be a $\BbbPk$-name such that

\Centerline{$a\VVdPk\,\dot K'=\dot K$,
\quad$1\Bsetminus a\VVdPk\,\dot K'=\{0,1\}^{\omega_1}$.}

\noindent By 553E, there is a negligible set $G\subseteq\{0,1\}^{\omega_1}$ such
that $\VVdPk\,\dot K'\cap\tilde G\ne\emptyset$.
Now there is a $\xi<\omega_1$ such that $G\subseteq H_{\xi}$, so that

\Centerline{$a\VVdPk\,\dot K\cap\tilde H_{\xi}
\supseteq\dot K'\cap\tilde G\ne\emptyset$.  \Qed}

\noindent By 531Vb,

\Centerline{$\VVdPk\,\omega_1$ does not have Haydon's property.}
}%end of proof of 553F

\leader{553G}{Lemma} Let $(\frak A,\bar\mu)$ be a probability algebra,
$\frak C$ a subalgebra of $\frak A$, and $\sequencen{e_n}$ a sequence in
$\frak A$ stochastically independent of each other and of $\frak C$.   Let
$I\subseteq\frak A$ be a finite set and $\frak C_I$ the subalgebra of
$\frak A$ generated by $\frak C\cup I$.   Then for every $\epsilon>0$ there
is an $n_0\in\Bbb N$ such that
$|\bar\mu(b\Bcap e_n)-\bar\mu b\cdot\bar\mu e_n|\le\epsilon\bar\mu b$
whenever $b\in\frak C_I$ and $n\ge n_0$.

\proof{{\bf (a)} The first step is to show that if
$u\in L^1(\frak A,\bar\mu)$ then

\inset{\noindent
for every $\epsilon>0$ there is an $n_0\in\Bbb N$ such that
$|\int_{b\Bcap e_n}u-\bar\mu e_n\cdot\int_bu|\le\epsilon\int|u|$
whenever $b\in\frak C$ and $n\ge n_0$.}

\noindent\Prf\ Consider the set
$U$ of those $u\in L^1(\frak A,\bar\mu)$ for which this true.
This is a linear subspace of $L^1(\frak A,\bar\mu)$.   Also it is
$\|\,\|_1$-closed, because if $\int|v-u|\le\bover13\int|u|$ and
$|\int_{b\Bcap e_n}v-\bar\mu e_n\cdot\int_bv|\le\bover14\epsilon\int|v|$
then $|\int_{b\Bcap e_n}u-\bar\mu e_n\cdot\int_bu|\le\epsilon\int|u|$.
If we take $\frak D_m$ to be the subalgebra of $\frak A$
generated by $\frak C\cup\{e_n:n\le m\}$, then
$\bar\mu(a\Bcap b\Bcap e_n)=\bar\mu(a\Bcap b)\cdot\bar\mu e_n$ whenever
$a\in\frak D_m$, $b\in\frak C$ and $n\ge m$,
so $\chi a\in U$ for every $a\in\frak D_m$.
Consequently $\chi a\in U$ for every $a\in\frak D$, where $\frak D$ is the
metric closure of $\bigcup_{m\in\Bbb N}\frak D_m$ in $\frak A$.
Identifying
$L^1(\frak D,\bar\mu\restrp\frak D)$ with the closed linear subspace of
$L^1(\frak A,\bar\mu)$ generated by $\{\chi a:a\in\frak D\}$
(365Q, 365F),
we see that $U\supseteq L^1(\frak D,\bar\mu\restrp\frak D)$.   Now suppose
that $u$ is any member of $L^1(\frak A,\bar\mu)$.   Then we have a
conditional expectation $Pu$ of $u$ in $L^1(\frak D,\bar\mu\restrp\frak D)$
(365Q), and

\Centerline{$|\int_{b\Bcap e_n}u-\bar\mu b\cdot\int_bu|
=|\int_{b\Bcap e_n}Pu-\bar\mu e_n\cdot\int_bPu|$}

\noindent for every $b\in\frak C$ and $n\in\Bbb N$, while $|Pu|\le P|u|$,
so $u\in U$ because $Pu\in U$.\ \Qed

\medskip

{\bf (b)} I show now, by induction on $\#(I)$, that if $a\in\frak A$ then

\inset{\noindent
for every $\epsilon>0$ there is an $n_0\in\Bbb N$ such that
$|\bar\mu(a\Bcap b\Bcap e_n)-\bar\mu(a\Bcap b)\cdot\bar\mu e_n|
\le\epsilon\mu a$ whenever $b\in\frak C_I$ and $n\ge n_0$.}

\noindent\Prf\ If $I$ is empty, we can apply (a) with $u=\chi a$.   For the
inductive step to $\#(I)=k+1$, express $I$ as $J\cup\{c\}$ where $\#(J)=k$.
Take $a\in\frak A$.   Let $n_0\in\Bbb N$ be such that

\Centerline{$|\bar\mu((a\Bcap c)\Bcap b\Bcap e_n)
  -\bar\mu((a\Bcap c)\Bcap b)\cdot\bar\mu e_n|
\le\epsilon\mu(a\Bcap c)$,}

\Centerline{$|\bar\mu((a\Bsetminus c)\Bcap b\Bcap e_n)
  -\bar\mu((a\Bsetminus c)\Bcap b)\cdot\bar\mu e_n|
\le\epsilon\mu(a\Bsetminus c)$}

\noindent whenever
$b\in\frak C_J$ and $n\ge n_0$.   Now take $b\in\frak C_I$ and $n\ge n_0$.
There are $b'$, $b''\in\frak C_J$ such that
$b=(b'\Bcap c)\Bcup(b''\Bsetminus c)$, so that

$$\eqalign{|\bar\mu(a\Bcap b\Bcap e_n)-\bar\mu(a\Bcap b)\cdot\bar\mu e_n|
&=|\bar\mu((a\Bcap c)\Bcap b'\Bcap e_n)
  -\bar\mu((a\Bcap c)\Bcap b')\cdot\bar\mu e_n\cr
&\mskip50mu+\bar\mu((a\Bsetminus c)\Bcap b''\Bcap e_n)
  -\bar\mu((a\Bsetminus c)\Bcap b'')\cdot\bar\mu e_n|\cr
&\le|\bar\mu((a\Bcap c)\Bcap b'\Bcap e_n)
  -\bar\mu((a\Bcap c)\Bcap b')\cdot\bar\mu e_n|\cr
&\mskip50mu+|\bar\mu((a\Bsetminus c)\Bcap b''\Bcap e_n)
  -\bar\mu((a\Bsetminus c)\Bcap b'')\cdot\bar\mu e_n|\cr
&\le\epsilon\bar\mu(a\Bcap c)+\epsilon\bar\mu(a\Bsetminus c)
=\epsilon\bar\mu a.\cr}$$

\noindent Thus the induction proceeds.\ \Qed

\medskip

{\bf (c)} Now the result as stated is just the case $a=1$ in (b).
}%end of proof of 553G

\leader{553H}{Theorem} If $\kappa>\frak c$, then

\Centerline{$\VVdPk$ there are no rapid $p$-point
ultrafilters, therefore no Ramsey filters on $\Bbb N$.}

%do we need "ultra"? yes, see 553Xc

\proof{ (See {\smc Jech 78}, \S38.)

\medskip

{\bf (a)} Let $\langle e_{\xi n}\rangle_{\xi<\kappa,n\in\Bbb N}$ be a
re-indexing of the standard generating family in $\frak B_{\kappa}$.
Let $\dot\Cal F$ be a $\BbbPk$-name for an ultrafilter, and set
$\hat a=\Bvalue{\dot\Cal F\text{ is a rapid }p\text{-point ultrafilter}}$.
\Quer\ Suppose, if possible, that $\hat a\ne 0$.
For each $f\in\BbbN^{\Bbb N}$,

\Centerline{$\hat a\VVdPk$ there is a $D\in\dot\Cal F$ such that
$\#(D\cap\check f(k))\le k$ for every $k$}

\noindent(538Ad);   let $\dot D_f$ be a $\BbbPk$-name for a subset of
$\Bbb N$ such that

\Centerline{$\hat a\VVdPk\dot D_f\in\dot\Cal F$}

\noindent and

\Centerline{$\hat a\VVdPk\#(\dot D_f\cap f(k)\varBspcheck)\le\check k$}

\noindent for every $k\in\Bbb N$.
Let $J\subseteq\kappa$ be a set with cardinal
at most $\frak c$ such that $\Bvalue{\check n\in\dot D_f}$ belongs to the closed
subalgebra $\frak C$ generated by
$\{e_{\xi i}:\xi\in J$, $i\in\Bbb N\}$ for
every $f\in\Bbb N^{\Bbb N}$ and every $n\in\Bbb N$, and $\hat a$ also
belongs to $\frak C$.

\medskip

{\bf (b)} Let $\zeta<\kappa$ be such that the ordinal sum $\zeta+k$ does
not belong to $J$ for any $k\in\Bbb N$.
For each $k\in\Bbb N$ let $\dot C_k$ be a $\BbbPk$-name for a subset of
$\Bbb N$ such that $\Bvalue{\check n\in\dot C_k}=e_{\zeta+k,n}$ for every
$n\in\Bbb N$.
Set $c_k=\Bvalue{\dot C_k\notin\dot\Cal F}$ and let $\dot A_k$ be a
$\BbbPk$-name for a subset of $\Bbb N$ such that

\Centerline{$c_k\VVdPk\dot A_k=\Bbb N\setminus\dot C_k\in\dot\Cal F$,
\quad$1\Bsetminus c_k\VVdPk\dot A_k=\dot C_k\in\dot\Cal F$.}

\noindent Then $\VVdPk\dot A_k\in\dot\Cal F$ for every $k$, and
$\Bvalue{\check n\in\dot A_k}=c_k\Bsymmdiff e_{\zeta+k,n}$ for every
$n\in\Bbb N$.

\medskip

{\bf (c)} For $k$, $n\in\Bbb N$ set

\Centerline{$b_{kn}=\Bvalue{\check n\in\bigcap_{i<\check k}\dot A_i}
=\inf_{i<k}\Bvalue{\check n\in\dot A_i}
=\inf_{i<k}c_i\Bsymmdiff e_{\zeta+i,n}$.}

\noindent Then we have a non-decreasing $f:\Bbb N\to\Bbb N$
such that
$\bar\nu_{\kappa}(c\Bcap b_{kn})\le(2^{-k+1}-2^{-2k})\bar\nu_{\kappa}c$
whenever $c\in\frak C$, $k\in\Bbb N$ and $n\ge f(k)$.   \Prf\  Define
$f$ inductively, as follows.   If
$k=0$ then (interpreting $\inf\emptyset$ as $1$) we have $b_{kn}=1$ for every
$n$ so we can take $f(0)=0$.   For the inductive step to $k+1$,
let $\frak C_k$ be the closed subalgebra of $\frak B_{\kappa}$ generated
by $\frak C\cup\{e_{\zeta+i,n}:i<k$, $n\in\Bbb N\}$ and $\frak D_k$ the
subalgebra generated by $\frak C_k\cup\{c_i:i\le k\}$.
Then $\frak C_k$ and
$\sequencen{e_{\zeta+k,n}}$ are stochastically independent, so Lemma 553G tells
us that there is an $f(k+1)\ge f(k)$ such that

\Centerline{$|\bar\nu_{\kappa}(d\Bcap e_{\zeta+k,n})-\Bover12\bar\nu_{\kappa}d|
\le\Bover1{24}\cdot 2^{-k}\bar\nu_{\kappa}d$ whenever $d\in\frak D_k$ and
$n\ge f(k+1)$.}

\noindent Take $n\ge f(k+1)$ and $c\in\frak C$.   Then

$$\eqalignno{\bar\nu_{\kappa}(c\Bcap b_{k+1,n})
&=\bar\nu_{\kappa}(c\Bcap b_{kn}\Bcap(c_k\Bsymmdiff e_{\zeta+k,n}))\cr
&=\bar\nu_{\kappa}(c\Bcap b_{kn}\Bcap c_k)
  -2\bar\nu_{\kappa}(c\Bcap b_{kn}\Bcap c_k\Bcap e_{\zeta+k,n})
  +\bar\nu_{\kappa}(c\Bcap b_{kn}\Bcap e_{\zeta+k,n})\cr
&\le 2|\bar\nu_{\kappa}(c\Bcap b_{kn}\Bcap c_k\Bcap e_{\zeta+k,n})
    -\Bover12\bar\nu_{\kappa}(c\Bcap b_{kn}\Bcap c_k)|\cr
&\mskip100mu+|\bar\nu_{\kappa}(c\Bcap b_{kn}\Bcap e_{\zeta+k,n})
    -\Bover12\bar\nu_{\kappa}(c\Bcap b_{kn})|
    +\Bover12\bar\nu_{\kappa}(c\Bcap b_{kn})\cr
&\le\Bover1{12}\cdot 2^{-k}\bar\nu_{\kappa}(c\Bcap b_{kn}\Bcap c_k)
    +\Bover1{24}\cdot 2^{-k}\bar\nu_{\kappa}(c\Bcap b_{kn})
    +\Bover12\bar\nu_{\kappa}(c\Bcap b_{kn})\cr
\displaycause{because all the elements
$c\Bcap b_{kn}$ and $c\Bcap b_{kn}\Bcap c_k$ belong to $\frak D_k$}
&\le(2^{-k-3}+\Bover12)\bar\nu_{\kappa}(c\Bcap b_{kn})\cr
&\le(2^{-k-3}+\Bover12)(2^{-k+1}-2^{-2k})\bar\nu_{\kappa}c\cr
\displaycause{because $n\ge f(k)$}
&\le(2^{-k}-2^{-2k-2})\bar\nu_{\kappa}c.\cr}$$

\noindent So the construction proceeds.\ \Qed

\medskip

{\bf (d)} Because

\Centerline{$\hat a\VVdPk\dot{\Cal F}$ is a $p$-point ultrafilter and
$\dot A_k\in\dot{\Cal F}$ for every $k$,}

\noindent there are a $\BbbPk$-name $\dot A$ for a subset of $\Bbb N$ and a
$\BbbPk$-name $\dot g$ for a function from $\Bbb N$ to itself such that

\Centerline{$\hat a\VVdPk\dot A\in\dot{\Cal F}$ and
$\dot A\setminus\dot A_i\subseteq\dot g(k)$ whenever $i<k\in\Bbb N$.}

\noindent Let $g$ (in the ordinary universe) be a non-decreasing
function such that $f(k)\le g(k)$ and
$\bar\nu_{\kappa}(\Bvalue{\dot g(\check k)>g(k)\varBspcheck})
\penalty-100\le 2^{-k-2}\bar\nu_{\kappa}\hat a$ for every $k$.   Set
$\hat a_1=\hat a\Bcap\Bvalue{\dot g\le\check g}$;  then
$\bar\nu_{\kappa}\hat a_1\ge\bover12\bar\nu_{\kappa}\hat a$.

\medskip

{\bf (e)} Take the function $g$ from (d) and the
name $\dot D_g$ from (a), and set
$d_n=\hat a\Bcap\Bvalue{\check n\in\dot D_g}\in\frak C$ for
every $n$.   Then

\Centerline{$\sum_{n=g(k)}^{g(k+1)-1}\bar\nu_{\kappa}(d_n\Bcap b_{kn})
\le 2^{-k+1}(k+1)$}

\noindent for every $k\in\Bbb N$.   \Prf\ Set $K=g(k+1)\setminus g(k)$.
We have

$$\eqalignno{\sum_{n=g(k)}^{g(k+1)-1}\bar\nu_{\kappa}(d_n\Bcap b_{kn})
&=\sum_{n\in K}\sum_{I\subseteq K}
  \bar\nu_{\kappa}(d_n\Bcap b_{kn}
    \Bcap\Bvalue{\check I=\dot D_g\cap\check K})\cr
&=\sum_{I\subseteq K}\sum_{n\in I}
  \bar\nu_{\kappa}(d_n\Bcap b_{kn}
    \Bcap\Bvalue{\check I=\dot D_g\cap\check K})\cr
\displaycause{because
$d_n\Bcap\Bvalue{\check I=\dot D_g\cap\check K}
\Bsubseteq\Bvalue{\check n\in\dot D_g}
  \Bcap\Bvalue{\check I=\dot D_g\cap\check K}$ is zero if
$n\notin I$}
&=\sum_{I\in[K]^{\le k+1}}\sum_{n\in I}
  \bar\nu_{\kappa}(d_n\Bcap b_{kn}
     \Bcap\Bvalue{\check I=\dot D_g\cap\check K})\cr
\displaycause{because
$d_n\Bcap\Bvalue{\check I=\dot D_g\cap K}
\Bsubseteq\hat a
  \Bcap\Bvalue{\check I\subseteq\dot D_g\cap g(k+1)\varBspcheck}$
is zero if $\#(I)>k+1$}
&\le 2^{-k+1}\sum_{I\in[K]^{\le k+1}}\sum_{n\in I}
  \bar\nu_{\kappa}
  (d_n\Bcap\Bvalue{\check I=\dot D_g\cap\check K})\cr
\displaycause{because
$d_n\Bcap\Bvalue{\check I=\dot D_g\cap\check K}\in\frak C$
for every $n$ and $I$, and we are looking only at $n\ge g(k)\ge f(k)$}
&\le 2^{-k+1}(k+1)\sum_{I\in[K]^{\le k+1}}
  \bar\nu_{\kappa}\Bvalue{\check I=\dot D_g\cap\check K}\cr
&\le 2^{-k+1}(k+1).  \text{  \Qed}\cr}$$

\woddheader{553H}{4}{2}{2}{30pt}

{\bf (f)} As $\hat a\ne 0$, $\hat a_1\ne 0$.   Let $m$ be such that
$\sum_{k=m}^{\infty}2^{-k+1}(k+1)$ is less than
$\bar\nu_{\kappa}\hat a_1$;  then

\Centerline{$\hat a_2
=\hat a_1\Bsetminus\sup_{k\ge m}\sup_{g(k)\le n<g(k+1)}(d_n\Bcap b_{kn})$}

\noindent is non-zero.   Let $\dot B$
be a $\BbbPk$-name for a subset of $\Bbb N$ such that
$\VVdPk\dot B=\dot A\Bcap\dot D_g\setminus g(m)\varBspcheck$.   Then
$\hat a\VVdPk\dot B\in\dot{\Cal F}$.   But
$\hat a_2\VVdPk\dot B=\emptyset$.   \Prf\ Take any $n\in\Bbb N$.   If
$n<g(m)$ then $\VVdPk n\notin\dot B$.   If $k\ge m$ and $g(k)\le n<g(k+1)$,
then

\Centerline{$\hat a_1\VVdPk\dot g(\check k)\le g(k)\varBspcheck$,
\quad$\hat a\VVdPk\dot A\setminus\dot A_i\subseteq\dot g(\check k)$
for every $i<\check k$,}

\noindent so

\Centerline{$\hat a_1\Bcap\Bvalue{\check n\in\dot B}
\Bsubseteq\hat a\Bcap\Bvalue{\check n\in\dot A\setminus\dot g(\check k)}
\Bsubseteq\Bvalue{\check n\in\bigcap_{i<\check k}\dot A_i}=b_{kn}$.}

\noindent Also, of course, $\VVdPk\dot B\subseteq\dot D_g$, so
$\hat a_1\Bcap\Bvalue{\check n\in\dot B}\Bsubseteq d_n\Bcap b_{kn}$ is
disjoint from $\hat a_2$.   But this means that
$\hat a_2\VVdPk\check n\notin\dot B$.
As $n$ is arbitrary, $\hat a_2\VVdPk\dot B=\emptyset$.\ \QeD\   Now

\Centerline{$\hat a_2\VVdPk\emptyset\in\dot{\Cal F}$,}

\noindent which is impossible.\ \Bang

\medskip

{\bf (g)} So $\hat a=0$, that is,

\Centerline{$\VVdPk\dot{\Cal F}$ is not a rapid $p$-point ultrafilter.}

\noindent As $\dot{\Cal F}$ is arbitrary,

\Centerline{$\VVdPk$ there are no rapid $p$-point ultrafilters.}

\medskip

{\bf (h)} Finally, by 538Fa,

\Centerline{$\VVdPk$ there are no Ramsey ultrafilters.}
}%end of proof of 553H

\vleader{72pt}{553I}{Lemma}
Suppose that $S\subseteq\omega_1^2$ is a set such that
whenever $n\in\Bbb N$ and
$\ofamily{\xi}{\omega_1}{I_{\xi}}$ is a family in $[\omega_1]^n$ such that
$I_{\xi}\cap\xi=\emptyset$ for every $\xi<\omega_1$, there are
$\xi<\omega_1$ and $\eta<\xi$ such that
$I_{\xi}\times I_{\eta}\subseteq S$.   Let $P$ be the set

\Centerline{$\{I:I\in[\omega_1]^{<\omega}$, $I\cap\xi\subseteq S[\{\xi\}]$ for
every $\xi\in I\}$,}

\noindent ordered by $\subseteq$.   Then $P$ is upwards-ccc.

\proof{ Let $\ofamily{\xi}{\omega_1}{J_{\xi}}$ be any family in $P$.
Then there are distinct $\xi$, $\eta<\omega_1$ such that
$J_{\xi}\cup J_{\eta}\in P$.   \Prf\ By the
$\Delta$-system Lemma (4A1Db), there is an uncountable set
$A_0\subseteq\omega_1$
such that $\family{\xi}{A_0}{J_{\xi}}$ is a $\Delta$-system with root $J$ say;
next, there is an $n\in\Bbb N$ such that $A_1=\{\xi:\xi\in A_0$,
$\#(J_{\xi}\setminus J)=n\}$ is uncountable.    If $n=0$ then
$J_{\xi}\cup J_{\eta}=J$ belongs to $P$ for any $\xi$, $\eta\in A_1$ and we can
stop.   Otherwise, there is an uncountable $A_2\subseteq A_1$ such that whenever
$\xi$, $\eta\in A_2$ and $\eta<\xi$ then
$\max J_{\eta}<\min(J_{\xi}\setminus J)$.
Re-enumerate $\family{\xi}{A_2}{J_{\xi}\setminus J}$ in
increasing order to get a family
$\ofamily{\xi}{\omega_1}{I_{\xi}}$ in $[\omega_1]^n$ such that
$\min I_{\xi}\ge\xi$ for every $\xi$.   Our hypothesis tells us that there are
$\eta<\xi$ such that $I_{\xi}\times I_{\eta}\subseteq S$.   Let $\xi'$,
$\eta'<\omega_1$ be such that $I_{\xi}=J_{\xi'}\setminus J$ and
$I_{\eta}=J_{\eta'}\setminus J$, and consider
$I=J\cup I_{\xi}\cup I_{\eta}$.   If $\alpha\in I$ and
$\beta\in I\cap\alpha$,

\inset{----- {\it either} $\alpha$, $\beta$ both belong to $J_{\eta'}$ so
$(\alpha,\beta)\in S$

----- {\it or} $\alpha$, $\beta$ both belong to $J_{\xi'}$ so
$(\alpha,\beta)\in S$

----- {\it or} $\alpha\in I_{\xi}$ and $\beta\in I_{\eta}$ so
$(\alpha,\beta)\in S$.}

\noindent So $J_{\xi'}\cup J_{\eta'}=I$ belongs to $S$.\ \Qed

Thus $P$ has no uncountable up-antichains and is upwards-ccc.
}%end of proof of 553I

\leader{553J}{Theorem} Let $\kappa$ be an infinite cardinal.   Then

\doubleinset{$\VVdPk$ there are two upwards-ccc partially ordered sets
whose product is not upwards-ccc.}

\cmmnt{\medskip

\noindent{\bf Remark} If $\kappa>\omega$ this is immediate from 552E, 537F
and 537G.   So we have a new result only if $\kappa=\omega$.}

\proof{{\bf (a)} Let $\family{\xi}{\kappa}{e_{\xi}}$ be the standard
generating family in $\frak B_{\kappa}$.   For $J\subseteq\kappa$ let
$\frak C_J$ be the
closed subalgebra of $\frak B_{\kappa}$ generated by
$\{e_{\xi}:\xi\in J\}$.   For $\xi<\omega_1$ let
$h_{\xi}:\xi\to\Bbb N$ be an injective function.

\medskip

{\bf (b)} Let $\dot S_0$ be a $\BbbPk$-name for a
subset of $\omega_1^2$ such that

$$\eqalign{\Bvalue{(\check\xi,\check\eta)\in\dot S_0}
&=e_{h_{\xi}(\eta)}\text{ if }\eta<\xi,\cr
&=0\text{ otherwise}.\cr}$$

\noindent Then

\doubleinset{$\VVdPk$ whenever $n\in\Bbb N$ and
$\ofamily{\xi}{\omega_1}{I_{\xi}}$ is a family in $[\omega_1]^n$ such that
$I_{\xi}\cap\xi=\emptyset$ for every $\xi<\omega_1$, there are
$\xi<\omega_1$ and $\eta<\xi$ such that
$I_{\xi}\times I_{\eta}\subseteq\dot S_0$.}

\noindent\Prf\Quer\ Suppose, if possible, otherwise.   Then we have an
$n\in\Bbb N$, an $a\in\frak B_{\kappa}^+$ and a family
$\ofamily{\xi}{\omega_1}{\dot I_{\xi}}$ of $\BbbPk$-names such that

\Centerline{$a\VVdPk\,\dot I_{\eta}\in[\omega_1\setminus\eta]^{\check n}$
and $\dot I_{\xi}\times\dot I_{\eta}\not\subseteq\dot S_0$ whenever
$\eta<\xi<\omega_1$.}

\noindent For each $\xi<\omega_1$ there are $a_{\xi}\in\frak B_{\kappa}^+$,
stronger than $a$, and $I_{\xi}\in[\omega_1\setminus\xi]^n$ such that
$a_{\xi}\VVdPk\,\dot I_{\xi}=\check I_{\xi}$.   By 525Tc
we can find an uncountable set
$A_0\subseteq\omega_1$ and an $\epsilon>0$ such that
$\bar\nu_{\kappa}(a_{\xi}\Bcap a_{\eta})\ge\epsilon$
whenever $\xi$, $\eta\in A_0$.  Next, there is an uncountable
set $A_1\subseteq A_0$ such that $I_{\eta}\subseteq\xi$ whenever $\xi\in A_1$
and $\eta\in\xi\cap A_1$;  consequently $I_{\eta}\cap I_{\xi}=\emptyset$
whenever $\xi$, $\eta\in A_1$ are distinct, and
$\bar\nu_{\kappa}
 \Bvalue{\check I_{\xi}\times\check I_{\eta}\subseteq\dot S_0}
=2^{-n^2}$.

Let $\delta>0$ be such that $2^{-n^2}(\epsilon-2\delta)-2\delta>0$.
For each $\xi\in A_1$ we can find a finite set $J_{\xi}\subseteq\kappa$ and an
$a'_{\xi}\in\frak C_{J_{\xi}}$ such that
$\bar\nu_{\kappa}(a_{\xi}\Bsymmdiff a'_{\xi})\le\delta$.   Let $m\in\Bbb N$ be
such that

\Centerline{$A_2=\{\xi:\xi\in A_1$, $J_{\xi}\cap\omega\subseteq m\}$}

\noindent is uncountable.   Let $\xi\in A_2$ be such that $A_2\cap\xi$ is
infinite.   In this case, $\family{\eta}{A_2\cap\xi}{h_{\zeta}[I_{\eta}]}$ is
disjoint for each $\zeta\in I_{\xi}$, so we have an $\eta\in A_2\cap\xi$ such
that $h_{\zeta}[I_{\eta}]\cap m=\emptyset$ for every $\zeta\in I_{\xi}$.
It follows that
$\Bvalue{\check I_{\xi}\times\check I_{\eta}\subseteq\dot S_0}
\in\frak C_{\omega\setminus m}$, while
$a'_{\xi}\Bcap a'_{\eta}\in\frak C_{m\cup(\kappa\setminus\omega)}$, so

$$\eqalign{\bar\nu_{\kappa}(a'_{\xi}\Bcap a'_{\eta}
\Bcap\Bvalue{\check I_{\xi}\times\check I_{\eta}\subseteq\dot S_0})
&=\bar\nu_{\kappa}(a'_{\xi}\Bcap a'_{\eta})
\cdot\bar\nu_{\kappa}
  \Bvalue{\check I_{\xi}\times\check I_{\eta}\subseteq\dot S_0}\cr
&=2^{-n^2}\bar\nu_{\kappa}(a'_{\xi}\Bcap a'_{\eta}).\cr}$$

\noindent If we set
$b=a_{\xi}\Bcap a_{\eta}
  \Bcap\Bvalue{\check I_{\xi}\times\check I_{\eta}\subseteq\dot S_0}$,

$$\eqalign{\bar\nu_{\kappa}b
&\ge 2^{-n^2}\bar\nu_{\kappa}(a'_{\xi}\Bcap a'_{\eta})-2\delta
\ge 2^{-n^2}(\bar\nu_{\kappa}(a_{\xi}\Bcap a_{\eta})-2\delta)-2\delta\cr
&\ge 2^{-n^2}(\epsilon-2\delta)-2\delta
>0.\cr}$$

\noindent But now we have $b\Bsubseteq a$ and

\Centerline{$b\VVdPk\,\dot I_{\xi}\times\dot I_{\eta}
=\check I_{\xi}\times\check I_{\eta}\subseteq\dot S_0$,}

\noindent which is supposed to be impossible.\ \Bang\Qed

\medskip

{\bf (c)} Let $\dot P_0$ be a $\BbbPk$-name for a partially ordered set defined
from $\dot S_0$ by the process of 553I, so that for a finite set
$I\subseteq\omega_1$

\Centerline{$\Bvalue{\check I\in\dot P_0}
=\inf_{\xi,\eta\in I,\eta<\xi}e_{h_{\xi}(\eta)}$.}

\noindent By 553I and (b) above,

\Centerline{$\VVdPk\,\dot P_0$ is upwards-ccc.}

\wheader{553J}{6}{2}{2}{48pt}

{\bf (d)} Similarly, if $\dot S_1$ is a $\BbbPk$-name for a
subset of $\omega_1^2$ such that

$$\eqalign{\Bvalue{(\check\xi,\check\eta)\in\dot S_1}
&=1\Bsetminus e_{h_{\xi}(\eta)}\text{ if }\eta<\xi,\cr
&=0\text{ otherwise},\cr}$$

\noindent and $\dot P_1$ is a $\BbbPk$-name for a partially ordered set
defined from $\dot S_1$ by the process of 553I, then

\Centerline{$\VVdPk\,\dot P_1$ is upwards-ccc.}

\noindent (The point is just that
$\ofamily{\xi}{\kappa}{1\Bsetminus e_{\xi}}$ also is a stochastically
independent family of elements of measure $\bover12$.)
But now observe that if $\eta<\xi<\omega_2$ then

\Centerline{$\Bvalue{\{\check\xi,\check\eta\}\in\dot P_0\cap\dot P_1}
=\Bvalue{(\check\xi,\check\eta)\in\dot S_0\cap\dot S_1}
=e_{h_{\xi}(\eta)}\Bcap(1\Bsetminus e_{h_{\xi}(\eta)})=0$.}

\noindent So

\doubleinset{$\VVdPk\,\{\{\{\xi\},\{\xi\}\}:\xi<\omega_1\}$
is an up-antichain in
$\dot P_0\times\dot P_1$, and $\dot P_0\times\dot P_1$ is not upwards-ccc.}

\noindent Thus we have the required example.
}%end of proof of 553J

\leader{553K}{}\cmmnt{ I extract an elementary step from the proof of the next
lemma.

\medskip

\noindent}{\bf Lemma} Let $\frak A$ be a Boolean algebra and
$\nu:\frak A\to\coint{0,\infty}$ a non-negative additive functional.   Then

\Centerline{$\sum_{i=0}^n\nu a_i
\le\nu(\sup_{i\le n}a_i)+\sum_{i<j\le n}\nu(a_i\Bcap a_j)$}

\noindent whenever $a_0,\ldots,a_n\in\frak A$.

\proof{ Let $d$ be any atom of the subalgebra of $\frak A$ generated by
$a_0,\ldots,a_n$.   Suppose that $\#(\{i:i\le n$, $d\Bsubseteq a_i\})=m$.
Then

$$\eqalign{\nu(d\Bcap\sup_{i\le n}a_i)+\sum_{i<j\le n}&\nu(d\Bcap a_i\Bcap a_j)
  -\sum_{i=0}^n\nu(d\Bcap a_i)\cr
&=0\text{ if }m\le 1,\cr
&=1+\Bover{m(m-1)}2-m=\Bover12(m-1)(m-2)\ge 0\text{ otherwise}.\cr}$$

\noindent Summing over $d$, we have the result.
}%end of proof of 553K

\vleader{72pt}{553L}{Lemma} Let $(\frak A,\bar\mu)$ be a
probability algebra, $I$ an uncountable set, $X$ a non-empty set and
$\Cal F$ an ultrafilter on $X$.
Let $\langle a_{ix}\rangle_{i\in I,x\in X}$ be a
family in $\frak A$ such that
$\inf_{i\in I}\lim_{x\to\Cal F}\bar\mu a_{ix}>0$.
Then there are an uncountable set $S\subseteq I$ and a family
$\family{i}{S}{b_i}$ in $\frak A\setminus\{0\}$ such that

\Centerline{$b_i\Bcap b_j
\Bsubseteq\sup_{x\in F}a_{ix}\Bcap a_{jx}$}

\noindent for all $i$, $j\in S$ and $F\in\Cal F$.

\proof{{\bf (a)} We can suppose that $I=\omega_1$.   For each
$\xi<\omega_1$ set $u_{\xi}=\lim_{x\to\Cal F}\chi a_{\xi x}$,
the limit being taken for the weak topology on $L^2(\frak A,\bar\mu)$
(\S366), so that

\Centerline{$\int_au_{\xi}
=\lim_{x\to\Cal F}\bar\mu(a\Bcap a_{\xi x})$}

\noindent for every $a\in\frak A$.   In particular,
$\int u_{\xi}\ge\epsilon$, where
$\epsilon=\inf_{\xi<\omega_1}\lim_{x\to\Cal F}\bar\mu a_{\xi x}>0$;  set
$b'_{\xi}=\Bvalue{u_{\xi}>\bover12\epsilon}$, so that $b'_{\xi}\ne 0$.

\medskip

{\bf (b)} For $\xi$, $\eta<\omega_1$ set

\Centerline{$c_{\xi\eta}
=\inf_{F\in\Cal F}\sup_{x\in F}a_{\xi x}\Bcap a_{\eta x}$.}

\noindent For $K\subseteq\omega_1$ set

\Centerline{$d_K
=\inf_{\xi\in K}b'_{\xi}\Bsetminus
\sup_{\xi,\eta\in K\text{ are distinct}}c_{\xi\eta}$.}

\noindent If $d_K\ne 0$ then
$\epsilon\#(K)<3$.   \Prf\ We may suppose that $K$ is finite and not
empty; set $n=\#(K)$.
We have $\int_{d_K}u_{\xi}>\bover12\epsilon\bar\mu d_K$ for every
$\xi\in K$, so

\Centerline{$F_0
=\{x:x\in X$,
$\bar\mu(d_K\Bcap a_{\xi x})\ge\bover12\epsilon\bar\mu d_K$ for every
$\xi\in K\}$}

\noindent belongs to $\Cal F$.   Let $F\in\Cal F$ be such that, setting
$c'_{\xi\eta}=\sup_{x\in F}a_{\xi x}\Bcap a_{\eta x}$,
$\bar\mu(c'_{\xi\eta}\Bsetminus c_{\xi\eta})\le\Bover{\bar\mu d_K}{n^2}$
for all $\xi$, $\eta\in K$.   Take any $x\in F\cap F_0$.
If $\xi$, $\eta\in K$ are distinct,

\Centerline{$\bar\mu(d_K\Bcap a_{\xi x}\Bcap a_{\eta x})
\le\bar\mu(c'_{\xi\eta}\Bsetminus c_{\xi\eta})\le\Bover{\bar\mu d_K}{n^2}$,}

\noindent so

$$\eqalignno{\Bover{n\epsilon}2\bar\mu d_K
&\le\sum_{\xi\in K}\bar\mu(d_K\Bcap a_{\xi x})
\le\bar\mu d_K+\sum_{\xi,\eta\in K,\xi<\eta}
\bar\mu(d_K\Bcap a_{\xi x}\Bcap a_{\eta x})\cr
\displaycause{553K}
&\le\bar\mu d_K+\Bover{n(n-1)}2\cdot\Bover{\bar\mu d_K}{n^2}
<\Bover32\bar\mu d_K\cr}$$

\noindent and $n\epsilon<3$.\ \Qed

\medskip

{\bf (c)} For each infinite $\xi<\omega_1$ there is therefore a maximal
subset $K_{\xi}$ of $\xi$ such that
$b_{\xi}=d_{K_{\xi}\cup\{\xi\}}$ is
non-zero.   Every $K_{\xi}$ is finite, so there is a
$K\in[\omega_1]^{<\omega}$ such that
$S=\{\xi:\omega\le\xi<\omega_1$, $K_{\xi}=K\}$ is
stationary.   \Prf\ By the Pressing-Down Lemma (4A1Cc), there is a
$\zeta<\omega_1$ such that
$\{\xi:\xi<\omega_1$, $\sup K_{\xi}=\zeta\}$ is stationary.
As $[\zeta+1]^{<\omega}$ is countable, there will be a $K\subseteq\zeta+1$
such that $\{\xi:K_{\xi}=K\}$ is stationary.\ \QeD\
Now suppose that $\eta$, $\xi\in S$ and $\eta<\xi$.
Then

\Centerline{$b_{\eta}\Bcap b_{\xi}\Bsetminus c_{\eta\xi}
=d_{K_{\eta}\cup\{\eta\}}\Bcap d_{K_{\xi}\cup\{\xi\}}
  \Bsetminus c_{\xi\eta}
=d_{K\cup\{\eta,\xi\}}=0$}

\noindent because $K\cup\{\eta\}$ is a subset
of $\xi$ properly including $K_{\xi}$.   So we have an appropriate
family $\family{\xi}{S}{b_{\xi}}$.
}%end of proof of 553L

\leader{553M}{Proposition}\cmmnt{ ({\smc Laver 87})}
If $\frak m>\omega_1$ and $\kappa$ is any infinite cardinal, then

\Centerline{$\VVdPk$ every Aronszajn tree is special, so Souslin's
hypothesis is true.}

\proof{{\bf (a)} By 5A1E(b-ii) and 5A1E(d-ii), it is enough to show that

\Centerline{$\VVdPk$ every Aronszajn tree ordering of $\omega_1$ included
in the usual ordering is special.}

\noindent
Let $\dot\preccurlyeq$ be a $\BbbPk$-name for an Aronszajn tree ordering of
$\omega_1$ included in the usual ordering of $\omega_1$.
For $\alpha$, $\beta<\omega_1$ set
$a_{\alpha\beta}=\Bvalue{\check\alpha\dot\preccurlyeq\check\beta}$;
note that
$a_{\alpha\alpha}=1$, $a_{\alpha\beta}=0$ if $\beta<\alpha$ and
$a_{\alpha\beta}\Bsupseteq a_{\alpha\gamma}\Bcap a_{\beta\gamma}$
whenever $\alpha\le\beta\le\gamma<\omega_1$.

If $\Cal F$ is an ultrafilter on $\omega_1$ containing
$\omega_1\setminus\zeta$ for every $\zeta<\omega_1$, then
$\lim_{\xi\to\Cal F}\bar\nu_{\kappa}a_{\alpha\xi}=0$
for all but countably many
$\alpha<\omega_1$.   \Prf\Quer\ Otherwise, there is an $\epsilon>0$ such
that $I=\{\alpha:\alpha<\omega_1$,
$\lim_{\xi\to\Cal F}\bar\mu a_{\alpha\xi}\ge\epsilon\}$ is uncountable.
By 553L, there are an uncountable $S\subseteq I$ and a family
$\family{\alpha}{S}{b_{\alpha}}$ in $\frak A\setminus\{0\}$ such that

\Centerline{$b_{\alpha}\Bcap b_{\beta}
\Bsubseteq\sup_{\xi\ge\beta}a_{\alpha\xi}\Bcap a_{\beta\xi}
\Bsubseteq a_{\alpha\beta}$}

\noindent whenever $\alpha$, $\beta\in S$ and $\alpha<\beta$.
Set $c=\inf_{\alpha<\omega_1}\sup_{\beta\in S\setminus\alpha}b_{\beta}$,
so that $c\ne 0$.   Let $\dot Y$ be a $\BbbPk$-name
for a subset of $\omega_1$
such that $\Bvalue{\check\alpha\in\dot Y}=b_{\alpha}$ for $\alpha\in S$,
$\Bvalue{\check\alpha\in\dot Y}=0$ for other $\alpha$.   Then

\Centerline{$\VVdPk\alpha\mskip5mu\dot\preccurlyeq\mskip5mu\beta$ whenever $\alpha$,
$\beta\in\dot Y$ and $\alpha<\beta$,}

\Centerline{$c\VVdPk\dot Y$ is uncountable;}

\noindent so

\Centerline{$c\VVdPk\dot Y$ is an uncountable branch in the Aronszajn
tree,}

\noindent which is impossible.\ \Bang\Qed

\medskip

{\bf (b)} Let $\ofamily{\xi}{\kappa}{e_{\xi}}$ be the standard
generating family in $\frak B_{\kappa}$.   Choose inductively a
non-decreasing family $\ofamily{\alpha}{\omega_1}{J_{\alpha}}$ of
countably infinite
subsets of $\kappa$ such that $a_{\beta\alpha}$ belongs to the
closed subalgebra $\frak C_{J_{\alpha}}$
of $\frak B_{\kappa}$ generated by
$\{e_{\xi}:\xi\in J_{\alpha}\}$ whenever $\beta\le\alpha<\omega_1$.
%Let $\frak E_{\alpha}$ be the (countable) subalgebra of $\frak B_{\kappa}$
%generated by $\{e_{\xi}:\xi\in J_{\alpha}\}$.

Let $P$ be the partially ordered set of functions $f$ such that

\inset{$\dom f$ is a finite subset of $\omega_1\times\omega$,

for every $(\alpha,n)\in\dom f$, $f(\alpha,n)\in\frak C_{J_{\alpha}}$ and
$\bar\nu_{\kappa}f(\alpha,n)>\bover12$,

$f(\alpha,n)\Bcap f(\beta,n)\Bcap a_{\beta\alpha}=0$
whenever $(\alpha,n)$, $(\beta,n)\in\dom f$ and $\beta<\alpha$.}

\noindent Say that $f\le g$ if $\dom f\subseteq\dom g$ and
$g(\alpha,n)\Bsubseteq f(\alpha,n)$
for every $(\alpha,n)\in\dom f$.   Then $\le$ is a partial order on $P$.

$P$ is upwards-ccc.   \Prf\ Let $\ofamily{\xi}{\omega_1}{f_{\xi}}$ be a
family in $P$.   Let $A_0\subseteq\omega_1$ be an uncountable set such that
$\family{\xi}{A_0}{\dom f_{\xi}}$ is a
$\Delta$-system with root $K$
say;  let $\epsilon>0$, $m\in\Bbb N$ be such that

$$\eqalign{A_1
&=\{\xi:\xi\in A_0,\,\#(\dom f_{\xi})=m+\#(K),\cr
&\mskip100mu
\bar\nu_{\kappa}f_{\xi}(\alpha,n)\ge\Bover12+2\epsilon
  \text{ whenever }(\alpha,n)\in\dom f_{\xi}\}\cr}$$

\noindent is uncountable.   Let $A_2\subseteq A_1$ be an uncountable set
such that
$\bar\mu(f_{\eta}(\alpha,n)\Bsymmdiff f_{\xi}(\alpha,n))\le\epsilon$
whenever $\xi$, $\eta\in A_2$ and $(\alpha,n)\in K$;  such a set exists
because $\frak C_{J_{\alpha}}$ is metrically separable for each $\alpha$.
Let $A_3\subseteq A_2$ be an uncountable set such that
$\beta<\alpha$ whenever $\eta\in A_3$, $\xi\in A_3$, $\eta<\xi$,
$(\beta,m)\in\dom f_{\eta}$ and
$(\alpha,n)\in(\dom f_{\xi})\setminus K$.

For $\xi\in A_3$, enumerate $(\dom f_{\xi})\setminus K$ as
$\ofamily{i}{m}{(\alpha_{\xi i},n_{\xi i})}$.   Let $\Cal F$ be an
ultrafilter on $\omega_1$ containing $A_3\setminus\zeta$ for every
$\zeta<\omega_1$, and for $i<m$ let $\Cal F_i$ be the ultrafilter
$\{F:F\subseteq\omega_1$, $\{\xi:\alpha_{\xi i}\in F\}\in\Cal F\}$.
By (a), we have an uncountable $A_4\subseteq A_3$ such that

\Centerline{$\lim_{\xi\to\Cal F_i}
  \bar\nu_{\kappa}a_{\alpha_{\eta j},\xi}=0$}

\noindent for every $i$, $j<m$ and every $\eta\in A_4$;  that is,

\Centerline{$\lim_{\xi\to\Cal F}
  \bar\nu_{\kappa}a_{\alpha_{\eta j},\alpha_{\xi i}}=0$}

\noindent whenever $i$, $j<m$ and $\eta\in A_4$.   But this means that we
can find $\eta\in A_4$ and $\xi\in A_3$ such that $\eta<\xi$ and
$\bar\nu_{\kappa}a_{\alpha_{\eta j},\alpha_{\xi i}}
\le\Bover{\epsilon}{m+1}$ for all $i$, $j<m$.   Now consider the function
$g$ with domain $\dom f_{\eta}\cup\dom f_{\xi}$ such that

$$\eqalign{g(\alpha,n)
&=f_{\eta}(\alpha,n)\Bcap f_{\xi}(\alpha,n)
  \text{ if }(\alpha,n)\in K,\cr
&=f_{\eta}(\alpha,n)
  \text{ if }(\alpha,n)\in\dom f_{\eta}\setminus K,\cr
&=f_{\xi}(\alpha_{\xi i},n_{\xi i})
  \Bsetminus\sup_{j<m}a_{\alpha_{\eta j},\alpha_{\xi i}}
  \text{ if }i<m\text{ and }(\alpha,n)=(\alpha_{\xi i},n_{\xi i}).\cr}$$

\noindent Then $g(\alpha,n)\in\frak C_{J_{\alpha}}$
and $\bar\nu_{\kappa}g(\alpha,n)\ge\bover12+\epsilon$ for every
$(\alpha,n)\in\dom g$.   If $(\alpha,n)$ and $(\beta,n)$ belong to $\dom g$
and $\beta<\alpha$, then

\inset{----- if both $(\beta,n)$ and $(\alpha,n)$ belong to
$\dom f_{\eta}$, then

\Centerline{$g(\beta,n)\Bcap g(\alpha,n)\Bcap a_{\beta\alpha}
\Bsubseteq f_{\eta}(\beta,n)\Bcap f_{\eta}(\alpha,n)\Bcap a_{\beta\alpha}
=0$;}

----- if both $(\beta,n)$ and $(\alpha,n)$ belong to
$\dom f_{\xi}$, then

\Centerline{$g(\beta,n)\Bcap g(\alpha,n)\Bcap a_{\beta\alpha}
\Bsubseteq f_{\xi}(\beta,n)\Bcap f_{\xi}(\alpha,n)\Bcap a_{\beta\alpha}
=0$;}

----- if $(\beta,n)=(\alpha_{\eta j},n_{\eta j})$ and
$(\alpha,n)=(\alpha_{\xi i},n_{\xi i})$ then
$g(\alpha,n)$ is disjoint from
$a_{\alpha_{\eta j},\alpha_{\xi i}}=a_{\beta\alpha}$ so
\ifnum\stylenumber=12\break\fi
$g(\beta,n)\Bcap g(\alpha,n)\Bcap a_{\beta\alpha}=0$.}

\noindent So $g\in P$ and is an upper bound for $f_{\eta}$ and $f_{\xi}$.
Thus $\ofamily{\xi}{\omega_1}{f_{\xi}}$ is not an up-antichain in $P$;
as $\ofamily{\xi}{\omega_1}{f_{\xi}}$ is arbitrary, $P$ is upwards-ccc.\
\Qed

\medskip

{\bf (c)} For each $\alpha<\omega_1$ let $C_{\alpha}$ be a countable
metrically dense subset of
$\{c:c\in\frak C_{J_{\alpha}}$, $\bar\nu_{\kappa}c\le\bover12\}$.
For $\alpha<\omega_1$ and $c\in C_{\alpha}$, set

$$\eqalign{Q_{\alpha c}
&=\{f:f\in P\text{ and there is some }n\in\Bbb N\text{ such that }
(\alpha,n)\in\dom f,\cr
&\mskip250mu c\Bsubseteq f(\alpha,n)\text{ and }
\bar\nu_{\kappa}f(\alpha,n)=\Bover12+\Bover13\bar\nu_{\kappa}c\}.\cr}$$

\noindent Then $Q_{\alpha c}$ is cofinal with $P$.   \Prf\ Because
$J_{\alpha}$ is infinite, $\frak C_{J_{\alpha}}$ is atomless and there is
an $a\in\frak C_{J_{\alpha}}$ such that $c\Bsubseteq a$ and
$\bar\nu_{\kappa}a=\bover12+\bover13\bar\nu_{\kappa}c$.   Now take $n$ so
large that $i<n$ whenever $(\alpha,i)\in\dom f$, and set
$g=f\cup\{((\alpha,n),a)\}$;  then $f\le g\in Q_{\alpha c}$.\ \Qed

\medskip

{\bf (d)} Because $\frak m>\omega_1$, there is an upwards-directed set
$R\subseteq P$ meeting $Q_{\alpha c}$ whenever $\alpha<\omega_1$ and
$c\in C_{\alpha}$.   Now, for $n\in\Bbb N$, let $\dot A_n$ be a
$\BbbPk$-name for a subset of $\omega_1$ such that, for every
$\alpha<\omega_1$,

$$\eqalign{\Bvalue{\check\alpha\in\dot A_n}
&=\inf\{f(\alpha,n):f\in R,\,(\alpha,n)\in\dom f\}
  \text{ if }(\alpha,n)\in\bigcup_{f\in R}\dom f,\cr
&=0\text{ otherwise}\cr}$$

\noindent Then $\dot A_n$ is a name for an up-antichain for the tree order
$\dot\preccurlyeq$.   \Prf\ If $\beta<\alpha<\omega_1$, then either
$\Bvalue{\check\beta\in\dot A_n}=0$ or
$\Bvalue{\check\alpha\in\dot A_n}=0$ or there
are $f$, $g\in R$ such that $(\alpha,n)\in\dom f$ and $(\beta,n)\in\dom g$.
In this case, because $R$ is upwards-directed, there is an $h\in R$ such
that both $(\alpha,n)$ and $(\beta,n)$ belong to $\dom h$, so that

\Centerline{$\Bvalue{\check\alpha\in\dot A_n}
  \Bcap\Bvalue{\check\beta\in\dot A_n}
  \Bcap\Bvalue{\check\beta\dot\preccurlyeq\check\alpha}
\Bsubseteq h(\alpha,n)\Bcap h(\beta,n)\Bcap a_{\beta\alpha}
=0$.}

\noindent Thus

\Centerline{$\VVdPk$ if $\alpha$, $\beta\in\dot A_n$ then they are
$\dot\preccurlyeq$-incompatible upwards.}

\noindent As $\alpha$ and $\beta$ are arbitrary,

\Centerline{$\VVdPk\dot A_n$ is an up-antichain.  \Qed}

\wheader{553M}{6}{2}{2}{36pt}

{\bf (e)} Finally,

\Centerline{$\VVdPk\bigcup_{n\in\Bbb N}\dot A_n=\omega_1$.}

\noindent\Prf\Quer\ Otherwise, there is an $\alpha<\omega_1$ such that
$a=1\Bsetminus\sup_{n\in\Bbb N}\Bvalue{\check\alpha\in\dot A_n}\ne 0$.
Observe at this point that
$\Bvalue{\check\alpha\in\dot A_n}\in\frak C_{J_{\alpha}}$ for every $n$.    So
$a\in\frak C_{J_{\alpha}}$.   Let $a'\in\frak C_{J_{\alpha}}$ be such that
$a'\Bsubseteq a$ and $0<\bar\nu_{\kappa}a'\le\bover12$, and let
$c\in C_{\alpha}$ be such that
$\bar\nu_{\kappa}(a'\Bsymmdiff c)\le\bover14\bar\nu_{\kappa}a'$, so that
$c\ne 0$ and
$\bar\nu_{\kappa}(c\Bsetminus a')\le\bover13\bar\nu_{\kappa}c$.
Since $R$ meets $Q_{\alpha c}$, there are $n\in\Bbb N$, $f\in R$ such that
$c\Bsubseteq f(\alpha,n)$ and
$\bar\nu_{\kappa}f(\alpha,n)=\bover12+\bover13\bar\nu_{\kappa}c$.

If $g\in P$ and $f\le g$, then $g(\alpha,n)\Bsubseteq f(\alpha,n)$ and
$\bar\nu_{\kappa}g(\alpha,n)>\bover12$, so

\Centerline{$\bar\nu_{\kappa}(c\Bsetminus g(\alpha,n))
\le\bar\nu_{\kappa}f(\alpha,n)-\bar\nu_{\kappa}g(\alpha,n)
\le\Bover13\bar\nu_{\kappa}c$.}

\noindent Because $R$ is upwards-directed,
$\{g(\alpha,n):g\in R$, $(\alpha,n)\in\dom g\}$ is downwards-directed, and

$$\eqalign{\bar\nu_{\kappa}(c\Bsetminus\Bvalue{\check\alpha\in\dot A_n})
&=\sup\{\bar\nu_{\kappa}(c\Bsetminus g(\alpha,n)):g\in R,\,
(\alpha,n)\in\dom g\}\cr
&=\sup\{\bar\nu_{\kappa}(c\Bsetminus g(\alpha,n)):g\in R,\,f\le g\}
\le\Bover13\bar\nu_{\kappa}c.\cr}$$

Accordingly

\Centerline{$\bar\nu_{\kappa}(a'\Bcap\Bvalue{\check\alpha\in\dot A_n})
\ge\bar\nu_{\kappa}(c\Bcap\Bvalue{\check\alpha\in\dot A_n})
   -\bar\nu_{\kappa}(c\Bsetminus a')
\ge\Bover23\bar\nu_{\kappa}c-\Bover13\bar\nu_{\kappa}c
>0$;}

\noindent but $a'\Bsubseteq a$ is supposed to be disjoint from
$\Bvalue{\check\alpha\in\dot A_n}$.\ \Bang\Qed

So $\sequencen{\dot A_n}$ is a name for a sequence of antichains covering
$\omega_1$, and

\Centerline{$\VVdPk(\omega_1,\dot\preccurlyeq)$ is special,}

\noindent as required.
}%end of proof of 553M

\leader{553N}{Proposition} Suppose that there is a medial limit\cmmnt{
(definition:  538Q)}, and that $\kappa$ is a cardinal.   Then

\Centerline{$\VVdPk$ there is a medial limit.}

\proof{{\bf (a)} Let $\theta:\Cal P\Bbb N\to[0,1]$ be a medial limit.
Let $Q$ be the
rationally convex hull of the usual basis of $\ell^1$, that is, the set of
functions $v:\Bbb N\to\Bbb Q\cap[0,1]$ such that $\{n:v(n)\ne 0\}$ is
finite and $\sum_{n=0}^{\infty}v(n)=1$.   Note that $Q$ is absolute in the
sense that

\Centerline{$\VVdP\,\check Q$ is the rationally convex hull of the usual
basis of $\ell^1$}

\noindent for every forcing notion $\Bbb P$.   Let $\Cal F$ be the filter
on $Q$ which is the trace of the weak* neighbourhood filter of $\theta$,
that is, the filter generated by sets of the form

\Centerline{$\{v:v\in Q$,
$|\sum_{n=0}^{\infty}v(n)u(n)-\dashint u(n)\theta(dn)|\le\epsilon\}$}

\noindent where $u\in\ell^{\infty}$ and $\epsilon>0$.   (Identifying
$Q\subseteq\ell^1$ with its image in $(\ell^{\infty})^*\cong(\ell^1)^{**}$,
the weak* closure of $Q$ is convex, so is equal to its bipolar (4A4Eg)
and is the set of positive linear functionals on $\ell^{\infty}$
taking the value $1$ on the order unit $\chi\Bbb N$.   See 363L and
538P for the notation $\dashint\ldots\theta(dn)$.)
Let $\vec{\Cal F}$ be the $\BbbPk$-name derived from $\Cal F$ and
$(\{0,1\}^{\kappa},\Tau_{\kappa},\Cal N_{\kappa})$ by the method of 551Rb,
so that

\Centerline{$\VVdPk\,\vec{\Cal F}$ is a filter on $\check Q$.}

\noindent Let $\dot\nu$ be a $\BbbPk$-name such that

\doubleinset{$\VVdPk\,\dot\nu$ is a bounded additive functional on
$\Cal P\Bbb N$, and identifying $\check Q$ with a subset of
$(\ell^{\infty})^*$, itself identified with the space $M(\Cal P\Bbb N)$ of
bounded additive functionals on $\Cal P\Bbb N$,
$\dot\nu$ is a cluster point of $\vec{\Cal F}$ for the weak* topology.}

\medskip

{\bf (b)} Suppose that $a\in\frak B_{\kappa}^+$ and that
$\dot{\frak e}$ is a $\BbbPk$-name such that

\Centerline{$a\VVdPk\,\dot{\frak e}$ is a sequence of Borel subsets of
$\{0,1\}^{\Bbb N}$.}

\noindent By 551Fb, we have for each $n\in\Bbb N$ a set
$W_n\in\Tau_{\kappa}\tensorhat\CalBa_{\Bbb N}$ such that

\Centerline{$a\VVdPk\,\dot{\frak e}(\check n)=\vec W_n$,}

\noindent where $\vec W_n$ is defined as in 551D.   Let
$\lambda=\nu_{\kappa}\times\nu_{\omega}$ be the product measure on
$\{0,1\}^{\kappa}\times\{0,1\}^{\Bbb N}$.   Because
$\theta$ is a medial limit,
$\idashint\chi W_n(x,y)\theta(dn)\lambda(d(x,y))$ is defined and equal to
$\dashint\lambda W_n\theta(dn)$;  that is, there are a conegligible Baire set
$W\subseteq\{0,1\}^{\kappa}\times\{0,1\}^{\Bbb N}$ and a Baire measurable
function $\psi:\{0,1\}^{\kappa}\times\{0,1\}^{\Bbb N}\to[0,1]$ such that

\Centerline{$\psi(x,y)
=\dashint\chi W_n(x,y)\theta(dn)
=\lim_{v\to\Cal F}\sum_{n=0}^{\infty}v(n)\chi W_n(x,y)$}

\noindent whenever $(x,y)\in W$, and

\Centerline{$\int\psi\,d\lambda
=\dashint\lambda W_n\theta(dn)
=\lim_{v\to\Cal F}\sum_{n=0}^{\infty}v(n)\lambda W_n$.}

\noindent Let $\vec W$ and $\vec\psi$ be the corresponding $\BbbPk$-names,
as in 551D and 551M, so that

\Centerline{$\VVdPk\,\vec W\in\CalBa_{\Bbb N}$ and
$\vec\psi:\{0,1\}^{\Bbb N}\to\Bbb R$ is Baire measurable.}

\noindent Moreover, since $\nu_{\kappa}$-almost every
vertical section of $W$ must be $\nu_{\omega}$-conegligible,

\Centerline{$\VVdPk\,\nu_{\omega}\vec W=1$}

\noindent (551I).

\medskip

{\bf (c)} Now suppose that $\dot s$ is a $\BbbPk$-name and that
$b\in\frak B_{\kappa}^+$ is stronger than $a$ and such that

\Centerline{$b\VVdPk\,\dot s\in\vec W$.}

\medskip

\quad{\bf (i)} By 551Cc, there is a $\Tau_{\kappa}$-measurable
$f:\{0,1\}^{\kappa}\to\{0,1\}^{\Bbb N}$ such that

\Centerline{$b\VVdPk\,\dot s=\vec f$.}

\noindent Expressing $b$ as $E^{\ssbullet}$ where
$E\in\Tau_{\kappa}\setminus\Cal N_{\kappa}$,
$(x,f(x))\in W$ for $\nu_{\kappa}$-almost every $x\in E$, by 551Ea.

For each $m\in\Bbb N$, consider

$$\eqalign{C_m
&=\{(x,v):x\in\{0,1\}^{\kappa},\,v\in Q,\,(x,f(x))\in W,\cr
&\mskip50mu
|\psi(x,f(x))-\sum_{n=0}^{\infty}v(n)\chi W_n(x,f(x))|\le 2^{-m}\}.\cr}$$

\noindent Then $C_m\in\Tau_{\kappa}\tensorhat\Cal PQ$;  and if
$x\in\{0,1\}^{\kappa}$ is such that $(x,f(x))\in W$,
$C_m[\{x\}]\in\Cal F$.   Consequently

\Centerline{$b\VVdPk\,\vec C_m\in\vec{\Cal F}$,}

\noindent where in this formula $\vec C_m$ is the $\BbbPk$-name defined by
the method of 551Ra.   At the same time,

\Centerline{$b\VVdPk\,
|\vec\psi(\dot s)-\sum_{n=0}^{\infty}v(n)\chi(\dot{\frak e}(n))(\dot s)|
\le 2^{-\check m}$ for every $v\in\vec C_m$.}

\noindent\Prf\ Suppose we have a $c$ stronger than $b$ and a $\BbbPk$-name
$\dot v$ such that $c\VVdPk\,\dot v\in\vec C_m$.   Then there are a
$G\in\Tau_{\kappa}\setminus\Cal N_{\kappa}$ and a $v\in Q$ such that
$G^{\ssbullet}\Bsubseteq c$, $G^{\ssbullet}\VVdPk\,\dot v=\check v$,
and $(x,v)\in C_m$ for every $x\in G$.   Setting

\Centerline{$h(x)=\psi(x,f(x))$,
\quad$h_n(x)=\chi W_n(x,f(x))$}

\noindent for $x\in\{0,1\}^{\kappa}$ and $n\in\Bbb N$, and interpreting
$\vec h$, $\vec h_n$ as in 551B,

\Centerline{$b\VVdPk\,
\vec h=\vec\psi(\vec f)=\vec\psi(\dot s)$,}

\noindent and

$$\eqalignno{b\VVdPk\,\vec h_n
&=(\chi W_n)\sspvec(\dot s)=(\chi\vec W_n)(\dot s)\cr
\displaycause{551Nd}
&=(\chi\dot{\frak e}(\check n))(\dot s)\cr}$$

\noindent for $n\in\Bbb N$.
For $x\in G$, moreover,
$|h(x)-\sum_{n=0}^{\infty}v(n)h_n(x)|\le 2^{-m}$, so

\Centerline{$G^{\ssbullet}\VVdPk\,
|\vec\psi(\dot s)
   -\sum_{n=0}^{\infty}\dot v(n)\chi(\dot{\frak e}(n))(\dot s)|
=|\vec h-\sum_{n=0}^{\infty}\check v(n)\vec h_n|
\le 2^{-\check m}$.}

\noindent As $c$ and $\dot v$ are arbitrary, we have the result.\ \Qed

\medskip

\quad{\bf (ii)} As $m$ is arbitrary,

\Centerline{$b\VVdPk\,\{v:v\in\check Q,\,
|\vec\psi(\dot s)
   -\sum_{n=0}^{\infty}v(n)\chi(\dot{\frak e}(n))(\dot s)|
   \le\epsilon\}
\in\vec{\Cal F}$ for every $\epsilon>0$,}

\noindent that is,

\Centerline{$b\VVdPk\,
\vec\psi(\dot s)=\lim_{v\to\vec{\Cal F}}
   \sum_{n=0}^{\infty}v(n)\chi(\dot{\frak e}(n))(\dot s)$.}

\noindent As $b$ and $\dot s$ are arbitrary,

\Centerline{$a\VVdPk\,
\vec\psi(y)=\lim_{v\to\vec{\Cal F}}
   \sum_{n=0}^{\infty}v(n)\chi(\dot{\frak e}(n))(y)$
for every $y\in\vec W$;}

\noindent since $\VVdPk\,\vec W$ is conegligible,

\Centerline{$a\VVdPk\,
\vec\psi\eae\lim_{v\to\vec{\Cal F}}
   \sum_{n=0}^{\infty}v(n)\chi(\dot{\frak e}(n))$.}

\noindent Looking back at the choice of $\dot\nu$, we see that

\Centerline{$a\VVdPk\,
\vec\psi(y)=\dashint\chi(\dot{\frak e}(n))(y)\dot\nu(dn)$ for
$\nu_{\omega}$-almost every $y$.}

\medskip

{\bf (d)} As for the integral of $\vec\psi$, 551Nf tells us that

\Centerline{$\VVdPk\,\int\vec\psi\,d\nu_{\omega}=\vec h$,}

\noindent where I now set $h(x)=\int\psi(x,y)\nu_{\omega}(dy)$ for
$x\in\{0,1\}^{\kappa}$.   Similarly, setting
$h_n(x)=\nu_{\omega}W_n[\{x\}]$, we have

\Centerline{$a\VVdPk\,\nu_{\omega}\dot{\frak e}(\check n)
=\vec h_n$.}

Set

\Centerline{$H=\{x:x\in\{0,1\}^{\kappa}$, $W[\{x\}]$ is
conegligible in $\{0,1\}^{\Bbb N}\}$;}

\noindent then $H$ is conegligible in $\{0,1\}^{\kappa}$.
Now remember that $\theta$ is a medial limit.   If $x\in H$ we have
$\psi(x,y)=\dashint\chi W_n(x,y)\theta(dn)$ for every $y$ in the
conegligible set $W[\{x\}]$, so

$$\eqalign{h(x)
&=\int\psi(x,y)\nu_{\omega}(dy)
=\idashint\chi W_n(x,y)\theta(dn)\nu_{\omega}(dy)\cr
&=\dashiint\chi W_n(x,y)\nu_{\omega}(dy)\theta(dn)
=\dashint\nu_{\omega}W_n[\{x\}]\theta(dn)
=\lim_{v\to\Cal F}\sum_{n=0}^{\infty}v(n)h_n(x).\cr}$$

\noindent So if, for $m\in\Bbb N$, we set

\Centerline{$C'_m
=\{(x,v):x\in\{0,1\}^{\kappa}$, $v\in Q$,
   $|h(x)-\sum_{n=0}^{\infty}v(n)h_n(x)|
       \le 2^{-m}\}$,}

\noindent we shall again have $\VVdPk\,\vec C'_m\in\vec{\Cal F}$;  and if
$G\in\Tau_{\kappa}\setminus\Cal N_{\kappa}$ and $v\in Q$ are such that
$G^{\ssbullet}$ is stronger than $p$ and
$G^{\ssbullet}\VVdPk\,\check v\in\vec C'_m$, then

\Centerline{$G^{\ssbullet}\VVdPk\,
|\int\vec\psi\,d\nu_{\omega}
  -\sum_{n=0}^{\infty}\check v(n)\nu_{\omega}\dot{\frak e}(n)|
\le 2^{-\check m}$.}

\noindent So

\Centerline{$a\VVdPk\,\{v:|\int\vec\psi\,d\nu_{\omega}
  -\sum_{n=0}^{\infty}\check v(n)\nu_{\omega}\dot{\frak e}(n)|
\le 2^{-\check m}\}\in\vec{\Cal F}$}

\noindent for every $m$, and

\Centerline{$a\VVdPk\,\int\vec\psi\,d\nu_{\omega}
=\lim_{v\to\vec{\Cal F}}
  \sum_{n=0}^{\infty}\check v(n)\nu_{\omega}\dot{\frak e}(n)
=\dashint\nu_{\omega}\dot{\frak e}(n)\dot\nu(dn)$.}

\medskip

{\bf (e)} As $p$ and $\dot{\frak e}$ are arbitrary, we see that

\Centerline{$\VVdPk\,\dot\nu$ satisfies condition (iv) of 538P, so is a
medial functional.}

\noindent It is now easy to check that

\Centerline{$\VVdPk\,\dot\nu\ge 0$,
$\dot\nu\Bbb N=1$ and $\dot\nu\{n\}=0$ for every
$n\in\Bbb N$, so $\dot\nu$ is a medial limit.}

\noindent This completes the proof.
}%end of proof of 553N

\leader{553O}{}\dvAnew{2010}\cmmnt{ For the most familiar classes of
`small' set --
the Lebesgue null ideal, or the meager ideal of $\Bbb R$, for instance --
it is easy to calculate the number of sets in the class;  because there is
a nowhere dense Lebesgue negligible set with cardinal $\frak c$, there must
be exactly $2^{\frakc}$ meager Lebesgue negligible sets, and therefore
there are just $2^{\frakc}$ Lebesgue measurable subsets of $\BbbR^r$ for
any $r\ge 1$.   But when we come to the ideal
$\Cal N_{\text{universal}}\normalsubgroup\Cal P\Bbb R$
of universally negligible sets, or the algebra
$\Sigma_{\text{um}}\subseteq\Cal P\Bbb R$ of universally
measurable sets, the position is much less clear.   In general, since by
Grzegorek's theorem
(439F) we know that there is a universally negligible subset of $\Bbb R$ of
cardinal $\non\Cal N(\nu_{\omega})$, we can say that

\Centerline{$\frak c\le 2^{\non\Cal N(\nu_{\omega})}
\le\#(\Cal N_{\text{universal}})
\le\#(\Sigma_{\text{um}})\le 2^{\frakc}$.}

\noindent It turns out that in random real models these inequalities may
well collapse to the lower bound, as in (b) of the next theorem.

\wheader{553O}{4}{2}{2}{60pt}

\noindent}{\bf Theorem}\cmmnt{ ({\smc Larson Neeman \& Shelah 10})}
Let $\kappa$ be an infinite cardinal.

(a)

\vskip-14pt plus 0pt minus 1pt

\doubleinset{$\VVdPk$ every universally measurable subset of
$\{0,1\}^{\Bbb N}$ is expressible as the union of at most
$\check{\frak c}$ Borel sets.}

(b) If the cardinal power $\kappa^{\frak c}$ is equal to $\kappa$, then

\Centerline{$\VVdPk$ there are exactly $\frak c$ universally measurable
subsets of $\{0,1\}^{\Bbb N}$.}

\proof{{\bf (a)(i)} It will save a moment later if I note at once that we
need consider only the case $\kappa>\frak c$.   \Prf\ If
$\kappa\le\frak c$, then $\kappa^{\omega}=\frak c$, so

\Centerline{$\VVdPk$ $\frak c=\check\frak c$}

\noindent by 552B.   But since we surely have

\doubleinset{$\VVdPk$ every universally measurable subset of
$\{0,1\}^{\Bbb N}$ is expressible as the union of at most
$\frak c$ singleton sets,}

\noindent we get the result.\ \Qed

So henceforth I will take it that $\kappa>\frak c$.   It will save
time to have a local notation:  if $M\subseteq\kappa$ and
$V\subseteq\{0,1\}^{\kappa}\times\{0,1\}^{\Bbb N}$, I will say that
$V$ is {\bf $M$- } if $(\tilde\omega,\omega')\in V$ whenever
$(\omega,\omega')\in V$ and $\tilde\omega\in\{0,1\}^{\kappa}$ is such
that $\tilde\omega\restr M=\omega\restr M$.

\medskip

\quad{\bf (ii)} (The testing measures.)  If
$E\in\CalBa_{\kappa}\setminus\Cal N(\nu_{\kappa})$,
$g:\{0,1\}^{\kappa}\to\{0,1\}^{\Bbb N}$ is
a $\CalBa_{\kappa}$-measurable function, and $I\subseteq\kappa$ is a set,
write $Q_{IEg}$ for the set of pairs $(V,h)$ where
$V\in\CalBa_{\kappa}\tensorhat\CalBa_{\Bbb N}$ and
$h:\{0,1\}^{\kappa}\to[0,1]$ is defined by saying that

\Centerline{$h(\omega)=\nu_{\kappa\setminus I}
  \{\omega':(\omega\restr I)\cup\omega'\in E$,
    $(\omega,g((\omega\restr I)\cup\omega'))\in V\}$}

\noindent for every $\omega\in\{0,1\}^{\kappa}$.   Then
$h$ is $\CalBa_{\kappa}$-measurable for every $(V,h)\in Q_{IEg}$.
\Prf\ The function

\Centerline{$(\omega,\omega')
\mapsto(\omega\restr I)\cup\omega':
  \{0,1\}^{\kappa}\times\{0,1\}^{\kappa\setminus I}
  \to\{0,1\}^{\kappa}$}

\noindent is $(\CalBa_{\kappa}\tensorhat\CalBa_{\kappa\setminus I},
\CalBa_{\kappa})$-measurable, because
$\CalBa_{\kappa}=\Tensorhat_{\kappa}\Cal P(\{0,1\})$ (4A3Na).   So

\Centerline{$\{(\omega,\omega'):(\omega\restr I)\cup\omega'\in E$,
    $(\omega,g((\omega\restr I)\cup\omega'))\in V\}
  \in\CalBa_{\kappa}\tensorhat\CalBa_{\kappa\setminus I}$}

\noindent and we can apply 252P.\ \Qed

We can therefore set

\Centerline{$\dot\mu_{IEg}
=\{((\vec V,\vec h),\Bbbone):(V,h)\in Q_{IEg}\}$,}

\noindent and $\dot\mu_{IEg}$ will be a $\BbbPk$-name, subject to the
conventions I use concerning the interpretation of the brackets in the
formula $((\vec V,\vec h),\Bbbone)$.

\medskip

\quad{\bf (iii)} If $I$, $E$, $g$ and $\dot\mu_{IEg}$ are as in (ii),
then

\Centerline{$\VVdPk\,\dot\mu$ is a $[0,1]$-valued function with domain
$\CalBa_{\Bbb N}$.}

\noindent\Prf\ Suppose that $(V_0,h_0)$,
$(V_1,h_1)\in Q_{IEg}$ and $p\in\BbbPk$ are such that
$p\VVdPk\,\vec V_0=\vec V_1$.
Express $p$ as $F^{\ssbullet}$ where $F\in\CalBa_{\kappa}$.   By 551Gb,
$F'=F\setminus\{\omega:V_0[\{\omega\}]=V_1[\{\omega\}]\}$ is negligible.
But for $\omega\in F\setminus F'$,

$$\eqalign{\{\omega':(\omega\restr I)\cup\omega'\in E,
   \,(&\omega,g((\omega\restr I)\cup\omega'))\in V_0\}\cr
&=\{\omega':(\omega\restr I)\cup\omega'\in E,
   \,(\omega,g((\omega\restr I)\cup\omega'))\in V_1\}\cr}$$

\noindent and $h_0(\omega)=h_1(\omega)$.   So $p\VVdPk\,\vec h_0=\vec h_1$.

Thus $\dot\mu_{IEg}$ satisfies the condition (ii) of 5A3Ea, and

\doubleinset{$\VVdPk\,\dot\mu_{IEg}$ is a function with domain
$\{(\vec V,\Bbbone):V\in\CalBa_{\kappa}\tensorhat\CalBa_{\Bbb N}\}
=\CalBa_{\Bbb N}$}

\noindent where the second $\CalBa_{\Bbb N}$ is interpreted in the forcing
language (551F).   Since

\Centerline{$\VVdPk\,\vec h\in[0,1]$}

\noindent whenever $h:\{0,1\}^{\kappa}\to[0,1]$ is a
$\CalBa_{\kappa}$-measurable function (551B),

\Centerline{$\VVdPk\,\dot\mu_{IEg}$ takes values in $[0,1]$.\ \Qed}

Next,

\Centerline{$\VVdPk\,\dot\mu_{IEg}$ is countably additive, so is a Borel
measure on $\{0,1\}^{\Bbb N}$.}

\noindent\Prf\ (Compare 551M-551N.)
Use the formulae of 551E.   If $\dot{\pmb{G}}$ is a
$\BbbPk$-name such that

\Centerline{$\VVdPk$ $\dot{\pmb{G}}$ is a disjoint sequence in
$\CalBa_{\Bbb N}$,}

\noindent then there is a sequence $\sequencen{W_n}$ in
$\CalBa_{\kappa}\tensorhat\CalBa_{\Bbb N}$ such that

\Centerline{$\VVdPk$ $\dot{\pmb{G}}=\sequencen{\vec W_n}$.}

\noindent If $m\ne n$, then

\Centerline{$\VVdPk$ $(W_m\cap W_n)\sspvec=\vec W_m\cap\vec W_n=\emptyset$,}

\noindent so
$\nu_{\kappa}\{\omega:W_m[\{\omega\}]\cap W_n[\{\omega\}]\ne\emptyset\}=0$
(551Ga);  accordingly $\sequencen{W_n[\{\omega\}]}$ is disjoint for
$\nu_{\kappa}$-almost every $\omega$.   Setting
$W=\bigcup_{n\in\Bbb N}W_n$,

\Centerline{$h_n(\omega)=\nu_{\kappa\setminus I}
  \{\omega':(\omega\restr I)\cup\omega'\in E$,
    $(\omega,g((\omega\restr I)\cup\omega'))\in W_n\}$,}

\Centerline{$h(\omega)=\nu_{\kappa\setminus I}
  \{\omega':(\omega\restr I)\cup\omega'\in E$,
    $(\omega,g((\omega\restr I)\cup\omega'))\in W\}$}

\noindent for $\omega\in\{0,1\}^{\kappa}$ and $n\in\Bbb N$,
we see that whenever $\sequencen{W_n[\{\omega\}]}$ is disjoint then

\Centerline{$\sequencen{\{\omega':(\omega\restr I)\cup\omega'\in E$,
    $(\omega,g((\omega\restr I)\cup\omega'))\in W_n\}}$}

\noindent is disjoint, with union

\Centerline{$\{\omega':(\omega\restr I)\cup\omega'\in E$,
    $(\omega,g((\omega\restr I)\cup\omega'))\in W\}$,}

\noindent so $h(\omega)=\sum_{n=0}^{\infty}h_n(\omega)$.
Accordingly

$$\eqalignno{\VVdPk\dot\mu_{IEg}(\bigcup_{n\in\Bbb N}\vec W_n)
&=\dot\mu_{IEg}\vec W\cr
\displaycause{551Ed}
&=\vec h
=\sum_{n=0}^{\infty}\vec h_n\cr
\displaycause{5A3L(c-iii), 5A3Ld}
&=\sum_{n=0}^{\infty}\dot\mu_{IEg}\vec W_n.\cr}$$

\noindent As $\dot{\pmb G}$ is arbitrary,

\Centerline{$\VVdPk$ $\dot\mu_{IEg}$ is countably additive.  \Qed}

\medskip

\quad{\bf (iv)} Still supposing that $I$, $E$, $g$ and $\dot\mu_{IEg}$
are as in (ii), let $J$, $M\subseteq\kappa$ be
such that $E$ and $g$ are determined by coordinates in $J$, and
$J\cap M=I$.   If
$V\in\CalBa_{\kappa}\tensorhat\CalBa_{\Bbb N}$ is $M$-determined in the
sense of (i) above, and

\Centerline{$\VVdPk\,\dot\mu_{IEg}(\vec V)=0$,}

\noindent then

\Centerline{$E^{\ssbullet}\VVdPk\,\vec g\notin\vec V$.}

\noindent\Prf\ We can suppose that $J\cup M=\kappa$, so that
$(I,J\setminus I,M\setminus I)$ is a partition of $\kappa$.   Identifying
$\{0,1\}^{\kappa}$ with
$\{0,1\}^I\times\{0,1\}^{J\setminus I}\times\{0,1\}^{M\setminus I}$,
we can find
$E'\in\CalBa_I\tensorhat\CalBa_{J\setminus I}$, a
$\CalBa_I\tensorhat\CalBa_{J\setminus I}$-measurable function
$g':\{0,1\}^I\times\{0,1\}^{J\setminus I}\to\{0,1\}^{\Bbb N}$ and
a set
$V'\in\CalBa_I\tensorhat\CalBa_{M\setminus I}\tensorhat\CalBa_{\Bbb N}$
such that

\Centerline{$E=\{(\omega_0,\omega_1,\omega_2):(\omega_0,\omega_1)\in E'$,
  $\omega_2\in\{0,1\}^{M\setminus I}\}$,}

\Centerline{$g(\omega_0,\omega_1,\omega_2)=g'(\omega_0,\omega_1)$ for all
$\omega_0\in\{0,1\}^I$, $\omega_1\in\{0,1\}^{J\setminus I}$,
$\omega_2\in\{0,1\}^{M\setminus I}$,}

\Centerline{$V=\{((\omega_0,\omega_1,\omega_2),x):
((\omega_0,\omega_2),x)\in V'$, $\omega_1\in\{0,1\}^{J\setminus I}\}$.}

\noindent The hypothesis

\Centerline{$\VVdPk\,\dot\mu_{IEg}(\vec V)=0$}

\noindent translates into `$h=0\,\,\nu_{\kappa}$-a.e.', where

\Centerline{$h(\omega)=\nu_{\kappa\setminus I}
  \{\omega':(\omega\restr I)\cup\omega'\in E$,
    $(\omega,g((\omega\restr I)\cup\omega'))\in V\}$,}

\noindent that is, identifying $\{0,1\}^{\kappa\setminus I}$ with
$\{0,1\}^{J\setminus I}\times\{0,1\}^{M\setminus I}$,

\Centerline{$h(\omega_0,\omega_1,\omega_2)
=\nu_{\kappa\setminus I}\{(\omega'_1,\omega'_2):
  (\omega_0,\omega'_1)\in E',\,
    ((\omega_0,\omega_2),g'(\omega_0,\omega'_1))\in V'\}$.}

\noindent So we see that

\Centerline{$\nu_{\kappa\setminus I}\{(\omega'_1,\omega'_2):
  (\omega_0,\omega'_1)\in E',\,
  ((\omega_0,\omega_2),g'(\omega_0,\omega'_1))\in V'\}=0$}

\noindent for $\nu_{\kappa}$-almost every $(\omega_0,\omega_1,\omega_2)$.
It follows that

\Centerline{$\nu_{J\setminus I}\{\omega'_1:
  (\omega_0,\omega'_1)\in E',\,
  ((\omega_0,\omega_2),g'(\omega_0,\omega'_1))\in V'\}=0$}

\noindent for $\nu_{\kappa}$-almost every $(\omega_0,\omega_1,\omega_2)$,
and therefore for $\nu_M$-almost every $(\omega_0,\omega_2)$, here
identifying $\{0,1\}^M$ with $\{0,1\}^I\times\{0,1\}^{M\setminus I}$.
Consequently

\Centerline{$W=\{(\omega_0,\omega'_1,\omega_2):
  (\omega_0,\omega'_1)\in E',\,
    ((\omega_0,\omega_2),g'(\omega_0,\omega'_1))\in V'\}$}

\noindent is $\nu_{\kappa}$-negligible.   But $W$ is also

$$\eqalign{&\{(\omega_0,\omega_1,\omega_2):
  (\omega_0,\omega_1,\omega_2)\in E,\,
   ((\omega_0,\omega_1,\omega_2),g(\omega_0,\omega_1,\omega_2))\in V\}\cr
&\mskip200mu
=\{\omega:\omega\in E,\,(\omega,g(\omega))\in V\}.\cr}$$

By 551Ea,
$\Bvalue{\vec g\in\vec V}=\{\omega:(\omega,g(\omega))\in V\}^{\ssbullet}$;
we have just seen that this is disjoint from $E^{\ssbullet}$ in
$\frak B_{\kappa}$, so

\Centerline{$E^{\ssbullet}\VVdPk\,\vec g\notin\vec V$,}

\noindent as required.\ \Qed

\medskip

\quad{\bf (v)} (The key.)
Once again taking $I$, $E$, $g$ and $\dot\mu_{IEg}$
as in (ii), let $J$, $M\subseteq\kappa$ be such that $J$ is countable,
$E$ and $g$ are determined by coordinates in $J$, $J\cap M=I$
and $M\setminus I$ is infinite.   Then there are an
$E'\in\CalBa_{\kappa}\setminus\Cal N(\nu_{\kappa})$ and a
$\CalBa_{\kappa}$-measurable function
$g':\{0,1\}^{\kappa}\to\{0,1\}^{\Bbb N}$, both determined by coordinates in
$M$, such that

\Centerline{$\VVdPk$ $\dot\mu_{IEg}=\dot\mu_{IE'g'}$.}

\noindent\Prf\ Because $J$ is countable, $I\subseteq M$ and
$M\setminus I$ is infinite,
there is a permutation $\alpha$ of $\kappa$ such that $\alpha(\xi)=\xi$
for every $\xi\in I$ and $\alpha[J]\subseteq M$.   Set

\Centerline{$E'
=\{\omega:\omega\in\{0,1\}^{\kappa}$, $\omega\alpha\in E\}$,
\quad$g'(\omega)=g(\omega\alpha)$ for every $\omega\in\{0,1\}^{\Bbb N}$.}

\noindent Because $\omega\mapsto\omega\alpha$ is an autohomeomorphism of
$\{0,1\}^{\kappa}$, $E'$ is a Baire set and $g'$ is Baire measurable.
If $\omega$, $\omega'\in\{0,1\}^{\kappa}$ and
$\omega\restr M=\omega'\restr M$, then
$\omega\alpha\restr J=\omega'\alpha\restr J$;  so $E'$ and $g'$ are both
determined by coordinates in $M$.

Take any $V\in\CalBa_{\kappa}\tensorhat\CalBa_{\Bbb N}$ and set

\Centerline{$h(\omega)=\nu_{\kappa\setminus I}
  \{\omega':(\omega\restr I)\cup\omega'\in E$,
    $(\omega,g((\omega\restr I)\cup\omega'))\in V\}$,}

\Centerline{$h'(\omega)=\nu_{\kappa\setminus I}
  \{\omega':(\omega\restr I)\cup\omega'\in E'$,
    $(\omega,g'((\omega\restr I)\cup\omega'))\in V\}$}

\noindent for $\omega\in\{0,1\}^{\kappa}$.
Consider the permutation
$\beta=\alpha\restr\kappa\setminus I$ of $\kappa\setminus I$, and set
$\hat\beta\omega'=\omega'\beta$ for
$\omega'\in\{0,1\}^{\kappa\setminus I}$.   Then, for any $\omega$,

$$\eqalign{\{\omega':(\omega\restr I)\cup\omega'\in E',\,
    (&\omega,g'((\omega\restr I)\cup\omega'))\in V\}\cr
&=\{\omega':(\omega\restr I)\cup\hat\beta(\omega')\in E,\,
    (\omega,g((\omega\restr I)\cup\hat\beta(\omega')))\in V\}\cr
&=\hat\beta^{-1}[\{\omega':(\omega\restr I)\cup\omega'\in E,\,
    (\omega,g((\omega\restr I)\cup\omega'))\in V\}].\cr}$$

\noindent As $\hat\beta$ is an automorphism of
$(\{0,1\}^{\kappa\setminus I},\nu_{\kappa\setminus I})$,
$\{\omega':(\omega\restr I)\cup\omega'\in E',\,
    (\omega,g'((\omega\restr I)\cup\omega'))\in V\}$ and
$\{\omega':(\omega\restr I)\cup\omega'\in E,\,
    (\omega,g((\omega\restr I)\cup\omega'))\in V\}$ have the same measure.
Thus $h=h'$, and

\Centerline{$\VVdPk\,\dot\mu_{IEg}(\vec V)
=\vec h=\vec{h'}=\dot\mu_{IE'g'}(\vec V)$.}

\noindent As $V$ is arbitrary,

\Centerline{$\VVdPk\,\dot\mu_{IEg}=\dot\mu_{IE'g'}$.}

\noindent So we have an appropriate pair $E'$, $g'$.\ \Qed

\medskip

\quad{\bf (vi)} I come at last to universally measurable sets.
Let $\dot A$ be a $\Bbb P_{\kappa}$-name such that

\Centerline{$\VVdPk\,\dot A$ is a universally measurable
subset of $\{0,1\}^{\Bbb N}$.}

\noindent Then there is a set $M\subseteq\kappa$ with cardinal $\frak c$
such that whenever $I\in[M]^{\le\omega}$,
$E\in\CalBa_{\kappa}\setminus\Cal N(\nu_{\kappa})$ and a Baire measurable
$g:\{0,1\}^{\kappa}\to\{0,1\}^{\Bbb N}$ are determined by coordinates in
$M$, then there are $M$-determined sets
$F$, $V\in\CalBa_{\kappa}\tensorhat\CalBa_{\Bbb N}$ such that

\Centerline{$\VVdPk\,\dot A\symmdiff\vec F\subseteq\vec V$ and
$\dot\mu_{IEg}\vec V=0$.}

\noindent\Prf\ If $I\in[\kappa]^{\le\omega}$,
$E\in\CalBa_{\kappa}\setminus\Cal N(\nu_{\kappa})$ and
$g:\{0,1\}^{\kappa}\to\{0,1\}^{\Bbb N}$ is Baire measurable, then

\doubleinset{$\VVdPk$ the completion of $\dot\mu_{IEg}$ measures $\dot A$,
so there are Borel sets $G$, $H\subseteq\{0,1\}^{\Bbb N}$ such that
$\dot A\symmdiff G\subseteq H$ and $\dot\mu_{IEg}H=0$.}

\noindent By 551F, as usual, there must be $F$,
$V\in\CalBa_{\kappa}\tensorhat\CalBa_{\Bbb N}$ such that

\Centerline{$\VVdPk$
$\dot A\symmdiff\vec F\subseteq\vec V$ and $\dot\mu_{IEg}\vec V=0$.}

\noindent Now there will be a countable set $K(I,E,g)\subseteq\kappa$ such
that $F$ and $V$ are both $K(I,E,g)$-determined.
If we build inductively a non-decreasing family
$\langle M_{\xi}\rangle_{\xi\le\omega_1}$ of
subsets of $\kappa$ with cardinal $\frak c$ such that whenever
$\xi<\omega_1$, $I\in[M_{\xi}]^{\le\omega}$,
$E'\in\CalBa_{\kappa}\setminus\Cal N(\nu_{\kappa})$ and a Baire measurable
$g':\{0,1\}^{\kappa}\to\{0,1\}^{\Bbb N}$ are determined by coordinates in
$M_{\xi}$, then $K(I,E,g)\subseteq M_{\xi+1}$ (which is possible because if
$\#(M_{\xi})=\frak c$ then there are just $\frak c$ possibilities for $I$,
$E$ and $g$), we shall be able to set $M=M_{\omega_1}$ to get a set of the
type we need.\ \Qed

Enumerate

\Centerline{$\{W:W\in\CalBa_{\kappa}\tensorhat\CalBa_{\Bbb N}$,
$W$ is $M$-determined$\}$}

\noindent as $\ofamily{\xi}{\frak c}{W_{\xi}}$.
Then

\Centerline{$\VVdPk\,\dot A=\bigcup\{\vec W_{\xi}:\xi<\check\frak c$,
$\vec W_{\xi}\subseteq\dot A\}$.}

\noindent\Prf\ Suppose that
$E\in\CalBa_{\kappa}\setminus\Cal N(\nu_{\kappa})$ and a $\BbbPk$-name
$\dot x$ are such that

\Centerline{$E^{\ssbullet}\VVdPk$ $\dot x\in\dot A$.}

\noindent Then there is a Baire measurable
function $g:\{0,1\}^{\kappa}\to\{0,1\}^{\Bbb N}$ such that

\Centerline{$E^{\ssbullet}\VVdPk\,\dot x=\vec g$}

\noindent (551Cc).
Let $J\in[\kappa]^{\le\omega}$ be such that $E$ and $g$ are both
determined by coordinates in $J$, and set $I=J\cap M$.
By (v) above, there are $E'$ and $g'$, both
determined by coordinates in $M$, such that

\Centerline{$\VVdPk\,\dot\mu_{IEg}=\dot\mu_{IE'g'}$.}

\noindent We therefore have $M$-determined $F$,
$V\in\CalBa_{\kappa}\tensorhat\CalBa_{\Bbb N}$ such that

\Centerline{$\VVdPk\,\dot A\symmdiff\vec F\subseteq\vec V$ and
$\dot\mu_{IEg}\vec V=\dot\mu_{IE'g'}\vec V=0$.}

\noindent Let $\xi<\frak c$ be such that $F\setminus V=W_{\xi}$.
By (iv),

\Centerline{$E^{\ssbullet}\VVdPk\,\vec g\notin\vec V$ and
$\dot x=\vec g\in\vec W_{\xi}\subseteq\dot A$.}

\noindent
As $E$ and $\dot x$ are arbitrary, we have the result.\ \Qed

Thus

\Centerline{$\VVdPk$ $\dot A$ is expressible as the union of
$\check\frak c$ Borel sets.}

\noindent As $\dot A$ is arbitrary, (a) is proved.

\medskip

{\bf (b)} Now all we have to do is count.   We surely have

\Centerline{$\VVdPk$ there are at least $\frak c$ universally measurable
subsets of $\{0,1\}^{\Bbb N}$}

\noindent just because singletons are universally measurable.   In the
other direction, because $\kappa^{\frak c}=\kappa$,

\Centerline{$\VVdPk$ $\frak c^{\check\frak c}
=(2^{\omega})^{\check\frak c}=2^{\check\frak c}=\check\kappa
=(\kappa^{\omega})\varBspcheck=\frak c$}

\noindent (552B), while (a) tells us that

\Centerline{$\VVdPk$ the number of universally measurable subsets of
$\{0,1\}^{\Bbb N}$ is at most $\frak c^{\check\frak c}$.}
}%end of proof of 553O

\exercises{\vleader{60pt}{553X}{Basic exercises (a)}(i)
%\spheader 553Xa
Suppose that $A\subseteq\{0,1\}^{\Bbb N}$ has strong measure zero,
and that $\kappa$ is a cardinal.   Show that

\Centerline{$\VVdPk\,\check A$ has strong measure zero in
$\{0,1\}^{\Bbb N}$.}

\noindent (ii) Repeat with $\Bbb R$ in place of $\{0,1\}^{\Bbb N}$.
(iii)\dvAnew{2010} Suppose that $\frak m=\frak c>\omega_1$.   Show that

\Centerline{$\Vdash_{\Bbb P_{\omega_1}}$ there is a set of strong measure
zero in $\Bbb R$ with cardinal greater than $\frakmctbl$.}
%552O 553B

\spheader 553Xb Let $W\subseteq\{0,1\}^{\omega}\times\{0,1\}^{\omega}$ be the
set

\Centerline{$\{(x,y):x(2n)=y(2n)$ for every $n\in\Bbb N\}$.}

\noindent Show that, for every $y\in\{0,1\}^{\omega}$,

\Centerline{$\VVdash_{\Bbb P_{\omega}}\,\vec W$ is homeomorphic to
$\{0,1\}^{\omega}$ and $\check y\notin\vec W$.}
%553E

\spheader 553Xc(i) Suppose that $\Cal F$ is a $p$-point filter on
$\Bbb N$, and that $\Bbb P$ is a ccc forcing notion.   Show
that

\Centerline{$\VVdP$ the filter on $\Bbb N$ generated by $\check{\Cal F}$
is a $p$-point filter.}

\noindent (ii) Suppose that $\Cal F$ is a rapid filter on $\Bbb N$,
and that $\kappa$ is a cardinal.   Show that

\Centerline{$\VVdPk$ the filter on $\Bbb N$ generated by $\check{\Cal F}$
is a rapid filter.}
%553H out of order query

\spheader 553Xd Let $\frak A$ be a Boolean algebra and
$\nu:\frak A\to\coint{0,\infty}$ a non-negative additive functional.   Show that
if $\familyiI{a_i}$ is a finite family in $\frak A$ then

\Centerline{$\nu(\sup_{i\in I}a_i)
\le\sum_{k=1}^m(-1)^{k+1}\sum_{J\in[I]^k}\nu(\inf_{i\in J}a_i)$
if $m\ge 1$ is odd,}

\Centerline{$\nu(\sup_{i\in I}a_i)
\ge\sum_{k=1}^m(-1)^{k+1}\sum_{J\in[I]^k}\nu(\inf_{i\in J}a_i)$
if $m\ge 1$ is even.}
%553K

\spheader 553Xe Let $\Bbb P$ be a forcing notion which satisfies Knaster's
condition.   (i) Show that if
$(P,\le)$ is an upwards-ccc partially ordered set then

\Centerline{$\VVdP\,(\check P,\check{\le})$ is upwards-ccc.}

\noindent (ii) Show that if $(T,\le)$ is a Souslin tree then

\Centerline{$\VVdP\,(\check T,\check{\le})$ is a Souslin tree.}
%553M

\leader{553Y}{Further exercises (a)}
%\spheader 553Ya
Let $\kappa$ be a cardinal, $\dot G$ a $\BbbPk$-name and
$a\in\frak B_{\kappa}^+$ such that

\Centerline{$a\VVdPk\,\dot G$ is a dense open subset of
$\{0,1\}^{\omega}$.}

\noindent Show that there is a
$W\in\Tau_{\kappa}\tensorhat\CalBa_{\omega}$ such that every vertical
section of $W$ is a dense open set and $a\VVdPk\,\dot G=\vec W$.
%551E

\spheader 553Yb Let $\kappa$ be a cardinal and
$W\in\Tau_{\kappa}\tensorhat\CalBa_{\omega}$ a set such that every
vertical section of $W$ is a dense open set.   Let $C$ be the space of
continuous functions from $\{0,1\}^{\kappa}$ to $\{0,1\}^{\omega}$ with the
compact-open topology.   Show that
$\{f:f\in C$, $\{x:(x,f(x))\in W\}$ is conegligible$\}$ is comeager in $C$.

\spheader 553Yc Show that

\Centerline{$\VVdash_{\Bbb P_{\omega}}
\,\frakmctbl\ge(\frakmctbl)\varBspcheck$.}

\noindent\Hint{work with the ideal of meager sets in the Polish space of
continuous functions from $\{0,1\}^{\omega}$ to itself.}
%553Ya 553Yb

\spheader 553Yd Let $\Cal K$ be the family of compact well-ordered subsets
of $\Bbb Q\cap\coint{0,\infty}$ containing $0$.
For $s$, $t\in\Cal K$ say that
$s\preccurlyeq t$ if $s=t\cap[0,\gamma]$ for some $\gamma\in\Bbb R$;  for
$s\in\Cal K$ and $\gamma\in\Bbb Q$, set
$A(s,\gamma)=\{t:t\in\Cal K$, $\max t=\gamma$, $s\preccurlyeq t\}$.
(i) Show that $(\Cal K,\preccurlyeq)$ is a tree, and that $\otp(t)=r(t)+1$ for
every $t\in\Cal K$.
(ii) Choose $\ofamily{\xi}{\omega_1}{\Cal K_{\xi}}$,
$\ofamily{\xi}{\omega_1}{T_{\xi}}$ inductively so that
$\Cal K_0=T_0=\{\{0\}\}$ and for $0<\xi<\omega_1$

\inset{$\Cal K_{\xi}=\{t:t\in \Cal K$, $r(t)=\xi$,
  $s\in\bigcup_{\eta<\xi}T_{\eta}$ whenever $s\prec t\}$,

$T_{\xi}\subseteq\Cal K_{\xi}$ is countable,

if $\eta<\xi$, $s\in T_{\eta}$ and $\gamma\in\Bbb Q$
are such that $\gamma>\max s$ and $A(s,\gamma)$ meets $\Cal K_{\xi}$, then
$A(s,\gamma)$ meets $T_{\xi}$.}

\noindent Show that if $\eta<\xi<\omega_1$, $s\in T_{\eta}$,
$\gamma\in\Bbb Q$ and $\gamma>\max s$, there is a $t\in T_{\xi}$ such that
$\max t=\gamma$ and $s\preccurlyeq t$.
(iii) Show that $T=\bigcup_{\xi<\omega_1}T_{\xi}$ is a
special Aronszajn tree.
%\Hint{for any $q\in\Bbb Q$, $\{t:\max t=q\}$ is an antichain.}
%553M

\spheader 553Ye Show that
$\VVdash_{\Bbb P_{\omega}}\,
\frak p\ge(\frak m_{\sigma\text{-linked}})\varBspcheck$.
%mt55bits
}%end of exercises

\leader{553Z}{Problem} Suppose that the generalized continuum hypothesis is
true.   Is it the case that

\Centerline{$\VVdash_{\Bbb P_{\omega_2}}$ there is a Borel lifting for
Lebesgue measure?}

\cmmnt{\noindent(Compare 554I.)}

\endnotes{
\Notesheader{553}
To my mind, the chief interest of the results of this section is that
they force us to explore aspects of the structures considered in new ways.
We know, for instance, that if a set has Rothberger's property
(in a separable metrizable space) this can be witnessed by a family of
$\frak d$ sequences.   The point of
553C is that (in random real models) any family of $\frak d$ sequences is
associated with a set $\dot Y$ with cardinal at most the cardinal power
$\frak d^{\omega}=\frak c$ (taken in the ordinary universe $V$),
such that $\dot Y$ must include the given set with Rothberger's property.
Remember that

\Centerline{$\VVdPk\,(\BbbN^{\Bbb N})\varBspcheck$ is cofinal with
$\BbbN^{\Bbb N}$}

\noindent (see the proof of 552C),
so there is no point in looking at `new' members of
$\BbbN^{\Bbb N}$ in part (a) of the proof.

In 553E, we need to distinguish between the $\Bbb P_{\kappa}$-names
$\tilde G$ and $\check G$.   It is quite possible to have

\Centerline{$\VVdPk\,\dot K\cap(\{0,1\}^{\lambda})\varBspcheck=\emptyset$;}

\noindent that is, we might have $\VVdPk\,\dot K=\vec W$ where
$W\subseteq\{0,1\}^{\kappa}\times\{0,1\}^{\lambda}$ has negligible
horizontal sections (553Xb).
The name $\tilde G$ refers not to a copy of the
set $G$ but to a re-interpretation of one (or any) of its descriptions as
an F$_{\sigma}$ set.

In 553H and 553M, we have to look quite deeply into the structure of
measure algebras.   Lemmas 553G and 553L are already not obvious,
and the combinatorial measure theory of the
proof of 553H is delicate.   553J is easier.
The idea here is to `randomize' a construction from {\smc Galvin 80},
where the continuum hypothesis was used to
build complementary sets $S_0$, $S_1$ with the property of 553I.

I give a bit of space to `Aronszajn trees' because the results here express yet
another contrast between random and Cohen forcing.   Cohen forcing creates
Souslin trees (554Yc).
Random forcing preserves old Souslin trees (553Xe) but does not
necessarily produce new ones (553M).

}%end of notes

\leaveitout{\leader{}{} Do random reals give $\shr(\CalSmz)=\omega_1$ or
something?

\leader{}{Problem} If the Borel conjecture is true does it follow that

\Centerline{$\VVdPk$ the Borel conjecture is true}

\noindent for every infinite $\kappa$?  (equivalently, for
$\kappa=\omega$).

See {\smc Bartoszy\'nski \& Judah 95}, 8.3B.
}%end of leaveitout


\discrpage
\frfilename{mt554.tex}
\versiondate{2.9.14}
\copyrightdate{2006}

\def\chaptername{Possible worlds}
\def\sectionname{Cohen reals}

\Loadfourteens

\def\BbbQk{\Bbb Q_{\kappa}}
\long\def\doubleinset#1{\inset{\inset{\parindent=-20pt #1}}}
\def\Fn{\mathop{\text{Fn}}\nolimits}
\def\VVdash{\mskip5mu\vrule height 7.5pt depth 2.5pt width 0.5pt
  \mskip2.5mu\vrule height 7.5pt depth 2.5pt width 0.5pt
  \vrule height 2.75pt depth -2.25pt width 4pt\mskip2mu}
\def\VVdP{\VVdash_{\Bbb P}}
\def\VVdQk{\VVdash_{\Bbb Q_{\kappa}}}

\newsection{554}

Parallel to the theory of random reals as described in \S\S552-553, we have
a corresponding theory based on category algebras rather than measure
algebras.   I start with the exactly matching result on cardinal arithmetic
(554B), and continue with Lusin sets (balancing the Sierpi\'nski
sets of 552E) and the cardinal functions of the
meager ideal of $\Bbb R$ (554C-554E, %554C 554D 554E
554F).   In the last third of the section I use the theory of Freese-Nation
numbers (\S518) to prove Carlson's theorem on Borel liftings (554I).

\leader{554A}{Notation} For any set $I$, I will write $\widehat{\Cal B}_I$
for the Baire-property algebra of $\{0,1\}^I$\cmmnt{, $\CalBa_I$ for the
Baire $\sigma$-algebra of $\{0,1\}^I$}, $\Cal M_I$ for the meager
ideal of $\{0,1\}^I$, $\frak G_I=\widehat{\Cal B}_I/\Cal M_I$ for the
category algebra of $\{0,1\}^I$, and $\Bbb Q_I$ for the forcing notion
$\frak G_I^+=\frak G_I\setminus\{\emptyset\}$ active downwards.
\cmmnt{$\Cal C_I$ will be the family of basic cylinder sets
$\{x:z\subseteq x\in\{0,1\}^I\}$ for $z\in\Fn_{<\omega}(I;\{0,1\})$, and
$C_I$ the corresponding set
$\{C^{\ssbullet}:C\in\Cal C_I\}\subseteq\frak G_I$;
then $C_I$ is order-dense in $\frak G_I$\cmmnt{ (because $\Cal C_I$ is a
$\pi$-base for the topology of $\{0,1\}^I$)}.   It follows that
$\tau(\frak G_I)\le\pi(\frak G_I)\le\max(\omega,\#(I))$.   (These
inequalities are of course equalities if $I$ is infinite.)}

\leader{554B}{Theorem} Suppose that $\lambda$ and $\kappa$ are infinite
cardinals.   Then

\Centerline{$\VVdQk\,2^{\check\lambda}=(\kappa^{\lambda})\varBspcheck$.}

\proof{ (Compare 552B.)

\medskip

{\bf (a)} Since $\frak G_{\kappa}$ is ccc and has an order-dense
subset $C_{\kappa}$ with cardinal $\kappa$,
$\#(\frak G_{\kappa})$ is at most the cardinal power $\kappa^{\omega}$.

If $\dot A$ is a $\BbbQk$-name for a subset of
$\check\lambda$, then we have a corresponding family
$\ofamily{\eta}{\lambda}{\Bvalue{\check\eta\in\dot A}}$ of truth values;
and if $\dot A$, $\dot B$ are two such names, and
$\Bvalue{\check\eta\in\dot A}=\Bvalue{\check\eta\in\dot B}$ for every
$\eta<\lambda$, then

\Centerline{$\VVdQk\,\dot A=\dot B$.}

\noindent So

\Centerline{$\VVdQk\,2^{\check\lambda}=\#(\Cal P\check\lambda)
\le\#((\frak G_{\kappa}^{\lambda})\varBspcheck)
=(\kappa^{\lambda})\varBspcheck$.}

\medskip

{\bf (b)} Consider first the case in which $\lambda\le\kappa$.
Let $F$ be the set of all functions from $\lambda$ to $\kappa$,
so that $\#(F)=\kappa^{\lambda}$.   As in part (b) of the proof of 552B,
there is a set $G\subseteq F$ such that
$\#(G)=\kappa^{\lambda}$ and $\{\eta:\eta<\lambda$, $f(\eta)\ne g(\eta)\}$
is infinite whenever $f$, $g\in G$ are distinct.
Let $\langle\zeta_{\xi\eta}\rangle_{\xi<\kappa,\eta<\lambda}$ be a
family of distinct elements of $\kappa$ and set
$E_{\xi\eta}=\{x:x\in\{0,1\}^{\kappa}$, $x(\zeta_{\xi\eta})=1\}$ for
$\xi<\kappa$ and $\eta<\lambda$.
For $f\in G$ let $\dot A_f$ be a $\BbbQk$-name for
a subset of $\lambda$ such that

\Centerline{$\Bvalue{\check\eta\in\dot A_f}=E^{\ssbullet}_{f(\eta),\eta}$}

\noindent for every $\eta<\lambda$.   If $f$, $g\in G$ are distinct, set
$I=\{\eta:f(\eta)\ne g(\eta)\}$;  then

\Centerline{$\Bvalue{\dot A_f\ne\dot A_g}
=\sup_{\eta<\lambda}E_{f(\eta),\eta}^{\ssbullet}\Bsymmdiff
   E_{g(\eta),\eta}^{\ssbullet}=\Bbbone$}

\noindent because
$\bigcup_{\eta\in I}E_{f(\eta),\eta}\symmdiff E_{g(\eta),\eta}$
is a dense open set in $\{0,1\}^{\kappa}$.

Thus in the forcing language we have a name for an injective function from
$\check G$ to $\Cal P\lambda$, corresponding to the map $f\mapsto\dot A_f$
from $G$ to names of subsets of $\lambda$.   So

\Centerline{$\VVdQk\,2^{\check\lambda}\ge\#(\check G)
=(\kappa^{\lambda})\varBspcheck$.}

\noindent Putting this together with (a), we have

\Centerline{$\VVdQk\,2^{\check\lambda}=(\kappa^{\lambda})\varBspcheck$.}

\medskip

{\bf (c)} If $\lambda>\kappa$, then
$2^{\lambda}=\kappa^{\lambda}$.   Now

\Centerline{$\VVdQk\,
  (\Cal P\lambda)\varBspcheck\subseteq\Cal P\check\lambda$,}

\noindent so

\Centerline{$\VVdQk\,(\kappa^{\lambda})\varBspcheck
=\#((\Cal P\lambda)\varBspcheck)
\le\#(\Cal P\check\lambda)=2^{\check\lambda}$,}

\noindent and again we have

\Centerline{$\VVdQk\,2^{\check\lambda}=(\kappa^{\lambda})\varBspcheck$.}
}%end of proof of 554B

\leader{554C}{Definition} If $X$ is a topological space, a subset of $X$
is a {\bf Lusin set} if it is uncountable but meets every meager set
in a countable set\cmmnt{;  equivalently, if it is uncountable but meets
every nowhere dense set in a countable set}.

\leader{554D}{Proposition}
Let $\kappa$ be a cardinal such that $\Bbb R$ has a
Lusin set with cardinal $\kappa$.

(a) Writing $\Cal M$ for the ideal of meager subsets of $\Bbb R$,
$\non\Cal M=\omega_1$ and $\frakmctbl\ge\kappa$.

(b) There is a point-countable family $\Cal A$ of
Lebesgue-conegligible subsets of $\Bbb R$ with $\#(\Cal A)=\kappa$.

(c) If $(\frak A,\bar\mu)$ is a semi-finite measure algebra which is not
purely atomic, $(\kappa,\omega_1)$ is not a precaliber pair of $\frak A$.

\proof{ Let $B\subseteq\Bbb R$ be a Lusin set with cardinal $\kappa$.

\medskip

{\bf (a)} By 522Sa, $\frakmctbl=\cov\Cal M$.
Any uncountable subset of $B$ is non-meager, so
$\non\Cal M=\omega_1$.
If $\Cal E$ is a cover of $\Bbb R$ by meager sets,
then each member of $\Cal E$ meets $B$ in a countable set, so

\Centerline{$\kappa=\#(B)\le\max(\omega,\#(\Cal E))$}

\noindent and $\#(\Cal E)\ge\kappa$;  thus $\cov\Cal M\ge\kappa$.

\medskip

{\bf (b)}
Let $E\subseteq\Bbb R$ be a conegligible meager set containing $0$, and set
$\Cal A=\{x+E:x\in B\}$.   Then $\Cal A$ is a family of conegligible sets.
If $y\in\Bbb R$, then $y-E$ is meager so
$\{x:x\in B$, $y\in x+E\}=B\cap(y-E)$ is countable;  thus $\Cal A$ is
point-countable.   Also, each
member of $\Cal A$ is meager, so meets $B$ in a
countable set, and (because $B\subseteq\bigcup\Cal A$)

\Centerline{$\kappa=\#(B)\le\max(\omega,\#(\Cal A))\le\#(\Cal A)\le\#(B)$,}

\noindent so $\#(\Cal A)=\kappa$.

\medskip

{\bf (c)} Let $K\subseteq E$ be a compact set of non-zero measure.   If
$\Gamma\subseteq B$ is uncountable,
$\bigcap_{x\in\Gamma}x+K=\emptyset$, $\{x+K:x\in\Gamma\}$ does not have the
finite intersection property and $\{(x+K)^{\ssbullet}:x\in\Gamma\}$ is not
centered in the measure algebra $\frak A_L$ of Lebesgue measure.   Thus
$\family{x}{B}{(x+K)^{\ssbullet}}$ witnesses that $(\kappa,\omega_1)$ is
not a precaliber pair of $\frak A_L$.

Since $(\frak A,\bar\mu)$ is semi-finite and not purely atomic, there is a
subalgebra of a principal ideal of $\frak A$ which is isomorphic to
$\frak A_L$, and $(\kappa,\omega_1)$ is not a precaliber pair of $\frak A$,
by 516Sa.

}%end of proof of 554D

\leader{554E}{Theorem} Let $\kappa$ be an uncountable cardinal.   Then

\Centerline{$\VVdQk$ there is a Lusin set $A\subseteq\Bbb R$ with cardinal
$\check\kappa$.}

\proof{{\bf (a)} (Compare 552E.)   Write $\Bbb P$ for
$\Bbb Q_{\kappa\times\omega}$.   For each
$\xi<\kappa$, let $f_{\xi}:\{0,1\}^{\kappa\times\omega}\to\{0,1\}^{\omega}$
be given by setting $f_{\xi}(x)(n)=x(\xi,n)$ for every
$x\in\{0,1\}^{\kappa\times\lambda}$ and $n<\omega$;  then, taking
$\vec f_{\xi}$ to be the $\Bbb P$-name defined by the process of 551Cb,

\Centerline{$\VVdP\vec f_{\xi}\in\{0,1\}^{\omega}$.}

\noindent If $\xi$, $\xi'<\kappa$ are distinct, then, by 551Cd,

$$\eqalign{\Bvalue{\vec f_{\xi}=\vec f_{\xi'}}
&=\{x:f_{\xi}(x)=f_{\xi'}(x)\}^{\ssbullet}\cr
&=\{x:x(\xi,n)=x(\xi',n)\text{ for every }n\}^{\ssbullet}
=0\cr}$$

\noindent because $\{x:x(\xi,n)=x(\xi',n)\text{ for every }n\}$ is closed
and nowhere dense.   So, taking $\dot A$ to be the $\VVdP$-name
$\{(\vec f_{\xi},\Bbbone):\xi<\kappa\}$, we have

\Centerline{$\VVdP\,\dot A\subseteq\{0,1\}^{\omega}$ has
cardinal $\check\kappa$.}

\medskip

{\bf (b)} Now suppose that $\dot W$ is a $\Bbb P$-name such that

\Centerline{$\VVdP\,\dot W$ is a nowhere dense zero set in
$\{0,1\}^{\omega}$.}

\noindent By 551Fb there is a
$W\in\widehat{\Cal B}_{\kappa\times\omega}\tensorhat\CalBa_{\omega}$
such that, in the language of 551D, $\VVdP\,\dot W=\vec W$.   Now $W$ is
meager in $\{0,1\}^{\kappa\times\omega}$.   \Prf\ For
$z\in\Fn_{<\omega}(\omega;\{0,1\})$ set
$V_z=\{(x,y):x\in\{0,1\}^{\kappa\times\omega}$,
$z\subseteq y\in\{0,1\}^{\omega}\}$.   By 551Ee,

\Centerline{$\VVdP\,\vec V_z
=\{y:\check z\subseteq y\in\{0,1\}^{\omega}\}$;}

\noindent and as $\widehat{\Cal B}_{\kappa\times\omega}$ is closed under
Souslin's operation (431Fb),

\Centerline{$\Bvalue{\vec W\cap\vec V_z=\emptyset}
=\{x:W[\{x\}]\cap\{y:y\supseteq z\}
  =\emptyset\}^{\ssbullet}$}

\noindent (551Ga).   Now we have

$$\eqalign{1
&=\Bvalue{\vec W\text{ is nowhere dense}}\cr
&=\inf_{z\in\Fn_{<\omega}(\omega;\{0,1\})}
  \sup_{z'\in\Fn_{<\omega}(\omega;\{0,1\}),z'\supseteq z}
  \Bvalue{\vec W\cap\{y:\check z'\subseteq y\}=\emptyset}\cr
&=\inf_{z\in\Fn_{<\omega}(\omega;\{0,1\})}
  \sup_{z'\in\Fn_{<\omega}(\omega;\{0,1\}),z'\supseteq z}
  \{x:W[\{x\}]\cap\{y:y\supseteq z'\}=\emptyset\}^{\ssbullet}
  \cr
&=\bigl(\bigcap_{z\in\Fn_{<\omega}(\omega;\{0,1\})}
  \bigcup_{z'\in\Fn_{<\omega}(\omega;\{0,1\}),z'\supseteq z}
  \{x:W[\{x\}]\cap\{y:y\supseteq z'\}=\emptyset\}
  \bigr)^{\ssbullet}\cr
&=\{x:W[\{x\}]\text{ is nowhere dense}\}^{\ssbullet}.\cr}$$

\noindent So $\{x:W[\{x\}]$ is meager$\}$ is comeager.
Because $W$ has the Baire property in
$\{0,1\}^{\kappa\times\omega}\times\{0,1\}^{\omega}$ (5A4E(b-ii)),
it must be meager, by the Kuratowski-Ulam theorem (527D).\ \Qed

\medskip

{\bf (c)} Continuing from (b), there
is a meager Baire set $W'\supseteq W$ (5A4E(c-ii)).
Let $J\subseteq\kappa$ be
a countable set such that $W'$ is determined by coordinates in
$(J\times\omega)\mskip5mu\dot\cup\mskip5mu\omega$, that is, if
$(x,y)\in W'$, $x'\in\{0,1\}^{\kappa\times\omega}$ and
$x'\restr J\times\omega=x\restr J\times\omega$ then
$(x',y)\in W'$.   Take any $\xi\in\kappa\setminus J$.
Set $L=(\kappa\setminus\{\xi\})\times\omega$ and

\Centerline{$V=\{(x\restr L,y):(x,y)\in W'\}$;}

\noindent then $V\subseteq\{0,1\}^L\times\{0,1\}^{\omega}$ is meager
(applying 527D to

\Centerline{$V\times\{0,1\}^{\{\xi\}\times\omega}
\subseteq\{0,1\}^L\times\{0,1\}^{\{\xi\}\times\omega}
\cong\{0,1\}^{\kappa\times\omega}\times\{0,1\}^{\omega}$).}

Now consider the map
$\phi:\{0,1\}^{\kappa\times\omega}\to\{0,1\}^L\times\{0,1\}^{\omega}$
defined by setting $\phi(x)=(x\restr L,f_{\xi}(x))$ for
$x\in\{0,1\}^{\kappa\times\omega}$.
Looking back at the definition of $f_{\xi}$, we see that
this is a homeomorphism.   So $\phi^{-1}[V]$ must be meager, and

$$\eqalignno{\Bvalue{\vec f_{\xi}\in\vec W}
&\Bsubseteq\Bvalue{\vec f_{\xi}\in\vec W'}
=\{x:(x,f_{\xi}(x))\in W'\}^{\ssbullet}\cr
\displaycause{551Ea}
&=\{x:(x\restr L,f_{\xi}(x))\in V\}^{\ssbullet}
=(\phi^{-1}[V])^{\ssbullet}
=0,\cr}$$

\noindent that is, $\VVdP\,\vec f_{\xi}\notin\vec W$.

This is true for every $\xi\in\kappa\setminus J$.   So

\Centerline{$\VVdP\,\dot A\cap\dot W
\subseteq\{\vec f_{\xi}:\xi\in\check J\}$ is countable.}

\noindent As $\dot W$ is arbitrary,

\Centerline{$\VVdP\,\dot A$ has countable intersection with every nowhere
dense zero set.}

\noindent It follows at once that

\Centerline{$\VVdP\,\dot A$ has countable intersection with every nowhere
dense set, and is a Lusin set.}

\noindent As $\Bbb P$ and $\BbbQk$ are isomorphic,

\Centerline{$\VVdQk\,\{0,1\}^{\omega}$ has a Lusin set with cardinal
$\check\kappa$.}

\medskip

{\bf (d)} The statement of the proposition referred to $\Bbb R$ rather than
to $\{0,1\}^{\omega}$.   But, writing $\Cal M$ for the ideal of meager
subsets of
$\Bbb R$ and $\Cal M_{\omega}$ for the ideal of meager subsets of
$\{0,1\}^{\omega}$,
$(\Bbb R,\Cal M)$ and $(\{0,1\}^{\omega},\Cal M_{\omega})$ are isomorphic
(522Wb), and one will have Lusin sets iff the other does.   So

\Centerline{$\VVdQk\,\Bbb R$ has a Lusin set with cardinal $\check\kappa$.}
}%end of proof of 554E

\leader{554F}{Corollary} Let $\kappa$ be a cardinal which is equal to the
cardinal power $\kappa^{\omega}$.   Write $\Cal M$ for the ideal of meager
subsets of $\Bbb R$.   Then

\Centerline{$\VVdQk\,\non\Cal M=\omega_1$ and
$\frakmctbl=\frak c$.}

\proof{ By 554B, $\VVdQk\,\frak c=\check\kappa$;  so we have only to put
554E and 554Da together.
}%end of proof of 554F

\leader{554G}{Theorem} Let $\kappa$ be an infinite cardinal such that
$\FN(\frak G_{\kappa})=\omega_1$.   Then

\Centerline{$\VVdQk\,\FN(\Cal P\Bbb N)=\omega_1$.}

\proof{{\bf (a)} We need to know that $\frak G_{\kappa}$ is isomorphic to
the simple power
algebra $\frak G_{\kappa}^{\Bbb N}$.  \Prf\ The algebra $\Cal E$
of open-and-closed subsets of $\{0,1\}^{\kappa}$ is isomorphic to a free
product of two-element algebras, so is homogeneous (316Q);
$\frak G_{\kappa}$ is isomorphic to the Dedekind
completion of $\Cal E$, so is homogeneous (316P).
Now we have a partition
of unity $\sequencen{p_n}$ in $\frak G_{\kappa}$ consisting of non-zero
elements, so that $\frak G_{\kappa}$ is isomorphic to the simple product of
the corresponding principal ideals (315F) and to
$\frak G_{\kappa}^{\Bbb N}$.\ \QeD\  There is therefore a Freese-Nation
function $\theta:
\frak G_{\kappa}^{\Bbb N}\to[\frak G_{\kappa}^{\Bbb N}]^{\le\omega}$.

For $\xi<\kappa$, set $E_{\xi}=\{x:x\in\{0,1\}^{\kappa}$, $x(\xi)=1\}$;
for $J\subseteq\kappa$, let $\frak C_J$ be the order-closed subalgebra of
$\frak G_{\kappa}$ generated by $\{E_{\xi}^{\ssbullet}:\xi\in J\}$,
and let $C_J$ be the set of elements of $\frak C_J$ of the form
$\inf_{\xi\in K}E_{\xi}^{\ssbullet}
\Bsetminus\sup_{\xi\in L}E_{\xi}^{\ssbullet}$ where $K$, $L$ are disjoint
finite subsets of $J$.

For $v\in\frak G_{\kappa}^{\Bbb N}$ let $\vec v$ be the $\BbbQk$-name
$\{(\check n,v(n)):n\in\Bbb N$, $v(n)\ne 0\}$;  then
$\VVdQk\,\vec v\subseteq\Bbb N$, and $\Bvalue{\check n\in\vec v}=v(n)$
for every $n\in\Bbb N$.

For any $\BbbQk$-name $\dot u$, let $J(\dot u)$ be a countable subset of
$\kappa$ such that $\Bvalue{\check n\in\dot u}\in\frak C_{J(\dot u)}$
for every $n\in\Bbb N$.

\medskip

{\bf (b)} Let $\dot X$ be a discriminating $\BbbQk$-name such that
$\VVdQk\,\dot X=\Cal P\Bbb N$ (5A3Ka).   For $\sigma=(\dot u,p)\in\dot X$
set

\Centerline{$\theta_1(\sigma)
=\bigcup_{e\in C_{J(\dot u)}}
  \theta(\sequencen{\Bvalue{\check n\in\dot u}\Bcap e)})
\cup\bigcup_{e\in C_{J(\dot u)}}
  \theta(\sequencen{\Bvalue{\check n\in\dot u}\Bcup(1\Bsetminus e)})
\in[\frak G_{\kappa}^{\Bbb N}]^{\le\omega}$,}

\Centerline{$\theta_2(\sigma)=\{(\vec v,p):v\in\theta_1(\sigma)\}$,}

\noindent so that $\theta_2(\sigma)$ is a $\BbbQk$-name and

\Centerline{$\VVdQk\,\theta_2(\sigma)$ is a countable subset of
$\Cal P\Bbb N$.}

\medskip

{\bf (c)} Set

\Centerline{$\dot\theta
=\{((\dot u,\theta_2(\dot u,p)),p):(\dot u,p)\in\dot X\}$.}

\noindent By 5A3Kb,

\Centerline{$\VVdQk\,\dot\theta$ is a function with domain
$\dot X=\Cal P\Bbb N$.}

\noindent Next,

\Centerline{$\VVdQk\,\dot\theta$ takes values in
$[\Cal P\Bbb N]^{\le\omega}$.}

\noindent\Prf\ Suppose that $\dot x$ is a $\BbbQk$-name and $p\in\frak G_{\kappa}^+$ is
such that

\Centerline{$p\VVdQk\,\dot x$ is a value of $\dot\theta$.}

\noindent Then there are a $(\dot u,q)\in\dot X$ and a $p'$ stronger than
both $p$ and $q$ such that

\Centerline{$p'\VVdQk\,\dot x=(\dot u,\theta_2(\dot u,q))$ has second
member $\theta_2(\dot u,q)\in[\Cal P\Bbb N]^{\le\omega}$.}

\noindent As $p$ and $\dot x$ are arbitrary,

\doubleinset{$\VVdQk$ every value of $\dot\theta$, being the second member
of an element of $\dot\theta$, is a countable subset of $\Cal P\Bbb N$.
\Qed}

\medskip

{\bf (d)} In fact,

\Centerline{$\VVdQk\,\dot\theta$ is a Freese-Nation function on
$\Cal P\Bbb N$.}

\noindent\Prf\ Suppose that $\dot A_1$, $\dot A_2$ are $\BbbQk$-names and
$p\in\frak G_{\kappa}^+$ is such that

\Centerline{$p\VVdQk\,\dot A_1\subseteq\dot A_2\subseteq\Bbb N$.}

\noindent Because $\VVdQk\,\dot X=\Cal P\Bbb N$, there must be
$(\dot u_1,q_1)$ and $(\dot u_2,q_2)\in\dot X$ and a $p_1$
stronger than $p$, $q_1$ and $q_2$ such that

\Centerline{$p_1\VVdQk\,\dot u_1=\dot A_1$ and $\dot u_2=\dot A_2$.}

\noindent In this case, for both $i$,
$((\dot u_i,\theta_2(\dot u_i,q_i)),q_i)\in\dot\theta$,
so we have

\Centerline{$p_1\VVdQk\,\dot\theta(\dot A_i)=\dot\theta(\dot u_i)
=\theta_2(\dot u_i,q_i)$.}

\noindent Let $e\Bsubseteq p_1$ be a member of $C_{\kappa}$, that
is, a member of $\frak G_{\kappa}$ which is the equivalence class of a
basic cylinder set.   We have

\Centerline{$e\VVdQk\,\dot u_1=\dot A_1\subseteq\dot A_2=\dot u_2$,}

\noindent so
$e\Bcap\Bvalue{\check n\in\dot u_1}\Bsubseteq\Bvalue{\check n\in\dot u_2}$
for every $n\in\Bbb N$.   Express $e$ as $e_1\Bcap e_2\Bcap e_3$ where
$e_1\in C_{J(\dot u_1)}$, $e_2\in C_{J(\dot u_2)}$ and
$e_3\in C_{\kappa\setminus K}$, where $K=J(\dot u_1)\cup J(\dot u_2)$.
For each $n\in\Bbb N$,

\Centerline{$e_1\Bcap e_2\Bcap\Bvalue{\check n\in\dot u_1}
\Bsetminus\Bvalue{\check n\in\dot u_2}$}

\noindent belongs to $\frak C_K$ and is disjoint from
$e_3\in\frak C_{\kappa\setminus K}\setminus\{0\}$, so must be zero;
we therefore have

\Centerline{$e_1\Bcap\Bvalue{\check n\in\dot u_1}
\Bsubseteq\Bvalue{\check n\in\dot u_2}\Bcup(1\Bsetminus e_2)$}

\noindent for every $n$, that is,

\Centerline{$\sequencen{\Bvalue{\check n\in\dot u_1}\Bcap e_1}
\Bsubseteq\sequencen{\Bvalue{\check n\in\dot u_2}\Bcup(1\Bsetminus e_2)}$}

\noindent in $\frak G_{\kappa}^{\Bbb N}$.   Because $\theta$ is a
Freese-Nation function, there is a sequence

\Centerline{$\sequencen{a_n}
\in\theta(\sequencen{\Bvalue{\check n\in\dot u_1}\Bcap e_1})
\cap\theta(\sequencen{\Bvalue{\check n\in\dot u_2}\Bcup(1\Bsetminus e_2)})$}

\noindent such that

\Centerline{$\Bvalue{\check n\in\dot u_1}\Bcap e_1
\Bsubseteq a_n
\Bsubseteq\Bvalue{\check n\in\dot u_2}\Bcup(1\Bsetminus e_2)$}

\noindent for every $n$.   Now $v=\sequencen{a_n}$ belongs to
$\theta_1(\dot u_1,p_1)\cap\theta_1(\dot u_2,p_2)$, so
$(\vec v,p_i)\in\theta_2(\dot u_i,p_i)$ and

\Centerline{$p_i\VVdQk\,\vec v\in\theta_2(\dot u_i,p_i)
=\dot\theta(\dot u_i)$}

\noindent for both $i$.   Returning to $e$, we have

\Centerline{$e\Bcap\Bvalue{\check n\in\dot u_1}\Bsubseteq e\Bcap a_n
\Bsubseteq e\Bcap\Bvalue{\check n\in\dot u_2}$}

\noindent for every $n$, because $e\Bsubseteq e_1\Bcap e_2$.   So

\Centerline{$e\VVdQk\,\dot u_1\Bsubseteq\vec v\Bsubseteq\dot u_2$.}

\noindent Also $e$ is stronger than $p$ and

\Centerline{$e\VVdQk\,\vec v
\in\dot\theta(\dot u_1)\cap\dot\theta(\dot u_2)
=\dot\theta(\dot A_1)\cap\dot\theta(\dot A_2)$.}

\noindent As $p$, $\dot A_1$ and $\dot A_2$ are arbitrary,

\doubleinset{$\VVdQk$ for any $A$, $B\subseteq\Bbb N$ there is a
$C\in\dot\theta(A)\cap\dot\theta(B)$ such that $A\subseteq C\subseteq B$;
that is, $\dot\theta$ is a Freese-Nation function.  \Qed}

\medskip

{\bf (e)} Putting (c) and (d) together, we have

\Centerline{$\VVdQk\,\FN(\Cal P\Bbb N)\le\omega_1$;}

\noindent and since the Freese-Nation number of $\Cal P\Bbb N$ is surely
uncountable (522U), this is enough.
}%end of proof of 554G

\leader{554H}{Corollary} Suppose that $\FN(\Cal P\Bbb N)=\omega_1$ and that
$\kappa$ is an infinite cardinal such that

\inset{($\alpha$) $\cff[\lambda]^{\le\omega}\le\lambda^+$ for every
cardinal $\lambda\le\kappa$,

($\beta$) $\square_{\lambda}$ is true for every uncountable cardinal
$\lambda\le\kappa$ of countable cofinality.}

\noindent Then $\VVdQk\,\FN(\Cal P\Bbb N)=\omega_1$.

\proof{ Any countably generated order-closed subalgebra $\frak C$ of
$\frak G_{\kappa}$ is (in the language of part (a) of the proof of 554G)
included in $\frak C_J$ for some countable $J\subseteq\kappa$, which has a
countable $\pi$-base $C_J$;  so $\frak C_J$ and $\frak C$ are
$\sigma$-linked, and $\FN(\frak C)\le\FN(\Cal P\Bbb N)=\omega_1$,
by 518D.
By 518I, the conditions ($\alpha$) and ($\beta$), together with the fact
that $\tau(\frak G_{\kappa})\le\kappa$, now ensure that
$\FN(\frak G_{\kappa})\le\omega_1$, so 554G gives the result.
}%end of proof of 554H

\leader{554I}{Theorem}\cmmnt{ ({\smc Carlson Frankiewicz \& Zbierski 94})}
Suppose that the continuum hypothesis is true.   Then

\Centerline{$\VVdash_{\Bbb Q_{\omega_2}}\,\frak c=\omega_2$ and
Lebesgue measure has a Borel lifting.}

\proof{ Of course the cardinal power $\omega_2^{\omega}$ (in the ordinary
universe) is equal to
$\max(\frak c,\penalty-100\cff[\omega_2]^{\le\omega})=\omega_2$.
From 554H and 554B we see that

\Centerline{$\VVdash_{\Bbb Q_{\omega_2}}\,
\FN(\Cal P\Bbb N)=\omega_1$ and $\frak c=\omega_2$.}

\noindent So 535E(b-ii) tells us that

\Centerline{$\VVdash_{\Bbb Q_{\omega_2}}$
Lebesgue measure has a Borel lifting.}
}%end of proof of 554I

\exercises{\leader{554X}{Basic exercises (a)}
%\spheader 554Xa
Show that $\#(\frak G_{\kappa})=\kappa^{\omega}$ for every
infinite cardinal $\kappa$.
%554A

\spheader 554Xb\dvAnew{2014} Show that if $I$ is any set, every
regular uncountable cardinal is a precaliber of $\frak G_I$.
%554A 4A1D

\spheader 554Xc\dvAnew{2014} Let $I$ be any set.   (i)
Show that $(\Cal C_I,\supseteq)$ is isomorphic to
$(\Fn_{<\omega}(I;\{0,1\}),\subseteq)$ (definition:  552A).   (ii)
Show that
$\frak G_I$ can be identified with the regular open algebra
$\RO^{\uparrow}(\Fn_{<\omega}(I;\{0,1\}))$.
%554A out of order \query

\spheader 554Xd\dvAnew{2011}
Let $\kappa$ be an infinite cardinal such that $\Bbb R$ has a Lusin set
with cardinal $\kappa$.   Show that there is a first-countable compact Hausdorff
space $X$ such that $\kappa\in\MahR(X)$.   \Hint{531N.}
%554D 531N out of order query

\spheader 554Xe Devise a definition of `strongly Lusin' set to match 537Ab,
and state and prove a result corresponding to 552E.
\Hint{527Xf.}
%554E

\spheader 554Xf
Describe Cicho\'n's diagram in the forcing universe
$V^{\Bbb Q_{\omega_2}}$ (i) if we start with $\frak c=\omega_1$
(ii) if we start with $\frak m=\frak c=\omega_2$.   %\Hint{522Sa, 554Da.}
%554E

\leader{554Y}{Further exercises (a)}
%\spheader 554Ya
For how many of the results of 552F-552J %552F 552G 552H 552I 552J
can you find equivalents with respect to Cohen real forcing?
\Hint{{\smc Bartoszy\'nski \& Judah 95}.}

\spheader 554Yb(i) Show that there is a family
$\ofamily{\xi}{\omega_1}{e_{\xi}}$ such that ($\alpha$) for each $\xi$,
$e_{\xi}\subseteq\xi\times\Bbb N$ is an injective function from $\xi$ to
$\Bbb N$ ($\beta$) if $\eta\le\xi<\omega_1$ then
$e_{\eta}\setminus e_{\xi}$ is finite.
\Hint{choose the $e_{\xi}$ inductively, taking care that
$\Bbb N\setminus e_{\xi}[\xi]$ is infinite for every $\xi$.}   (ii) Set
$T=\{e_{\xi}\restr\eta:\eta$, $\xi<\omega_1$ are successor ordinals$\}$.
Show that $T\cup\{\emptyset\}$,
ordered by $\subseteq$, is a special Aronszajn
tree.   \Hint{for any $n\in\Bbb N$, $\{t:t(\max(\dom t))=n\}$ is an
antichain.}
%553M

\spheader 554Yc ({\smc Todor\v{c}evi\'c 87}) %p 292
Let $\kappa$ be an infinite cardinal.
Take $\ofamily{\xi}{\omega_1}{e_{\xi}}$ as in 554Yb.
Let $\dot\preccurlyeq$ be the $\Bbb Q_{\kappa}$-name

\Centerline{$\{((\check\eta,\check\xi),p):
  \eta\le\xi<\omega_1,\,p\in\Bbb Q_{\kappa},\,
  p\Bsubseteq\{x:xe_{\eta}\subseteq xe_{\xi}\}^{\ssbullet}\}$.}

\noindent Show that
$\VVdQk\,(\omega_1,\dot\preccurlyeq)$ is a Souslin tree.
%554Yb mt55bits
}%end of exercises


\endnotes{
\Notesheader{554}
The original theories of Cohen and random reals were developed in
parallel;  see {\smc Kunen 84} for an account of the special properties of
null and meager ideals which made this possible.   Thus the Sierpi\'nski
sets of random real models become Lusin sets in Cohen real models, and
the horizontal gap which appears in Cicho\'n's diagram if we add random
reals becomes a vertical gap if we add Cohen reals
(552F-552I, %552F 552G 552H 552I
554F).   I give a
very much briefer account of Cohen reals because I am restricting attention
to results which have consequences in measure theory, as in 554Dc and 554I,
and (except in 554Yc/553M)
I make no attempt to look for reflections of the patterns in \S553,
which are mostly there for the illumination they throw on the structure of
measure algebras.
But I do not seek out the shortest route in every case.   In particular, I
spell out some of the theory of Freese-Nation numbers (554G-554H) for its
own sake as well as to provide a proof of Carlson's theorem 554I.   Let me
remind you that $\omega_2$ has a very special place in the arguments here;
see 518Rb and 535Zb.

I have written this section in terms of forcing with category algebras,
partly in order to emphasize the connexion with random reals, and partly to
be able to quote from \S551.   But of course it can equally be regarded as
a fragment of the theory of forcing with partially ordered sets
$\Fn_{<\omega}(I;\{0,1\})$ (554Xc), and there are many places
(e.g.\ 554Yc) where this simplifies the details.

}%end of notes
\discrpage



\frfilename{mt555.tex}
\versiondate{23.10.14}
\copyrightdate{2005}

\def\chaptername{Possible worlds}
\def\sectionname{Solovay's construction of \rvm\ cardinals}

\Loadfourteens

\def\BbbPk{\Bbb P_{\kappa}}
\def\BbbQk{\Bbb Q_{\kappa}}
\long\def\doubleinset#1{\inset{\inset{\parindent=-20pt #1}}}
\def\Etwovmc{$\exists$2vmc}
\def\Eqmc{$\exists$qmc}
\def\Eamc{$\exists$amc}
\def\Fn{\mathop{\text{Fn}}\nolimits}
\def\pssqa{power set $\sigma$-quotient algebra}
\def\VVdash{\mskip5mu\vrule height 7.5pt depth 2.5pt width 0.5pt
  \mskip2.5mu\vrule height 7.5pt depth 2.5pt width 0.5pt
  \vrule height 2.75pt depth -2.25pt width 4pt\mskip2mu}
\def\VVdPk{\VVdash_{\Bbb P_{\kappa}}}
\def\VVdQk{\VVdash_{\Bbb Q_{\kappa}}}
\def\VVdP{\VVdash_{\Bbb P}}

\newsection{555}

While all the mathematical ideas of Chapter 54 were
expressed as arguments in ZFC, many would be of little interest
if it appeared that there could be no \am\ cardinals.   In this section I
present R.M.Solovay's theorem that if there is a \2vm\ cardinal in the
original universe, then there is a forcing notion $\Bbb P$ such that

\Centerline{$\VVdP$ there is an \am\ cardinal}

\noindent(555D).   Varying $\Bbb P$ we find that we can force models with
other kinds of \qm\ cardinal (555G, 555K);  starting from a stronger
hypothesis we can reach the normal measure axiom (555N).

\leader{555A}{Notation}\cmmnt{ As in \S\S552-553,} I will write
$(\frak B_{\kappa},\bar\nu_{\kappa})$ for the measure algebra of the usual
measure on $\{0,1\}^{\kappa}$, and $\BbbPk$ for the forcing notion
$\frak B_{\kappa}^+=\frak B_{\kappa}\setminus\{0\}$, active downwards.
\cmmnt{In this context, as in 525A,}
$\ofamily{\eta}{\kappa}{e_{\eta}}$ will be the standard
generating family in $\frak B_{\kappa}$.

\cmmnt{As in \S554, }I will write $\frak G_{\kappa}$ for the category
algebra of $\{0,1\}^{\kappa}$, and $\BbbQk$ for the forcing notion
$\frak G_{\kappa}^+$, active downwards.
\cmmnt{Recall that }$\frak G_{\kappa}$ is isomorphic to the regular
open algebra $\RO(\{0,1\}^{\kappa})$\cmmnt{ (514If)}.

\leader{555B}{Theorem} Suppose that $X$ is a set,
and $\Cal I$ a proper $\sigma$-ideal of subsets of
$X$ containing singletons.
Let $\Bbb P=(P,\le,\Bbbone,\updownarrows)$ be a ccc forcing notion, and
$\dot{\Cal I}$ a $\Bbb P$-name such that

\Centerline{$\VVdP\,\dot{\Cal I}
=\{J:$ there is an $I\in\check{\Cal I}$ such that $J\subseteq I\}$.}

\noindent Then

(a)(i) If $\dot J$ is a $\Bbb P$-name and $p\in P$ is such that
$p\VVdP\,\dot J\in\dot{\Cal I}$, there is an $I\in\Cal I$ such that
$p\VVdP\,\dot J\subseteq\check I$.

\quad(ii)

\doubleinset{$\VVdP\,\dot{\Cal I}$ is the ideal of subsets of $\check X$
generated by $\check{\Cal I}$;  it is a proper
$\sigma$-ideal containing singletons.}

(b) $\VVdP\,\add\dot{\Cal I}=(\add\Cal I)\varBspcheck$.

(c) If $\Cal I$ is $\omega_1$-saturated in $\Cal PX$, then

\Centerline{$\VVdP\,\dot{\Cal I}$ is $\omega_1$-saturated in
$\Cal P\check X$, so $\Cal P\check X/\dot{\Cal I}$ is ccc and
Dedekind complete.}

(d) If $X=\lambda$ is a regular uncountable
cardinal and $\Cal I$ is a normal ideal on $\lambda$, then

\Centerline{$\VVdP\,\dot{\Cal I}$ is a normal ideal on $\check\lambda$.}

\proof{{\bf (a)(i)} We have

\Centerline{$p\VVdP$ there is an $I\in\check{\Cal I}$ such that
$\dot J\subseteq I$.}

\noindent Set

\Centerline{$A=\{q:$ there is an $I\in\Cal I$ such that
$q\VVdP\,\dot J\subseteq\check I\}$.}

\noindent If $p'$ is stronger than $p$, there is a $q\in A$ stronger than
$p'$.   Let $A'\subseteq A$ be a maximal
antichain.   Then $A'$ is countable and for each $q\in A'$ there is an
$I_q\in\Cal I$ such that $q\VVdP\,\dot J\subseteq\check I_q$.   Set
$I=\bigcup_{q\in A}I_q$;  because $\Cal I$ is a $\sigma$-ideal,
$I\in\Cal I$.   Now $q\VVdP\,\dot J\subseteq\check I$ for every
$q\in A'$.   If $p'$ is stronger than $p$ there is a $q\in A'$ which is
compatible with $p'$,
so $p\VVdP\,\dot J\subseteq\check I$, as required.

\medskip

\quad{\bf (ii)} Because

\Centerline{$\VVdP\,\check{\Cal I}$ is a family of subsets of $\check X$
closed under finite unions,}

\noindent we have

\Centerline{$\VVdP\,\dot{\Cal I}$ is an ideal of subsets of $\check X$.}

\noindent Because

\Centerline{$\VVdP\,\check I\in\dot{\Cal I}$}

\noindent whenever $I\in\Cal I$,

\Centerline{$\VVdP\,\check{\Cal I}\subseteq\dot{\Cal I}$ and
$\dot{\Cal I}$ is the ideal generated by $\check{\Cal I}$.}

Since $X\notin\Cal I$, (i) tells us that

\Centerline{$\VVdP\,\check X\notin\dot{\Cal I}$.}

\noindent Since $\{x\}\in\Cal I$ for every $x\in X$,

\Centerline{$\VVdP\,\{x\}\in\dot{\Cal I}$ for every $x\in\check X$.}

Thus

\Centerline{$\VVdP\,\dot{\Cal I}$ is a proper ideal of $\Cal P\check X$
containing singletons.}

\noindent I defer the final step to (b-i) below.

\medskip

{\bf (b)} Set $\theta=\add\Cal I$.

\medskip

\quad{\bf (i)} Suppose that $p\in P$ and that $\dot{\Cal A}$ is a
$\Bbb P$-name such that

\Centerline{$p\VVdP\,\dot{\Cal A}\subseteq\dot{\Cal I}$ and
$\#(\dot{\Cal A})<\check\theta$.}

\noindent Then there are a $q$ stronger than $p$, a $\delta<\theta$ and a
family $\ofamily{\xi}{\delta}{\dot A_{\xi}}$ of $\Bbb P$-names such that

\Centerline{$q\VVdP\,\dot{\Cal A}=\{\dot A_{\xi}:\xi<\check\delta\}$.}

\noindent For each $\xi<\delta$, $q\VVdP\,\dot A_{\xi}\in\dot{\Cal I}$,
so we have an $I_{\xi}\in\Cal I$ such that
$q\VVdP\,\dot A_{\xi}\subseteq\check I_{\xi}$.   Set
$I=\bigcup_{\xi<\delta}I_{\xi}\in\Cal I$.   Then

\Centerline{$q\VVdP\,\dot A_{\xi}\subseteq\check I$ for every
$\xi<\check\delta$, so $\bigcup\dot{\Cal A}\subseteq\check I$ and
$\bigcup\dot{\Cal A}\in\dot{\Cal I}$.}

\noindent As $p$ and $\dot{\Cal A}$ are arbitrary,

\Centerline{$\VVdP\,\add\dot{\Cal I}\ge\check\theta$.}

In particular, since we certainly have
$\theta\ge\omega_1$,

\Centerline{$\VVdP\,\dot{\Cal I}$ is a $\sigma$-ideal.}

\medskip

\quad{\bf (ii)} In the other direction, there is a family
$\ofamily{\xi}{\theta}{I_{\xi}}$ in $\Cal I$ with no upper bound in
$\Cal I$.    Now $\VVdP\,\check I_{\xi}\in\dot\Cal I$ for every
$\xi<\theta$.   \Quer\ If $p\in P$ is such that

\Centerline{$p\VVdP\,\bigcup_{\xi<\theta}\check I_{\xi}\in\dot\Cal I$,}

\noindent then there is an $I\in\Cal I$ such that

\Centerline{$p\VVdP\,\bigcup_{\xi<\check\theta}\check I_{\xi}
  \subseteq\check I$}

\noindent and $\bigcup_{\xi<\theta}I_{\xi}\subseteq I\in\Cal I$.\
\BanG\  So

\Centerline{$\VVdP\,
\bigcup_{\xi<\check\theta}\check I_{\xi}\notin\dot\Cal I$
and $\add\dot\Cal I\le\check\theta$.}

\medskip

{\bf (c)} Let $p\in P$ and a family
$\ofamily{\eta}{\omega_1}{\dot A_{\eta}}$ of $\Bbb P$-names be such
that

\Centerline{$p\VVdP\,\ofamily{\eta}{\omega_1}{\dot A_{\eta}}$ is a
disjoint family of subsets of $\check X$.}

\noindent For each $x\in X$,
$\ofamily{\eta}{\omega_1}{\widehat{p}
\Bcap\Bvalue{\check x\in\dot A_{\eta}}}$
is a disjoint family in $\RO(\Bbb P)$, where

\Centerline{$\widehat{p}
=\interior\overline{\{q:q\text{ is stronger than }p\}}$}

\noindent is the regular open set corresponding to $p$.   So there is an
$\alpha_x<\omega_1$ such that
$\widehat{p}\Bcap\Bvalue{\check x\in\dot A_{\eta}}=0$ for every
$\eta\ge\alpha_x$, that is, $p\VVdP\,\check x\notin\dot A_{\eta}$ for every
$\eta\ge\alpha_x$.    Because $\Cal I$ is $\omega_1$-saturated, therefore
$\omega_2$-additive (542B-542C), there is an $\alpha<\omega_1$ such that
$I=\{x:x\in X$, $\alpha_x\ge\alpha\}$ belongs to $\Cal I$.
Now $p\VVdP\,\check x\notin\dot A_{\alpha}$
for every $x\in X\setminus I$, that is,

\Centerline{$p\VVdP\,\dot A_{\alpha}\subseteq\check I$ and
$\dot A_{\alpha}\in\dot{\Cal I}$.}

\noindent As $p$ and $\ofamily{\eta}{\omega_1}{\dot A_{\eta}}$ are
arbitrary, $\VVdP\,\dot{\Cal I}$ is $\omega_1$-saturated.

Thus $\VVdP\,\Cal P\check X/\dot{\Cal I}$ is ccc.   But since we know from
(b) that $\VVdP\,\dot{\Cal I}$ is a $\sigma$-ideal, and of course
$\VVdP\,\Cal P\check X$ is Dedekind complete, we have

\Centerline{$\VVdP\,\Cal P\check X/\dot{\Cal I}$ is Dedekind
$\sigma$-complete, therefore Dedekind complete.}

\medskip

{\bf (d)} Suppose that $p\in P$ and that
$\ofamily{\xi}{\lambda}{\dot A_{\xi}}$ is a $\Bbb P$-name such that

\Centerline{$p\VVdP\,\dot A_{\xi}\in\dot{\Cal I}$ for every
$\xi<\check\lambda$.}

\noindent For each $\xi<\lambda$ we have an $I_{\xi}\in\Cal I$ such that
$p\VVdP\,\dot A_{\xi}\subseteq\check I_{\xi}$;  let $I$ be the diagonal
union

\Centerline{$\{\xi:\xi<\lambda$, $\xi\in\bigcup_{\eta<\xi}I_{\eta}\}$.}

\noindent Because $\Cal I$ is a normal ideal on $\lambda$, $I\in\Cal I$.
Now suppose that $q$ is stronger than $p$ and that $\xi<\lambda$ is such
that

\Centerline{$q\VVdP\,\check\xi\in\bigcup_{\eta<\check\xi}\dot A_{\eta}$.}

\noindent Then

\Centerline{$q\VVdP\,\check\xi\in\bigcup_{\eta<\check\xi}\check I_{\eta}$,}

\noindent so $\xi\in\bigcup_{\eta<\xi}I_{\eta}$ and $\xi\in I$ and
$q\VVdP\,\check\xi\in\check I$.   As $q$ and $\xi$ are arbitrary,

\Centerline{$p\VVdP$ the diagonal union of
$\ofamily{\xi}{\check\lambda}{\dot A_{\xi}}$ is included in $\check I$ and
belongs to $\dot{\Cal I}$.}

\noindent As $p$ and $\ofamily{\xi}{\lambda }{\dot A_{\xi}}$ are arbitrary,

\Centerline{$\VVdP\,\dot{\Cal I}$ is normal.}
}%end of proof of 555B

\leader{555C}{Theorem} Let $(X,\Cal PX,\mu)$ be a probability space such
that $\mu\{x\}=0$ for every $x\in X$, and $\Cal N$ the null ideal of
$\mu$.   Let $\kappa>0$ be a cardinal.   Then we can find a $\BbbPk$-name
$\dot\mu$ such that

(i) $\VVdPk\,\dot\mu$ is a probability
measure with domain $\Cal P\check X$, zero on singletons;

(ii) if $\dot{\Cal N}$ is a $\BbbPk$-name for the ideal of subsets
of $\check X$ generated by $\check{\Cal N}$\cmmnt{, as in 555B}, then

\Centerline{$\VVdPk\,\dot{\Cal N}$ is the null ideal of $\dot\mu$.}

\proof{{\bf (a)} For each function $\sigma:X\to\frak B_{\kappa}$,
write $\vec\sigma$ for the $\BbbPk$-name

\Centerline{$\{(\check x,\sigma(x)):x\in X$, $\sigma(x)\ne 0\}$.}

\noindent Then

\Centerline{$\VVdPk\,\vec\sigma\subseteq\check X$}

\noindent and $\Bvalue{\check x\in\vec\sigma}=\sigma(x)$ for every
$x\in X$.   Moreover, if $\dot A$ is any $\BbbPk$-name such that
$\VVdPk\,\dot A\subseteq\check X$, then $\VVdPk\,\dot A=\vec\sigma$, where
$\sigma(x)=\Bvalue{\check x\in\dot A}$ for $x\in X$.

\medskip

{\bf (b)} For $\sigma\in\frak B_{\kappa}^X$, the functional

\Centerline{$a\mapsto
\int\bar\nu_{\kappa}(\sigma(x)\Bcap a)\,\mu(dx)$}

\noindent is additive and dominated by $\bar\nu_{\kappa}$, so
there is a unique
$u_{\sigma}\in L^{\infty}(\frak B_{\kappa})$ such that

\Centerline{$\int_au_{\sigma}\,d\bar\nu_{\kappa}
=\int\bar\nu_{\kappa}(\sigma(x)\Bcap a)\mu(dx)$}

\noindent for every $a\in\frak B_{\kappa}$
(365E, 365D(d-ii)), and $0\le u_{\sigma}\le\chi 1$.
Observe that if $\sigma$, $\tau\in\frak B_{\kappa}^X$,
$a\in\frak B_{\kappa}^+$ and $a\VVdPk\vec\sigma=\vec\tau$, then
$a\Bcap\sigma(x)=a\Bcap\tau(x)$ for every $x\in X$, so that
$u_{\sigma}\times\chi a=u_{\tau}\times\chi a$.

\medskip

{\bf (c)} For $u\in L^{\infty}(\frak B_{\kappa})$
let $\vec u$ be the corresponding
$\BbbPk$-name for a real number (5A3L-5A3M, identifying $\frak B_{\kappa}$ with
$\RO(\BbbPk)$ as usual).   Consider the $\BbbPk$-name

\Centerline{$\dot\mu=\{((\vec\sigma,\vec u_{\sigma}),\Bbbone):
\sigma\in\frak B_{\kappa}^X\}$.}

\noindent Then

\Centerline{$\VVdPk\,\dot\mu$ is a function from
$\Cal P\check X$ to $[0,1]$.}

\noindent\Prf\ Because $0\le u_{\sigma}\le\chi 1$,
$\VVdPk\,\vec u_{\sigma}\in[0,1]$ for each $\sigma$, and
$\VVdPk\,\dot\mu\subseteq\Cal P\check X\times[0,1]$.
If $\sigma$, $\tau\in\frak B_{\kappa}^X$ and $a\in\frak B_{\kappa}^+$
are such that $a\VVdPk\,\vec\sigma=\vec\tau$, then
$u_{\sigma}\times\chi a=u_{\tau}\times\chi a$, by (b);
but this means that
$a\VVdPk\,\vec u_{\sigma}=\vec u_{\tau}$ (5A3M).   So
$\VVdPk\,\dot\mu$ is a function (5A3Ea).   If $a\in P$ and
$\dot A$ are such that $a\VVdPk\,\dot A\subseteq\check X$, then there is a
$\sigma\in\frak B_{\kappa}^X$ such that
$a\VVdPk\,\dot A=\vec\sigma$, so that
$\VVdPk\,\dom\dot\mu=\Cal P\check X$ (5A3Eb).\ \Qed

\medskip

{\bf (d)} Now we have to check the properties of $\dot\mu$.

\medskip

\quad{\bf (i)} $\VVdPk\,\dot\mu$ is countably additive.
\Prf\ Suppose that $\sequencen{\dot A_n}$ is a sequence of
$\BbbPk$-names and $a\in\frak B_{\kappa}^+$ is such that

\Centerline{$a\VVdPk\,\sequencen{\dot A_n}$ is a disjoint sequence of
subsets of $\check X$.}

\noindent For each $n\in\Bbb N$, let
$\sigma_n\in\frak B_{\kappa}^X$ be such that
$a\VVdPk\,\dot A_n=\vec\sigma_n$;  then $\sequencen{a\Bcap\sigma_n(x)}$
is a disjoint sequence in $\frak B_{\kappa}$ for each $x\in X$.   Set
$\sigma(x)=\sup_{n\in\Bbb N}\sigma_n(x)$ for each $x$.   Then
$\Bvalue{\check x\in\vec\sigma}
=\sup_{n\in\Bbb N}\Bvalue{\check x\in\vec\sigma_n}$
for each $x$, so $\VVdPk\,\vec\sigma=\bigcup_{n\in\Bbb N}\vec\sigma_n$.
Now for any $b\Bsubseteq a$,

$$\eqalign{\int_bu_{\sigma}d\bar\nu_{\kappa}
&=\int\bar\nu_{\kappa}(b\Bcap\sigma(x))\mu(dx)
=\int\sum_{n=0}^{\infty}\bar\nu_{\kappa}(b\Bcap\sigma_n(x))\mu(dx)\cr
&=\sum_{n=0}^{\infty}\int\bar\nu_{\kappa}(b\Bcap\sigma_n(x))\mu(dx)
=\sum_{n=0}^{\infty}\int_bu_{\sigma_n}d\bar\nu_{\kappa}.\cr}$$

\noindent So

\Centerline{$\chi a\times u_{\sigma}
=\sup_{n\in\Bbb N}\chi a\times\sum_{i=0}^nu_{\sigma_i}$}

\noindent in $L^0(\frak B_{\kappa})$, and

\Centerline{$a\VVdPk\,\dot\mu(\bigcup_{n\in\Bbb N}\dot A_n)
=\dot\mu(\bigcup_{n\in\Bbb N}\vec\sigma_n)
=\dot\mu(\vec\sigma)
=\vec u_{\sigma}
=\sum_{n=0}^{\infty}\vec u_{\sigma_n}
=\sum_{n=0}^{\infty}\dot\mu(\dot A_n)$.}

\noindent As $a$ and $\sequencen{\dot A_n}$ are arbitrary,
$\VVdPk\,\dot\mu$ is countably additive.\ \Qed

\medskip

\quad{\bf (ii)} Suppose that
$\dot y$ is a $\BbbPk$-name and $a\in\frak B_{\kappa}^+$ is such
that $a\VVdPk\,\dot y\in\check X$.   Take any $b$ stronger than
$a$ and $y\in X$ such that $b\VVdPk\,\dot y=\check y$.
Set $\sigma(y)=1$ and
$\sigma(x)=0$ for $x\in X\setminus\{y\}$.   Then

\Centerline{$\int u_{\sigma}d\bar\nu_{\kappa}=\mu\{y\}=0$,}

\noindent so

\Centerline{$b\VVdPk\,\dot\mu(\{\dot y\})
=\dot\mu(\{\check y\})=\dot\mu(\vec\sigma)
=\vec u_{\sigma}=0$.}

\noindent Thus

\Centerline{$\VVdPk\,\dot\mu$ is zero on singletons.}

\medskip

\quad{\bf (iii)} If $\sigma(x)=1$ for every $x\in X$, then
$u_{\sigma}=\chi 1$ so

\Centerline{$\VVdPk\dot\mu(\check X)=\vec{\chi 1}=1$.}

\medskip

{\bf (e)} $\VVdPk\,\dot{\Cal N}=\{A:\dot\mu A=0\}$.   \Prf\ Let
$a\in\frak B_{\kappa}^+$ and a $\BbbPk$-name $\dot A$ be such that
$a\VVdPk\,\dot A\subseteq\check X$.   Let
$\sigma\in\frak B_{\kappa}^X$ be such that
$a\VVdPk\,\dot A=\vec\sigma$;  then
$a\VVdPk\,\dot\mu(\dot A)=\vec u_{\sigma}$.

\medskip

\quad{\bf (i)} If $a\VVdPk\,\dot A\in\dot\Cal N$, then there is an
$I\in\Cal N$ such that
$a\VVdPk\,\vec\sigma\subseteq\check I$ (555B(a-i)),
that is, $a\Bcap\sigma(x)=0$ for
every $x\in X\setminus I$.   But this means that

\Centerline{$\int_au_{\sigma}d\bar\nu_{\kappa}
=\int\bar\nu_{\kappa}(\sigma(x)\Bcap a)\mu(dx)
\le\mu I=0$,}

\noindent and $\chi a\times u_{\sigma}=0$.   So

\Centerline{$a\VVdPk\,\dot\mu(\dot A)=\vec u_{\sigma}=0$.}

\medskip

\quad{\bf (ii)} If $a\VVdPk\,\dot\mu(\dot A)=0$, then
$\chi a\times u_{\sigma}=0$, so

\Centerline{$\int\bar\nu_{\kappa}(\sigma(x)\Bcap a)\mu(dx)
=\int_au_{\sigma}d\bar\nu_{\kappa}=0$.}

\noindent Set $I=\{x:a\Bcap\sigma(x)\ne 0\}$;  then $\mu I=0$ and

\Centerline{$a\VVdPk\,\dot A\subseteq\check I$ so $\dot A\in\dot{\Cal N}$.}

\noindent Putting these together we have what we need.\ \Qed
}%end of proof of 555C

\leader{555D}{Corollary}\cmmnt{ ({\smc Solovay 71})}
Suppose that $\lambda$ is a \2vm\ cardinal and
that $\kappa\ge\lambda$ is a cardinal.   Then

\Centerline{$\VVdPk\,\check\lambda$ is \am.}

\proof{ Putting 555C and 555B together,

\doubleinset{$\VVdPk$ there is a probability
measure $\mu$ with domain $\Cal P\check\lambda$, zero on singletons,
such that the null ideal of $\mu$ is $\check\lambda$-additive.}

\noindent By 552B and 543Bc,

\Centerline{$\VVdPk\,\check\lambda\le\frak c$ is a \rvm\ cardinal, so is
\am.}
}%end of proof of 555D

\leader{555E}{Theorem} Let $\lambda$ be a \2vm\ cardinal, and $\Cal I$
a $\lambda$-additive maximal proper ideal of $\Cal P\lambda$ containing
singletons;  let $\mu$ be the $\{0,1\}$-valued probability measure on
$\lambda$ with null ideal $\Cal I$.
Let $\kappa\ge\lambda$ be a cardinal, and define $\dot\mu$ from $\mu$ as
in\cmmnt{ Theorem} 555C.   Set
$\theta=\Tr_{\Cal I}(\lambda;\kappa)$\cmmnt{ (definition:  5A1Ma)}.
Then

\Centerline{$\VVdPk\,\dot\mu$ is \Mth\ with Maharam type
$\check\theta$.}

\proof{{\bf (a)}
Let $\ofamily{\alpha}{\theta}{g_{\alpha}}$ be a family in
$\kappa^{\lambda}$ such that
$\{\xi:g_{\alpha}(\xi)=g_{\beta}(\xi)\}\in\Cal I$ whenever
$\alpha<\beta<\theta$ (541F).   Because $\lambda\le\kappa$, we can suppose
that all the $g_{\alpha}$ are injective.   (Just arrange that
$g_{\alpha}(\xi)$ always belongs to some $J_{\xi}\in[\kappa]^{\kappa}$
where $\ofamily{\xi}{\lambda}{J_{\xi}}$ is disjoint.)
For $\alpha<\theta$ and $\xi<\lambda$ set
$\sigma_{\alpha}(\xi)=e_{g_{\alpha}(\xi)}$.   For
$\sigma\in\frak B_{\kappa}^{\lambda}$ let $\vec\sigma$ be the corresponding
$\BbbPk$-name for a subset of $\lambda$ as in the prood of 555C.
Then for any non-empty finite $K\subseteq\theta$ we have

\Centerline{$\VVdPk\,
\dot\mu(\bigcap_{\alpha\in\check K}\vec\sigma_{\alpha})
=(2^{-\#(K)})\varBspcheck$.}

\noindent\Prf\ Set $\sigma(\xi)=\inf_{\alpha\in K}\sigma_{\alpha}(\xi)$ for
each $\xi$, so that

\Centerline{$\VVdPk\,
\vec\sigma=\bigcap_{\alpha\in\check K}\vec\sigma_{\alpha}$.}

\noindent Set

\Centerline{$I
=\bigcup_{\alpha,\beta\in K\text{ are different}}
  \{\xi:\xi<\lambda,\,g_{\alpha}(\xi)=g_{\beta}(\xi)\}$;}

\noindent then $I\in\Cal I$.
If $a\in\frak B_{\kappa}$, let $J\in[\kappa]^{\le\omega}$
be such that $a$ belongs to the closed subalgebra of $\frak B_{\kappa}$
generated by $\{e_{\eta}:\eta\in J\}$.   Then

\Centerline{$\{\xi:\xi<\lambda,\,\bar\nu_{\kappa}(\sigma(\xi)\Bcap a)
   \ne 2^{-\#(K)}\bar\nu_{\kappa}(a)\}
\subseteq I\cup\bigcup_{\alpha\in K}g_{\alpha}^{-1}[J]
\in\Cal I$,}

\noindent so

\Centerline{$\int\bar\nu_{\kappa}(\sigma(\xi)\Bcap a)\mu(d\xi)
=2^{-\#(K)}\bar\nu_{\kappa}(a)$.}

\noindent This means that $u_{\sigma}$, as defined in the proof of 555C, is
just $2^{-\#(K)}\chi 1$ and
$\VVdPk\,\vec u_{\sigma}=(2^{-\#(K)})\varBspcheck$, that is,

\Centerline{$\VVdPk\,
\dot\mu(\bigcap_{\alpha\in\check K}\vec\sigma_{\alpha})
=(2^{-\#(K)})\varBspcheck$.  \Qed}

Thus

\doubleinset{$\VVdPk\,\ofamily{\alpha}{\check\theta}{\vec\sigma_{\alpha}}$
is a
stochastically independent family in $\Cal P\lambda$ of elements of measure
$\bover12$, and every principal ideal of the measure algebra of $\dot\mu$
has Maharam type at least $\check\theta$}

\noindent (331Ja).

\medskip

{\bf (b)} In the other direction, suppose that $a\in\frak B_{\kappa}^+$,
$\delta$ is a cardinal, $t>0$ is a rational number
and $\ofamily{\alpha}{\delta}{\dot A_{\alpha}}$ is a family of
$\BbbPk$-names such that

\Centerline{$a\VVdPk\,\dot A_{\alpha}\subseteq\check\lambda$,
$\dot\mu(\dot A_{\alpha}\symmdiff\dot A_{\beta})\ge 3\check t$
whenever $\alpha<\beta<\check\delta$.}

\noindent For each $\alpha<\delta$ let
$\sigma_{\alpha}\in\frak B_{\kappa}^{\lambda}$ be such that
$a\VVdPk\,\vec\sigma_{\alpha}=\dot A_{\alpha}$;  then

\Centerline{$\int\bar\nu_{\kappa}
(a\Bcap(\sigma_{\alpha}(\xi)\Bsymmdiff\sigma_{\beta}(\xi)))\mu(d\xi)
\ge 3t\bar\nu_{\kappa}a$}

\noindent whenever $\alpha<\beta<\delta$.   Let
$D\subseteq\frak B_{\kappa}$ be a set with cardinal $\kappa$ which is dense for
the measure-algebra topology (521E(a-ii)), and for $\alpha<\delta$,
$\xi<\kappa$ take $d_{\alpha}(\xi)\in D$ such that
$\bar\nu_{\kappa}(d_{\alpha}(\xi)\Bsymmdiff\sigma_{\alpha}(\xi))
\le t\bar\nu_{\kappa}(a)$.   Then

\Centerline{$\int\bar\nu_{\kappa}
  (d_{\alpha}(\xi)\Bsymmdiff d_{\beta}(\xi))\mu(d\xi)
>0$}

\noindent and $\{\xi:d_{\alpha}(\xi)\ne d_{\beta}(\xi)\}\notin\Cal I$
whenever $\alpha<\beta<\delta$;  as $\Cal I$ is a maximal ideal,
$\{\xi:d_{\alpha}(\xi)=d_{\beta}(\xi)\}\in\Cal I$
whenever $\alpha<\beta<\delta$, and $\ofamily{\alpha}{\delta}{d_{\alpha}}$
witnesses that

\Centerline{$\delta\le\Tr_{\Cal I}(\lambda;D)=\Tr_{\Cal I}(\lambda;\kappa)
=\theta$.}

\noindent As
$a$, $t$ and $\ofamily{\alpha}{\delta}{\dot A_{\alpha}}$ are arbitrary,

\Centerline{$\VVdPk$ the Maharam type of $\dot\mu$ is at most
$\check\theta$;}

\noindent with (a), this means that

\Centerline{$\VVdPk\,\dot\mu$ is \Mth\ with Maharam type
$\check\theta$.}
}%end of proof of 555E

\leader{555F}{Proposition} Let $\lambda$ be a \2vm\ cardinal
and $\kappa>0$.   Let $\mu$ be a normal
witnessing probability on $\lambda$ and
$\dot\mu$ the corresponding $\BbbPk$-name for a measure on
$\check\lambda$, as in 555C.   Then

\doubleinset{
$\VVdPk$ the covering number of the null ideal of the product measure
$\dot\mu^{\Bbb N}$ on $\check\lambda^{\Bbb N}$ is $\check\lambda$.}

\proof{{\bf (a)} It may save a moment's thought later on if I remark now
that if $(Y,\Tau,\nu)$ is any measure space and $W\subseteq Y^{\Bbb N}$ is
negligible for the product measure $\nu^{\Bbb N}$, then there is a family
$\langle F_{ij}\rangle_{j\le i\in\Bbb N}$ in $\Tau$ such that

\Centerline{$W
\subseteq\bigcap_{n\in\Bbb N}\bigcup_{i\ge n}
  \{y:y\in Y^{\Bbb N}$, $y(j)\in F_{ij}$ for every $j\le i\}$,
\quad$\sum_{i=0}^{\infty}\prod_{j=0}^i\nu F_{ij}\le 1$.}

\noindent\Prf\ For each $k\in\Bbb N$, let $\sequence{i}{C_{ki}}$ be a
sequence of measurable cylinders such that
$W\subseteq\bigcup_{i\in\Bbb N}C_{ki}$ and
$\sum_{i=0}^{\infty}\lambda^{\Bbb N}C_{ki}\le 2^{-k-1}$;  let
$\sequence{i}{C_i}$ be a re-listing of the double family
$\langle C_{ki}\rangle_{k,i\in\Bbb N}$ with enough copies of the empty set
interleaved to ensure that $C_i$ is determined by coordinates less than or
equal to $i$ for each $i$;  express each $C_i$ as
$\{y:y(j)\in F_{ij}$ for $j\le i\}$.\ \Qed

\medskip

{\bf (b)} It will also help to be able to do some calculations with
sequences of $\BbbPk$-names for subsets of $\lambda$.
For $J\subseteq\kappa$ let $\frak C_J$ be the closed subalgebra
of $\frak B_{\kappa}$ generated by $\{e_{\eta}:\eta\in J\}$, and
$P_J:L^{\infty}(\frak B_{\kappa})\to L^{\infty}(\frak C_J)$ the
corresponding conditional
expectation operator;  see 242J, 254R and 365Q\formerly{3{}65R}
for the basic manipulations of these
operators.   Let $\Cal F$ be the normal ultrafilter $\{F:\mu F=1\}$.

\medskip

\quad{\bf (i)}
Let $\langle\dot A_{ij}\rangle_{j\le i\in\Bbb N}$ be a family of
$\BbbPk$-names such that

\Centerline{$\VVdPk\,\dot A_{ij}\subseteq\check\lambda$}

\noindent whenever $j\le i\in\Bbb N$.
For $j\le i\in\Bbb N$ and $\xi<\lambda$ set
$\sigma_{ij}(\xi)=\Bvalue{\check\xi\in\dot A_{ij}}$.
Then there is a countable
set $I_{\xi}\subseteq\kappa$ such that
$\sigma_{ij}(\xi)\in\frak C_{I_{\xi}}$ whenever
$j\le i\in\Bbb N$.   By 541Rb, %this is where we need "normal ideal"
there are an $F_0\in\Cal F$ and a countable
set $I\subseteq\kappa$ such that $I_{\xi}\cap I_{\xi'}\subseteq I$ for all
distinct $\xi$, $\xi'\in F_0$.

\medskip

\quad{\bf (ii)} Set $u_{ij\xi}=P_I(\chi\sigma_{ij}(\xi))$ for
$\xi<\lambda$.   Because $\#(L^{\infty}(\frak C_I))\le\frak c<\lambda$ and
$\Cal F$ is a $\lambda$-complete ultrafilter, there are
$u_{ij}\in L^{\infty}(\frak C_I)$ such that
$\{\xi:u_{ij\xi}=u_{ij}\}$ belongs to
$\Cal F$ whenever $j\le i\in\Bbb N$.   Set

\Centerline{$F
=F_0\cap\{\xi:u_{ij\xi}=u_{ij}$ whenever $j\le i\in\Bbb N\}$,}

\noindent so that $F\in\Cal F$.

\medskip

\quad{\bf (iii)} We have

\Centerline{$\VVdPk\,\dot\mu\dot A_{ij}=\vec u_{ij}$}

\noindent whenever $j\le i\in\Bbb N$.   \Prf\ If $a\in\frak B_{\kappa}$,
there is a countable $J\subseteq\kappa$ such that $a$ belongs to the
closed subalgebra $\frak C_J$ generated by $\{e_{\eta}:\eta\in J\}$;  we
may suppose that $I\subseteq J$.   Now
$\family{\xi}{F}{I_{\xi}\setminus I}$ is disjoint, so
$\{\xi:\xi\in F$, $I_{\xi}\cap J\not\subseteq I\}$ is countable, and
$F'=\{\xi:\xi\in F$, $I_{\xi}\cap J\subseteq I\}$ belongs to $\Cal F$.
For $\xi\in F'$ we have

\Centerline{$P_J(\chi\sigma_{ij}(\xi))=P_JP_{I_{\xi}}(\chi\sigma_{ij}(\xi))
=P_{J\cap I_{\xi}}(\chi\sigma_{ij}(\xi))\in\frak C_I$}

\noindent so that

\Centerline{$P_J(\chi\sigma_{ij}(\xi))=P_IP_J(\chi\sigma_{ij}(\xi))
=P_I(\chi\sigma_{ij}(\xi))=u_{ij}$;}

\noindent consequently

\Centerline{$\int_au_{ij}d\bar\nu_{\kappa}
=\int_a\chi\sigma_{ij}(\xi)d\bar\nu_{\kappa}
=\bar\nu_{\kappa}(a\Bcap\sigma_{ij}(\xi))$.}

\noindent Because $F'$ is $\mu$-conegligible,

\Centerline{$\int_au_{ij}d\bar\nu_{\kappa}
=\int\bar\nu_{\kappa}(a\Bcap\sigma_{ij}(\xi))\mu(d\xi)$.}

\noindent As this is true for every $a\in\frak B_{\kappa}$, the
construction in 555C gives $\VVdPk\,\dot\mu\dot A_{ij}=\vec u_{ij}$.\ \Qed

\medskip

\quad{\bf (iv)} Note also that if $i\in\Bbb N$ and
$\xi_0,\ldots,\xi_j\in F$ are distinct, then

\Centerline{$\int\prod_{j=0}^iu_{ij}d\bar\nu_{\kappa}
=\bar\nu_{\kappa}(\inf_{j\le i}\sigma_{ij}(\xi_j))$.}

\noindent\Prf\ The algebras $\frak C_{I\cup\{\eta\}}$, for
$\eta\in\kappa\setminus I$, are relatively stochastically independent over
$\frak C_I$ in the sense of 458L;  by 458H/458Le, the algebras
$\frak C_{I\cup I_{\xi}}$, for $\xi\in F$, are relatively stochastically
independent over $\frak C_I$;  but, disentangling the definitions, this is
exactly what we need to know.\ \Qed

\medskip

{\bf (c)} We are now ready for the central idea of the proof.
Suppose that $\dot W$ is a $\BbbPk$-name such that

\Centerline{$\VVdPk\,\dot W\subseteq\check\lambda^{\Bbb N}$ and
$\dot\mu^{\Bbb N}\dot W=0$.}

\noindent Then there is an $F\in\Cal F$ such that

\Centerline{$\VVdPk\,\dot W$ is disjoint from
$(F^{\Bbb N}\setminus\Delta)\varBspcheck$}

\noindent where
$\Delta=\bigcup_{j<k\in\Bbb N}\{x:x\in\lambda^{\Bbb N}$, $x(j)=x(k)\}$.
\Prf\ By (a), we have a family
$\langle\dot A_{ij}\rangle_{j\le i\in\Bbb N}$ of $\Bbb P_{\kappa}$-names
such that

\Centerline{$\VVdPk\,\dot A_{ij}\subseteq\check\lambda$ whenever
$j\le i\in\Bbb N$,}

\Centerline{$\VVdPk\,\dot W\subseteq\bigcap_{n\in\Bbb N}\bigcup_{i\ge n}
  \{x:x\in\check\lambda^{\Bbb N}$,
  $x(j)\in\dot A_{ij}$ for every $j\le i\}$,}

\Centerline{$\VVdPk\,\sum_{i=0}^{\infty}\prod_{j=0}^i\dot\mu\dot A_{ij}
\le 1$.}

\noindent Take $\sigma_{ij}(\xi)$, $I\in[\kappa]^{\le\omega}$,
$u_{ij}\in\frak C_I$ and $F\in\Cal F$ as in (b).   Suppose that
$x\in F^{\Bbb N}$ and $x(j)\ne x(k)$ for distinct $j$, $k\in\Bbb N$.   Then

\Centerline{$\Bvalue{\check x\in\dot W}
\Bsubseteq\inf_{n\in\Bbb N}\sup_{i\ge n}\inf_{j\le i}
  \Bvalue{x(j)\varBspcheck\in\dot A_{ij}}
=\inf_{n\in\Bbb N}\sup_{i\ge n}\inf_{j\le i}\sigma_{ij}(x(j))$.}

\noindent So

\Centerline{$\bar\nu_{\kappa}\Bvalue{\check x\in\dot W}
\le\inf_{n\in\Bbb N}\sum_{i=n}^{\infty}
  \bar\nu_{\kappa}(\inf_{j\le i}\sigma_{ij}(x(j)))
=\inf_{n\in\Bbb N}\sum_{i=n}^{\infty}
  \int\prod_{j\le i}u_{ij}d\bar\nu_{\kappa}$}

\noindent by (b-iv).   On the other hand, setting
$v_i=\prod_{j\le i}u_{ij}$ for $i\in\Bbb N$
and $w=\sup_{m\in\Bbb N}\sum_{i=0}^mv_i$, we have

\Centerline{$\VVdPk\,\vec w=\sum_{i=0}^{\infty}\vec v_i
=\sum_{i=0}^{\infty}\prod_{j=0}^i\dot\mu\dot A_{ij}
\le 1$.}

\noindent So $w\le\chi 1$ and

\Centerline{$\sum_{i=0}^{\infty}\int\prod_{j\le i}u_{ij}d\bar\nu_{\kappa}
=\int w\,d\bar\nu_{\kappa}\le 1$.}

\noindent Putting these together, we see that
$\bar\nu_{\kappa}\Bvalue{\check x\in\dot W}=0$ and
$\VVdPk\,\check x\notin\dot W$.   As $x$ is arbitrary,

\Centerline{$\VVdPk\,\dot W$ is disjoint from
$(F^{\Bbb N}\setminus\Delta)\varBspcheck$. \Qed}

\medskip

{\bf (d)} We are nearly home.   Suppose that $a\in\frak B_{\kappa}^+$
and that $\dot{\Cal W}$ is a $\BbbPk$-name such that

\Centerline{$a\VVdPk\,\dot\Cal W$ is a family of negligible sets in
$\check\lambda^{\Bbb N}$ and $\#(\dot\Cal W)<\check\lambda$.}

\noindent Take any $b$ stronger than $a$, $\theta<\lambda$ and family
$\ofamily{\zeta}{\theta}{\dot W_{\zeta}}$ of $\BbbPk$-names such that

\Centerline{$b\VVdPk\,\dot\Cal W=\{\dot W_{\zeta}:\zeta<\check\theta\}$.}

\noindent For each $\zeta<\theta$ let $\dot W'_{\zeta}$ be a $\BbbPk$-name
such that

\Centerline{$\VVdPk\,\dot W'_{\zeta}\subseteq\check\lambda^{\Bbb N}$ is
negligible,
\quad$b\VVdPk\,\dot W'_{\zeta}=\dot W_{\zeta}$.}

\noindent By (c), we have an $F_{\zeta}\in\Cal F$ such that

\Centerline{$\VVdPk\,\dot W'_{\zeta}
\cap(F_{\zeta}^{\Bbb N}\setminus\Delta)\varBspcheck=\emptyset$.}

\noindent Because $\theta<\lambda$, $\bigcap_{\zeta<\theta}F_{\zeta}$
belongs to $\Cal F$ and is infinite, and there is an
$x\in\bigcap_{\zeta<\theta}F_{\zeta}^{\Bbb N}\setminus\Delta$.   But now

\Centerline{$\VVdPk\,
\check x\notin\bigcup_{\zeta<\check\theta}\dot W'_{\zeta}$}

\noindent and

\Centerline{$b\VVdPk\,\check x\notin\bigcup\dot\Cal W$, so $\dot\Cal W$
does not cover $\check\lambda^{\Bbb N}$.}

\noindent As $b$ is arbitrary,

\Centerline{$a\VVdPk\,\dot\Cal W$ does not cover $\check\lambda^{\Bbb N}$;}

\noindent as $a$ and $\dot\Cal W$ are arbitrary,

\Centerline{
$\VVdPk$ the covering number of the null ideal of the product measure
on $\check\lambda^{\Bbb N}$ is at least $\check\lambda$.}

\noindent The reverse inequality is trivial, since

\Centerline{$\VVdPk\,\dot\mu\{\xi\}=0$}

\noindent for every $\xi<\check\lambda$;  so the proposition is proved.
}%end of proof of 555F

\leader{555G}{Cohen \dvrocolon{forcing}}\cmmnt{ If
we allow ourselves
to start from a measurable cardinal, we can find forcing constructions for
a variety of \pssqa{}s besides the probability algebras provided by Theorem
555C.   In view of \S554, an obvious construction is the following.

\medskip

\noindent}{\bf Theorem} Let $\lambda$ be a \2vm\ cardinal and
$\kappa\ge\lambda$ a cardinal.   Let $\Cal I$ be a
$\lambda$-additive maximal proper ideal of subsets of $\lambda$, and
$\dot{\Cal I}$ a $\Bbb Q_{\kappa}$-name for the ideal of subsets of
$\check\lambda$ generated by $\check{\Cal I}$\cmmnt{, as in 555B}.   Set
$\theta=\Tr_{\Cal I}(\lambda;\kappa)$.   Then

\Centerline{$\VVdQk\,
\Cal P\lambda/\dot{\Cal I}\cong\frak G_{\check\theta}$.}

\proof{{\bf (a)} For $\eta<\kappa$ let $e_{\eta}\in\frak G_{\kappa}$
be the equivalence class of $\{x:x\in\{0,1\}^{\kappa}$, $x(\eta)=1\}$;
for $L\subseteq\kappa$ let $\frak C_L$ be the closed subalgebra of
$\frak G_{\kappa}$ generated by $\{e_{\eta}:\eta\in L\}$.
For $\sigma\in\frak G_{\kappa}^{\lambda}$ let
$\vec\sigma$ be the $\BbbPk$-name
$\{(\check\xi,\sigma(\xi)):\xi<\lambda$, $\sigma(\xi)\ne 0\}$, so that
$\VVdQk\,\vec\sigma\subseteq\check\lambda$, and
$\Bvalue{\check\xi\in\vec\sigma}=\sigma(\xi)$ for any $\xi<\lambda$.

Write
$\Cal F=\{\lambda\setminus I:I\in\Cal I\}=\Cal P\lambda\setminus\Cal I$,
so that $\Cal F$ is a $\lambda$-complete ultrafilter on $\lambda$.

\medskip

{\bf (b)} For $z\in\Fn_{<\omega}(\kappa;\{0,1\})$ set

\Centerline{$v_z=\{x:z\subseteq x\in\{0,1\}^{\kappa}\}^{\ssbullet}
\in\frak G_{\kappa}$.}

\noindent Then $\{v_z:z\in\Fn_{<\omega}(\kappa;\{0,1\})\}$ is order-dense
in $\frak G_{\kappa}$.   For $A\subseteq\lambda$ and
$\tau\in\Fn_{<\omega}(\kappa;\{0,1\})^A$, set
$\sigma_{\tau}(\xi)=v_{\tau(\xi)}$ if $\xi\in A$, $0$ if
$\xi\in\lambda\setminus A$.   Note that

\Centerline{$\VVdQk\,\vec{\sigma}_{\tau}\subseteq(\dom\tau)\varBspcheck$.}

\noindent Now if $a\in\frak G_{\kappa}^+$
and $\dot C$ is a $\BbbQk$-name such that
$a\VVdQk\,\dot C\subseteq\check\lambda$, there is a countable set
$T\subseteq\bigcup_{A\subseteq\lambda}\Fn_{<\omega}(\kappa;\{0,1\})^A$
such that $a\VVdQk\,\dot C=\bigcup_{\tau\in\check T}\vec\sigma_{\tau}$.
\Prf\ Set $D=\{\xi:\xi<\lambda$, $\VVdQk\,\check\xi\notin\dot C\}$.
For each $\xi\in\lambda\setminus D$,
choose a sequence $\sequence{n}{\tau_n(\xi)}$ in
$\Fn_{<\omega}(\kappa;\{0,1\})$ such that
$\Bvalue{\check\xi\in\dot C}=\sup_{n\in\Bbb N}v_{\tau_n(\xi)}$.
If $\xi\in D$, then $\xi\notin\dom\tau_n$ and
$\sigma_{\tau_n}(\xi)=0$ for every $n$;  accordingly

\Centerline{$\Bvalue{\check\xi\in\dot C}
=0=\sup_{n\in\Bbb N}\Bvalue{\check\xi\in\vec\sigma_{\tau_n}}$.}

\noindent While if $\xi\in\lambda\setminus D$,

\Centerline{$\Bvalue{\check\xi\in\dot C}
=\sup_{n\in\Bbb N}v_{\tau_n(\xi)}
=\sup_{n\in\Bbb N}\sigma_{\tau_n}(\xi)
=\sup_{n\in\Bbb N}\Bvalue{\check\xi\in\vec\sigma_{\tau_n}}$.}

\noindent So

\Centerline{$a\VVdQk\,\dot C
=\bigcup_{n\in\Bbb N}\vec\sigma_{\tau_n}$}

\noindent and we can set $T=\{\tau_n:n\in\Bbb N\}$.\ \Qed

\medskip

{\bf (c)} There is a family $G_0\subseteq\kappa^{\lambda}$, with cardinal
$\theta$, such that
$\{\xi:g(\xi)=g'(\xi)\}\in\Cal I$ whenever $g$, $g'\in G$ are distinct
(541F again),
and we can suppose that every member of $G_0$ is injective (see
part (a) of the proof of 555E).   Let $G\supseteq G_0$
be a maximal family such that
$\{\xi:g(\xi)=g'(\xi)\}$ belongs to $\Cal I$ whenever $g$, $g'\in G$ are
distinct.   Then $\#(G)\le\theta$, by the definition of
$\Tr_{\Cal I}(\lambda;\kappa)$, so in fact $\#(G)=\theta$.
Enumerate $G$ as $\ofamily{\alpha}{\theta}{g_{\alpha}}$, and for
$\alpha<\theta$, $\xi<\lambda$ set
$\rho_{\alpha}(\xi)=e_{g_{\alpha}(\xi)}\in\frak G_{\kappa}$.

\medskip

{\bf (d)} Suppose that $\alpha<\theta$ and
$a\in\frak G_{\kappa}^+$ are such that

\Centerline{$a\VVdQk\,\vec\rho_{\alpha}^{\mskip4mu\ssbullet}$
is neither $0$ nor $1$ in $\Cal P\check\lambda/\dot{\Cal I}$.}

\noindent Then $g_{\alpha}^{-1}[\{\eta\}]\notin\Cal F$ for
every $\eta<\kappa$.   \Prf\Quer\ Otherwise, take $b$ to be one of
$a\Bcap e_{\eta}$, $a\Bsetminus e_{\eta}$ and non-zero, and any
$\xi\in g_{\alpha}^{-1}[\{\eta\}]$.
If $b\Bsubseteq e_{\eta}$ then $b\Bsubseteq\rho_{\alpha}(\xi)$ and
$b\VVdQk\,\check\xi\in\vec\rho_{\alpha}$.   If
$b\Bcap e_{\eta}=0$ then $\rho_{\alpha}(\xi)\Bcap b=0$ and
$b\VVdQk\,\check\xi\notin\vec\rho_{\alpha}$.   So

\doubleinset{$b\VVdQk\,g_{\alpha}^{-1}[\{\eta\}]\varBspcheck$ is either
included in or disjoint
from $\vec\rho_{\alpha}$, and $\vec\rho_{\alpha}^{\mskip4mu\ssbullet}$
is either $1$ or $0$ in $\Cal P\check\lambda/\dot{\Cal I}$.}

\noindent But this contradicts the assumption on $a$.\ \Bang\Qed

\medskip

{\bf (e)} $\VVdQk\,
\{\vec\rho_{\alpha}^{\mskip4mu\ssbullet}:
\alpha<\check\theta\}\setminus\{0,1\}$
is a Boolean-independent family in $\Cal P\check\lambda/\dot{\Cal I}$.

\Prf\Quer\ Otherwise, there must be disjoint finite sets $J$,
$K\subseteq\theta$ and $a\in\frak G_{\kappa}^+$ such that

\Centerline{$a\VVdQk\,\vec\rho_{\alpha}^{\mskip4mu\ssbullet}$
is neither $0$ nor $1$ in $\Cal P\check\lambda/\dot{\Cal I}$}

\noindent for every $\alpha\in J\cup K$, but

\Centerline{$a\VVdQk\,
\inf_{\alpha\in\check J}\vec\rho_{\alpha}^{\mskip4mu\ssbullet}
\Bsetminus\sup_{\alpha\in\check K}\vec\rho_{\alpha}^{\mskip4mu\ssbullet}
=0$.}

\noindent In this case,

\Centerline{$I=\{\xi:\xi<\lambda$,
there are distinct $\alpha$, $\beta\in J\cup K$ such that
$g_{\alpha}(\xi)=g_{\beta}(\xi)\}$}

\noindent belongs to $\Cal I$.   For $\xi<\lambda$ define
$\sigma(\xi)\in\frak G_{\kappa}$ by saying that

$$\eqalign{\sigma(\xi)
&=1\text{ if }\xi\in I,\cr
&=\inf_{\alpha\in J}e_{g_{\alpha}(\xi)}
  \Bsetminus\sup_{\alpha\in K}e_{g_{\alpha}(\xi)}
\text{ otherwise}.\cr}$$

\noindent For $\alpha\in J$ and $\xi\in\lambda\setminus I$,
$\sigma(\xi)\Bsubseteq\rho_{\alpha}(\xi)$;  so

\Centerline{$\VVdQk\,\vec\sigma\setminus\vec\rho_{\alpha}$ is included in
$\check I$ and belongs to $\dot{\Cal I}$, that is,
$\vec\sigma^{\ssbullet}\Bsubseteq\vec\rho_{\alpha}^{\mskip4mu\ssbullet}$.}

\noindent On the other hand, if $\alpha\in K$ and
$\xi\in\lambda\setminus I$,
$\sigma(\xi)\Bcap\rho_{\alpha}(\xi)=0$;  so

\Centerline{$\VVdQk\,\vec\sigma\cap\vec\rho_{\alpha}$ is included in
$\check I$ and belongs to $\dot{\Cal I}$, that is,
$\vec\sigma^{\ssbullet}\Bcap\vec\rho_{\alpha}^{\mskip4mu\ssbullet}=0$.}

\noindent So we have

\Centerline{$\VVdQk\,\vec\sigma^{\ssbullet}
\Bsubseteq\inf_{\alpha\in\check J}\vec\rho_{\alpha}^{\mskip4mu\ssbullet}
\Bsetminus\sup_{\alpha\in\check K}\vec\rho_{\alpha}^{\mskip4mu\ssbullet}$}

\noindent and

\Centerline{$a\VVdQk\,\vec\sigma\in\dot{\Cal I}$.}

Let $I'\in\Cal I$ be such that $a\VVdQk\,\vec\sigma\subseteq\check I'$
(see part (a) of the proof of 555B), and $L\in[\kappa]^{\le\omega}$ such
that $a\in\frak C_L$.    By (d), $g_{\alpha}^{-1}[\{\eta\}]\in\Cal I$ for
every $\alpha\in J\cup K$ and
$\eta<\kappa$, so there must be a $\xi\in\lambda\setminus(I'\cup I)$
such that $g_{\alpha}(\xi)\notin L$ for every $\alpha\in J\cup K$.
In this case, $\sigma(\xi)\in\frak C_{\kappa\setminus L}$ so
$a\Bcap\sigma(\xi)$ is non-zero.   But
$\sigma(\xi)=\Bvalue{\check\xi\in\vec\sigma}$ and
$a\VVdQk\,\check\xi\notin\vec\sigma$, so this is impossible.\ \Bang\Qed

\medskip

{\bf (f)} $\VVdQk$ the subalgebra of $\Cal P\check\lambda/\dot{\Cal I}$
generated by
$\{\vec\rho_{\alpha}^{\mskip4mu\ssbullet}:\alpha<\check\theta\}$
is order-dense in $\Cal P\check\lambda/\dot{\Cal I}$.

\Prf\ If $a\in\frak G_{\kappa}^+$ and $\dot c$ is a $\Bbb Q_{\kappa}$-name
such that

\Centerline{$a\VVdQk\,
\dot c\in(\Cal P\check\lambda/\dot{\Cal I})\setminus\{0\}$,}

\noindent there is a $\BbbQk$-name $\dot C$ such that

\Centerline{$a\VVdQk\,\dot C\subseteq\check\lambda$,
$\dot C\notin\dot{\Cal I}$ and $\dot c=\dot C^{\ssbullet}$.}

\noindent Take a countable set
$T\subseteq\bigcup_{A\subseteq\lambda}\Fn_{<\omega}(\kappa;\{0,1\})^A$
such that $a\VVdQk\,\dot C=\bigcup_{\tau\in\check T}\vec\sigma_{\tau}$,
as described in (b).
Since $\VVdQk\,\dot{\Cal I}$ is a $\sigma$-ideal, there are a $b$ stronger
than $a$ and a $\tau\in T$ such that
$b\VVdQk\,\vec\sigma_{\tau}\notin\dot{\Cal I}$.
Set $F_0=\dom\tau$;  since $\VVdQk\,\vec\sigma_{\tau}\subseteq\check F_0$,
$F_0\notin\Cal I$..   Since $\Cal I$ is a
$\sigma$-ideal, there is an $n\in\Bbb N$ such that
$F_1=\{\xi:\xi\in F_0$, $\#(\tau(\xi))=n\}\notin\Cal I$, and
$F_1\in\Cal F$.

Let $\langle h_i\rangle_{i<n}$ be a finite
sequence of functions from $\lambda$ to
$\kappa$ such that $\dom\tau(\xi)=\{h_i(\xi):i<n\}$ for every
$\xi\in F_1$.   As $G$ was maximal, there is for
each $i<n$ an $\alpha_i<\theta$ such that
$\{\xi:g_{\alpha_i}(\xi)=h_i(\xi)\}$ belongs to $\Cal F$;  set
$F_2=\{\xi:\xi\in F_1$, $g_{\alpha_i}(\xi)=h_i(\xi)$ for every $i<n\}$.
Note that if $i<j<n$ then $g_{\alpha_i}(\xi)\ne g_{\alpha_j}(\xi)$
for any $\xi\in F_1$, so
$\alpha_i\ne\alpha_j$.   Next, there must be an $L\subseteq n$ such that

$$\eqalign{F_3
=\{\xi:\xi\in F_2,\,\,&\tau(\xi)(g_{\alpha_i}(\xi))=1
  \text{ for every }i\in L,\cr
&\tau(\xi)(g_{\alpha_i}(\xi))=0
  \text{ for every }i\in n\setminus L\}\cr}$$

\noindent belongs to $\Cal F$.   Set $J=\{\alpha_i:i\in L\}$ and
$K=\{\alpha_i:i\in n\setminus L\}$;  of course $J\cap K=\emptyset$, because
all the $\alpha_i$ are different.
Then, for $\xi\in F_3$, $\dom\tau_{\xi}=\{g_{\alpha_i}(\xi):i<n\}$ and

$$\eqalign{\Bvalue{\check\xi\in\vec\sigma_{\tau}}
&=\sigma_{\tau}(\xi)
=v_{\tau(\xi)}
=\inf_{i\in L}e_{g_{\alpha_i}(\xi)}
   \Bsetminus\sup_{i\in n\setminus L}e_{g_{\alpha_i}(\xi)}\cr
&=\inf_{\alpha\in J}e_{g_{\alpha}(\xi)}
   \Bsetminus\sup_{\alpha\in K}e_{g_{\alpha}(\xi)}
=\inf_{\alpha\in J}\Bvalue{\check\xi\in\vec\rho_{\alpha}}
  \Bsetminus\sup_{\alpha\in K}\Bvalue{\check\xi\in\vec\rho_{\alpha}}.\cr}$$

\noindent Accordingly

\Centerline{$\VVdQk\,\vec\sigma_{\tau}
\symmdiff(\bigcap_{\alpha\in\check J}\vec\rho_{\alpha}
  \setminus\bigcup_{\alpha\in\check K}\vec\rho_{\alpha})$
is disjoint from $\check F_3$ and belongs to $\dot{\Cal I}$,}

\noindent so

\Centerline{$b\VVdQk\,\inf_{\alpha\in\check J}
     \vec\rho_{\alpha}^{\mskip4mu\ssbullet}
  \Bsetminus\sup_{\alpha\in\check K}\vec\rho_{\alpha}^{\mskip4mu\ssbullet}
  =\vec\sigma_{\tau}^{\ssbullet}\Bsubseteq\dot c$.}

\noindent As $a$ and $\dot c$ are arbitrary, this proves the result.\ \Qed

\medskip

{\bf (g)} If $\alpha<\theta$ is such that $g_{\alpha}\in G_0$, then

\Centerline{$\VVdQk\,\vec\rho_{\alpha}^{\mskip4mu\ssbullet}\ne 0$.}

\noindent\Prf\Quer\ Otherwise, there is a non-zero
$a\in\frak G_{\kappa}$ such that

\Centerline{$a\VVdQk\,\vec\rho_{\alpha}^{\mskip4mu\ssbullet}=0$,}

\noindent that is,

\Centerline{$a\VVdQk\,\{\xi:\vec\rho_{\alpha}(\xi)\ne 0\}\in\dot{\Cal I}$,}

\noindent and there is an $I\in\Cal I$ such that

\Centerline{$a\VVdQk\,
\{\xi:\vec\rho_{\alpha}(\xi)\ne 0\}\subseteq\check I$.}

\noindent In this case, for $\xi\in\lambda\setminus I$,

\Centerline{$a\VVdPk\,\vec\rho_{\alpha}(\xi)=0$,}

\noindent that is,

\Centerline{$0=a\Bcap\rho_{\alpha}(\xi)=a\Bcap e_{g_{\alpha}(\xi)}$.}

\noindent But $\lambda\setminus I$ is infinite and $g_{\alpha}$ is
injective, so $\{g_{\alpha}(\xi):\xi\in\lambda\setminus I\}$ is infinite
and $a=0$.\ \Bang\Qed

Similarly,

\Centerline{$\VVdQk\,\vec\rho_{\alpha}^{\mskip4mu\ssbullet}\ne 1$.}

\noindent As this is true whenever $g_{\alpha}\in G_0$, and
$\#(G_0)=\theta$, we see that

\Centerline{$\VVdQk\,\#(\{\alpha:\vec\rho_{\alpha}^{\mskip4mu\ssbullet}
  \notin\{0,1\}\})=\check\theta$.}

\medskip

{\bf (h)} Putting (e)-(g) together, and using 555Bc, 515Nc and 514Ih,

\doubleinset{$\VVdQk\,\Cal P\check\lambda/\dot{\Cal I}$ has a
Boolean-independent family with cardinal $\check\theta$
generating an order-dense subalgebra;  being Dedekind complete, it is
isomorphic to $\RO(\{0,1\}^{\check\theta})\cong\frak G_{\check\theta}$.}

}%end of proof of 555G

\leader{555H}{Corollary} Suppose that $\lambda$ is a \2vm\ cardinal and
$\kappa=2^{\lambda}$.   Then

\Centerline{$\VVdQk$ there is a
non-trivial atomless $\sigma$-centered \pssqa.}

\proof{{\bf (a)} Note first that $\{0,1\}^{\frak c}$ is separable
(4A2B(e-ii)), so $\frak G_{\frak c}\cong\RO(\{0,1\}^{\frak c})$ is
$\sigma$-centered (514H(b-iii));  also, of course, it is atomless.

\medskip

{\bf (b)} Taking $\Cal I$ and $\dot{\Cal I}$ as in 555G, we have

\Centerline{$\VVdQk\,\Cal P\check\lambda/\dot{\Cal I}
\cong\frak G_{\check\theta}$}

\noindent where $\theta=\Tr_{\Cal I}(\lambda;\kappa)$.
But since $\theta$ lies between $\kappa$ and the cardinal power
$\kappa^{\lambda}=\kappa^{\omega}=\kappa$, we have

\Centerline{$\VVdQk\,\check\theta=\check\kappa=\frak c$}

\noindent (554B), and

\Centerline{$\VVdQk\,\Cal P\check\lambda/\dot{\Cal I}
\cong\frak G_{\frak c}$ is $\sigma$-centered and atomless}

\noindent by (a).
}%end of proof of 555H

\leader{555I}{}\cmmnt{ The next example relies on some interesting
facts which I have not yet had any
compelling reason to spell out.   I must begin with
a definition which has so far been confined to the exercises.

\medskip

\noindent}{\bf Definition} A Boolean algebra $\frak A$ has the
{\bf Egorov property} if whenever
$\sequencen{A_n}$ is a sequence of countable partitions of unity in
$\frak A$ then there is a countable partition $B$ of unity such that
$\{a:a\in A_n$, $a\Bcap b\ne 0\}$ is finite for every $b\in B$ and
$n\in\Bbb N$.

\leader{555J}{Lemma} (a) Suppose that $X$ is a set and $\#(X)<\frak b$.
Then $\Cal PX$ has the Egorov property.

(b) Let $\frak A$ be a Dedekind $\sigma$-complete
Boolean algebra with the Egorov property and $I$
a $\sigma$-ideal of $\frak A$.   Then $\frak A/I$ has the Egorov property.

(c) A ccc Boolean algebra has the Egorov property iff it is \wsid.

\proof{{\bf (a)} Let $\sequencen{A_n}$ be a sequence of
countable partitions of $X$;  enumerate each $A_n$ as
$\ofamily{i}{N_n}{a_{ni}}$ where $N_n\in\Bbb N\cup\{\omega\}$ for each
$n$.   For each $x\in X$ and $n\in\Bbb N$, let $f_x(n)<N_n$ be such that
$x\in a_{n,f_x(n)}$.   Because $\#(X)<\frak b$, there is an
$f:\Bbb N\to\Bbb N$ such that $\{n:f(n)<f_x(n)\}$ is finite for every
$x\in X$ (522C);  set $g(x)=\sup\{n:f(n)<f_x(n)\}$.   Now set
$b_n=\{x:x\in X$, $n=\max_{k\le g(x)}f_x(k)\}$
for each $n$, and
$B=\{b_n:n\in\Bbb N\}$;  then $B$ is a partition of $X$, and for any
$m$, $n\in\Bbb N$ we have $b_n\cap a_{mi}=\emptyset$ whenever
$\max(n,f(m))<i<N_m$.
So $\{a:a\in A_m$, $b_n\cap a\ne\emptyset\}$ is finite for
all $m$, $n\in\Bbb N$.   As $\sequencen{A_n}$ is arbitrary,
$\Cal PX$ has the Egorov property.

\medskip

{\bf (b)} Let $\sequencen{C_n}$ be a sequence of countable partitions of
unity in $\frak A/I$.   For each $n\in\Bbb N$, we can choose a countable
disjoint family $A_n\subseteq\frak A$ such that
$C_n=\{a^{\ssbullet}:a\in A_n\}$;  set
$A'_n=A_n\cup\{1\Bsetminus\sup A_n\}$, so that $A'_n$ is a countable
partition of unity in $\frak A$.   Let $B$ be a countable
partition of unity in
$\frak A$ such that $\{a:a\in A'_n,\,a\Bcap b\ne 0\}$ is finite for every
$n\in\Bbb N$.   Then $D=\{b^{\ssbullet}:b\in B\}$ is a countable partition
of unity in $\frak A/I$ and $\{c:c\in C_n$, $c\Bcap d\ne 0\}$ is finite for
every $d\in D$ and $n\in\Bbb N$.

\medskip

{\bf (c)} This is elementary, because every partition of unity in $\frak A$
is countable, so the Egorov property exactly matches (ii) of 316H.
}%end of proof of 555J

\leader{555K}{G{\l}\'owczy\'nski's
example}\cmmnt{ ({\smc G{\l}\'owczy\'nski 91}, {\smc Balcar Jech \&
Paz\'ak 05}, {\smc G{\l}\'owczy\'nski 08})} Let $\lambda$ be a
\2vm\ cardinal, and $\Bbb P$ a ccc forcing notion such that

\Centerline{$\VVdP\,\check\lambda<\frak m$\dvro{.}{}}

\noindent\cmmnt{(5A3P). }Then, taking $\Cal I$ to be the null ideal of a
witnessing measure on $\lambda$, and $\dot\Cal I$ to be a $\Bbb P$-name for
the ideal of subsets of $\check\lambda$ generated by
$\check{\Cal I}$\cmmnt{, as in 555B},

\doubleinset{$\VVdP\,\Cal P\check\lambda/\dot{\Cal I}$ is ccc, atomless,
Dedekind complete, \wsid, has Maharam type $\omega$ and is not a Maharam
algebra.}

\proof{ We know from 555B that

\Centerline{$\VVdP\,\dot{\Cal I}$ is a $\sigma$-ideal, and the
quotient $\Cal P\check\lambda/\dot{\Cal I}$ is ccc and
Dedekind complete.}

\noindent By 541P, because $\frak m\le\frak c$,

\Centerline{$\VVdP\,\Cal P\check\lambda/\dot{\Cal I}$ is atomless.}

\noindent By 555J, because $\frak m\le\frak b$,

\doubleinset{$\VVdP\,\Cal P\check\lambda$ has the Egorov property, so
$\Cal P\check\lambda/\dot{\Cal I}$ has the Egorov property and is \wsid.}

\noindent Moreover, because $\frak m\le\frak p$, 517Rc tells us that

\doubleinset{$\VVdP\,\Cal P\check\lambda$ is $\sigma$-generated by a
countable
set, so $\Cal P\check\lambda/\dot{\Cal I}$ is $\sigma$-generated by a
countable set and has countable Maharam type.}

\noindent Finally, since we certainly have

\doubleinset{$\VVdP\,\check\lambda<\frak m\le\frakmctbl\le\frak c$, so
there is a separable metrizable topology on $\check\lambda$,}

\noindent 539I shows that

\Centerline{$\VVdP$ there is no non-zero Maharam submeasure on
$\Cal P\check\lambda/\dot{\Cal I}$.}
}%end of proof of 555K

\leader{555L}{Supercompact cardinals and the normal measure
\dvrocolon{axiom}}\cmmnt{ If we allow ourselves to
go a good deal farther than `measurable
cardinal' we can use similar techniques to find a forcing language in which
NMA is true.

\medskip

\noindent}{\bf Definition} An uncountable cardinal $\kappa$ is
{\bf supercompact} if for every set $X$
there is a $\kappa$-additive maximal proper ideal $\Cal I$ of subsets of
$S=[X]^{<\kappa}$ such that

\inset{($\alpha$) $\{s:s\in S$, $x\notin s\}\in\Cal I$ for every $x\in X$,

($\beta$) if $A\subseteq S$, $A\notin\Cal I$ and $f:A\to X$ is such that
$f(s)\in s$ for every $s\in A$, then there is an $x\in X$ such that
$\{s:s\in A$, $f(s)=x\}\notin\Cal I$.}

\noindent\cmmnt{(Compare 545D.)}

\leader{555M}{Proposition} A supercompact cardinal is \2vm.

\proof{ If $\kappa$ is supercompact, let $\Cal I$ be a
$\kappa$-additive maximal ideal of subsets of
$S=[\kappa]^{<\kappa}$ satisfying ($\alpha$) of 555L.
Define $f:S\to\kappa$ by setting $f(s)=\min(\kappa\setminus s)$ for
$s\in S$.   Then $\Cal J=\{J:J\subseteq\kappa$, $f^{-1}[J]\in\Cal I\}$ is a
$\kappa$-additive maximal ideal of subsets of $\kappa$.   If $\xi<\kappa$,
then

\Centerline{$f^{-1}[\{\xi\}]=\{s:f(s)=\xi\}\subseteq\{s:\xi\notin s\}
\in\Cal I$,}

\noindent so $\{\xi\}\in\Cal J$.  Thus
$\Cal J$ contains all singletons, and witnesses that $\kappa$ is \2vm.
}%end of proof of 555M

\leader{555N}{Theorem}\cmmnt{ ({\smc Prikry 75},
{\smc Fleissner 91})} Suppose that
$\kappa$ is a supercompact cardinal.   Then

\Centerline{$\VVdPk$ the normal measure axiom and the product measure
extension axiom are true.}

\proof{{\bf (a)} Life will be a little easier if I start by pointing out
that we can work with a variation of NMA as stated in 545D.  First, for a
set $X$ and an uncountable cardinal $\lambda$
let $\ddagger(X,\lambda)$ be the statement

\inset{\noindent there is a $\lambda$-additive
probability measure $\nu$ on $S=[X]^{<\lambda}$, with domain $\Cal PS$,
such that

($\alpha$) $\nu\{s:x\in s\in S\}=1$ for every $x\in X$,

($\beta$) if $g:S\setminus\{\emptyset\}\to X$ is such that
$g(s)\in s$ for every $s\in S\setminus\{\emptyset\}$, then there is a
countable set $K\subseteq X$ such that
$\nu g^{-1}[K]=1$.}

\noindent Now the point is that if $\ddagger(\alpha,\frak c)$
is true for every
ordinal $\alpha$, then the normal measure axiom is true.   \Prf\ Let $X$ be
any set.   Since, as always, we are working with the
axiom of choice, $X$ is equipollent with some ordinal and
$\ddagger(X,\frak c)$ is true;  let $\nu$ be a measure on
$S=[X]^{<\frak c}$ as above.   Given $A\subseteq S$ and a function
$f:A\to X$ which is regressive in the sense of ($\beta$) in 545D, then we
can extend $f$ to a function $g:S\setminus\{\emptyset\}\to X$ which is
regressive in the sense of ($\beta$) here.   If $K$ is a countable set such
that $g^{-1}[K]$ is conegligible, and $A$ is not negligible, then there
must be a $\xi\in K$ such that $A\cap g^{-1}[\{\xi\}]=f^{-1}[\{\xi\}]$ is
not negligible, as required in 545D.\ \Qed

\medskip

{\bf (b)} For the time being (down to the end of (d) below),
fix an ordinal $\alpha$.   Let $\Cal I$ be a
$\kappa$-additive maximal ideal of subsets of $S=[\alpha]^{<\kappa}$ as in
555L, and $\nu$ the corresponding measure on $S$, setting $\nu A=0$ and
$\nu(S\setminus A)=1$ if $A\in\Cal I$.   By 555C, we
have a corresponding $\BbbPk$-name $\dot\nu$ for a measure on $\check S$.
Now

\Centerline{$\VVdPk\,\check S\subseteq[\check{\alpha}]^{<\check{\kappa}}$,}

\noindent so we have a $\BbbPk$-name $\dot\mu$ such that

\doubleinset{$\VVdPk\,\dot\mu$ is a measure with domain
$\Cal P([\check{\alpha}]^{<\check{\kappa}})$ and
$\dot\mu(D)=\dot\nu(D\cap\check S)$ for
every $D\subseteq[\check{\alpha}]^{<\check{\kappa}}$.}

\noindent By 555C,

\doubleinset{$\VVdPk\,\dot\nu$ is a $\check{\kappa}$-additive
probability measure,
so $\dot\mu$ is a $\check{\kappa}$-additive probability measure.}

\medskip

{\bf (c)} $\VVdPk\,
\dot\mu\{s:\xi\in s\in[\check{\alpha}]^{<\check{\kappa}}\}=1$ for every
$\xi<\check{\alpha}$.

\Prf\ If $a\in\frak B_{\kappa}^+$ and $\dot\xi$ are such that
$a\VVdP\,\dot\xi<\check{\alpha}$, take any $b$ stronger than $a$ and
$\xi<\alpha$ such that $b\VVdPk\,\dot\xi=\check\xi$.   Now
$I=\{s:s\in S$, $\xi\notin s\}\in\Cal I$ so

$$\eqalign{b\VVdPk\,&0=\dot\nu\check I
=\dot\nu\{s:s\in\check S,\,\check\xi\notin s\}
=\dot\mu\{s:s\in[\check\alpha]^{<\check\kappa},\,\check\xi\notin s\}\cr
&\text{ and }
1=\dot\mu\{s:s\in[\check\alpha]^{<\check\kappa},\,\check\xi\in s\}.\cr}$$

\noindent As $b$ and $\xi$ are arbitrary,

\Centerline{$a\VVdPk\,
\dot\mu\{s:\dot\xi\in s\in[\check\alpha]^{<\check\kappa}\}=1$;}

\noindent as $a$ and $\dot\xi$ are arbitrary,

\Centerline{$\VVdPk\,
\dot\mu\{s:\xi\in s\in[\check{\alpha}]^{<\check{\kappa}}\}=1$ for every
$\xi<\check{\alpha}$.  \Qed}

\medskip

{\bf (d)} Suppose that $a\in\frak B_{\kappa}^+$ and that $\dot f$ is a
$\BbbPk$-name such that

\Centerline{$a\VVdPk\,
\dot f:[\check\alpha]^{<\check\kappa}\setminus\{\emptyset\}
\to\check\alpha$ is a function
and $\dot f(s)\in s$ whenever
$\emptyset\ne s\in[\check\alpha]^{<\check\kappa}$.}

\noindent For each $s\in S\setminus\{\emptyset\}$, we have

\Centerline{$a\VVdPk\,\dot f(\check s)\in\check s$;}

\noindent because $\BbbPk$ is ccc, there is a non-empty countable set
$J_s\subseteq s$ such that $a\VVdPk\,\dot f(\check s)\in\check J_s$
(5A3Nc).   Let $\sequencen{h_n(s)}$ be a sequence running over $J_s$.
For each $n\in\Bbb N$, we have a $\beta_n<\alpha$ such that
$\{s:s\in S\setminus\{\emptyset\}$, $h_n(s)\ne\beta_n\}\in\Cal I$.
Set $K=\{\beta_n:n\in\Bbb N\}$;
since $\Cal I$ is a $\sigma$-ideal containing $\{\emptyset\}$,
$I=\{s:s\in S\setminus\{\emptyset\}$, $J_s\not\subseteq K\}
\cup\{\emptyset\}$ belongs to $\Cal I$.
But in this case, $\VVdPk\,\dot\mu\check I=\dot\nu\check I=0$ and

\Centerline{$a\VVdPk\,\dot f(\check s)\in\check J_s\subseteq\check K$}

\noindent whenever $s\in S\setminus I$, so

\Centerline{$a\VVdPk\,\dot\mu(\dot f^{-1}[\check K])
=\dot\nu(\check S\cap\dot f^{-1}[\check K])
\ge\dot\nu(\check S\setminus\check I)=1$,}

\noindent while of course $\VVdPk\,\check K$ is countable.

\medskip

{\bf (e)} What this means is that

\Centerline{$\VVdPk\,\ddagger(\check\alpha,\check\kappa)$}

\noindent for every ordinal $\alpha$;  since forcing adds no new ordinals
(5A3Na),

\Centerline{$\VVdPk\,\ddagger(\alpha,\check\kappa)$ for every ordinal
$\alpha$.}

\noindent But 552B, with 555M and 5A1Fc, tells us that

\Centerline{$\VVdPk\,\frak c=\check\kappa$, so $\ddagger(\alpha,\frak c)$
for every ordinal $\alpha$;}

\noindent with (a) above and 545E, we get

\Centerline{$\VVdPk$ NMA and PMEA.}
}%end of proof of 555N

\leader{555O}{}\cmmnt{ All forcing constructions of \qm\ cardinals
start from \2vm\ cardinals, and there is a reason for this.

\medskip

\noindent}{\bf Theorem}\cmmnt{ ({\smc Solovay 71})}
If $\kappa$ is an uncountable cardinal and $\Cal I$
is a proper $\kappa$-saturated $\kappa$-additive ideal of $\Cal P\kappa$
containing singletons, then

\doubleinset{$L(\Cal I)\,\vDash\,\kappa$ is \2vm\ and the generalized
continuum hypothesis is true.}

\cmmnt{\medskip

\noindent{\bf Remarks}
The proof employs techniques not used elsewhere in this treatise,
so I omit it entirely, to the point of not explaining what
$L(\Cal I)$ is or what the symbol $\vDash$ means;  I remark only
that $L(\Cal I)$ is a proper class containing every ordinal and the set
$\Cal I$, and that the
theorem says that the axioms of ZFC, together with `$\kappa$ is \2vm' and
the generalized continuum hypothesis, are true when relativized
appropriately to the class $L(\Cal I)$.   For more,
see {\smc Jech 78}, p.\ 416, Theorem 82a.

}%end of comment

\exercises{\leader{555X}{Basic exercises (a)}
%\spheader 555Xa
Suppose that $\lambda$ is a \rvm\ cardinal with witnessing
probability $\nu$, and $\kappa$ a cardinal.   Let $\dot\mu$ be the
$\BbbPk$-name for a measure on $\check\lambda$ as defined in 555C.
Show that

\Centerline{$\VVdPk\,\dot\mu\check A=(\nu A)\varBspcheck$}

\noindent for any $A\subseteq\lambda$.
%555C

\spheader 555Xb Suppose that $\lambda$ is a \2vm\ cardinal.   Set
$\kappa=2^{\lambda}$.   Show that

\doubleinset{$\VVdPk\,\check\lambda$ is an \am\ cardinal with a witnessing
probability with Maharam type $\frak c=2^{\check\lambda}$.}
%555E

\spheader 555Xc Suppose that $\lambda$ is a \2vm\ cardinal and that
$\kappa=(2^{\lambda})^{(+\omega)}$.   Show that

\doubleinset{$\VVdPk\,\check\lambda$ is an \am\ cardinal with a witnessing
probability with Maharam type less than $\frak c$.}
%555E

\spheader 555Xd Suppose that $\lambda$ is a \2vm\ cardinal and
$\kappa=2^{\lambda}$.   Show that

\Centerline{$\VVdPk$ there is a non-trivial atomless $\sigma$-linked
\pssqa.}
%555E

\spheader 555Xe Let $\frak A$ be a Dedekind $\sigma$-complete Boolean
algebra.   Show that
$\frak A$ has the Egorov property iff for every sequence $\sequencen{u_n}$
in $L^0=L^0(\frak A)$ there is a sequence $\sequencen{\alpha_n}$ in
$\ooint{0,\infty}$ such that $\{\alpha_nu_n:n\in\Bbb N\}$ is order-bounded
in $L^0$.
%555J

\leader{555Y}{Further exercises (a)}
%\spheader 555Ya
Suppose that $X$ is a set,
and $\Cal I$ a proper ideal of subsets of $X$ containing singletons.
Let $\Bbb P$ be a forcing notion such that $\sat\Bbb P\le\add\Cal I$,
and $\dot{\Cal I}$ a $\Bbb P$-name for the ideal of subsets of $\check X$
generated by $\check{\Cal I}$, as in 555B.
(i) Show that

\Centerline{$\VVdP\,\add\dot{\Cal I}=(\add\Cal I)\varBspcheck$.}

\noindent
(ii) Suppose that $\sat(\Cal PX/\Cal I)<\add\Cal I$.   Set
$\theta=\max(\sat\Bbb P,\sat(\Cal PX/\Cal I))$.   Show that

\Centerline{$\VVdP\,\check\theta$ is a cardinal,
$\dot{\Cal I}$ is $\check\theta$-saturated in
$\Cal P\check X$ and $\Cal P\check X/\dot{\Cal I}$ is Dedekind complete.}

\noindent(iii) Show that if $X=\lambda$ is a regular uncountable cardinal
and $\Cal I$ is a normal ideal on $\lambda$, then

\Centerline{$\VVdP\,\dot{\Cal I}$ is a normal ideal on $\check\lambda$.}
%555B

\spheader 555Yb
In 555B, show that if $\Cal I$ is $\theta$-saturated in $\Cal PX$,
where $\theta$ is
an uncountable cardinal such that $\cff[\theta]^{\le\omega}<\add\Cal I$,
then

\Centerline{$\VVdP\,\dot{\Cal I}$ is $\check\theta$-saturated in
$\Cal P\check X$.}
%555B

\spheader 555Yc Suppose that $\lambda$ is a \2vm\ cardinal, and that
$\Bbb P$ is a forcing notion with $\#(\Bbb P)<\lambda$.   Show that
$\VVdP\,\check\lambda$ is a \2vm\ cardinal.

\spheader 555Yd Suppose that $\kappa$ is a \2vm\ cardinal, and that
$\frak m=\frak c$.   Show that

\doubleinset{$\VVdPk\,\frak c$ is \rvm,
$\frak b=\frak d=\check\frak c$
and the shrinking number of the Lebesgue null ideal is at least
$\check\frak c$.}

\spheader 555Ye\dvAnew{2014} Let $\kappa$ be a cardinal.   Suppose that
for every set $X$
there is a $\kappa$-additive maximal proper ideal $\Cal I$ of subsets of
$S=[X]^{<\kappa}$ such that

\inset{($\alpha$) $\{s:s\in S$, $x\notin s\}\in\Cal I$ for every $x\in X$,

($\beta$) if $A\subseteq S$, $A\notin\Cal I$ and $f:A\to X$ is such that
$f(s)\in s$ for every $s\in A$, then there is an $x\in X$ such that
$\{s:s\in A$, $f(s)=x\}\notin\Cal I$.}

\noindent Show that $\kappa$ is supercompact.
%555L

\spheader 555Yf Let $\kappa$ be a supercompact cardinal.   Show that
$\square_{\lambda}$ is false for every $\lambda\ge\kappa$.
%555L  mt55bits Jech 03, 27.3

\spheader 555Yg\dvAnew{2016} Suppose that $\lambda$ is a
\2vm\ cardinal, and $\kappa>\lambda$ a cardinal.   Show that

\doubleinset{$\VVdPk$ there is a probability measure on $\omega_1$ with
Maharam type greater than the least \am\ cardinal.}
%555E 555D 552Hb out of order query

\spheader 555Yh\dvAnew{2016} In 555C, suppose that $X=\kappa$ and that
$\mu$ witnesses that $\kappa$ is \2vm, that is, $\mu$ is a
$\kappa$-additive $\{0,1\}$-valued measure with domain
$\Cal P\kappa$.   For $J\subseteq\kappa$ let
$P_J:L^{\infty}(\frak B_{\kappa})\to L^{\infty}(\frak B_{\kappa})$ be the
corresponding conditional expectation as in part (b) of the proof of 555F.
Show that for every $\sigma\in\frak B_{\kappa}^{\kappa}$
there is a countable set $J\subseteq\kappa$ such that
$\mu\{\xi:\xi<\kappa$, $u_{\sigma}=P_J(\chi\sigma(\xi))\}=1$.
%555C
}%end of exercises

\leader{555Z}{Problems (a)}
%\spheader 555Za
In 555B, what can we say about the $\pi$-weight of
$\Cal P\check X/\dot{\Cal I}$?
%555K

\spheader 555Zb
Suppose that $\lambda$ is an \am\ cardinal with a normal witnessing
probability.   Let $\ofamily{\eta}{\omega_1}{A_{\eta}}$ be a
family of non-negligible subsets of $\lambda$.   Must there be a countable
set meeting every $A_{\eta}$?\cmmnt{   (See 555F and 521Xi.)}
%555F

\endnotes{
\Notesheader{555}
The point of Solovay's theorems 555D and 555O is that they are relative
consistency results.   Continuing the discussion in the notes to \S541,
write `\Etwovmc', `\Eqmc', `\Eamc' for the sentences `there is a
\2vm\ cardinal', `there is a \qm\
cardinal' and `there is an \am\ cardinal'.   I have already noted that
there are fundamental metamathematical reasons why
we cannot have a proof, in ZFC, that

\Centerline{if ZFC is consistent then ZFC + \Eqmc\ is
consistent}

\noindent unless ZFC is actually {\it inconsistent}.
But 555D tells us that

\Centerline{if ZFC + \Etwovmc\ is consistent, then ZFC + \Eamc\ is
consistent}

\noindent and 555O that

\Centerline{if ZFC + \Eqmc\ is consistent, then ZFC + \Etwovmc\ is
consistent.}

\noindent Since \Eqmc\ is actually a consequence of both \Etwovmc\ and \Eamc,
we see that

\Centerline{if one of ZFC + \Etwovmc, ZFC + \Eamc, ZFC + \Eqmc\ is
consistent, so are the others;}

\noindent that is, \Etwovmc, \Eamc\ and \Eqmc\ are equiconsistent in ZFC.
Of course they are not in general {\it equiveridical}
(unless all are disprovable);
as noted in 555O, if we start from a universe in which \Eqmc\ is true, we
can move to one in which \Etwovmc\ and CH are both true,
so that \Eamc\ is false, and
there are easier arguments to show that if we start with \Eamc, we can
move to

\Centerline{\Eamc\ + `there are no strongly inaccessible cardinals',}

\noindent so that we have \Eamc\ true but \Etwovmc\ false.

The reason for working through these
equiconsistency results is to show that assertions like NMA, PMEA and
\Eamc, which are of interest in measure theory and general topology, are no
more dangerous than appropriate assertions about large cardinals which have
been explored in depth ({\smc Jech 78}, chap.\ 6;
{\smc Kanamori 03}, \S22; {\smc Jech 03}, \S20),
and for which we can have corresponding confidence
that either they are consistent with ZFC, or that an eventual contradiction
would lead to an earthquake, and rescue (if it came) would be from outside
measure theory.

In \S544 I examined some of the consequences of supposing that there is an
\am\ cardinal;  for instance, that there are many Sierpi\'nski sets (544G).
It is not an accident that we get similar properties of random real models
(552E).   If we want to know if something might be implied by the
existence of an \am\ cardinal, the first step is to look at what can be
determined in the forcing models of 555C.   This is often straightforward;
for
instance, since $\frak d$ is not changed by random real forcing, and since
$\frak d$ must be much lower than any \2vm\ cardinal, it must be much lower
than any \am\ cardinal created by random real forcing.   But it is quite
unclear that the same can be said about \am\ cardinals in general (544Zd).
I offer 555F as another example of a phenomenon which appears in Solovay's
models but which is not known to be true for all \am\ cardinals (555Zb).

In \S\S546-547 %546E 547F
I gave some of the Gitik-Shelah results showing that non-trivial
`simple' algebras cannot be \pssqa s.   Of course
this depends a good deal on what
we mean by `simple'.   Looking at the basic cardinal functions, we see that
(at least if there can be measurable cardinals) then there can be
non-trivial ccc \pssqa s which are $\sigma$-centered
or have countable Maharam
type (555H, 555K).   But they are still very far away from any algebra
which can be specified without (perhaps implicitly)
using a \2vm\ cardinal at some stage.
We cannot have a non-trivial \pssqa\ with countable $\pi$-weight (547Xa),
but I do not see how to rule out `small' $\pi$-weight in general (547Zb,
555Za).


}%end of notes

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\frfilename{mt556.tex}
\versiondate{3.1.15}
\copyrightdate{2007}

\def\chaptername{Forcing}
\def\sectionname{Forcing with Boolean subalgebras}

\newsection{556}

I propose
now to describe a completely different way in which forcing can be
used to throw light on problems in measure theory.   Rather than finding
forcing models of new mathematical universes, we look for models which will
express structures of the ordinary universe in new ways.   The problems to
which this approach seems to be most relevant are those centered on
invariant algebras:  in ergodic theory, fixed-point algebras;  in the
theory of relative independence, the core $\sigma$-algebras.

Most of the section is taken up with development of basic machinery.
The strategic plan is straightforward enough:  given a
specific Boolean algebra
$\frak C$ which seems to be central to a question in hand, force with
$\frak C\setminus\{0\}$, and translate the question into a question
in the forcing language.   In order to do this, we need an efficient scheme
for automatic translation.   This is what
556A-556L % 556A 556B 556C 556D 556E 556G 556H 556I 556J 556K 556L
and 556O are setting up.   The translation has to work both ways, since we
need to be able to deduce properties of the ground model from properties of
the forcing model.

There are four actual theorems for which I offer proofs by these methods.
The first three are 556M
(a strong law of large numbers), 556N (Dye's theorem on orbit-isomorphic
measure-preserving transformations) and
556P (Kawada's theorem on invariant measures).   In each of these,
the aim is
to prove a general form of the theorem from the classical special case in
which the algebra $\frak C$ is trivial.   In the final example 556S
(I.Farah's description of the Dedekind completion of the asymptotic density
algebra $\frak Z$), we have a natural subalgebra $\frak C$
of $\frak Z$ and a structure in the corresponding
forcing universe to which we can apply Maharam's theorem.

\vleader{60pt}{556A}{Forcing with Boolean subalgebras}
Let $\frak A$ be a Boolean algebra, not $\{0\}$, and
$\frak C$ a subalgebra of $\frak A$.   Let $\Bbb P$ be the forcing notion
$\frak C^+=\frak C\setminus\{0\}$, active downwards.

\spheader 556Aa If $a\in\frak A$, the
{\bf forcing name for $a$ over $\frak C$} will be the $\Bbb P$-name

\Centerline{$\dot a
=\{(\check b,p):p\in\frak C^+$, $b\in\frak A$, $p\Bcap b\Bsubseteq a\}$.}

\spheader 556Ab If $\frak B$ is a Boolean subalgebra of $\frak A$ including
$\frak C$, then the {\bf forcing name for $\frak B$ over $\frak C$}
will be the
$\Bbb P$-name $\{(\dot b,1):b\in\frak B\}$, where here
$1=1_{\frak A}=1_{\frak B}=1_{\frak C}$.

\spheader 556Ac For each of the binary
operations $\frmedcirc=\Bcap$, $\Bcup$, $\Bsymmdiff$,
$\Bsetminus$ on $\frak A$, the {\bf forcing name for $\frmedcirc$ over
$\frak C$} will be the $\Bbb P$-name

\Centerline{$\dot\frmedcirc=\{(((\dot a,\dot b),(a\frmedcirc b)^{\centerdot}),1):
a$, $b\in\frak A\}$.}

\spheader 556Ad The {\bf forcing name for $\Bsubseteq$ over $\frak C$}
will be the $\Bbb P$-name

\Centerline{$\dot{\Bsubseteq}=\{((\dot a,\dot b),1):a$, $b\in\frak A$,
$a\Bsubseteq b\}$.}

\spheader 556Ae Let $\pi:\frak A\to\frak A$ be a ring
homomorphism such that $\pi c\Bsubseteq c$
for every $c\in\frak C$.   In this case, I
will say that the {\bf forcing name for $\pi$ over $\frak C$} is the
$\Bbb P$-name $\{((\dot a,(\pi a)^{\centerdot}),1):a\in\frak A\}$.

\spheader 556Af Now suppose that $\frak A$ is Dedekind $\sigma$-complete.
For $u\in L^0(\frak A)$, the {\bf forcing name for $u$ over $\frak C$} will
be the $\Bbb P$-name $\{((\check\alpha,\Bvalue{u>\alpha}^{\centerdot}),1):
\alpha\in\Bbb Q\}$.

\medskip

\noindent{\bf Remark}\cmmnt{ We shall need to agree on what it is that
the formula $L^0(\frak A)$
abbreviates.}   The primary definition in 364Aa speaks of functions from
$\Bbb R$ to $\frak A$.   Because $\Bbb R$ is inadequately absolute this is
not convenient here, and I will move to the alternative version\cmmnt{ in
364Af}:   a member $u$ of
$L^0(\frak A)$ is a family $\family{\alpha}{\Bbb Q}{\Bvalue{u>\alpha}}$
in $\frak A$ such that

\Centerline{$\Bvalue{u>\alpha}
=\sup_{\beta\in\Bbb Q,\beta>\alpha}\Bvalue{u>\beta}$ for every
$\alpha\in\Bbb Q$,}

\Centerline{$\inf_{n\in\Bbb N}\Bvalue{u>n}=0$,\quad
$\sup_{n\in\Bbb N}\Bvalue{u>-n}=1$.}

\vleader{72pt}{556B}{Theorem}
Let $\frak A$ be a Boolean algebra, not $\{0\}$, and
$\frak C$ a subalgebra of $\frak A$.   Let $\Bbb P$ be the forcing notion
$\frak C^+$, active downwards, and
$\dot\frak A$ the forcing name for $\frak A$ over $\frak C$.

(a) If $p\in\frak C^+$, $a$, $b\in\frak A$ and $\dot a$, $\dot b$ are the
forcing names of $a$, $b$ over $\frak C$, then

\Centerline{$p\VVdP\,\dot a=\dot b$}

\noindent iff $\upr(p\Bcap(a\Bsymmdiff b),\frak C)=0$, that is,
for every $q$
stronger than $p$ there is an $r$ stronger than $q$ such that
$r\Bcap a=r\Bcap b$.   In particular,

\Centerline{$p\VVdP\,\dot a=\dot b$}

\noindent whenever $p\Bcap a=p\Bcap b$.

(b) Writing $\dot\frmedcirc$ for the forcing name for $\frmedcirc$ over $\frak C$,

\Centerline{$\VVdP\,\dot{\frmedcirc}$ is a binary operation on
$\dot\frak A$ and $\dot a\,\dot\frmedcirc\,\dot b=(a\frmedcirc b)^{\centerdot}$}

\noindent for each of the binary
operations $\frmedcirc=\Bcap$, $\Bcup$, $\Bsymmdiff$
and $\Bsetminus$ and all $a$, $b\in\frak A$.

(c) All the standard identities translate.   For instance,

\Centerline{$\VVdP\,x\dot\Bcap(y\dot\Bsymmdiff z)
  =(x\dot\Bcap y)\dot\Bsymmdiff(x\dot\Bcap z)$ for all $x$, $y$,
  $z\in\dot\frak A$.}

(d)

\doubleinset{$\VVdP\,\dot\frak A$, with the operations $\dot\Bsymmdiff$,
$\dot\Bcap$, $\dot\Bcup$ and $\dot\Bsetminus$, is a Boolean algebra,
with zero $\dot 0$ and identity $\dot 1$.}

(e)(i) Writing $\dot{\Bsubseteq}$ for the forcing name for $\Bsubseteq$
over $\frak C$,

\Centerline{$\VVdP\,\dot{\Bsubseteq}$ is the inclusion relation in the
Boolean algebra $\dot\frak A$.}

\quad(ii) For $a$, $b\in\frak A$ and $p\in\frak C^+$,

\Centerline{$p\VVdP\,\dot a\dot{\Bsubseteq}\dot b$}

\noindent iff $\upr(p\Bcap a\Bsetminus b,\frak C)=0$.

(f) If $\frak B$ is a Boolean subalgebra of $\frak A$ including $\frak C$,
then

\Centerline{$\VVdP\,\dot\frak B$ is a Boolean subalgebra of $\dot\frak A$.}

\proof{{\bf (a)(i)} Recall that
$\upr(a,\frak C)=\inf\{c:a\Bsubseteq c\in\frak C\}$ if the infimum is
defined in $\frak C$ (313S).
So $\upr(a,\frak C)=0$ iff for every
non-zero $c\in\frak C$ there is a $c'\in\frak C$ such that $a\Bsubseteq c'$
and $c\notBsubseteq c'$;  that is, for every non-zero $c\in\frak C$ there
is a non-zero $c'\in\frak C$ such that $c'\Bsubseteq c\Bsetminus a$.   In
the present context, we see that for $p\in\frak C^+$ and $a$,
$b\in\frak A$, $\upr(p\Bcap(a\Bsymmdiff b),\frak C)=0$ iff
for every $q$ stronger than $p$ there is an $r$ stronger than $q$ such that
$r\Bcap(a\Bsymmdiff b)=0$.

\medskip

\quad{\bf (ii)} Suppose that $p\Bcap a=p\Bcap b$,
that $q\in\frak C^+$ is stronger than $p$, and that $\dot x$ is a
$\Bbb P$-name such that $q\VVdP\,\dot x\in\dot a$.   Then there are an
$r\in\frak C^+$, a $d\in\frak A$ such that $(\check d,r)\in\dot a$, and an
$s$ stronger than both $r$ and $q$ such that $s\VVdP\,\dot x=\check d$.
In this case

\Centerline{$s\Bcap d
\Bsubseteq p\Bcap r\Bcap d\Bsubseteq p\Bcap a\Bsubseteq b$,}

\noindent so $(\check d,s)\in\dot b$ and

\Centerline{$s\VVdP\,\dot x=\check d\in\dot b$.}

\noindent As $q$ and $\dot x$ are arbitrary,

\Centerline{$p\VVdP\,\dot a$ is a subset of $\dot b$;}

\noindent similarly,

\Centerline{$p\VVdP\,\dot b$ is a subset of $\dot a$ and $\dot b=\dot a$.}

\medskip

\quad{\bf (iii)} If $\upr(p\Bcap(a\Bsymmdiff b),\frak C)=0$, then for every
$q$ stronger than $p$ there is an $r$ stronger than $q$ such that
$r\Bcap a=r\Bcap b$ and $r\VVdP\,\dot a=\dot b$, by (ii).
As $q$ is arbitrary, $p\VVdP\,\dot a=\dot b$.

\medskip

\quad{\bf (iv)} Now suppose that $p\VVdP\,\dot a=\dot b$ and that $q$ is
stronger than $p$.   Then $(\check a,q)\in\dot a$, so
$q\VVdP\,\check a\in\dot a=\dot b$.   There must therefore be a
$(\check d,r)\in\dot b$ and an $s$ stronger than both $r$ and $q$ such
that $s\VVdP\,\check a=\check d$;  in this case $d=a$,
$s\Bcap a\Bsubseteq r\Bcap d\Bsubseteq b$ and $s\Bcap a\Bsetminus b=0$.

As $q$ is arbitrary, $\upr(p\Bcap(a\Bsetminus b),\frak C)=0$.   Similarly,
$\upr(p\Bcap(b\Bsetminus a),\frak C)=0$.   By 313Sb,
$\upr(p\Bcap(a\Bsymmdiff b),\frak C)=0$.

\medskip

{\bf (b)} Of course

\Centerline{$\VVdP\,\dot{\frmedcirc}
\subseteq(\dot\frak A\times\dot\frak A)\times\dot\frak A$,}

\noindent just because

\Centerline{$\VVdP\,\dot a\in\dot\frak A$}

\noindent for every $a\in\frak A$.   To see that $\dot\frmedcirc$ is a name for
a function with domain $\dot A\times\dot A$, use 5A3Ea.   If
$(((\dot a_1,\dot b_1),(a_1\frmedcirc b_1)^{\centerdot}),1)$ and
$(((\dot a_2,\dot b_2),(a_2\frmedcirc b_2)^{\centerdot}),1)$ are two members of
$\dot\frmedcirc$, and $p\in\frak C^+$ is such that

\Centerline{$p\VVdP\,(\dot a_1,\dot b_1)=(\dot a_2,\dot b_2)$,}

\noindent then

$$\eqalign{\upr(p\Bcap((a_1\frmedcirc b_1)\Bsymmdiff(a_2\frmedcirc b_2)))
&\Bsubseteq\upr(p\Bcap((a_1\Bsymmdiff a_2)\Bcup(b_1\Bsymmdiff b_2)))\cr
&=\upr(p\Bcap(a_1\Bsymmdiff a_2))\Bcup\upr(p\Bcap(b_1\Bsymmdiff b_2))
=0\cr}$$

\noindent by (a) above and 313Sb again.   So

\Centerline{$p\VVdP\,(a_1\frmedcirc b_1)^{\centerdot}
=(a_2\frmedcirc b_2)^{\centerdot}$}

\noindent by (a) in the other direction.   Thus the condition of
5A3E(a-ii) is satisfied, and

\Centerline{$\VVdP\,\dot\frmedcirc$ is a function,}

\noindent while of course

\Centerline{$\VVdP\,\dot a\dot\frmedcirc\dot b=(a\frmedcirc b)^{\centerdot}$}

\noindent for all $a$, $b\in\frak A$.   Moreover, setting
$\dot A=\{((\dot a,\dot b),1):a$, $b\in\frak A\}$, 5A3Eb tells us that

\Centerline{$\VVdP\,\dom\dot\frmedcirc=\dot A=\dot\frak A\times\dot\frak A$,
so $\dot\frmedcirc$ is a binary operation on $\dot\frak A$.}

\woddheader{556B}{4}{2}{2}{30pt}

{\bf (c)} I work through only the given example.
Suppose that $p\in\frak C^+$ and that $\dot x$, $\dot y$ and
$\dot z$ are $\Bbb P$-names such that

\Centerline{$p\VVdP\,\dot x$, $\dot y$, $\dot z\in\dot\frak A$.}

\noindent If $q$ is stronger than $p$, there are an $r$ stronger than
$q$ and $a$, $b$, $c\in\frak A$ such
that

\Centerline{$r\VVdP\,\dot x=\dot a$, $\dot y=\dot b$ and $\dot z=\dot c$.}

\noindent Then

\Centerline{$r\VVdP\,\dot y\dot\Bsymmdiff\dot z=\dot b\dot\Bsymmdiff\dot c
  =(b\Bsymmdiff c)^{\centerdot}$,}

\Centerline{$r\VVdP\,
  \dot x\dot\Bcap(\dot y\dot\Bsymmdiff\dot z)
  =(a\Bcap(b\Bsymmdiff c))^{\centerdot}
  =((a\Bcap b)\Bsymmdiff(a\Bcap c))^{\centerdot}
  =(\dot x\dot\Bcap\dot y)\dot\Bsymmdiff(\dot x\dot\Bcap\dot z)$.}

\noindent As $q$ is arbitrary,

\Centerline{$p\VVdP\,
  \dot x\dot\Bcap(\dot y\dot\Bsymmdiff\dot z)
  =(\dot x\dot\Bcap\dot y)\dot\Bsymmdiff(\dot x\dot\Bcap\dot z)$;}

\noindent as $p$, $\dot x$, $\dot y$ and $\dot z$ are arbitrary,

\Centerline{$\VVdP\,x\dot\Bcap(y\dot\Bsymmdiff z)
  =(x\dot\Bcap y)\dot\Bsymmdiff(x\dot\Bcap z)$ for all $x$, $y$,
  $z\in\dot\frak A$.}

\medskip

{\bf (d)} This is now elementary, amounting to repeated use of the
technique in (c).

\medskip

{\bf (e)(i)} It will be enough to show that

\Centerline{$\VVdP$ for all $x$, $y\in\dot\frak A$,
$x\dot{\Bsubseteq}y\iff x\dot\Bcap y=x$.}

\noindent\Prf\ Suppose that $p\in\frak C^+$ and that $\dot x$,
$\dot y$ are $\Bbb P$-names such that

\Centerline{$p\VVdP\,\dot x$, $\dot y\in\dot\frak A$.}

\noindent($\alpha$) Suppose that $p\VVdP\,\dot x\dot{\Bsubseteq}\dot y$.
If $q$ is stronger than $p$, there are an $r$ stronger than $q$ and $a$,
$b\in\frak A$ such that $a\Bsubseteq b$ and

\Centerline{$r\VVdP\,\dot x=\dot a$ and $\dot y=\dot b$.}

\noindent Now

\Centerline{$r\VVdP\,\dot x\dot\Bcap\dot y
=(a\Bcap b)^{\centerdot}=\dot a=\dot x$;}

\noindent as $q$ is arbitrary, $p\VVdP\,\dot x\dot\Bcap\dot y=\dot x$.
($\beta$) Conversely, suppose that
$p\VVdP\,\dot x\dot\Bcap\dot y=\dot x$.   If $q$ is stronger than $p$ there
are $r$ stronger than $q$ and $a$, $b\in\frak A$ such that

\Centerline{$r\VVdP\,\dot x=\dot a$, $\dot y=\dot b$,
  $(a\Bcap b)^{\centerdot}=\dot a$;}

\noindent now $(((a\Bcap b)^{\centerdot},\dot b),1)\in\dot{\Bsubseteq}$,
so $\VVdP\,(a\Bcap b)^{\centerdot}\dot{\Bsubseteq}\dot b$ and

\Centerline{$r\VVdP\,\dot x=\dot a
=(a\Bcap b)^{\centerdot}\dot{\Bsubseteq}\dot b=\dot y$.}

\noindent As $q$ is arbitrary. $p\VVdP\,\dot x\dot{\Bsubseteq}\dot y$.

As $p$, $\dot x$ and $\dot y$ are arbitrary,

\Centerline{$\VVdP$ for $x$, $y\in\dot\frak A$,
$x\dot{\Bsubseteq}y\iff x\dot\Bcap y=x$.  \Qed}

\medskip

\quad{\bf (ii)} Now, for $a$, $b\in\frak A$ and $p\in\frak C^+$,

$$\eqalign{p\VVdP\,&\dot a\dot{\Bsubseteq}\dot b\cr
&\text{iff }p\VVdP\,\dot a\dot\Bcap\dot b=\dot a\cr
&\text{iff }p\VVdP\,(a\Bcap b)^{\centerdot}=\dot a\cr
&\text{iff }\upr(p\Bcap(a\Bsymmdiff(a\Bcap b)),\frak C)=0\cr
&\text{iff }\upr(p\Bcap a\Bsetminus b,\frak C)=0.\cr}$$

\medskip

{\bf (f)} This should now be easy.   As $\dot\frak B\subseteq\dot\frak A$,
$\VVdP\,\dot\frak B\subseteq\dot\frak A$.
If $p\in\frak C^+$ and $\dot x$, $\dot y$
are $\Bbb P$-names such that $p\VVdP\,\dot x$, $\dot y\in\dot\frak B$, then
for every $q$ stronger than $p$ there are $r$ stronger than $q$ and $a$,
$b\in\frak B$ such that $r\VVdP\,\dot x=\dot a$ and $\dot y=\dot b$.   In
this case

\Centerline{$r\VVdP\,
\dot x\dot\Bcap\dot y=(a\Bcap b)^{\centerdot}\in\dot\frak B$,
$\dot x\dot\Bsymmdiff\dot y=(a\Bsymmdiff b)^{\centerdot}\in\dot\frak B$;}

\noindent as $q$ is arbitrary,

\Centerline{$p\VVdP\,
\dot x\dot\Bcap\dot y$, $\dot x\dot\Bsymmdiff\dot y\in\dot\frak B$.}

\noindent As $p$, $\dot x$ and $\dot y$ are arbitrary,

\Centerline{$\VVdP\,\dot\frak B$ is a subring of $\dot\frak A$;}

\noindent as we also have
$\VVdP\,\dot 1\in\dot\frak B$, we get

\Centerline{$\VVdP\,\dot\frak B$ is a subalgebra
of $\dot\frak A$.}
}%end of proof of 556B

\leader{556C}{Theorem} Let $\frak A$ be a Boolean algebra, not $\{0\}$, and
$\frak C$ a subalgebra of $\frak A$.   Let $\Bbb P$ be the forcing notion
$\frak C^+$, active downwards, $\dot\frak A$ the
forcing name for $\frak A$ over $\frak C$, and
$\pi:\frak A\to\frak A$ a ring homomorphism such that
$\pi c\Bsubseteq c$ for every $c\in\frak C$;  write $\dot\pi$ for
the forcing name for $\pi$ over $\frak C$.

(a)(i)

\Centerline{$\VVdP\,\dot\pi$ is a ring homomorphism from $\dot\frak A$
to itself}

\noindent and

\Centerline{$\VVdP\,\dot\pi(\dot a)=(\pi a)^{\centerdot}$}

\noindent for every $a\in\frak A$.

\quad(ii) If $\pi$ is injective, $\VVdP\,\dot\pi$ is injective.

\quad(iii) If $\phi:\frak A\to\frak A$ is another ring homomorphism such
that $\phi c\Bsubseteq c$ for every $c\in\frak C$, with corresponding
forcing name $\dot\phi$, then

\Centerline{$\VVdP\,\dot\pi\dot\phi=(\pi\phi)^{\centerdot}$.}

(b) Now suppose that $\pi$ is a Boolean homomorphism.

\quad(i) $\pi c=c$ for every $c\in\frak C$.

\quad(ii) $\VVdP\,\dot\pi$ is a Boolean homomorphism.

\quad(iii) If $\pi$ is surjective, $\VVdP\,\dot\pi$ is surjective.

\quad(iv) If $\pi\in\Aut\frak A$ then

\Centerline{$\VVdP\,\dot\pi$ is a Boolean automorphism and
$(\dot\pi)^{-1}=(\pi^{-1})^{\centerdot}$.}

\quad(v) If the fixed-point subalgebra of $\pi$ is $\frak C$ exactly, then

\Centerline{$\VVdP$ the fixed-point subalgebra of $\dot\pi$
is $\{0,1\}$.}

\proof{{\bf (a)(i)}\grheada\
It will help to note straight away that
$\pi c=c\Bcap\pi 1$ for every $c\in\frak C$.   \Prf\
The hypothesis is that
$\pi c\Bsubseteq c$;  because $\pi$ is a ring homomorphism,
$\pi c\Bsubseteq\pi 1$, so $\pi c\Bsubseteq c\Bcap\pi 1$.
Since also

\Centerline{$\pi c=\pi 1\Bsetminus\pi(1\Bsetminus c)
\Bsupseteq\pi 1\Bsetminus(1\Bsetminus c)
=c\Bcap\pi 1$,}

\noindent we have equality.\ \QeD\  Consequently

\Centerline{$c\Bcap\pi a=c\Bcap\pi(1\Bcap a)=c\Bcap\pi 1\Bcap\pi a
=\pi c\Bcap\pi a=\pi(c\Bcap a)$}

\noindent whenever $c\in\frak C$ and $a\in\frak A$.

\medskip

\qquad\grheadb\ $\VVdP\,\dot\pi$ is a function from $\dot\frak A$ to
itself.   \Prf\ Of course
$\VVdP\,\dot\pi\subseteq\dot\frak A\times\dot\frak A$.
Suppose that $p\in\frak C^+$ and that
$((\dot a,(\pi a)^{\centerdot}),1)$,
$((\dot b,(\pi b)^{\centerdot}),1)$ are two members of $\dot\pi$ such that
$p\VVdP\,\dot a=\dot b$.   Then for every $q$ stronger than $p$ there is an
$r$ stronger than $q$ such that $r\Bcap a=r\Bcap b$ (556Ba), in which case

\Centerline{$r\Bcap\pi a=\pi(r\Bcap a)=\pi(r\Bcap b)=r\Bcap\pi b$.}

\noindent This shows that
$p\VVdP\,(\pi a)^{\centerdot}=(\pi b)^{\centerdot}$, by 556Ba in the other
direction.   As $a$ and $b$ are arbitrary, the condition of 5A3Ea is
satisfied, with $\dot A$ there exactly equal to $\dot\frak A$ here,
and

\Centerline{$\VVdP\,\dot\pi$ is a function with domain $\dot\frak A$.
\text{\Qed}}

If $a\in\frak A$ then $((\dot a,(\pi a)^{\centerdot}),1)\in\dot\pi$ so

\Centerline{$\VVdP\,(\dot a,(\pi a)^{\centerdot})\in\dot\pi$ and
$\dot\pi(\dot a)=(\pi a)^{\centerdot}$.}

\woddheader{556C}{4}{2}{2}{42pt}

\qquad\grheadc\ $\VVdP\,\dot\pi$ is a ring homomorphism.   \Prf\
Writing $\frmedcirc$ for either $\Bcap$ or $\Bsymmdiff$,

$$\eqalignno{\VVdP\,\dot\pi(\dot a\,\dot\frmedcirc\,\dot b)
&=\dot\pi(a\frmedcirc b)^{\centerdot}
=(\pi(a\frmedcirc b))^{\centerdot}\cr
&=(\pi a\frmedcirc\pi b)^{\centerdot}
=(\pi a)^{\centerdot}\,\dot\frmedcirc\,(\pi b)^{\centerdot}
=(\dot\pi\dot a)\,\dot\frmedcirc\,(\dot\pi\dot b)\cr}$$

\noindent for all $a$, $b\in\frak A$.   If now $p\in\frak C^+$ and
$\dot x$, $\dot y$ are $\Bbb P$-names such that

\Centerline{$p\VVdP\,\dot x,\,\dot y\in\dot\frak A$,}

\noindent then for any $q$ stronger than $p$ there are $r$ stronger than
$q$ and $a$, $b\in\frak A$ such that

\Centerline{$r\VVdP\,\dot x=\dot a$ and $\dot y=\dot b$,}

\noindent in which case

\Centerline{$r\VVdP\,\dot\pi(\dot x\,\dot\frmedcirc\,\dot b)
=\dot\pi(\dot a\,\dot\frmedcirc\,\dot b)
=\dot\pi(\dot a)\,\dot\frmedcirc\,\dot\pi(\dot b)
=\dot\pi(\dot x)\,\dot\frmedcirc\,\dot\pi(\dot y)$.}

\noindent As $q$ is arbitrary,

\Centerline{$p\VVdP\,\dot\pi(\dot x\,\dot\frmedcirc\,\dot y)
=\dot\pi(\dot x)\,\dot\frmedcirc\,\dot\pi(\dot y)$;}

\noindent as $p$, $\dot x$ and $\dot y$ are arbitrary,

\Centerline{$\VVdP\,\dot\pi(x\,\dot\frmedcirc\,y)
=(\dot\pi x)\,\dot\frmedcirc\,(\dot\pi y)$ for all $x$, $y\in\dot\frak A$.}

\noindent As this is true for both $\dot\frmedcirc=\dot\Bcap$ and
$\dot\frmedcirc=\dot\Bsymmdiff$,

\Centerline{$\VVdP\,\dot\pi$ is a ring homomorphism.  \Qed}

\medskip

\quad{\bf (ii)}
Let $p\in\frak C^+$ and a $\Bbb P$-name $\dot x$ be such that

\Centerline{$p\VVdP\,\dot x\in\dot\frak A$ and $\dot\pi\dot x=0$.}

\noindent For any $q$ stronger than $p$, there are an $r$ stronger than $q$
and an $a\in\frak A$ such that

\Centerline{$r\VVdP\,\dot a=\dot x$, therefore
$0=\dot\pi\dot a=(\pi a)^{\centerdot}$.}

\noindent By 556Ba, there is an $s$ stronger than $r$ such that
$s\Bcap\pi a=0$;  since $\pi s\Bsubseteq s$,
$\pi(s\Bcap a)=0$.   As $\pi$ is injective, $s\Bcap a=0$ and (using 556Ba
again)

\Centerline{$s\VVdP\,\dot x=\dot a=0$.}

\noindent As $q$ is arbitrary, $p\VVdP\,\dot x=0$;  as $p$ and $\dot x$ are
arbitrary, $\VVdP\,\dot\pi$ is injective.

\medskip

\quad{\bf (iii)} Suppose that $p\in\frak C^+$ and that
$\dot x$ is a $\Bbb P$-name such
that $p\VVdP\,\dot x\in\dot\frak A$.   For any $q$ stronger than $p$,
there are an $r$ stronger than $q$ and an
$a\in\frak A$ such that $r\VVdP\,\dot x=\dot a$, so that

\Centerline{$r\VVdP\,\dot\pi(\dot\phi(\dot x))
=\dot\pi(\dot\phi(\dot a))
=\dot\pi((\phi a)^{\centerdot})
=(\pi\phi a)^{\centerdot}
=(\pi\phi)^{\centerdot}(\dot a)
=(\pi\phi)^{\centerdot}(\dot x)$.}

\noindent As $q$ is arbitrary,

\Centerline{$p\VVdP\,\dot\pi(\dot\phi(\dot x))
=(\pi\phi)^{\centerdot}(\dot x)$;}

\noindent as $p$ and $\dot x$ are arbitrary,
$\VVdP\,\dot\pi\dot\phi=(\pi\phi)^{\centerdot}$.

\medskip

{\bf (b)(i)} I observed in (a-i-$\alpha$) above that $\pi c=c\Bcap\pi 1$
for every $c\in\frak C$, so if $\pi 1=1$ we shall have $\pi c=c$ for every
$c\in\frak C$.

\medskip

\quad{\bf (ii)} I pointed out in 556Bd that

\Centerline{$\VVdP\,\dot 1$ is the identity of $\dot\frak A$,}

\noindent and we now have

\Centerline{$\VVdP\,\dot\pi(\dot 1)=(\pi 1)^{\centerdot}=\dot 1$,
so $\dot\pi$ is a Boolean homomorphism.}

\medskip

\quad{\bf (iii)} Let $p\in\frak C^+$ and a $\Bbb P$-name $\dot x$
be such that $p\VVdP\,\dot x\in\dot\frak A$.
For any $q$ stronger than $p$, there are an $r$ stronger than $q$
and an $a\in\frak A$
such that $r\VVdP\,\dot a=\dot x$.   Now there is a $b\in\frak A$ such that
$a=\pi b$, in which case

\Centerline{$r\VVdP\,\dot x=\dot a=\dot\pi\dot b\in\dot\pi[\dot\frak A]$.}

\noindent As $q$ is arbitrary, $p\VVdP\,\dot x\in\dot\pi[\dot\frak A]$;
as $p$ and $\dot x$ are arbitrary, $\VVdP\,\dot\pi$ is surjective.

\medskip

\quad{\bf (iv)} By (a-iii),

\Centerline{$\VVdP\,\dot\pi(\pi^{-1})^{\centerdot}
=(\pi^{-1})^{\centerdot}\dot\pi=\dot\iota$}

\noindent where $\iota:\frak A\to\frak A$ is the identity automorphism.
But

\Centerline{$\VVdP\,\dot\iota$ is the identity on $\dot\frak A$.}

\noindent\Prf\ If $p\in\frak C^+$ and a $\Bbb P$-name $\dot x$ are such
that
$p\VVdP\,\dot x\in\dot\frak A$, then for any $q$ stronger than $p$ there
are an $r$ stronger than $q$ and an $a\in\frak A$ such that

\Centerline{$r\VVdP\,\dot x=\dot a=(\iota a)^{\centerdot}
=\dot\iota\dot a=\dot\iota\dot x$.}

\noindent As $q$ is arbitrary, $p\VVdP\,\dot\iota\dot x=\dot x$;  as
$p$ and $\dot x$ are arbitrary, $\VVdP\,\dot\iota$ is the identity.\ \QeD\
Putting these together,

\Centerline{$\VVdP\,(\pi^{-1})^{\centerdot}$ is the inverse of $\dot\pi$.}

\medskip

\quad{\bf (v)}
Suppose that $p\in\frak C^+$ and a $\Bbb P$-name $\dot x$ are such that

\Centerline{$p\VVdP\,\dot x\in\dot\frak A$ and $\dot\pi\dot x=\dot x$.}

\noindent For any $q$ stronger than $p$ there are an $r$ stronger than $q$
and an $a\in\frak A$ such that

\Centerline{$r\VVdP\,\dot a=\dot x=\dot\pi\dot x=\dot\pi\dot a
=(\pi a)^{\centerdot}$,}

\noindent and an $s$ stronger than $r$ such that
$s\Bcap a=s\Bcap\pi a$.   Now
$s\Bcap a=\pi(s\Bcap a)$ and
$s\Bcap a\in\frak C$.   If $s\Bcap a=0$, set $s'=s$;
then $s'\VVdP\,\dot x=0$.
Otherwise, set $s'=s\Bcap a$;  then $s'\VVdP\,\dot x=1$.   Thus in either
case we have an $s'$ stronger than $q$ such that
$s'\VVdP\,\dot x\in\{0,1\}$.   As $q$ is arbitrary,
$p\VVdP\,\dot x\in\{0,1\}$;  as $p$ and $\dot x$ are arbitrary,
$\VVdP\,\dot\pi$ has fixed-point subalgebra $\{0,1\}$.
}%end of proof of 556C

\leader{556D}{Regularly embedded \dvrocolon{subalgebras}}\cmmnt{ I am
trying to set these results out in maximal
generality, as usual.   However it seems that we need to move almost at
once to the case in which our subalgebra is regularly embedded, and we
have more effective versions of 556Ba and 556B(e-ii).

\medskip

\noindent}{\bf Proposition} Let $\frak A$ be a Boolean algebra,
not $\{0\}$, and   $\frak C$ a regularly embedded subalgebra of $\frak A$.
Let $\Bbb P$ be the
forcing notion $\frak C^+$, active downwards, and
for $a\in\frak A$ let $\dot a$ be the forcing name for $a$ over $\frak C$.

(a) For $p\in\frak C^+$ and $a$, $b\in\frak A$,

\Centerline{$p\VVdP\,\dot a=\dot b$}

\noindent iff $p\Bcap a=p\Bcap b$.

(b) Let $\dot{\Bsubseteq}$ be the forcing name for $\Bsubseteq$ over
$\frak C$.   Then for $p\in\frak C^+$ and $a$, $b\in\frak A$,

\Centerline{$p\VVdP\,\dot a\dot{\Bsubseteq}\dot b$}

\noindent iff $p\Bcap a\Bsubseteq b$.

\proof{ The point is just that $\upr(a,\frak C)=0$ only when $a=0$, because
infima in $\frak C$ are also infima in $\frak A$ (313N);  so that
$\upr(p\Bcap a\Bsetminus b,\frak C)=0$ iff $p\Bcap a\Bsubseteq b$, and
$\upr(p\Bcap(a\Bsymmdiff b),\frak C)=0$ iff $p\Bcap a=p\Bcap b$.
}%end of proof of 556D

\leader{556E}{Proposition} Let $\frak A$ be a Boolean algebra,
not $\{0\}$, $\frak C$ a
regularly embedded subalgebra of $\frak A$, $\Bbb P$ the forcing notion
$\frak C^+$, active downwards,
and $\dot\frak A$ the forcing name for $\frak A$ over
$\frak C$;  for $a\in\frak A$, write $\dot a$ for the forcing name for $a$
over $\frak C$.

(a) Let $\dot A$ be a $\Bbb P$-name, and set

\Centerline{$B
=\{q\Bcap a:q\in\frak C^+$, $a\in\frak A$, $q\VVdP\,\dot a\in\dot A\}$.}

\noindent Then for $d\in\frak A$ and $p\in\frak C^+$,

\Centerline{$p\VVdP\,\dot d$ is an upper bound for
$\dot A\cap\dot\frak A$}

\noindent iff $p\Bcap b\Bsubseteq d$ for every $b\in B$, and

\Centerline{$p\VVdP\,\dot d=\sup(\dot A\cap\dot\frak A)$}

\noindent iff $p\Bcap d=\sup_{b\in B}p\Bcap b$.

\wheader{556E}{0}{0}{0}{36pt}

(b)(i) If $\familyiI{a_i}$ is a family in $\frak A$ with supremum $a$, then

\Centerline{$\VVdP\,\dot a
=\sup_{i\in\check I}\dot a_i$.\footnote{See 5A3F for a note on the
interpretation of formulae of this kind.}}

\quad(ii) If $\familyiI{a_i}$ is a family in $\frak A$ with infimum $a$,
then

\Centerline{$\VVdP\,\dot a=\inf_{i\in\check I}\dot a_i$.}


(c) $\VVdP\,\sat(\dot\frak A)
\le\sat(\frak A)\varBspcheck$.\footnote{\smallerfonts{Of course I am not
asserting here that `$\eightVVdash_{\Bbb P}\,\sat(\frak A)\varBspcheck$
is a cardinal', only that
`$\eightVVdash_{\Bbb P}\,\sat(\dot\frak A)$ is a cardinal and
$\sat(\frak A)\varBspcheck$ is an ordinal'.}}

(d) $\VVdP\,\tau(\dot\frak A)\le\tau(\frak A)\varBspcheck$.

\wheader{556E}{0}{0}{0}{36pt}
\proof{{\bf (a)(i)} \Quer\ Suppose, if possible, that $b\in B$,
$p\Bcap b\notBsubseteq d$ and

\Centerline{$p\VVdP\,\dot d$ is an upper bound for
$\dot A\cap\dot\frak A$.}

\noindent Let $q\in\frak C^+$,
$a\in\frak A$ be such that $b=q\Bcap a$ and $q\VVdP\,\dot a\in\dot A$.
Then $p\Bcap q\ne 0$, so $p\Bcap q\in\frak C^+$ and

\Centerline{$p\Bcap q\VVdP\,\dot a\in\dot A\cap\dot\frak A$, therefore
$\dot a\dot{\Bsubseteq}\dot d$.}

\noindent It follows that $p\Bcap q\Bcap a\Bsubseteq d$ (556Db);  but
this contradicts the choice of $p$ and $b$.\ \Bang

Thus $p\Bcap b\Bsubseteq d$ whenever $b\in B$ and $p\VVdP\,\dot d$ is an
upper bound for $\dot A\cap\dot\frak A$.

\medskip

\quad{\bf (ii)} Next, suppose that $p\in\frak C^+$ and $d\in\frak A$ are
such that $p\Bcap b\Bsubseteq d$ for every $b\in B$.
Suppose that $q$ is stronger than $p$ and that $\dot x$ is a $\Bbb P$-name
such that $q\VVdP\,\dot x\in\dot A\cap\dot\frak A$.
If $r$ is stronger than $q$, there
are an $s$ stronger than $r$ and an $a\in\frak A$ such that
$s\VVdP\,\dot x=\dot a$.   In this case, $s\Bcap a\in B$ and
$s\Bcap a=p\Bcap s\Bcap a\Bsubseteq d$, so

\Centerline{$s\VVdP\,\dot x=\dot a
=(s\Bcap a)^{\centerdot}\dot{\Bsubseteq}\dot d$.}

\noindent As $r$ is arbitrary, $q\VVdP\,\dot x\dot{\Bsubseteq}\dot d$;  as
$q$ and $\dot x$ are arbitrary,

\Centerline{$p\VVdP\,\dot d$ is an upper bound for $\dot A$.}

\medskip

\quad{\bf (iii)} Putting these together, we see that
$p\Bcap b\Bsubseteq d$ for every $b\in B$
iff $p\VVdP\,\dot d$ is an upper bound for $\dot A\cap\dot\frak A$.

\medskip

\quad{\bf (iv)} Now suppose that
$p\VVdP\,\dot d=\sup(\dot A\cap\dot\frak A)$.   We know that $d$, and
therefore $p\Bcap d$, is an upper bound of $\{p\Bcap b:b\in B\}$.
If $e$ is any other upper bound of $\{p\Bcap b:b\in B\}$, then

\Centerline{$p\VVdP\,\dot e$ is an upper bound of $\dot A$, so
$\dot d\dot{\Bsubseteq}\dot e$}

\noindent and $p\Bcap d\Bsubseteq e$, by 556Db again;  thus
$p\Bcap d=\sup_{b\in B}p\Bcap b$.

\medskip

\quad{\bf (v)} Finally, suppose that $p\Bcap d=\sup_{b\in B}p\Bcap b$.
Suppose that $q$ is stronger than $p$ and that $\dot x$ is a
$\Bbb P$-name such that

\Centerline{$q\VVdP\,\dot x\in\dot\frak A$ is an upper bound of
$\dot A\cap\dot\frak A$.}

\noindent If $r$ is stronger than $q$, there are a
$s$ stronger than $r$
and a $c\in\frak A$ such that $s\VVdP\,\dot x=\dot c$.   In this case, by
(i), we must have $s\Bcap b\Bsubseteq c$ for every $b\in B$;
accordingly $s\Bcap d\Bsubseteq c$ (313Ba), so that

\Centerline{$s\VVdP\,\dot d\dot{\Bsubseteq}\dot c=\dot x$.}

\noindent As $r$ is arbitrary,

\Centerline{$q\VVdP\,\dot d\dot{\Bsubseteq}\dot x$;}

\noindent as $q$ and $\dot x$ are arbitrary,

\Centerline{$p\VVdP\,\dot d=\sup(\dot A\cap\dot\frak A)$.}

\medskip

{\bf (b)(i)} Of course

\Centerline{$\VVdP\,\dot a_i\dot{\Bsubseteq}\dot a$}

\noindent for every $i\in I$, so that

\Centerline{$\VVdP\,\dot a$ is an upper bound for
$\{\dot a_i:i\in\check I\}$.}

\noindent (Formally speaking:  if $p\in\frak C^+$ and $\dot x$ is a
$\Bbb P$-name such that $p\VVdP\,\dot x\in\{\dot a_i:i\in\check I\}$, then
for any $q$ stronger than $p$ there are an $r$ stronger than $q$ and an
$i\in I$ such that $r\VVdP\,\dot x=\dot a_i\dot{\Bsubseteq}\dot a$;  hence
$p\VVdP\,\dot x\dot{\Bsubseteq}\dot a$.)
In the other direction, suppose that $p\in\frak C^+$ and that
$\dot x$ is a $\Bbb P$-name such that

\Centerline{$p\VVdP\,\dot a_i\Bsubseteq\dot x\in\dot\frak A$ for every
$i\in\check I$.}

\noindent For any $q$ stronger than $p$ there are an $r$ stronger than $q$
and a $b\in\frak A$ such that $r\VVdP\,\dot x=\dot b$.   Now, for any
$i\in I$,

\Centerline{$r\VVdP\,\varchecki\in\check I$,
$\dot a_i\dot{\Bsubseteq}\dot x=\dot b$}

\noindent and therefore $r\Bcap a_i\Bsubseteq b$.   As $i$ is arbitrary,
$r\Bcap a\Bsubseteq b$ and $r\VVdP\,\dot a\dot{\Bsubseteq}\dot x$.
As $q$ is arbitrary, $p\VVdP\,\dot a\dot{\Bsubseteq}\dot x$; as $p$ and
$\dot x$ are arbitrary,

\Centerline{$\VVdP\,\dot a$ is the least upper bound of
$\{\dot a_i:i\in\check I\}$.}

\medskip

\quad{\bf (ii)} Now

$$\eqalign{\VVdP\,\inf_{i\in\check I}\dot a_i
&=1\dot\Bsetminus\sup_{i\in\check I}(1\dot\Bsetminus a_i)
=1\dot\Bsetminus\sup_{i\in\check I}(1\Bsetminus a_i)^{\centerdot}\cr
&=1\dot\Bsetminus(\sup_{i\in I}(1\Bsetminus a_i))^{\centerdot}
=(1\Bsetminus(\sup_{i\in I}(1\Bsetminus a_i)))^{\centerdot}
=(\inf_{i\in I}a_i)^{\centerdot}.\cr}$$

\medskip

{\bf (c)} \Quer\ Otherwise, there are a $p\in\frak C^+$ and a family
$\ofamily{\xi}{\kappa}{\dot x_{\xi}}$, where $\kappa=\sat(\frak A)$,
such that

\Centerline{$p\VVdP\,\dot x_{\xi}\in\dot\frak A^+$ for every
$\xi<\check\kappa$ and $\dot x_{\xi}\dot\Bcap\dot x_{\eta}=0$ whenever
$\xi<\eta<\check\kappa$.}

\noindent For each $\xi<\kappa$ choose $q_{\xi}$ stronger than $p$ and
$a_{\xi}\in\frak A$ such that $q_{\xi}\VVdP\,\dot x_{\xi}=\dot a_{\xi}$.
Then $q_{\xi}\VVdP\,\dot a_{\xi}\ne 0$, so $b_{\xi}=q_{\xi}\Bcap a_{\xi}$
is non-zero.   As $\sat(\frak A)=\kappa$, there must be $\xi<\eta<\kappa$
such that $b_{\xi}\Bcap b_{\eta}\ne 0$.   Set $r=q_{\xi}\Bcap q_{\eta}$;
then $r\in\frak C^+$ is stronger than $p$ and

\Centerline{$r\VVdP\,\dot x_{\xi}\dot\Bcap\dot x_{\eta}
  =\dot a_{\xi}\dot\Bcap\dot a_{\eta}
  =(a_{\xi}\Bcap a_{\eta})^{\centerdot}
  \ne 0$}

\noindent by 556Da, because $r\Bcap a_{\xi}\Bcap a_{\eta}\ne 0$.\ \Bang

\medskip

{\bf (d)} Let $A\subseteq\frak A$ be a set with cardinal $\kappa=\tau(\frak A)$
which $\tau$-generates $\frak A$.   Let $\dot A$ be the $\Bbb P$-name
$\{(\dot a,1):a\in A\}$;  then

\Centerline{$\VVdP\,\dot A\,\,\tau$-generates $\dot\frak A$ and
$\#(\dot A)\le\check\kappa$.}

\noindent\Prf\ (i) Suppose that $p\in\frak C^+$ and $\dot x$, $\dot\frak B$
are $\Bbb P$-names such that

\Centerline{$p\VVdP\,\dot\frak B$ is an order-closed subalgebra of
$\dot\frak A$ including $\dot A$, and $\dot x\in\dot\frak A$.}

\noindent Consider
$\frak D=\{a:a\in\frak A$, $p\VVdP\,\dot a\in\dot\frak B\}$.   Then
$\frak D$ is a subalgebra of $\frak A$, by 556Bb,
and is order-closed by (b)
here;  also $A\subseteq\frak D$, so $\frak D=\frak A$.   Next, for any $q$
stronger than $p$ there are an $r$ stronger than $q$ and an $a\in\frak A$
such that $r\VVdP\,\dot x=\dot a$;  since $a\in\frak D$,
$p\VVdP\,\dot a\in\dot\frak B$ and

\Centerline{$r\VVdP\,\dot x\in\dot\frak B$.}

\noindent As $p$, $\dot x$ and $\dot\frak B$ are arbitrary,

\Centerline{$\VVdP\,\dot A\,\,\tau$-generates $\dot\frak A$.}

\noindent (ii) If $\ofamily{\xi}{\kappa}{a_{\xi}}$ enumerates $A$, then

\Centerline{$\VVdP\,\dot A=\{\dot a_{\xi}:\xi<\check\kappa\}$ and
$\#(\dot A)\le\#(\check\kappa)\le\check\kappa$.  \Qed}

\noindent Accordingly

\Centerline{$\VVdP\,\tau(\dot\frak A)\le\check\kappa$.}
}%end of proof of 556E

\leader{556F}{Quotient \dvrocolon{forcing}}\cmmnt{ In 556A-556B I have
gone to pains to describe names
$\dot\frak A,\dot\Bsymmdiff,\dot\Bcap,\dot 0,\dot 1$ constituting a Boolean
algebra.   Of course we also have much simpler names
$\check\frak A,\check\Bsymmdiff,\check\Bcap,\check 0,\check 1$
also constituting
a Boolean algebra in the forcing language, and these must obviously be
related in some way to the construction here.
I think the details are worth bringing into the open.

\medskip

\noindent}{\bf Proposition}
Let $\frak A$ be a Boolean algebra, not $\{0\}$, and
$\frak C$ a subalgebra of $\frak A$.   Let $\Bbb P$ be the forcing notion
$\frak C^+$, active downwards, and $\dot\frak A$ the
forcing name for $\frak A$ over $\frak C$.

(a) Consider the $\Bbb P$-names

\Centerline{$\dot\psi=\{((\check a,\dot a),1):a\in\frak A\}$,
\quad$\dot{\Cal I}
=\{(\check a,p):p\in\frak C^+$, $a\in\frak A$, $p\Bcap a=0\}$.}

\noindent Then

\Centerline{$\VVdP\,\dot\psi$ is a Boolean homomorphism from
$\check\frak A$ onto
$\dot\frak A$, and its kernel is $\dot{\Cal I}$.}

(b) Now suppose that $\frak C$ is regularly embedded in
$\frak A$.   Set
$\dot\Bbb Q=(\dot\frak A^+,\dot{\Bsubseteqshort},\dot 1,\check{\downarrow})$
and let $\Bbb P*\dot\Bbb Q$ be the iterated forcing notion\cmmnt{ (5A3O)}.
Then $\RO(\Bbb P*\dot\Bbb Q)$ is isomorphic to the Dedekind
completion of $\frak A$.

(c) Suppose that $\frak C$ is regularly embedded in $\frak A$ and that
$\frak B$ is a Boolean algebra such that

\Centerline{$\VVdP\,\dot\frak A\cong\check\frak B$.}

\noindent Then the Dedekind completion $\widehat{\frak A}$ of $\frak A$ is
isomorphic to the Dedekind completion $\frak C\tensorhat\frak B$ of the
free product $\frak C\otimes\frak B$ of $\frak C$ and $\frak B$.

\wheader{556F}{0}{0}{0}{36pt}

\proof{{\bf (a)(i)} It is elementary that

\Centerline{$\VVdP\,\dot\psi:\check{\frak A}\to\dot\frak A$ is a
surjective function}

\noindent just because $\dot\frak A=\{(\dot a,1):a\in\frak A\}$.
By 556Bb,

\Centerline{$\VVdP\,\dot\psi$ is a ring homomorphism;
being surjective, it is a Boolean homomorphism.}

\medskip

\quad{\bf (ii)}\grheada\ If $p\in\frak C^+$ and
$\dot x$ is a $\Bbb P$-name such that $p\VVdP\,\dot x\in\dot{\Cal I}$, then
there are a $q\in\frak C^+$, an $a\in\frak A$, and an $r$ stronger than
both $p$ and $q$ such that

\Centerline{$q\Bcap a=0$,
\quad$r\VVdP\,\dot x=\check a$.}

\noindent In this case, $r\Bcap a=0$ so, by 556Ba,

\Centerline{$r\VVdP\,0=\dot a=\dot\psi(\check a)=\dot\psi(\dot x)$.}

\noindent As $p$ and $\dot x$ are arbitrary,

\Centerline{$\VVdP\,\dot{\Cal I}$ is included in the kernel of $\dot\psi$.}

\medskip

\qquad\grheadb\ If $p\in\frak C^+$ and $\dot x$ is a $\Bbb P$-name such
that

\Centerline{$p\VVdP\,\dot x\in\check\frak A$ and $\dot\psi(\dot x)=0$,}

\noindent then there are an
$a\in\frak A$ and a $q$ stronger than $p$ such that

\Centerline{$q\VVdP\,\dot x=\check a$ and
$\dot a=\dot\psi(\check a)=\dot\psi(\dot x)=0$.}

\noindent Now there is an $r$ stronger than $q$ such that $r\Bcap a=0$, so
that $(\check a,r)\in\dot\Cal I$ and

\Centerline{$r\VVdP\,\dot x=\check a\in\dot\Cal I$.}

\noindent As $p$ and $\dot x$ are arbitrary,

\Centerline{$\VVdP$ the kernel of $\dot\psi$ is included in $\dot{\Cal I}$,
so they coincide.}

\medskip

{\bf (b)(i)} In order to use the description of iterated forcing in 5A3O,
we need to set out an exact $\Bbb P$-name
for $\dot\frak A^+$.   If we say that $\dot\frak A^+$ abbreviates
$\{x:x\in\dot\frak A$, $x\ne 0\}$, and use the formulation of
Comprehension in {\smc Kunen 80}, Theorem 4.2, we get

\Centerline{$\dot\frak A^+
=\{(\dot x,p):\dot x\in\dom\dot\frak A$, $p\in\frak C^+$,
$p\VVdP\,\dot x\in\dot\frak A$ and $\dot x\ne 0\}$.}

\noindent Now 556Ab specifies $\dom\dot\frak A$ to be
$\{\dot a:a\in\frak A\}$, so we get

$$\eqalign{\dot\frak A^+
&=\{(\dot a,p):a\in\frak A,\,p\in\frak C^+,\,
   p\VVdP\,\dot a\in\dot\frak A\text{ and }\dot a\ne 0\}\cr
&=\{(\dot a,p):a\in\frak A,\,p\in\frak C^+,\,
   p\VVdP\,\dot a\ne 0\},\cr}$$

\Centerline{$\dom\dot\frak A^+
=\{\dot a:a\in\frak A$, $\notVVdash_{\Bbb P}\,\dot a=0\}
=\{\dot a:a\in\frak A^+\}$}

\noindent by 556Da.

\medskip

\quad{\bf (ii)} 5A3O now tells us that the underlying set of
$\Bbb P*\dot\Bbb Q$ is to be

\Centerline{$P
=\{(p,\dot a):p\in\frak C^+$, $a\in\frak A^+$, $p\VVdP\,\dot a\ne 0\}$.}

\noindent For $p\in\frak C^+$ and $a\in\frak A$,

$$\eqalignno{p\VVdP\,\dot a\ne 0
&\iff\text{ for every }q\text{ stronger than }p,
  \,q\notVVdash_{\Bbb P}\,\dot a=0\cr
&\iff\text{ for every non-zero }q\Bsubseteq p,
  \,q\Bcap a\ne 0\cr}$$

\noindent (556Da).   So $P$ is just

\Centerline{$\{(p,\dot a):p\in\frak C^+$, $a\in\frak A$, $q\Bcap a\ne 0$
  whenever $q\in\frak C$ and $0\ne q\Bsubseteq p\}$.}

\noindent Next, for $(p,\dot a)$, $(q,\dot b)\in P$,

$$\eqalign{(p,\dot a)\text{ is stronger than }(q,\dot b)
&\iff p\Bsubseteq q\text{ and }p\VVdP\,\dot a\dot{\Bsubseteq}\dot b\cr
&\iff p\Bsubseteq q\text{ and }p\Bcap a\Bsubseteq b\cr}$$

\noindent (556Db).

\medskip

\quad{\bf (iii)} We can define a function $f:P\to\frak A^+$ by setting

\Centerline{$f(p,\dot a)=p\Bcap a$}

\noindent whenever $(p,\dot a)\in P$.   \Prf\ If you look at the definition
of $\dot a$ in 556A, you will see that $((b,1),1)=(\check b,1)$
belongs to $\dot a$ iff
$b\Bsubseteq a$, so that $\dot a=\dot b$ only when $a=b$;  thus $f$ is a
function from $P$ to $\frak A$.
And the definition of $P$ ensures that $f(p,\dot a)\ne 0$
whenever $(p,\dot a)\in P$.\ \Qed

\medskip

\quad{\bf (iv)}\grheada\
If $(p,\dot a)$ is stronger than $(q,\dot b)$ in $P$, then
$p\Bsubseteq q$ and $p\Bcap a\Bsubseteq b$, so
$f(p,\dot a)\Bsubseteq f(q,b)$.

\medskip

\qquad\grheadb\ If $a\in\frak A^+$, then (because $\frak C$ is regularly
embedded) $C=\{q:a\Bsubseteq q\in\frak C\}$ does not have infimum $0$ in
$\frak C$;  let $p\in\frak C^+$ be a lower bound for $C$.   Then
$(p,\dot a)\in P$, and $f(p,\dot a)\Bsubseteq a$.   Thus $f[P]$ is
order-dense in $\frak A$.

\medskip

\qquad\grheadc\ If $(p,\dot a)$, $(q,\dot b)$ are incompatible in
$P$, then $f(p,\dot a)\Bcap f(q,\dot b)=0$.   \Prf\Quer\ Otherwise,
$c=p\Bcap a\Bcap q\Bcap b$ is non-zero.   Let $r\in\frak C^+$ be such that
$(r,\dot c)\in P$.   Since $r\Bsetminus(p\Bcap q)\VVdP\,\dot c=0$,
$r\Bsubseteq p\Bcap q$;  since $c\Bsubseteq a\Bcap b$,
$(r,\dot c)$ is stronger than both $(p,\dot a)$ and
$(q,\dot b)$, which is supposed to be impossible.\ \Bang\Qed

\medskip

\quad{\bf (v)} Thinking of $\frak A^+$ as an order-dense subset of
$\widehat{\frak A}$, and of $f$ as a function from $P$ to
$\widehat{\frak A}^+$, 514Sa tells us that

\Centerline{$\RO(\Bbb P*\dot\Bbb Q)
=\RO(P)\cong\widehat{\frak A}$,}

\noindent as claimed.

\medskip

{\bf (c)} For free products of Boolean algebras, see \S315;  for Dedekind
completions, see \S314.   This part can be regarded as a corollary of (b)
(see 556Ya-556Yb), but can also be approached directly, as follows.

\medskip

\quad{\bf (i)} Let $\dot\theta$ be a $\Bbb P$-name such that

\Centerline{$\VVdP\,\dot\theta:\dot\frak A\to\check\frak B$ is an
isomorphism.}

\noindent Set

\Centerline{$R=\{(p,b,a):p\in\frak C^+$, $b\in\frak B^+$, $a\in\frak A^+$,
$a\Bsubseteq p$ and $p\VVdP\,\dot\theta(\dot a)=\check b\}$,}

\noindent and give $R$ the ordering induced by the product partial ordering
of $\frak C^+\times\frak B^+\times\frak A^+$.

\medskip

\quad{\bf (ii)} $\RO^{\downarrow}(R)\cong\frak C\tensorhat\frak B$.   \Prf\
Define $f:R\to(\frak C\tensorhat\frak B)^+$ by setting
$f(p,b,a)=p\otimes b$.

\medskip

\qquad\grheada\ Of course $f(p,b,a)\Bsubseteq f(p',b',a')$ whenever
$(p,b,a)\le(p',b',a')$.

\medskip

\qquad\grheadb\ If $(p_0,b_0,a_0)$ and $(p_1,b_1,a_1)$ belong to $R$ and
$f(p_0,b_0,a_0)\Bcap f(p_1,b_1,a_1)\ne 0$, set $p=p_0\Bcap p_1$,
$b=b_0\Bcap b_1$ and $a=a_0\Bcap a_1$.   Then $p\in\frak C^+$,
$b\in\frak B^+$, $a\Bsubseteq p$ and

\Centerline{$p\VVdP\,\dot\theta(\dot a)
=\dot\theta(\dot a_0\dot\Bcap\dot a_1)
=\dot\theta(\dot a_0)\check\Bcap\dot\theta(\dot a_1)
=\check b_0\check\Bcap\check b_1
=\check b$.}

\noindent As $p\VVdP\,\dot\theta(\dot a)\ne\check 0$, $a$ cannot be $0$,
and $(p,b,a)\in R$;  so $(p_0,b_0,a_0)$ and $(p_1,b_1,a_1)$ are downwards
compatible in $R$.

\medskip

\qquad\grheadc\ If $d\in(\frak C\tensorhat\frak B)^+$, there are
$p_0\in\frak C^+$, $b\in\frak B^+$ such that
$p_0\otimes b\Bsubseteq d$.   Now there is a $\Bbb P$-name $\dot x$ such
that

\Centerline{$\VVdP\,\dot x\in\dot\frak A$ and
$\dot\theta(\dot x)=\check b$.}

\noindent Let $p$ stronger than $p_0$ and $a_0\in\frak A$ be such that
$p\VVdP\,\dot a_0=\dot x$, and set $a=p\Bcap a_0$;  then

\Centerline{$p\VVdP\,\dot a=\dot a_0$ so
$\dot\theta(\dot a)=\dot\theta(\dot a_0)=\check b$.}

\noindent As in ($\beta$), it follows that $a\ne 0$, so that
$(p,b,a)\in R$;  now $f(p,b,a)\Bsubseteq d$.   As $d$ is arbitrary,
$f[R]$ is order-dense in $\frak C\tensorhat\frak B$.

\medskip

\qquad\grheadd\ Thus $f$ satisfies the conditions of 514Sa
and $\RO^{\downarrow}(R)\cong\frak C\tensorhat\frak B$.\ \Qed

\medskip

\quad{\bf (iii)} $\RO^{\downarrow}(R)\cong\widehat{\frak A}$.   \Prf\
Define $g:R\to\widehat{\frak A}^+$ by setting $g(p,b,a)=a$ for
$(p,b,a)\in R$.

\medskip

\qquad\grheada\ Of course $g(p,b,a)\Bsubseteq g(p',b',a')$ whenever
$(p,b,a)\le(p',b',a')$ in $R$.

\medskip

\qquad\grheadb\ Suppose that $(p_0,b_0,a_0)$, $(p_1,b_1,a_1)\in R$ and that
$a=a_0\Bcap a_1\ne 0$.   Set $p=p_0\Bcap p_1$ and $b=b_0\Bcap b_1$.   Then
$p\Bsupseteq a\ne 0$ and

\Centerline{$p\VVdP\,\dot\theta(\dot a)=\check b$}

\noindent as in (ii-$\beta$).   Since $p\Bcap a\ne 0$,
$p\notVVdash_{\Bbb P}\,\dot a=\dot 0$ (556Da), so
$p\notVVdash_{\Bbb P}\,\check b=\check 0$ and $b\ne 0$.   Thus
$(p,b,a)\in R$ and $(p_0,b_0,a_0)$, $(p_1,b_1,a_1)$ are compatible
downwards in $R$.

\medskip

\qquad\grheadc\ If $d\in\widehat{\frak A}^+$, there is an $a_0\in\frak A^+$
such that $a_0\Bsubseteq d$.   In this case,
$\notVVdash_{\Bbb P}\,\dot a_0=\dot 0$ so
there is a $p_0\in\frak C^+$
such that $p_0\VVdP\,\dot a_0\ne\dot 0$.   Now
$p_0\VVdP\,\dot\theta(\dot a_0)\in\check\frak B$ so there are a $p$
stronger than $p_0$ and a $b\in\frak B$ such that
$p\VVdP\,\dot\theta(\dot a_0)=\check b$.   Set $a=p\Bcap a_0$;  then

\Centerline{$p\VVdP\,\dot a=\dot a_0\ne\dot 0$ and
$\check b=\dot\theta(\dot a)\ne\check 0$.}

\noindent Consequently $a$ and $b$ are both non-zero and $(p,b,a)\in R$,
while $g(p,b,a)\Bsubseteq d$.

\medskip

\qquad\grheadd\ Thus $g$ satisfies the conditions of 514Sa
and $\RO^{\downarrow}(R)\cong\widehat{\frak A}$.\ \Qed

\medskip

\quad{\bf (iv)} Putting these together, $\frak C\tensorhat\frak B$ and
$\widehat{\frak A}$ are isomorphic.
}%end of proof of 556F

\leader{556G}{Proposition} Let $\frak A$ be a
Dedekind complete Boolean algebra, not $\{0\}$,
$\frak C$ a regularly embedded
subalgebra of $\frak A$, $\Bbb P$ the forcing notion
$\frak C^+$, active downwards,
and $\dot\frak A$ the forcing name for $\frak A$ over $\frak C$.

(a) Whenever $p\in\frak C^+$ and $\dot x$ is a $\Bbb P$-name such that

\Centerline{$p\VVdP\,\dot x\in\dot\frak A$,}

\noindent there is an $a\in\frak A$ such that

\Centerline{$p\VVdP\,\dot x=\dot a$,}

\noindent where $\dot a$ is the forcing name for $a$ over $\frak C$.

(b) $\VVdP\,\dot\frak A$ is Dedekind complete.

\proof{{\bf (a)} Set

\Centerline{$B=\{q\Bcap b:q\in\frak C^+$ is stronger than $p$, $b\in\frak A$,
$q\VVdP\,\dot b=\dot x\}$,
\quad$a=\sup B$.}

\noindent Then 556Ea tells us that

\Centerline{$p\VVdP\,\dot a=\sup\{\dot x\}=\dot x$.}

\medskip

{\bf (b)} Suppose that $p\in\frak C^+$ and that $\dot A$ is a
$\Bbb P$-name such that $p\VVdP\,\dot A\subseteq\dot\frak A$.   Set

\Centerline{$B=\{q\Bcap a:a\in\frak A$, $q\in\frak C^+$ and
$q\VVdP\,\dot a\in\dot A\}$,
\quad$d=\sup B$.}

\noindent Then $p\Bcap d=\sup_{b\in B}p\Bcap b$, so
$p\VVdP\,\dot d=\sup\dot A$,
by 556Ea.   As $p$ and $\dot A$ are arbitrary,

\Centerline{$\VVdP\,\dot\frak A$ is Dedekind complete.}
}%end of proof of 556G

\leader{556H}{$L^0(\frak A)$:  Proposition} Let $\frak A$ be a
Dedekind complete Boolean algebra, not $\{0\}$,
$\frak C$ a regularly embedded
subalgebra of $\frak A$, $\Bbb P$ the forcing notion
$\frak C^+$, active downwards,
and $\dot\frak A$ the forcing name for $\frak A$ over
$\frak C$.   For $a\in\frak A$ let $\dot a$ be the forcing name for $a$
over $\frak C$.

(a)(i) For every $u\in L^0(\frak A)$,

\Centerline{$\VVdP\,\dot u\in L^0(\dot\frak A)$}

\noindent where $\dot u$ is the forcing name for $u$ over $\frak C$.

\quad(ii) If $u$, $v\in L^0(\frak A)$ and $\VVdP\,\dot u=\dot v$, then
$u=v$.

(b) For $u$, $v\in L^0(\frak A)$ and $\alpha\in\Bbb R$,

$$\eqalign{\VVdP\,\dot u+\dot v&=(u+v)^{\centerdot},\cr
-\dot u&=(-u)^{\centerdot}, \cr
\dot u\vee\dot v&=(u\vee v)^{\centerdot},\cr
\dot u\times\dot v&=(u\times v)^{\centerdot},\cr
\check\alpha\dot u&=(\alpha u)^{\centerdot}.\cr}$$

\noindent If $u\le v$, then $\VVdP\,\dot u\le\dot v$.

(c) If $\familyiI{u_i}$ is a family in $L^0(\frak A)$
with supremum $u\in L^0(\frak A)$, then

\Centerline{$\VVdP\,\dot u=\sup_{i\in\check I}\dot u_i$
in $L^0(\dot\frak A)$.}

(d) If $p\in\frak C^+$ and $\dot w$ is a
$\Bbb P$-name such that $p\VVdP\,\dot w\in L^0(\dot\frak A)$, then there is
a $u\in L^0(\frak A)$ such that

\Centerline{$p\VVdP\,\dot w=\dot u$.}

(e) If $\sequencen{u_n}$ is a sequence in $L^0(\frak A)$, then the
following are equiveridical:

\quad(i) $\sequencen{u_n}$ is order*-convergent to
$0$\cmmnt{ (definition: 367A)},

\quad(ii) $\VVdP\,\sequencen{\dot u_n}$ is order*-convergent to $0$.

\proof{{\bf (a)(i)} Examining the definition in 556Af, we see that we have

\Centerline{$\VVdP\,\dot u$ is a function from $\Bbb Q$ to $\dot\frak A$
and $\dot u(\check\alpha)=\Bvalue{u>\alpha}^{\centerdot}$}

\noindent for every $\alpha\in\Bbb Q$.   Now 556Eb tells us that,
for every $\alpha\in\Bbb Q$,

$$\eqalignno{\VVdP\,\dot u(\check\alpha)
&=\Bvalue{u>\alpha}^{\centerdot}
=(\sup_{\beta\in\Bbb Q,\beta>\alpha}\Bvalue{u>\beta})^{\centerdot}
=\sup_{\beta\in\Bbb Q,\beta>\check\alpha}\Bvalue{u>\beta}^{\centerdot}
=\sup_{\beta\in\Bbb Q,\beta>\check\alpha}\dot u(\check\beta),\cr
0
&=(\inf_{n\in\Bbb N}\Bvalue{u>n})^{\centerdot}
=\inf_{n\in\Bbb N}\Bvalue{u>n}^{\centerdot}
=\inf_{n\in\Bbb N}\dot u(\check n),\cr
1
&=(\sup_{n\in\Bbb N}\Bvalue{u>-n})^{\centerdot}
=\sup_{n\in\Bbb N}\Bvalue{u>-n}^{\centerdot}
=\sup_{n\in\Bbb N}\dot u(-\check n),\cr}$$

\noindent so

\Centerline{$\VVdP\,\dot u\in L^0(\dot\frak A)$,}

\noindent and I can write $\Bvalue{\dot u>\check\alpha}$ for the
$\Bbb P$-name $\dot u(\check\alpha)$, so that

\Centerline{$\VVdP\,\Bvalue{\dot u>\check\alpha}
=\Bvalue{u>\alpha}^{\centerdot}$}

\noindent for every $\alpha\in\Bbb Q$.

\medskip

\quad{\bf (ii)} For any $\alpha\in\Bbb Q$,

\Centerline{$\VVdP\,\Bvalue{u>\alpha}^{\centerdot}
=\Bvalue{\dot u>\check\alpha}
=\Bvalue{\dot v>\check\alpha}
=\Bvalue{u>\alpha}^{\centerdot}$.}

\noindent By 556Da, $\Bvalue{u>\alpha}=\Bvalue{v>\alpha}$.   As $\alpha$ is
arbitrary, $u=v$.

\medskip

{\bf (b)(i)} Suppose $u$, $v\in L^0(\frak A)$.   By 364D,
we have

\Centerline{$\Bvalue{u+v>\alpha}
=\sup_{\beta\in\Bbb Q}\Bvalue{u>\beta}\Bcap\Bvalue{v>\alpha-\beta}$}

\noindent for every $\alpha\in\Bbb Q$.   If $\alpha$, $\beta\in\Bbb Q$,

\Centerline{$\VVdP\,\Bvalue{\dot u>\check\beta}
   \dot\Bcap\Bvalue{\dot v>\check\alpha-\check\beta}
=\Bvalue{u>\beta}^{\centerdot}\dot\Bcap\Bvalue{v>\alpha-\beta}^{\centerdot}
=(\Bvalue{u>\beta}\Bcap\Bvalue{v>\alpha-\beta})^{\centerdot}$.}

\noindent Taking the supremum over $\beta$, as in 556E(b-i),

$$\eqalign{\VVdP\,\Bvalue{\dot u+\dot v>\check\alpha}
&=\sup_{\beta\in\Bbb Q}\Bvalue{\dot u>\beta}
  \dot\Bcap\Bvalue{\dot v>\check\alpha-\beta}
=\sup_{\beta\in\check\Bbb Q}\Bvalue{\dot u>\beta}
  \dot\Bcap\Bvalue{\dot v>\check\alpha-\beta}\cr
&=(\sup_{\beta\in\Bbb Q}\Bvalue{u>\beta}
   \Bcap\Bvalue{v>\alpha-\beta})^{\centerdot}
=\Bvalue{u+v>\alpha}^{\centerdot}
=\Bvalue{(u+v)^{\centerdot}>\check\alpha}\cr}$$

\noindent for every $\alpha\in\Bbb Q$, and

\Centerline{$\VVdP\,\dot u+\dot v=(u+v)^{\centerdot}$.}

\medskip

\quad{\bf (ii)} Concerning $u\vee v$, we have

$$\eqalign{\VVdP\,\Bvalue{(u\vee v)^{\centerdot}>\check\alpha}
&=\Bvalue{u\vee v>\alpha}^{\centerdot}
=\Bvalue{u>\alpha}^{\centerdot}
  \dot\Bcup\Bvalue{v>\alpha}^{\centerdot}\cr
&=\Bvalue{\dot u>\check\alpha}\dot\Bcup\Bvalue{\dot v>\check\alpha}
=\Bvalue{\dot u\vee\dot v>\check\alpha}\cr}$$

\noindent for every $u$, $v\in L^0(\frak A)$ and $\alpha\in\Bbb Q$, so

\Centerline{$\VVdP\,\dot u\vee\dot v=(u\vee v)^{\centerdot}$;}

\noindent it follows that if $u\ge 0$ then
$\VVdP\,\dot u=\dot u\vee 0\ge 0$.

\medskip

\quad{\bf (iii)}
If $u$, $v\in L^0(\frak A)^+$, $\alpha\in\Bbb Q$
and $\alpha\ge 0$, then, just as in (i),

$$\eqalign{\VVdP\,\Bvalue{\dot u\times\dot v>\check\alpha}
&=\sup_{\beta\in\Bbb Q,\beta>0}\Bvalue{\dot u>\beta}
  \dot\Bcap\Bvalue{\dot v>\Bover{\check\alpha}{\beta}}
=(\sup_{\beta\in\Bbb Q,\beta>0}\Bvalue{u>\beta}
   \Bcap\Bvalue{v>\Bover{\alpha}{\beta}})^{\centerdot}\cr
&=\Bvalue{u\times v>\alpha}^{\centerdot}
=\Bvalue{(u\times v)^{\centerdot}>\check\alpha};\cr}$$

\noindent so $\VVdP\,\dot u\times\dot v=(u\times v)^{\centerdot}$.
Using the
distributive law we see that the same is true for all $u$,
$v\in L^0(\frak A)$.

\wheader{556H}{6}{2}{2}{36pt}

\quad{\bf (iv)} Take $\alpha\in\Bbb R$ and set
$w=\alpha\chi 1\in L^0(\frak A)$.   If $\beta\in\Bbb Q$ and $\beta<\alpha$,
then

\Centerline{$\VVdP\,\Bvalue{\check\alpha\chi 1>\check\beta}
=1=\dot 1=\Bvalue{w>\beta}^{\centerdot}
=\Bvalue{\dot w>\check\beta}$;}

\noindent while if $\beta\ge\alpha$,

\Centerline{$\VVdP\,\Bvalue{\check\alpha\chi 1>\check\beta}
=0=\dot 0=\Bvalue{w>\beta}^{\centerdot}
=\Bvalue{\dot w>\check\beta}$.}

\noindent So

\Centerline{$\VVdP\,\Bvalue{\check\alpha\chi 1>\beta}
=\Bvalue{\dot w>\beta}\text{ for every }
  \beta\in\Bbb Q,\text{ and }
\check\alpha\chi 1=\dot w=(\alpha\chi 1)^{\centerdot}$.}

\noindent Putting this together with (iii), we have

\Centerline{$\VVdP\,\check\alpha\dot u
=(\check\alpha\chi 1)\times\dot u
=(\alpha\chi 1)^{\centerdot}\times\dot u
=(\alpha\chi 1\times u)^{\centerdot}
=(\alpha u)^{\centerdot}$}

\noindent for every $u\in L^0(\frak A)$.   In particular, taking
$\alpha=-1$, $\VVdP\,-\dot u=(-u)^{\centerdot}$.

\medskip

\quad{\bf (v)} Finally, if $u\le v$ then $u\vee v=v$, so

\Centerline{$\VVdP\,\dot u\vee\dot v=\dot v$ and $\dot u\le\dot v$.}

\medskip

{\bf (c)(i)} It will help to note that the criterion in
364L(a-ii)

\inset{if $A\subseteq L^0(\frak A)$ is
non-empty, then $v\in L^0(\frak A)$ is the supremum of $A$ in
$L^0(\frak A)$ iff $\Bvalue{v>\alpha}=\sup_{u\in A}\Bvalue{u>\alpha}$ in
$\frak A$ for every $\alpha\in\Bbb R$}

\noindent can be replaced by

\inset{if $A\subseteq L^0(\frak A)$ is
non-empty, then $v\in L^0(\frak A)$ is the supremum of $A$ in
$L^0(\frak A)$ iff $\Bvalue{v>\alpha}=\sup_{u\in A}\Bvalue{u>\alpha}$ in
$\frak A$ for every $\alpha\in\Bbb Q$.}

\noindent\Prf\ If the weaker condition is satisfied, and $\alpha\in\Bbb R$,
then

$$\eqalign{\Bvalue{v>\alpha}
&=\sup_{\beta\in\Bbb Q,\beta\ge\alpha}\Bvalue{v>\beta}
=\sup_{\beta\in\Bbb Q,\beta\ge\alpha}\sup_{u\in A}\Bvalue{u>\beta}\cr
&=\sup_{u\in A}\sup_{\beta\in\Bbb Q,\beta\ge\alpha}\Bvalue{u>\beta}
=\sup_{u\in A}\Bvalue{u>\alpha}.  \text{ \Qed}\cr}$$

\medskip

\quad{\bf (ii)} Now 556E(b-i) tells us that

\Centerline{$\VVdP\,\Bvalue{\dot u>\check\alpha}
=\sup_{i\in\check I}\Bvalue{\dot u_i>\check\alpha}$}

\noindent for every $\alpha\in\Bbb Q$, so

\Centerline{$\VVdP\,\dot u=\sup_{i\in\check I}\dot u_i$.}

\medskip

{\bf (d)} For each $\alpha\in\Bbb Q$ we have an $a_{\alpha}\in\frak A$ such
that $p\VVdP\,\dot a_{\alpha}=\Bvalue{\dot w>\check\alpha}$ (556Ga);  since
$p\VVdP\,\dot a=(a\Bcap p)^{\centerdot}$ for every $a\in\frak A$, we can
suppose that $a_{\alpha}\Bsubseteq p$ for every $\alpha$.   Now we find
that if $\alpha\in\Bbb Q$ and
$b_{\alpha}=\sup_{\beta\in\Bbb Q,\beta>\alpha}a_{\alpha}$, then

\Centerline{$p\VVdP\,\dot b_{\alpha}
=\sup_{\beta\in\Bbb Q,\beta>\check\alpha}\Bvalue{\dot w>\beta}
=\dot a_{\alpha}$,}

\noindent so $a_{\alpha}=b_{\alpha}$.   Similarly, if
$b=\inf_{n\in\Bbb N}a_n$ and $c=\sup_{n\in\Bbb N}a_{-n}$,

$$\eqalign{p\VVdP\,\dot b
&=\inf_{n\in\Bbb N}\Bvalue{\dot w>n}=0,\cr
\dot c&=\sup_{n\in\Bbb N}\Bvalue{\dot w>-n}=1\cr}$$

\noindent and $b=0$, $c=p$.   It is now easy to check that
there is a $u\in L^0(\frak A)$ such that

$$\eqalign{\Bvalue{u>\alpha}
&=a_{\alpha}\text{ if }\alpha\in\Bbb Q
  \text{ and }\alpha>0,\cr
&=a_{\alpha}\Bcup(1\Bsetminus p)
  \text{ for other }\alpha\in\Bbb Q,\cr}$$

\noindent and that $p\VVdP\,\dot u=\dot w$.

\medskip

{\bf (e)} Recall that
$\sequencen{u_n}$ order*-converges to $0$ iff $\sequencen{u_n}$ is
order-bounded and
$0=\inf_{n\in\Bbb N}\sup_{m\ge n}|u_n|$ (367G);  and we shall have a
similar formulation in the forcing language.   So if $\sequencen{u_n}$
order*-converges to $0$, then

\Centerline{$\VVdP\,\sup_{m\ge\check n}|\dot u_m|
=(\sup_{m\ge n}|u_m|)^{\centerdot}$}

\noindent for every $n\in\Bbb N$, and

\Centerline{$\VVdP\,\inf_{n\in\Bbb N}\sup_{m\ge n}|\dot u_m|
=(\inf_{n\in\Bbb N}\sup_{m\ge n}|u_m|)^{\centerdot}=0$, so
$\sequencen{\dot u_n}\to^*0$.}

Conversely, if $\VVdP\,\sequencen{\dot u_n}$ order*-converges to
$0$, then $\VVdP\,\sequencen{\dot u_n}$ is order-bounded, and there is a
$\Bbb P$-name $\dot w$ such that

\Centerline{$\VVdP\,\dot w\in L^0(\dot\frak A)$,
$|\dot u_n|\le\dot w$ for every $n\in\Bbb N$.}

\noindent By (d), there is a $v\in L^0(\frak A)$ such that
$\VVdP\,\dot w=\dot v$, so that

\Centerline{$\VVdP\,(v\vee|u_n|)^{\centerdot}=\dot w\vee|\dot u_n|=\dot v$
for every $n\in\Bbb N$}

\noindent and $|u_n|\le v$ for every $n$ (use (a-ii)).
We can therefore repeat the calculation just above to see that

\Centerline{$\VVdP\,(\inf_{n\in\Bbb N}\sup_{m\ge n}|u_m|)^{\centerdot}
=\inf_{n\in\Bbb N}\sup_{m\ge n}|\dot u_m|=0$,}

\noindent so that $\inf_{n\in\Bbb N}\sup_{m\ge n}|u_m|=0$ and
$\sequencen{u_n}$ order*-converges to $0$.
}%end of proof of 556H

\leader{556I}{Proposition} Let $\frak A$ be a Boolean algebra,
not $\{0\}$, and $\frak C$ a regularly embedded
subalgebra of $\frak A$.   Let $\Bbb P$ be the forcing notion
$\frak C^+$, active downwards, and
$\pi:\frak A\to\frak A$ a Boolean homomorphism fixing every point of
$\frak C$;  let $\dot\pi$ be the forcing name for $\pi$ over $\frak C$.

(a) $\pi$ is injective iff $\VVdP\,\dot\pi$ is injective.

(b) If $\pi$ is order-continuous, then

\Centerline{$\VVdP\,\dot\pi$ is order-continuous.}

(c) If $\pi$ has a support $\supp\pi$\cmmnt{ (definition:  381Bb)}, then

\Centerline{$\VVdP\,(\supp\pi)^{\centerdot}$ is the support of $\dot\pi$.}

\proof{{\bf (a)} We saw in 556C(a-ii) that if $\pi$ is injective then
$\VVdP\,\dot\pi$ is injective.   Now suppose
that $\pi$ is not injective;  let $a\in\frak A^+$ be such that $\pi a=0$.
Then $\VVdP\,\dot\pi\dot a=0$.
$1\Bcap a\ne 0$, so $\notVVdash_{\Bbb P}\,\dot a=0$, by 556Da, and

\Centerline{$\notVVdash_{\Bbb P}\,\dot\pi$ is injective.}

\medskip

{\bf (b)} Take $p\in\frak C^+$ and a $\Bbb P$-name $\dot A$ such that

\Centerline{$p\VVdP\,\dot A\subseteq\dot\frak A$ and $\sup\dot A=1$.}

\noindent Set
$B=\{q\Bcap a:q\in\frak C^+$, $a\in\frak A$, $q\VVdP\,\dot a\in\dot A\}$.
Then $\sup_{b\in B}p\Bcap b=p\Bcap 1=p$, by 556Ea.
Because $\pi$ is order-continuous,

\Centerline{$p\Bcap 1=p
=\pi p=\sup_{b\in B}\pi(p\Bcap b)=\sup_{b\in B}p\Bcap\pi b$.}

\noindent Consider

\Centerline{$C=\{q\Bcap a:q\in\frak C^+$, $a\in\frak A$,
$q\VVdP\,\dot a\in\dot\pi[\dot A]\}$.}

\noindent Then $\pi[B]\subseteq C$.   \Prf\ If $q\in\frak C^+$, $a\in\frak A$
and $q\VVdP\,\dot a\in\dot A$, then

\Centerline{$q\VVdP\,(\pi a)^{\centerdot}
=\dot\pi\dot a\in\dot\pi[\dot A]$}

\noindent so

\Centerline{$\pi(q\Bcap a)=q\Bcap\pi a\in C$.  \Qed}

\noindent Accordingly

\Centerline{$\{p\Bcap c:c\in C\}\supseteq\{p\Bcap\pi b:b\in B\}$}

\noindent must have supremum $p$, and $p\VVdP\,\sup\dot\pi[\dot A]=1$.

As $p$ and $\dot A$ are arbitrary,

\Centerline{$\VVdP\,\sup\dot\pi[\dot A]=1$ whenever
$\dot A\subseteq\frak A$ and $\sup\dot A=1$,
so $\dot\pi$ is order-continuous}

\noindent (313L(b-iii)).

\medskip

{\bf (c)(i)} $\VVdP$ if $x\in\dot\frak A$ and
$x\dot\Bcap(\supp\pi)^{\centerdot}=0$ then $\dot\pi x=x$.
\Prf\ Take $p\in\frak C^+$ and a $\Bbb P$-name $\dot x$ such that

\Centerline{$p\VVdP\,\dot x\in\dot\frak A$
and $\dot x\dot\Bcap(\supp\pi)^{\centerdot}=0$.}

\noindent For any $q$ stronger than $p$ there are an $r$ stronger than $q$
and an $a\in\frak A$ such that

\Centerline{$r\VVdP\,\dot x=\dot a$, $(a\Bcap\supp\pi)^{\centerdot}=0$;}

\noindent now $r\Bcap a\Bcap\supp\pi=0$ (556Da).   In this case,

\Centerline{$r\VVdP\,\dot\pi\dot x=\dot\pi(r\Bcap a)^{\centerdot}
=(\pi(r\Bcap a))^{\centerdot}=(r\Bcap a)^{\centerdot}=\dot x$.}

\noindent As $q$ is arbitrary, $p\VVdP\,\dot\pi\dot x=\dot x$;  as $p$ and
$\dot x$ are arbitrary, we have the result.\ \Qed

Now 381Ei, applied in the forcing language, tells us that

\Centerline{$\VVdP\,(\supp\pi)^{\centerdot}$ supports $\dot\pi$.}

\medskip

\quad{\bf (ii)} $\VVdP$ if $x\in\dot\frak A$ supports $\dot\pi$, then
$x\dot{\Bsupseteq}(\supp\pi)^{\centerdot}$.
\Prf\ Take $p\in\frak C^+$ and a $\Bbb P$-name $\dot x$ such that

\Centerline{$p\VVdP\,\dot x\in\dot\frak A$ supports $\dot\pi$.}

\noindent Then for any $q$ stronger than $p$ we have an $r$ stronger than
$q$ and an $a\in\frak A$ such that $r\VVdP\,\dot x=\dot a$.
Set $b=a\Bcup(1\Bsetminus r)$;  then
$r\VVdP\,\dot b=\dot a$ supports $\dot\pi$.   \Quer\ If $b$ does not
support $\pi$, then there is a non-zero $d\Bsubseteq 1\Bsetminus b$ such
that $d\Bcap\pi d=0$ (381Ei again).   Since $r\Bcap d=d\ne 0$, there is an
$s$ stronger than $r$ such that $s\VVdP\,\dot d\ne 0$.   Now

\Centerline{$s\VVdP\,\dot d\dot\Bcap\dot\pi\dot d
=(d\Bcap\pi d)^{\centerdot}=0$, while $\dot d\dot\Bcap\dot a=0$ and
$\dot a$ supports $\dot\pi$,}

\noindent which is impossible.\ \Bang

So $b\Bsupseteq\supp\pi$ and

\Centerline{$r\VVdP\,\dot x
=\dot b\dot{\Bsupseteq}(\supp\pi)^{\centerdot}$.}

\noindent As $q$ is arbitrary,

\Centerline{$p\VVdP\,\dot x\dot{\Bsupseteq}(\supp\pi)^{\centerdot}$;}

\noindent as $p$ and $\dot x$ are arbitrary, we have the result.\ \Qed

Putting this together with (i),

\Centerline{$\VVdP\,(\supp\pi)^{\centerdot}$ is the least element of
$\dot\frak A$
supporting $\dot\pi$, and is the support of $\dot\pi$.}
}%end of proof of 556I

\leader{556J}{Theorem} Let $\frak A$ be a Dedekind complete
Boolean algebra, not $\{0\}$, and $\frak C$ a regularly embedded
subalgebra of $\frak A$.   Let $\Bbb P$ be the forcing notion
$\frak C^+$, active downwards, and $\dot\frak A$ the
forcing name for $\frak A$ over $\frak C$.

(a) If $\dot\theta$ is a $\Bbb P$-name such that

\Centerline{$\VVdP\,\dot\theta$ is a ring homomorphism from
$\dot\frak A$ to itself,}

\noindent then there is a unique
ring homomorphism $\pi:\frak A\to\frak A$ such that $\pi c\Bsubseteq c$ for
every $c\in\frak C$ and

\Centerline{$\VVdP\,\dot\theta=\dot\pi$,}

\noindent where $\dot\pi$ is the forcing name for $\pi$ over $\frak C$.

(b)(i) If

\Centerline{$\VVdP\,\dot\theta$ is a Boolean homomorphism,}

\noindent then $\pi$ is a Boolean homomorphism, and $\pi c=c$ for every
$c\in\frak C$.

\quad(ii) If

\Centerline{$\VVdP\,\dot\theta$ is a Boolean automorphism,}

\noindent that $\pi$ is a Boolean automorphism.

\wheader{556J}{0}{0}{0}{48pt}

\proof{{\bf (a)(i)} For each $a\in\frak A$, 556Ga tells us that there is
a $b\in\frak A$ such that

\Centerline{$\VVdP\,\dot\theta(\dot a)=\dot b$;}

\noindent by 556Da, this defines $b$ uniquely, so we have a
unique function $\pi:\frak A\to\frak A$ defined by the rule

\Centerline{for every $a\in\frak A$, $\VVdP\,\dot\theta(\dot a)
=(\pi a)^{\centerdot}$.}

\medskip

\quad{\bf (ii)} Now, for $\frmedcirc=\Bsymmdiff$ or $\frmedcirc=\Bcap$,
and $a$, $b\in\frak A$,

$$\eqalignno{\VVdP\,(\pi(a\frmedcirc b))^{\centerdot}
&=\dot\theta((a\frmedcirc b)^{\centerdot})
=\dot\theta(\dot a\,\dot\frmedcirc\,\dot b)\cr
&=\dot\theta\dot a\,\dot\frmedcirc\,\dot\theta\dot b
=(\pi a)^{\centerdot}\,\dot\frmedcirc\,(\pi b)^{\centerdot}
=(\pi a\frmedcirc\pi b)^{\centerdot}\cr}$$

\noindent and $\pi(a\frmedcirc b)=\pi a\frmedcirc\pi b$.
So $\pi$ is a ring homomorphism.

\medskip

\quad{\bf (iii)} If $c\in\frak C$ then $\pi c\Bsubseteq c$.
\Prf\ If $c=1$ this is trivial.   Otherwise,

\Centerline{$1\Bsetminus c\VVdP\,\dot c=0$,
$(\pi c)^{\centerdot}=\dot\theta 0=0$,}

\noindent so $(1\Bsetminus c)\Bcap\pi c=0$ and $\pi c\Bsubseteq c$.\ \Qed

\medskip

\quad{\bf (iv)} We can therefore speak of the forcing name $\dot\pi$
(556Ae, 556C).
If $p\in\frak C^+$ and $\dot x$ is a $\Bbb P$-name such that
$p\VVdP\,\dot x\in\dot\frak A$, let $a\in\frak A$ be such that
$p\VVdP\,\dot x=\dot a$ (556Ga again);  then

\Centerline{$p\VVdP\,\dot\theta(\dot x)=\dot\theta(\dot a)
=(\pi a)^{\centerdot}=\dot\pi(\dot a)=\dot\pi(\dot x)$.}

\noindent As $p$ and $\dot x$ are arbitrary,

\Centerline{$\VVdP\,\dot\theta=\dot\pi$.}

\medskip

{\bf (b)(i)} If $\VVdP\,\dot\theta$ is a Boolean homomorphism, then

\Centerline{$\VVdP\,(\pi 1)^{\centerdot}=\dot\theta\dot 1=\dot 1$}

\noindent and $\pi 1=1$.   Now

\Centerline{$\pi c\Bsubseteq c=1\Bsetminus(1\Bsetminus c)
\Bsubseteq 1\Bsetminus\pi(1\Bsetminus c)
=\pi 1\Bsetminus(\pi 1\Bsetminus\pi c)
=\pi c$}

\noindent so $\pi c=c$ for every $c\in\frak C$.

\medskip

\quad{\bf (ii)} If $\VVdP\,\dot\theta$ is a Boolean automorphism, then the same
arguments tell us that there is a Boolean homomorphism
$\phi:\frak A\to\frak A$
such that $\phi c=c$ for every $c\in\frak C$ and
$\VVdP\,\dot\phi=\dot\theta^{-1}$.   But in this case

\Centerline{$\VVdP\,(\pi\phi)^{\centerdot}=\dot\pi\dot\phi
=\dot\theta\dot\theta^{-1}=\dot\iota$}

\noindent where $\iota$ is the identity automorphism on $\frak A$;  by the
uniqueness of the representing homomorphisms of $\frak A$, $\pi\phi=\iota$.
Similarly, $\phi\pi=\iota$ and $\phi=\pi^{-1}$, so that $\pi$ is an
automorphism.
}%end of proof of 556J

\leader{556K}{Theorem} Let $(\frak A,\bar\mu)$ be a probability algebra,
and $\frak C$ a closed subalgebra of $\frak A$;  let $\Bbb P$ be the
forcing notion $\frak C^+$,
active downwards, and $\dot\frak A$ the forcing name for $\frak A$
over $\frak C$.   We can identify $\frak C$ with the regular open algebra
$\RO(\Bbb P)$\cmmnt{ (514Sb)}.
For $u\in L^0(\frak C)$ write $\vec u$ for the
corresponding $\Bbb P$-name for a real number as described in 5A3L.

(a)(i) For each $a\in\frak A$ there is a
$u_a\in L^1(\frak C,\bar\mu\restrp\frak C)$ defined by saying that
$\int_cu_a=\bar\mu(a\Bcap c)$ for every $c\in\frak C$.

\quad(ii) If $p\in\frak C^+$ and $a$, $b\in\frak A$ are such that

\Centerline{$p\VVdP\,\dot a=\dot b$}

\noindent(where $\dot a$, $\dot b$ are the forcing names for $a$, $b$ over
$\frak C$), then

\Centerline{$p\VVdP\,\vec u_a=\vec u_b$.}

\woddheader{556K}{0}{0}{0}{30pt}

(b) There is a $\Bbb P$-name $\dot{\bar\mu}$ such that

\Centerline{$\VVdP\,(\dot\frak A,\dot{\bar\mu})$ is a probability algebra,}

\noindent and

\Centerline{$\VVdP\,\dot{\bar\mu}\dot a=\vec u_a$}

\noindent whenever $a\in\frak A$ and $\dot a$ is the corresponding
forcing name over $\frak C$.

(c) If $\pi:\frak A\to\frak A$ is a measure-preserving Boolean
homomorphism such that $\pi c=c$ for every $c\in\frak C$, and
$\dot\pi$ the corresponding forcing name over $\frak C$,
then

\Centerline{$\VVdP\,\dot\pi:\dot\frak A\to\dot\frak A$ is
measure-preserving.}

(d) If $\dot\phi$ is a $\Bbb P$-name such that

\Centerline{$\VVdP\,\dot\phi:\dot\frak A\to\dot\frak A$ is a
measure-preserving Boolean automorphism}

\noindent then there is a measure-preserving Boolean automorphism
$\pi:\frak A\to\frak A$ such that $\pi c=c$ for every $c\in\frak C$ and

\Centerline{$\VVdP\,\dot\phi=\dot\pi$.}

(e) If $v\in L^1(\frak A,\bar\mu)$ and
$u\in L^1(\frak C,\bar\mu\restrp\frak C)$ is its conditional expectation
on $\frak C$, then

\Centerline{$\VVdP\,\dot v\in L^1(\dot\frak A,\dot{\bar\mu})$ and
$\int\dot v\,d\dot{\bar\mu}=\vec u$.}

\proof{{\bf (a)(i)} This is just the Radon-Nikod\'ym theorem (365E).

\medskip

\quad{\bf (ii)} If $p\VVdP\,\dot a=\dot b$, then $p\Bcap a=p\Bcap b$
(556Da).   Consequently

\Centerline{$\int_cu_a\times\chi p
=\int_{c\Bcap p}u_a=\bar\mu(c\Bcap p\Bcap a)
=\bar\mu(c\Bcap p\Bcap b)=\int_cu_b\times\chi p$}

\noindent whenever $c\in\frak C$, and
$u_a\times\chi p=u_b\times\chi p$;  by 5A3M,

\Centerline{$p\VVdP\,\vec u_a=\vec u_b$.}

\medskip

{\bf (b)(i)} Note first the elementary properties of the conditional
expectation
$a\mapsto u_a:\frak A\to L^1(\frak C,\bar\mu\restrp\frak C)$:
it is additive and positive and
order-continuous, and $0\le u_a\le\chi 1$ for every $a$.   (To extract
these facts efficiently from the presentation in \S365, note that
$u_a=P(\chi a)$, where
$P:L^1(\frak A,\bar\mu)\to L^1(\frak C,\bar\mu\restrp\frak C)$ is the
conditional expectation operator of 365Q\formerly{3{}65R}.)
In particular,

\Centerline{$\VVdP\,\vec u_a\in[0,1]$}

\noindent for every $a\in\frak A$.   It is also worth observing that
if $c\in\frak C$ and $a\in\frak A$ then $u_{a\Bcap c}=u_a\times\chi c$
(see 365Oc\formerly{3{}65Pc}).

\medskip

\quad{\bf (ii)} Now consider the $\Bbb P$-name

\Centerline{$\dot{\bar\mu}=\{((\dot a,\vec u_a),1):a\in\frak A\}$.}

\noindent We have quite a lot to check, of course.   First,
$\dot{\bar\mu}$ is a name for a function with domain $\dot\frak A$.
\Prf\ If $((\dot a,\vec u_a),1)$ and $((\dot b,\vec u_b),1)$ are two
members of $\dot{\bar\mu}$, and $p\in\frak C^+$ is such that
$p\VVdP\,\dot a=\dot b$, then $p\Bcap a=p\Bcap b$, so
$p\VVdP\,\vec u_a=\vec u_b$, by (a-ii) above.
By 5A3E, $\VVdP\,\dot{\bar\mu}$ is a function.   Also
$\VVdP\,\dom\dot{\bar\mu}=\dot A$, where
$\dot A=\{(\dot a,1):a\in\frak A\}=\dot\frak A$.\ \Qed

\medskip

\quad{\bf (iii)} We have

\Centerline{$\VVdP\,\dot{\bar\mu}\dot a=\vec u_a\in[0,1]$}

\noindent for every $a\in\frak A$, so

\Centerline{$\VVdP\,\dot{\bar\mu}$ is a function from $\dot\frak A$ to
$[0,1]$.}

\noindent Since $u_1=\chi 1$,

\Centerline{$\VVdP\,\dot{\bar\mu} 1=\vec u_1=1$.}

\medskip

\quad{\bf (iv)} Next, $\VVdP\,\dot{\bar\mu}$ is additive.   \Prf\ Suppose that
$p\in\frak C^+$ and $\dot x$, $\dot y$ are $\Bbb P$-names such that

\Centerline{$p\VVdP\,\dot x$, $\dot y\in\dot\frak A$ are disjoint.}

\noindent By 556Ga there are $a$, $b\in\frak A$ such that

\Centerline{$p\VVdP\,\dot x=\dot a$, $\dot y=\dot b$,
$(a\Bcap b)^{\centerdot}=\dot x\dot\Bcap\dot y=0$.}

\noindent So $p\Bcap a\Bcap b=0$ and

\Centerline{$\chi p\times u_{a\Bcup b}
=u_{p\Bcap(a\Bcup b)}
=u_{p\Bcap a}+u_{p\Bcap b}
=\chi p\times u_a+\chi p\times u_b
=\chi p\times(u_a+u_b)$;}

\noindent it follows that

$$\eqalignno{p\VVdP\,\dot{\bar\mu}(\dot x\dot\Bcup\dot y)
&=\dot{\bar\mu}(\dot a\dot\Bcup\dot b)
=\dot{\bar\mu}(a\Bcup b)^{\centerdot}
=\vec u_{a\Bcup b}
=(u_a+u_b)\sspvec\cr
\displaycause{5A3M}
&=\vec u_a+\vec u_b
=\dot{\bar\mu}\dot x+\dot{\bar\mu}\dot y.\cr}$$

\noindent As $p$, $\dot x$ and $\dot y$ are arbitrary,

\Centerline{$\VVdP\,\dot{\bar\mu}$ is additive.  \Qed}

\medskip

\quad{\bf (v)} Suppose that $p\in\frak C^+$ and that $\dot A$ is a
$\Bbb P$-name such that

\Centerline{$p\VVdP\,\dot A\subseteq\dot\frak A$ is closed under
$\dot\Bcup$ and has supremum $1$.}

\noindent Then for every rational number $\alpha<1$ there are an
$r\in\frak C^+$, stronger than $p$, and a $d\in\frak A$ such that

\Centerline{$r\VVdP\,\dot d\in\dot A$ and $\dot{\bar\mu}\dot d
\ge\check\alpha$.}

\noindent\Prf\ Set

\Centerline{$B=\{q\Bcap a:q\in\frak C^+$, $a\in\frak A$,
$q\VVdP\,\dot a\in\dot A\}$,}

\noindent so that $p\Bsubseteq\sup B$ (556Ea).   Because $\bar\mu$ is
completely additive, there are $b_0,\ldots,b_{n-1}\in B$ such that
$\bar\mu(p\Bcap\sup_{i<n}b_i)>\alpha\bar\mu p$.   Express each
$b_i$ as $q_i\Bcap a_i$ where $q_i$ is stronger than $p$ and
$q_i\VVdP\,\dot a_i\in\dot A$.   For $J\subseteq n$ set

\Centerline{$c_J
=p\Bcap\inf_{i\in J}q_i\Bsetminus\sup_{i\in n\setminus J}q_i$,
\quad$d_J=\sup_{i\in J}a_i$;}

\noindent then $\langle c_J\rangle_{J\subseteq n}$ is disjoint and

\Centerline{$p\Bcap\sup_{i<n}b_i
=\sup_{\emptyset\ne J\subseteq n}c_J\Bcap d_J$.}

\noindent Accordingly

\Centerline{$\sum_{\emptyset\ne J\subseteq n}\bar\mu(c_J\Bcap d_J)
>\alpha\bar\mu p$}

\noindent and there must be a non-empty $J\subseteq n$ such that
$c_J\ne 0$ and

\Centerline{$\alpha\bar\mu c_J<\bar\mu(c_J\Bcap d_J)=\int_{c_J}u_{d_J}$.}

\noindent So $r=c_J\Bcap\Bvalue{u_{d_J}\ge\alpha}$ is non-zero.
Set $d=d_J$;  then

\Centerline{$r\VVdP\,\dot a_i\in\dot A$ for every $i\in\check J$, therefore
$\dot d=\sup_{i\in\check J}\dot a_i\in\dot A$,}

\noindent and

\Centerline{$r\VVdP\,\dot{\bar\mu}\dot d=\vec u_{d_J}\ge\check\alpha$.
\Qed}

\medskip

\quad{\bf (vi)} It follows that

\Centerline{$\VVdP\,\dot{\bar\mu}$ is completely additive.}

\noindent\Prf\ Suppose that $p\in\frak C^+$ and that $\dot A$ is a
$\Bbb P$-name such that

\Centerline{$p\VVdP\,\dot A\subseteq\dot\frak A$ is closed under
$\dot\Bcup$ and has supremum $1$.}

\noindent Then for every rational $\alpha<1$ and every $q$ stronger than
$p$ there is an $r$ stronger than $q$ such that

\Centerline{$r\VVdP$ there is an $x\in\dot A$ such that
$\dot{\bar\mu} x\ge\check\alpha$;}

\noindent as $q$ is arbitrary,

\Centerline{$p\VVdP$ there is an $x\in\dot A$ such that
$\dot{\bar\mu} x\ge\check\alpha$;}

\noindent as $\alpha$ is arbitrary,

\Centerline{$p\VVdP\,\sup_{x\in\dot A}\dot{\bar\mu} x=1$.}

\noindent As $p$ and $\dot A$ are arbitrary,

\Centerline{$\VVdP\,\sup_{x\in A}\dot{\bar\mu} x=1$ whenever
$A\subseteq\dot\frak A$ is closed under $\dot\Bcup$ and has supremum $1$.}

\noindent We know that

\Centerline{$\VVdP\,\dot{\bar\mu}$ is additive and $\dot{\bar\mu} 1=1$,}

\noindent so we can turn this over to get

\doubleinset{$\VVdP\,\inf_{x\in A}\dot{\bar\mu} x=0$ whenever
$A\subseteq\dot\frak A$ is closed under $\dot\Bcap$ and has infimum $0$,
therefore $\dot{\bar\mu}$ is completely additive.  \Qed}

\medskip

\quad{\bf (vii)} Since we already know that

\Centerline{$\VVdP\,\dot\frak A$ is Dedekind complete}

\noindent (556Gb), we have all the elements needed for

\Centerline{$\VVdP\,(\dot\frak A,\dot{\bar\mu})$ is a probability algebra.}

\medskip

{\bf (c)} The point is that if $a\in\frak A$ then $u_{\pi a}=u_a$.
\Prf\ For any $c\in\frak C$,

\Centerline{$\int_cu_{\pi a}
=\bar\mu(c\Bcap\pi a)
=\bar\mu(\pi(c\Bcap a))
=\bar\mu(c\Bcap a)
=\int_cu_a$.  \Qed}

\noindent Now suppose that $p\in\frak C^+$ and $\dot x$ is a
$\Bbb P$-name such that $p\VVdP\,\dot x\in\dot\frak A$.
Then there is an $a\in\frak A$ such that $p\VVdP\,\dot a=\dot x$
(556Ga again), and

\Centerline{$p\VVdP\,\dot{\bar\mu}(\dot\pi\dot x)
=\dot{\bar\mu}(\dot\pi\dot a)
=\dot{\bar\mu}(\pi a)^{\centerdot}
=\vec u_{\pi a}
=\vec u_a
=\dot{\bar\mu}\dot x$.}

\noindent As $p$ and $\dot x$ are arbitrary,

\Centerline{$\VVdP\,\dot\pi$ is measure-preserving.}

\medskip

{\bf (d)} By 556J, there is a unique $\pi\in\Aut\frak A$
such that $\pi c=c$ for every $c\in\frak C$ and
$\VVdP\,\dot\phi=\dot\pi$.   In this case, for any $a\in\frak A$,

\Centerline{$\VVdP\,\vec u_a
=\dot{\bar\mu}\dot a
=\dot{\bar\mu}(\dot\phi\dot a)
=\dot{\bar\mu}(\dot\pi\dot a)
=\dot{\bar\mu}(\pi a)^{\centerdot}
=\vec u_{\pi a}$.}

\noindent So $u_a=u_{\pi a}$ (5A3M again) and

\Centerline{$\bar\mu a=\int u_a=\int u_{\pi a}=\bar\mu(\pi a)$.}

\noindent Thus $\pi$ is measure-preserving.

\medskip

{\bf (e)} Let
$P:L^1(\frak A,\bar\mu)\to L^1(\frak C,\bar\mu\restrp\frak C)$ be the
conditional expectation operator, and let $U$ be the set of those
$v\in L^1(\frak A,\bar\mu)$ such that

\Centerline{$\VVdP\,\dot v\in L^1(\dot\frak A,\dot{\bar\mu})$ and
$\int\dot v\,d\dot{\bar\mu}=\vec{Pv}$.}

\noindent By (a), $\chi a\in U$ for every $a\in\frak A$;   by 556Hb, $U$
is closed under addition and rational multiplication;  by 556Hc
$\sup_{n\in\Bbb N}v_n\in U$ for every non-decreasing sequence
$\sequencen{v_n}$ in $U$.   So $U=L^1(\frak A,\bar\mu)$, as required.
}%end of proof of 556K

\leader{556L}{Relatively independent subalgebras} Let $(\frak A,\bar\mu)$
be a probability algebra and $\frak C$ a closed subalgebra
of $\frak A$;  let
$\Bbb P$ be the forcing notion $\frak C^+$, active downwards.
Let $\dot{\bar\mu}$ be the
forcing name for $\bar\mu$ described in 556K, so that
$\VVdP\,(\dot\frak A,\dot{\bar\mu})$ is a probability algebra.

(a) For a
subalgebra $\frak B$ of $\frak A$ including $\frak C$, let $\dot\frak B$ be
the forcing name for $\frak B$ over $\frak C$.
If $\familyiI{\frak B_i}$ is a family of subalgebras of $\frak A$
including $\frak C$, then $\familyiI{\frak B_i}$ is relatively
independent over $\frak C$\cmmnt{ (definition:  458La)} iff

\Centerline{$\VVdP\,\family{i}{\check I}{\dot\frak B_i}$ is stochastically
independent in $\dot\frak A$.}

(b) If $\familyiI{v_i}$ is a family in $L^0(\frak A)$ which is relatively
independent over $\frak C$, then

\Centerline{$\VVdP\,\family{i}{\check I}{\dot v_i}$ is
stochastically independent}

\noindent (writing $\dot v_i$ for the forcing name for $v_i$ over
$\frak C$).

\proof{{\bf (a)(i)} Suppose that $\familyiI{\frak B_i}$ is relatively
independent over $\frak C$.   Let $p\in\frak C^+$ and $\dot J$,
$\family{j}{\dot J}{\dot x_j}$ be $\Bbb P$-names such that

\Centerline{$p\VVdP\,\dot J\in[\check I]^{<\omega}$ is non-empty,
$\dot x_j\in\dot\frak B_j$ for every $j\in\dot J$.}

\noindent Then for every $q$ stronger than $p$ there are
an $r$ stronger than $q$ and a family $\family{j}{J}{b_j}$ such that $J$ is
a non-empty
finite subset of $I$, $b_j\in\frak B_j$ for every $j\in J$, and

\Centerline{$r\VVdP\,\dot J=\check J$ and $\dot x_j=\dot b_j$ for every
$j\in\check J$.}

\noindent Set $a=\inf_{j\in J}b_j$.
For each $j\in J$ let
$u_{b_j}\in L^1(\frak C,\bar\mu\restrp\frak C)$ be the conditional
expectation of $\chi b_j$ on $\frak C$, as in 556Ka;  then
$u_a=\prod_{i\in J}u_{b_j}$, because $\familyiI{\frak B_i}$ is relatively
stochastically independent.   But this means that

$$\eqalign{r\VVdP\,\dot{\bar\mu}(\inf_{j\in\dot J}\dot x_j)
&=\dot{\bar\mu}(\inf_{j\in\check J}\dot b_j)
=\dot{\bar\mu}\dot a
=\vec u_a\cr
&=\prod_{j\in\check J}\vec u_{b_j}
=\prod_{j\in\check J}\dot{\bar\mu}\dot b_j
=\prod_{j\in\dot J}\dot{\bar\mu}\dot x_j.\cr}$$

\noindent As $q$ is arbitrary,

\Centerline{$p\VVdP\,\dot{\bar\mu}(\inf_{j\in\dot J}\dot x_j)
=\prod_{j\in\dot J}\dot{\bar\mu}\dot x_j;$}

\noindent as $p$ and $\family{j}{\dot J}{\dot x_j}$ are arbitrary,

\Centerline{$\VVdP\,\family{i}{\check I}{\dot\frak B_i}$ is independent.}

\medskip

\quad{\bf (ii)} Now suppose that

\Centerline{$\VVdP\,\family{i}{\check I}{\dot\frak B_i}$ is independent.}

\noindent Take a finite set $J\subseteq I$ and
$\family{j}{J}{b_j}\in\prod_{j\in J}\frak B_j$.   Again set
$a=\inf_{j\in J}b_j$ and let $u_{b_j}$ be
the conditional expectation of $\chi b_j$ on $\frak C$ for each $j$.
Then

$$\eqalign{\VVdP\,(\prod_{j\in J}u_{b_j})\sspvec
&=\prod_{j\in\check J}\vec u_{b_j}
=\prod_{j\in\check J}\dot{\bar\mu}\dot b_j\cr
&=\dot{\bar\mu}(\inf_{j\in\check J}\dot b_j)
=\dot{\bar\mu}(\inf_{j\in J}b_j)^{\centerdot}
=\vec u_a,\cr}$$

\noindent so $\prod_{j\in J}u_{b_j}=u_a$.   As $\family{j}{J}{b_j}$ is
arbitrary, $\familyiI{\frak B_i}$ is relatively independent over $\frak C$.

\medskip

{\bf (b)} For each $i\in I$, let $\frak A_i$ be the closed subalgebra of
$\frak A$ generated by $\{\Bvalue{v_i>\alpha}:\alpha\in\Bbb Q\}$, and
$\frak B_i$ the closed subalgebra of $\frak A$ generated by
$\frak A_i\cup\frak C$.   Then $\familyiI{\frak B_i}$ is relatively
independent over $\frak C$ (458Ld), so
$\VVdP\,\family{i}{\check I}{\dot\frak B_i}$ is independent, by (a) here.
Now we have

\Centerline{$\VVdP\,\Bvalue{\dot v_i>\alpha}
=\Bvalue{v_i>\alpha}^{\centerdot}
\in\dot\frak B_i$}

\noindent whenever $\alpha\in\Bbb Q$ and $i\in I$, so

\Centerline{$\VVdP\,\Bvalue{\dot v_i>\alpha}\in\dot\frak B_i$
for every $\alpha\in\Bbb Q$ and $i\in\check I$, and
$\family{i}{\check I}{\dot v_i}$ is independent.}
}%end of proof of 556L

\leader{556M}{Laws of large numbers}\cmmnt{ As an elementary example
to show that we can
use this machinery to extend a classical result, I give the
following.}   Consider the two statements

\inset{($\ddagger$) Let $(X,\Sigma,\mu)$ be a probability space, $\Tau$ a
$\sigma$-subalgebra of $\Sigma$, and $\sequencen{f_n}$ a sequence in
$\eusm L^2(\mu)$ such that $\sequencen{f_n}$ is relatively independent
over $\Tau$ and $\int_Ff_nd\mu=0$ for every $n\in\Bbb N$ and every
$F\in\Tau$.   Suppose that $\sequencen{\beta_n}$ is a non-decreasing
sequence in $\ooint{0,\infty}$, diverging to $\infty$, such that
$\sum_{n=0}^{\infty}\Bover1{\beta_n^2}\|f_n\|_2^2<\infty$.
Then $\lim_{n\to\infty}\Bover1{\beta_n}\sum_{i=0}^nf_i=0$ a.e.}

\noindent and

\inset{($\dagger$) Let $(X,\Sigma,\mu)$ be a probability space and
$\sequencen{f_n}$ an independent sequence in
$\eusm L^2(\mu)$ such that $\int f_nd\mu=0$ for every $n\in\Bbb N$.
Suppose that $\sequencen{\beta_n}$ is a non-decreasing
sequence in $\ooint{0,\infty}$, diverging to $\infty$, such that
$\sum_{n=0}^{\infty}\Bover1{\beta_n^2}\|f_n\|_2^2<\infty$.
Then $\lim_{n\to\infty}\Bover1{\beta_n}\sum_{i=0}^nf_i=0$ a.e.}

\noindent\cmmnt{In 273D I presented ($\dagger$) as the basic strong
law of large
numbers from which the other standard forms could be derived.
($\ddagger$) may be found in Volume 4 as an exercise
(458Ye).   }\dvro{Then if there is a proof of ($\dagger$), there must be
a proof of ($\ddagger$).}{What I propose to do is to show how
($\ddagger$) can be deduced\cmmnt{, not exactly from
($\dagger$), but} from ($\dagger$) in a forcing model\cmmnt{;  relying
on the
fundamental theorem of forcing to confirm that if ($\dagger$) is true in
its ordinary sense, then its interpretation in any forcing language will
again be true}.}

\proof{{\bf (a)} In order to avoid
explanations involving names for real numbers, it seems helpful to re-word
$(\ddagger)$.   Consider the version

\inset{$(\ddagger)_1$ Let $(X,\Sigma,\mu)$ be a probability space, $\Tau$ a
$\sigma$-subalgebra of $\Sigma$, and $\sequencen{f_n}$ a sequence in
$\eusm L^2(\mu)$ such that $\sequencen{f_n}$ is relatively independent
over $\Tau$ and $\int_Ff_nd\mu=0$ for every $n\in\Bbb N$ and every
$F\in\Tau$.   Suppose that $\sequencen{\beta_n}$ is a non-decreasing
sequence in $\Bbb Q\cap\ooint{0,\infty}$, diverging to $\infty$, such that
$\sum_{n=0}^{\infty}\Bover1{\beta_n^2}\|f_n\|_2^2<\infty$.
Then $\lim_{n\to\infty}\Bover1{\beta_n}\sum_{i=0}^nf_i=0$ a.e.}

\noindent Then $(\ddagger)_1$ implies ($\ddagger$).   \Prf\ Given the
structure of ($\ddagger$), with general $\beta_n>0$, let
$\delta_n\in\Bbb Q\cap\ocint{0,\beta_n}$ be such that

\Centerline{$\Bover1{\delta_n^2}\|f_n\|_2^2
\le\Bover1{\beta_n^2}\|f_n\|_2^2+2^{-n}$}

\noindent for every $n$.   Set $\gamma_n=\sup_{m\le n}\delta_m$ for each
$n$;  then $\sequencen{\gamma_n}$ is a non-decreasing sequence in
$\Bbb Q\cap\ooint{0,\infty}$ and
$\sum_{n=0}^{\infty}\Bover1{\gamma_n^2}\|f_n\|_2^2$ is finite, so
$(\ddagger)_1$ tells us that

\Centerline{$\lim_{n\to\infty}\Bover1{\beta_n}\sum_{i=0}^nf_i
=\lim_{n\to\infty}\Bover{\gamma_n}{\beta_n}\cdot\Bover1{\gamma_n}
\sum_{i=0}^nf_i=0$ a.e.   \Qed}

\medskip

{\bf (b)} Now formulate the assertions $(\ddagger)_1$ and
($\dagger$) in terms of measure algebras;  we get

\inset{$(\ddagger)'$ Let $(\frak A,\bar\mu)$ be a probability algebra,
$\frak C$ a closed subalgebra of $\frak A$, and $\sequencen{v_n}$ a
sequence in $L^2(\frak A,\bar\mu)$ such that $\sequencen{v_n}$ is
relatively independent over $\frak C$ and $Pv_n=0$ for every
$n\in\Bbb N$, where
$P:L^1(\frak A,\bar\mu)\to L^1(\frak C,\bar\mu\restrp\frak C)$
is the conditional expectation operator.
Suppose that $\sequencen{\beta_n}$ is a non-decreasing
sequence in $\Bbb Q\cap\ooint{0,\infty}$, diverging to $\infty$, such that
$\sum_{n=0}^{\infty}\Bover1{\beta_n^2}\|v_n\|_2^2<\infty$.
Then $\sequencen{\Bover1{\beta_n}\sum_{i=0}^nv_i}$ is order*-convergent to
$0$.}

\noindent and

\inset{$(\dagger)'$ Let $(\frak A,\bar\mu)$ be a probability algebra
and $\sequencen{v_n}$ an
independent sequence in $L^2(\frak A,\bar\mu)$ such that
$\int v_nd\bar\mu=0$ for every $n$.
Suppose that $\sequencen{\beta_n}$ is a non-decreasing
sequence in $\ooint{0,\infty}$, diverging to $\infty$, such that
$\sum_{n=0}^{\infty}\Bover1{\beta_n^2}\|v_n\|_2^2<\infty$.
Then $\sequencen{\Bover1{\beta_n}\sum_{i=0}^nv_i}$ is order*-convergent to
$0$.}

\noindent (As usual, the conversions are just a matter of applying the
Loomis-Sikorski theorem, with 367F to translate order*-convergence in $L^0$
into almost-everywhere convergence of functions.)

\medskip

{\bf (c)} Assuming $(\dagger)'$, take a structure
$(\frak A,\bar\mu,\frak C,\sequencen{v_n},\sequencen{\beta_n})$ as in
$(\ddagger)'$, let $\Bbb P$ be the forcing notion $\frak C^+$, active
downwards, and consider the corresponding forcing names $\dot\frak A$,
$\dot{\bar\mu}$ and $\sequencen{\dot v_n}$.   Let
$P:L^1(\frak A,\bar\mu)\to L^1(\frak C,\bar\mu\restrp\frak C)$ be the
conditional expectation operator.   For each $n\in\Bbb N$,

\Centerline{$\VVdP\,\dot v_n\times\dot v_n
=(v_n\times v_n)^{\centerdot}\in L^1(\dot\frak A,\dot{\bar\mu})$,
\quad$\|\dot v_n\|_2^2
=\int\dot v_n^2d\dot{\bar\mu}=(P(v_n^2))\sspvec$}

\noindent by 556Hb and 556Ke.   Now

\Centerline{$\sum_{n=0}^{\infty}
  \Bover1{\beta_n}\int P(v_n^2)d(\bar\mu\restrp\frak C)
\le\sum_{n=0}^{\infty}\Bover1{\beta_n}\|v_n\|_2^2
<\infty$,}

\noindent so

\Centerline{$v=\sum_{i=n}^{\infty}\Bover1{\beta_n}P(v_n^2)$}

\noindent is defined in $L^0(\frak C)$, and

\Centerline{$\VVdP\,\sum_{n=0}^{\infty}\Bover1{\check\beta_n}
   \|\dot v_n\|_2^2
\le\vec v$ is finite.}

\noindent At the same time,

\Centerline{$\VVdP\,\int\dot v_nd\dot{\bar\mu}=\vec{Pv_n}=0$ for
every $n\in\Bbb N$,}

\noindent and by 556Lb

\Centerline{$\VVdP\,\sequencen{\dot v_n}$ is independent.}

Applying $(\dagger)'$ in the forcing language,

\Centerline{$\VVdP\,
\sequencen{(\Bover1{\check\beta_n}\sum_{i=0}^nv_i)^{\centerdot}}
=\sequencen{\Bover1{\check\beta_n}\sum_{i=0}^n\dot v_i}$
order*-converges to $0$ in $L^0(\dot\frak A)$,}

\noindent so $\sequencen{\Bover1{\beta_n}\sum_{i=0}^nv_i}$ order*-converges
to $0$ in $L^0(\frak A)$, by 556He.

Thus $(\ddagger)'$ is true, and we're home.
}%end of proof of 556M

\leader{556N}{Dye's theorem}\cmmnt{ Now for something from Volume 3.}
Let me state two versions of Dye's theorem\cmmnt{ (388L)}:  the
`full' version

\inset{($\ddagger$)
Let $(\frak A,\bar\mu)$ be a probability algebra of countable Maharam type,
$\frak C$ a closed subalgebra of $\frak A$, and $\pi_1$, $\pi_2$ two
measure-preserving automorphisms of $\frak A$ with fixed-point algebra
$\frak C$.   Then there is a measure-preserving automorphism $\phi$ of
$\frak A$ such that $\phi c=c$ for every $c\in\frak C$ and
$\pi_1$ and $\phi\pi_2\phi^{-1}$ generate the same
full subgroups of $\Aut\frak A$.}

\noindent and the `simple' version

\inset{($\dagger$) Let $(\frak A,\bar\mu)$ be a probability algebra of
countable Maharam type, and $\pi_1$, $\pi_2$ two ergodic
measure-preserving automorphisms of $\frak A$.
Then there is a measure-preserving automorphism $\phi$ of
$\frak A$ such that $\pi_1$ and $\phi\pi_2\phi^{-1}$ generate the same
full subgroups of $\Aut\frak A$.}

\noindent Here\cmmnt{ also} the machinery of this section provides a
proof of ($\ddagger$) from ($\dagger$).

\proof{{\bf (a)} Assume ($\dagger$).   Take $(\frak A,\bar\mu)$,
$\frak C$, $\pi_1$ and $\pi_2$ as in ($\ddagger$).   Let
$\Bbb P$ be the forcing notion $\frak C^+$, active downwards,
and let $\dot\frak A$, $\dot\pi_1$ and $\dot\pi_2$ be
the forcing names for $\frak A$, $\pi_1$ and $\pi_2$ over $\frak C$.
By 556C(b-iv), 556C(b-v), 556Gb, 556Kb and 556Kc, and using 372Pc,

\doubleinset{$\VVdP$ there is a measure on $\dot\frak A$ with respect to
which it is a probability measure and $\dot\pi_1$ and $\dot\pi_2$ are
measure-preserving automorphisms with fixed-point subalgebra $\{0,1\}$,
so are ergodic, because $\dot\frak A$ is Dedekind complete.}

\noindent By 556Ed,

\Centerline{$\VVdP\,\dot\frak A$ has countable Maharam type.}

\noindent By ($\dagger$), applied in the forcing universe,

\doubleinset{$\VVdP$ there is a measure-preserving automorphism $\theta$ of
$\dot\frak A$ such that $\dot\pi_1$ and $\theta\dot\pi_2\theta^{-1}$
generate the same full subgroups of $\Aut\dot\frak A$.}

\noindent Let $\dot\theta$ be a $\Bbb P$-name such that

\doubleinset{$\VVdP\,\dot\theta$ is a measure-preserving automorphism of
$\dot\frak A$ such that $\dot\pi_1$ and
$\dot\theta\dot\pi_2\dot\theta^{-1}$
generate the same full subgroups of $\Aut\dot\frak A$.}

\noindent
By 556Kd, there is a $\phi\in\Aut_{\bar\mu}\frak A$ such that
$\phi c=c$ for every $c\in\frak C$ and
$\VVdP\,\dot\theta=\dot\phi$, so that, setting $\pi_3=\phi\pi_2\phi^{-1}$,

\Centerline{$\VVdP\,\dot\pi_1$ and $\dot\pi_3$ generate the same full
subgroups of $\Aut\dot\frak A$}

\noindent (using 556C(a-iii) and 556C(b-iv)).

\medskip

{\bf (b)} Since

\Centerline{$\VVdP\,\dot\pi_3$ belongs to the full subgroup of
$\Aut\dot\frak A$ generated by $\dot\pi_1$,}

\noindent we can apply 381I(c-iv) in the forcing language to get

\Centerline{$\VVdP\,\inf_{n\in\Bbb Z}\supp(\dot\pi_1^n\dot\pi_3)=0$.}

\noindent Now by 556Ic we know that

\Centerline{$\VVdP\,\supp(\dot\pi_1^n\dot\pi_3)
=(\supp(\pi_1^n\pi_3))^{\centerdot}$}

\noindent for every $n\in\Bbb Z$ (of course we need to check that
$\VVdP\,\dot\pi_1^n\dot\pi_3=(\pi_1^n\pi_3)^{\centerdot}$;  but this is
easily deduced from 556C(a-iii),
an induction on $n$ for $n\ge 0$, and 556C(b-iv)).
So

$$\eqalignno{\VVdP\,(\inf_{n\in\Bbb Z}\supp(\pi_1^n\pi_3))^{\centerdot}
&=\inf_{n\in\Bbb Z}(\supp(\pi_1^n\pi_3)^{\centerdot})\cr
\displaycause{556E(b-ii)}
&=\inf_{n\in\Bbb Z}\supp(\pi_1^n\pi_3)^{\centerdot}
=\inf_{n\in\Bbb Z}\supp(\dot\pi_1^n\dot\pi_3)
=0.\cr}$$

\noindent By 556Da, as usual,
$\inf_{n\in\Bbb Z}\supp(\pi_1^n\pi_3)=0$;  by 381I(c-iv), in the other
direction and in the ordinary universe, $\pi_3$ belongs to the full
subgroup of $\Aut\frak A$ generated by $\pi_1$.   Similarly, $\pi_1$
belongs to the full subgroup generated by $\pi_3$, so $\pi_1$ and $\pi_3$
generate the same full subgroups, as required by ($\ddagger$).
}%end of proof of 556N

\leader{556O}{}\cmmnt{ For the next result, I prepare the ground with a
note on `full local semigroups' as defined in
\S395.

\medskip

\noindent}{\bf Lemma} Let $\frak A$ be a Dedekind complete Boolean algebra,
not $\{0\}$, and $\frak C$ a regularly embedded subalgebra of
$\frak A$;  let $\Bbb P$ be the forcing notion $\frak C^+$, active
downwards.   Let $\dot\frak A$
be the forcing name for $\frak A$ over $\frak C$, and for a ring
homomorphism $\pi:\frak A\to\frak A$ such that $\pi c\Bsubseteq c$ for
every $c\in\frak C$ let $\dot\pi$ be the forcing name for
$\pi$ over $\frak C$.   Let $G$ be a subgroup of $\Aut\frak A$ such that
every point of $\frak C$ is fixed by every member of $G$,
and $\dot G$ the $\Bbb P$-name $\{(\dot\pi,1):\pi\in G\}$.

(a) $\VVdP\,\dot G$ is a subgroup of $\Aut\dot\frak A$.

(b) If $\phi:\frak A\to\frak A$ is a ring homomorphism such that
$\phi c\Bsubseteq c$ for every $c\in\frak C$, and

\Centerline{$\VVdP\,\dot\phi$ belongs to the full local semigroup generated
by $\dot G$,}

\noindent then $\phi$ belongs to the full local semigroup generated by $G$.

\proof{{\bf (a)(i)} If $p\in\frak C^+$ and $\dot x$ is a $\Bbb P$-name such
that
$p\VVdP\,\dot x\in\dot G$, then for every $q$ stronger than $p$ there must
be an $r$ stronger than $q$ and a $\pi\in G$ such that

\Centerline{$r\VVdP\,\dot x=\dot\pi\in\Aut\dot\frak A$.}

\noindent (556C(b-iv)).
As $q$ is arbitrary, $p\VVdP\,\dot x\in\Aut\dot\frak A$;  as $p$
and $\dot x$ are arbitrary,

\Centerline{$\VVdP\,\dot G\subseteq\Aut\dot\frak A$.}

\medskip

\quad{\bf (ii)} Writing $\iota$ for the identity automorphism of $\frak A$,

\Centerline{$\VVdP\,\dot\iota\in\dot G$ is the identity automorphism of
$\dot\frak A$}

\noindent (see part (b-iv) of the proof of 556C).
If $p\in\frak C^+$ and $\dot x$, $\dot y$ are $\Bbb P$-names
such that $p\VVdP\,\dot x$, $\dot y\in\dot G$, then for every $q$ stronger
than $p$ there are $r$ stronger than $q$ and $\pi_1$, $\pi_2\in G$ such
that

\Centerline{$r\VVdP\,\dot x=\dot\pi_1$, $\dot y=\dot\pi_2$,
$\dot x\cdot\dot y=\dot\pi_1\dot\pi_2=(\pi_1\pi_2)^{\centerdot}
\in\dot G$,
$\dot x^{-1}=(\dot\pi_1)^{-1}=(\pi_1^{-1})^{\centerdot}\in\dot G$}

\noindent (556C(a-iii), 556C(b-iv)),
because $\pi_1\pi_2$ and $\pi_1^{-1}$ belong to $G$.   As $q$ is arbitrary,

\Centerline{$p\VVdP\,\dot x\cdot\dot y$ and $\dot x^{-1}$ belong to
$\dot G$;}

\noindent as $p$, $\dot x$ and $\dot y$ are arbitrary,

\Centerline{$\VVdP\,\dot G$ is a subgroup of $\Aut\dot\frak A$.}

\medskip

{\bf (b)} Take any non-zero $a\in\frak A$.   Then there is a
$p\in\frak C^+$ such that $p\VVdP\,\dot a\ne 0$ (556Da).   Since

\Centerline{$p\VVdP\,\dot\phi$ belongs to the full local semigroup
generated by $\dot G$,}

\noindent there must be $\Bbb P$-names $\dot x$, $\dot\theta$ such that

\Centerline{$p\VVdP\,\dot x\in\dot\frak A\setminus\{0\}$,
$\dot x\dot{\Bsubseteq}\dot a$,
$\dot\theta\in\dot G$,
$\dot\theta y=\dot\phi y$ whenever $y\dot{\Bsubseteq}\dot x$}

\noindent (395B(a-ii)).   Now there are a $q$ stronger than $p$ and
$b\in\frak A$, $\pi\in G$ such that

\Centerline{$q\VVdP\,\dot b=\dot x$, $\dot\pi=\dot\theta$.}

\noindent Since $q\VVdP\,\dot b\ne 0$, $q\Bcap b\ne 0$.   Suppose that
$d\Bsubseteq q\Bcap b$.   Then

\Centerline{$q\VVdP\,\dot d\dot{\Bsubseteq}\dot x$, so
$(\pi d)^{\centerdot}=\dot\pi\dot d
=\dot\theta\dot d=\dot\phi\dot d=(\phi d)^{\centerdot}$}

\noindent and

$$\eqalignno{\pi d
&=q\Bcap\pi d\cr
\displaycause{see (a-i-$\alpha$) of the proof of 556C}
&=q\Bcap\phi d\cr
\displaycause{556Da, because $\phi d\Bsubseteq\phi q\Bsubseteq q$}
&=\phi d.\cr}$$

\noindent Thus $\pi$ and $\phi$ agree on the principal ideal
$\frak A_{q\Bcap b}$, while $q\Bcap b\Bsubseteq a$ is non-zero.   As $a$
is arbitrary, $\phi$ belongs to the full local semigroup generated by
$G$, by 395B(a-ii) in the other direction.
}%end of proof of 556O

\vleader{72pt}{556P}{Kawada's theorem} In the same way as in 556M and 556N,
we have two versions of 395P:

\inset{($\ddagger$) Let $\frak A$ be a Dedekind complete Boolean
algebra and $G$ a fully non-paradoxical subgroup of $\Aut\frak A$, with
fixed-point subalgebra $\frak C$, such that $\frak C$ is a measurable
algebra.   Then there is a strictly positive $G$-invariant countably
additive real-valued functional on $\frak A$.}

\noindent and

\inset{($\dagger$) Let $\frak A$ be a Dedekind
complete Boolean algebra such that $\Aut\frak A$ has a subgroup $G$
which is ergodic and fully non-paradoxical.
Then there is a strictly positive $G$-invariant countably
additive real-valued functional on $\frak A$.}

\noindent Once again,\cmmnt{ I claim that} we can prove ($\ddagger$) from
($\dagger$).

\proof{{\bf (a)} Take $\frak A$, $G$ and $\frak C$ as in ($\ddagger$).
If $\frak A=\{0\}$, the result is trivial;  so let us suppose from now on
that $\frak A\ne\{0\}$.
Let $\bar\lambda$ be a functional such that $(\frak C,\bar\lambda)$ is a
probability algebra.
Let $\Bbb P$ be the forcing notion $\frak C^+$, active downwards,
and let $\dot\frak A$ be the forcing name for
$\frak A$ over $\frak C$;  for $\pi\in\Aut\frak A$ let $\dot\pi$ be the
forcing name for $\pi$ over $\frak C$.   Let $\dot G$ be the
$\Bbb P$-name $\{(\dot\pi,1):\pi\in G\}$.

\medskip

{\bf (b)} $\VVdP\,\dot G$ is an ergodic subgroup of $\Aut\dot\frak A$.
\Prf\ I noted in 556Oa that

\Centerline{$\VVdP\,\dot G$ is a subgroup of $\Aut\dot\frak A$.}

\noindent  For its ergodicity, copy the argument of 556C(b-v).
Suppose that $p\in\frak C^+$ and $\dot x$ is a $\Bbb P$-name
such that

\Centerline{$p\VVdP\,\dot x\in\dot\frak A$ and $\theta(\dot x)=\dot x$ for
every $\theta\in\dot G$.}

\noindent For any $q$ stronger than $p$ there are an $r$ stronger than $q$
and an $a\in\frak A$ such that $r\VVdP\,\dot x=\dot a$.   Take any
$\pi\in G$.   Then

\Centerline{$r\VVdP\,\dot\pi\in\dot G$, $(\pi a)^{\centerdot}
=\dot\pi\dot x=\dot x=\dot a$,}

\noindent so

\Centerline{$\pi(r\Bcap a)=r\Bcap\pi a=r\Bcap a$}

\noindent (556Da).   As $\pi$ is arbitrary, $r\Bcap a\in\frak C$.   If
$r\Bcap a\ne 0$, then $r\Bcap a\VVdP\,\dot a=1$;  if $r\Bcap a=0$, then
$r\VVdP\,\dot a=0$.   In either case, we have an $s$ stronger than $r$
such that $s\VVdP\,\dot x\in\{0,1\}$.   As $q$ is arbitrary,
$p\VVdP\,\dot x\in\{0,1\}$;  as $p$ and $\dot x$ are arbitrary,

\doubleinset{$\VVdP\,\dot G$ has fixed-point subalgebra $\{0,1\}$, so
is ergodic, because $\dot\frak A$ is Dedekind complete}

\noindent (556Gb, 395Gf).  \Qed

\medskip

{\bf (c)} $\VVdP\,\dot G$ is fully non-paradoxical.

\Prf\ {\bf (i)} \Quer\ Otherwise,

\Centerline{$\notVVdash_{\Bbb P}\,\dot G$ satisfies condition (i) of 395D,}

\noindent and there must be a $p\in\frak C^+$ and $\Bbb P$-names
$\dot\theta$, $\dot x$ such that

\doubleinset{$p\VVdP\,\dot x\in\dot\frak A\setminus\{1\}$,
$\dot\theta$ is a Boolean homomorphism from
$\dot\frak A$ to the principal ideal generated by $\dot x$, and
$\dot\theta$ belongs to the full local semigroup generated by $\dot G$.}

\noindent In order to apply 556J and 556O as stated
we need a $\Bbb P$-name $\dot\theta_1$
such that $\VVdP\,\dot\theta_1$ is a ring homomorphism.
If $p=1$, take
$\dot\theta_1=\dot\theta$;  otherwise, take $\dot\theta_1$ such that

\Centerline{$p\VVdP\,\dot\theta_1=\dot\theta$,
\quad$1\Bsetminus p\VVdP\,\dot\theta_1$ is the identity automorphism.}

\noindent Then

\doubleinset{$\VVdP\,\dot\theta_1:\dot\frak A\to\dot\frak A$ is a ring
homomorphism belonging to the full local semigroup
generated by $\dot G$.}

\medskip

\quad{\bf (ii)} By 556J there is a unique ring homomorphism
$\phi:\frak A\to\frak A$ such that $\phi c\Bsubseteq c$ for every
$c\in\frak C$ and $\VVdP\,\dot\theta_1=\dot\phi$.   By 556Ob, $\phi$
belongs to the full local semi-group generated by $G$.   Since $G$ is fully
non-paradoxical, $\phi 1=1$ and $\VVdP\,\dot\theta_11=\dot\phi 1=1$.
But $p\VVdP\,\dot\theta_11=\dot x\ne 1$.\ \Bang\Qed

\medskip

{\bf (d)} Applying ($\dagger$) in the forcing language, we see that

\doubleinset{$\VVdP$ there is a strictly positive $\dot G$-invariant
countably additive functional on $\dot\frak A$, therefore there is a
there is a strictly positive $\dot G$-invariant
countably additive functional on $\dot\frak A$ taking values in $[0,1]$.}

\noindent Let $\dot\nu$ be a $\Bbb P$-name such that

\doubleinset{$\VVdP\,\dot\nu$ is a strictly positive $\dot G$-invariant
$[0,1]$-valued countably additive functional on $\dot\frak A$.}

\noindent For each $a\in\frak A$,
$\VVdP\,\dot\nu\dot a\in[0,1]$, so there
is a unique $u_a\in L^0(\frak C)^+$ such that

\Centerline{$\VVdP\,\dot\nu\dot a=\vec u_a$}

\noindent (5A3M once more), and $0\le u_a\le\chi 1$.
Set $\mu a=\int u_ad\bar\lambda$ for $a\in\frak A$.

\medskip

{\bf (e)} $\mu$ is a strictly positive $G$-invariant
countably additive functional on $\frak A$.

\medskip

\Prf\ {\bf (i)} If $a$, $b\in\frak A$ are disjoint,

\Centerline{$\VVdP\,\dot a\dot{\Bcap}\dot b=0$, so
$\vec u_a+\vec u_b=\dot\nu\dot a+\dot\nu\dot b
=\dot\nu(\dot a\dot\Bcup\dot b)=\dot\nu(a\Bcup b)^{\centerdot}
=\vec u_{a\Bcup b}$}

\noindent (using 556Bb);  it follows that $u_a+u_b=u_{a\Bcup b}$ and
$\mu a+\mu b=\mu(a\Bcup b)$.   Thus $\mu$ is additive.

\medskip

\quad{\bf (ii)} If $\sequencen{a_n}$ is a
non-decreasing family in $\frak A$ with supremum $a$, then, by 556Be and
556Eb,

\doubleinset{$\VVdP\,\sequencen{\dot a_n}$ is a non-decreasing sequence
in $\dot\frak A$ with supremum $\dot a$, so
$\sequencen{\vec u_{a_n}}=\sequencen{\dot\nu\dot a_n}$ is a non-decreasing
sequence in $[0,1]$ with supremum $\vec u_a=\dot\nu\dot a$.}

\noindent Now $\sequencen{u_{a_n}}$ is a non-decreasing sequence in
$L^0(\frak C)$ with supremum $u_a$ (5A3Ld), so
$\mu a=\sup_{n\in\Bbb N}\mu a_n$.   Thus $\mu$ is countably additive.

\wheader{556P}{6}{2}{2}{36pt}
\quad{\bf (iii)} Because $u_a\ge 0$, $\mu a\ge 0$ for every $a\in\frak A$.
If $\mu a=0$, then $u_a=0$ so

\Centerline{$\VVdP\,\dot\nu\dot a=\vec u_a=0$, therefore $\dot a=0$,
because $\dot\nu$ is strictly positive,}

\noindent and $a=0$ (556Da).   Thus $\mu$ is strictly positive.

\medskip

\quad{\bf (iv)} Suppose that $\pi\in G$ and $a\in\frak A$.   Then

\Centerline{$\VVdP\,\dot\pi\in\dot G\text{ and }\dot\nu\text{ is }
  \dot G\text{-invariant, so }
\vec u_{\pi a}
=\dot\nu(\pi a)^{\centerdot}
=\dot\nu(\dot\pi\dot a)
=\dot\nu\dot a
=\vec u_a$.}

\noindent So $u_{\pi a}=u_a$ and $\mu(\pi a)=\mu a$.   Thus $\mu$ is
$G$-invariant.\ \Qed

Accordingly $\mu$ is a functional as required by ($\ddagger$).
}%end of proof of 556P

\leader{556Q}{}\cmmnt{ For the final application of the methods of this
section, I turn to a result of a quite different kind.   Here the structure
under consideration, the asymptotic density algebra $\frak Z$, is off the
main line of this treatise, but has some important measure-theoretic
properties (see \S491);  and it turns out that there is a remarkable
identification of its Dedekind completion (556S) which can be established
by applying Maharam's theorem in a suitable forcing universe of the kind
considered here.   I start with a couple of easy lemmas, one just a
restatement of ideas from Volume 3, and the other a straightforward
property of a basic class of forcing notions.

\medskip

\noindent}{\bf Lemma} (a) Let $\frak A$ be a Boolean algebra and
$\bar\mu:\frak A\to[0,1]$ a strictly positive additive functional such that
$\bar\mu 1=1$.   Suppose that
whenever $\sequencen{a_n}$ is a non-increasing sequence in $\frak A$,
there is an $a\in\frak A$ such that $a\Bsubseteq a_n$ for every $n$ and
$\bar\mu a=\inf_{n\in\Bbb N}\bar\mu a_n$.   Then $(\frak A,\bar\mu)$ is a
probability algebra.

(b) Let $(\frak A,\bar\mu)$ be a probability algebra.   Suppose that
$\kappa\ge\tau(\frak A)$ is an infinite cardinal and that
$\ofamily{\xi}{\kappa}{e_{\xi}}$
is a family in $\frak A$ such that
$\bar\mu(\inf_{\xi\in K}e_{\xi})=2^{-\#(K)}$ for every finite
$K\subseteq I$.   Then $(\frak A,\bar\mu)$ is isomorphic to the measure
algebra $(\frak B_{\kappa},\bar\nu_{\kappa})$ of the usual measure on
$\{0,1\}^{\kappa}$.

\proof{{\bf (a)} Let $A\subseteq\frak A$ be a non-empty
countable set.   Let $\sequencen{a_n}$ be a sequence running over $A$, and
set $b_n=\inf_{i\le n}a_i$ for each $n$.
There is a $b\in\frak A$, a lower
bound for $\{b_n:n\in\Bbb N\}$ and therefore for $A$, such that
$\bar\mu b=\inf_{n\in\Bbb N}\bar\mu b_n$.
If $c\in\frak A$ is any lower bound for $A$, then $b\Bcup c\Bsubseteq b_n$
for every $n$, so

\Centerline{$\bar\mu b+\bar\mu(c\Bsetminus b)
=\bar\mu(b\Bcup c)\le\inf_{n\in\Bbb N}\bar\mu b_n=\bar\mu b$,}

\noindent and $\bar\mu(c\Bsetminus b)=0$;
as $\bar\mu$ is strictly positive,
$c\Bsubseteq b$.   Thus $b=\inf A$.   As $A$ is
arbitrary, $\frak A$ is Dedekind $\sigma$-complete.   But this is the only
clause missing from the definition of `probability algebra'.

\medskip

{\bf (b)} By 331Ja,
$\tau(\frak A_d)\ge\kappa$ for every non-zero $d\in\frak A$.   So
$\frak A$ is \Mth, with Maharam type $\kappa$, and
$(\frak A,\bar\mu)\cong(\frak B_{\kappa},\bar\mu_{\kappa})$ (331I).
}%end of proof of 556Q

\leader{556R}{Proposition} Let $\Bbb P$ be a countably closed forcing
notion.   Then, for any set $I$, writing $(\frak B_I,\bar\nu_I)$ for the
measure algebra of the usual measure on $\{0,1\}^I$,

\Centerline{$\VVdP\,\,(\frak B_{\check I},\bar\nu_{\check I})
\cong(\check\frak B_I,\check{\bar\nu}_I)$.}

\proof{ If $I$ is finite, this is elementary (and does not rely on $\Bbb P$
being countably closed), so I shall suppose that $I$ is infinite.

\medskip

{\bf (a)}

\doubleinset{$\VVdP$ if $\sequencen{x_n}$ is a non-increasing sequence in
$\check\frak B_I$, there is an $x\in\check\frak B_I$ such that
$x\check{\Bsubseteq}x_n$ for every $n$ and
$\check{\bar\nu}_I(x)=\inf_{n\in\Bbb N}\check{\bar\nu}_I(x_n)$.}

\noindent\Prf\ Let $p$ be a condition of $\Bbb P$ and
$\sequencen{\dot x_n}$ a sequence of $\Bbb P$-names such that

\Centerline{$p\VVdP\,\dot x_n\in\check\frak B_I$ and
$\dot x_{n+1}\check{\Bsubseteq}\dot x_n$}

\noindent for every $n$.   If $q$ is stronger
than $p$, we can choose $\sequencen{q_n}$, $\sequencen{b_n}$
inductively so that $q_0=q$ and, for each $n$, $q_{n+1}$ is stronger than
$q_n$, $b_n\in\frak B_I$ and $q_{n+1}\VVdP\,\dot x_n=\check b_n$.
In this case,

\Centerline{$q_{n+1}\VVdP\,\check b_{n+1}=\dot x_{n+1}
  \check{\Bsubseteq}\dot x_n=\check b_n$,}

\noindent so $b_{n+1}\Bsubseteq b_n$ for each $n$.   Setting
$b=\inf_{n\in\Bbb N}b_n$, $\bar\nu_Ib=\inf_{n\in\Bbb N}\bar\nu_Ib_n$.
Also, because $\Bbb P$ is countably closed, there is a condition $r$
stronger than any $q_n$.   So

\Centerline{$r\VVdP\,\check b\check{\Bsubseteq}\check b_n=\dot x_n$
for every $n\in\Bbb N$,
$\check{\bar\nu}_I(\check b)
=\inf_{n\in\Bbb N}\check{\bar\nu}_I(\dot x_n)$.}

\noindent As $q$ is arbitrary,

\doubleinset{$p\VVdP$ there is an $x\in\check\frak B_I$ such that
$x\check{\Bsubseteq}\dot x_n$ for every $n$ and
$\check{\bar\nu}_I(x)=\inf_{n\in\Bbb N}\check{\bar\nu}_I(\dot x_n)$.}

\noindent As $p$ and $\sequencen{\dot x_n}$ are arbitrary,

\doubleinset{$\VVdP$ if $\sequencen{x_n}$ is a non-increasing sequence in
$\check\frak B_I$, there is an $x\in\check\frak B_I$ such that
$x\check{\Bsubseteq}x_n$ for every $n$ and
$\check{\bar\nu}_I(x)=\inf_{n\in\Bbb N}\check{\bar\nu}_I(x_n)$.  \Qed}

Since we certainly have

\doubleinset{$\VVdP\,\check\frak B_I$ is a Boolean algebra and
$\check{\bar\nu}:\check\frak B_I\to[0,1]$ is a strictly positive additive
functional such that $\check{\bar\nu}_I1=1$,}

\noindent 556Qa, applied in the forcing universe, tells us that

\Centerline{$\VVdP\,(\check\frak B,\check{\bar\nu}_I)$ is a
probability algebra.}

\medskip

{\bf (b)} Let $\familyiI{e_i}$ be the standard generating family in
$\frak B_I$.   Then
$\bar\nu_I(\inf_{i\in K}e_i)=2^{-\#(K)}$ for every finite set
$K\subseteq I$, so

\Centerline{$\VVdP\,
\check{\bar\nu}_I(\inf_{i\in K}\check e_i)=2^{-\#(K)}$
for every finite set $K\subseteq\check I$.}

\noindent Next, if $\frak D$ is the
subalgebra of $\frak B_I$ generated by $\{e_i:i\in I\}$, then $\frak D$ is
dense in $\frak B_I$ for the measure metric.   Now

\doubleinset{$\VVdP\,\check\frak D$
is the subalgebra of $\check\frak B_I$ generated by
$\{\check e_i:i\in\check I\}$ and $\check\frak D$ is metrically dense in
$\check\frak B_I$, so $\tau(\check\frak B_I)\le\#(\check I)$.
By 556Q, $(\check\frak B_I,\check{\bar\nu}_I)
\cong(\frak B_{\#(\check I)},\bar\nu_{\#(\check I)})
\cong(\frak B_{\check I},\bar\nu_{\check I})$,}

\noindent as required.
}%end of proof of 556R

\leader{556S}{Theorem}\cmmnt{ ({\smc Farah 06})} Let
$\Cal Z$ be the ideal of subsets of $\Bbb N$
with asymptotic density $0$ and $\frak Z$ the asymptotic density algebra
$\Cal P\Bbb N/\Cal Z$.   Then the Dedekind completion of
$\frak Z$ is isomorphic to the Dedekind
completion of the free product
$(\Cal P\Bbb N/[\Bbb N]^{<\omega})\otimes\frak B_{\frakc}$.

\proof{{\bf (a)} For $n\in\Bbb N$, set $I_n=\{i:2^n\le i<2^{n+1}\}$, so
that $\sequencen{I_n}$ is a partition of $\Bbb N\setminus\{0\}$, and
$\#(I_n)=2^n$ for every $n\in\Bbb N$.   Recall that

\Centerline{$\Cal Z
=\{J:J\subseteq\Bbb N$, $\lim_{n\to\infty}2^{-n}\#(J\cap I_n)=0\}$}

\noindent (491Ab).   The notation of this proof will be slightly less
appalling if I write $b_J$ for  $J^{\ssbullet}\in\frak Z$ when
$J\subseteq\Bbb N$ and $c_K$ for
$(\bigcup_{n\in K}I_n)^{\ssbullet}$ when $K\subseteq\Bbb N$.

Set

\Centerline{$\frak C=\{c_K:K\subseteq\Bbb N\}$.}

\noindent Because $K\mapsto c_K:\Cal P\Bbb N\to\frak Z$ is a Boolean
homomorphism, $\frak C$ is a subalgebra of $\frak Z$.   Now
$\frak C\cong\Cal P\Bbb N/[\Bbb N]^{<\omega}$.   \Prf\
If $K\subseteq\Bbb N$, then

\Centerline{$c_K=0
\iff\bigcup_{n\in K}I_n\in\Cal Z
\iff K$ is finite.}

\noindent So the Boolean homomorphism $K\mapsto c_K$ induces a Boolean
isomorphism
$\pi:\Cal P\Bbb N/[\Bbb N]^{<\omega}\to\frak C$ defined by setting
$\pi(K^{\ssbullet})=c_K$ for every $K\subseteq\Bbb N$.\ \Qed

\woddheader{556S}{0}{0}{0}{36pt}

For $p\in\frak C^+$, set

\Centerline{$\Cal F_p=\{K:K\subseteq\Bbb N$, $p\Bsubseteq c_K\}$,}

\noindent so that $\Cal F_p$ is a filter on $\Bbb N$ containing every
cofinite set.   Note that if $p\Bsubseteq q$ then $\Cal F_p$ is finer than
$\Cal F_q$.

\medskip

{\bf (b)} $\frak C$ is regularly embedded in $\frak Z$.   \Prf\ Suppose
that $A\subseteq\frak C$ has infimum $0$ in $\frak C$, and that
$b\in\frak Z^+$.   Let $J_0\in\Cal P\Bbb N\setminus\Cal Z$ be such that
$b=b_{J_0}$.   Then $\limsup_{n\to\infty}2^{-n}\#(J_0\cap I_n)>0$,
so there is an $\epsilon>0$ such that
$K=\{n:\#(J_0\cap I_n)\ge 2^n\epsilon\}$ is infinite.   $c_K$ cannot be a
lower bound of $A$ in $\frak C$, so there is an $L\subseteq\Bbb N$ such
that $c_L\in A$ and $c_K\notBsubseteq c_L$, that is, $K\setminus L$ is
infinite.   Set $J=\bigcup_{n\in K\setminus L}J_0\cap I_n$;  then
$\#(J\cap I_n)\ge 2^n\epsilon$ for infinitely many $n$, so
$J\notin\Cal Z$ and $0\ne b_J\Bsubseteq b$.   On the other hand,
$b_J\Bcap c_L=0$.   So $b\notBsubseteq c_L$ and $b$ is not a lower bound of
$A$ in $\frak Z$.   As $b$ is arbitrary, $A$ has infimum $0$ in $\frak Z$;
as $A$ is arbitrary, the embedding $\frak C\embedsinto\frak Z$ is
order-continuous (313L(b-v)), and $\frak C$ is regularly embedded in
$\frak Z$.\ \Qed.

\medskip

{\bf (c)} Let $\Bbb P$ be the forcing notion $\frak C^+$, active downwards.
Then $\Bbb P$ is countably closed.   \Prf\ Let $\sequence{m}{p_m}$ be a
non-increasing sequence in $\frak C^+$.   For each $m\in\Bbb N$, let
$K_m\subseteq\Bbb N$ be such that $p_m=c_{K_m}$.
Then $K_{m+1}\setminus K_m$ is finite for each $m$.   Let
$\sequence{k}{n_k}$ be a strictly increasing sequence such that
$n_k\in K_m$ whenever $m\le k\in\Bbb N$, and set
$K=\{n_k:k\in\Bbb N\}$;  then $c_K$ belongs to
$\frak C^+$, and $c_K\Bsubseteq p_m$ for every $m\in\Bbb N$.\ \Qed

\medskip

{\bf (d)(i)} Let $\dot\frak Z$ be the forcing name for $\frak Z$ over
$\frak C$, and for $b\in\frak Z$ let $\dot b$ be the forcing name for $b$
over $\frak C$.   Let $\dot\nu$ be the $\Bbb P$-name

\Centerline{$\{((\dot b_J,\check\alpha),p):
  p\in\frak C^+,\,J\subseteq\Bbb N,\,
  \lim_{n\to\Cal F_p}2^{-n}\#(J\cap I_n)$ is defined and equal to
  $\alpha\}$.}

\medskip

\quad{\bf (ii)} $\VVdP\,\dot\nu$ is a function.   \Prf\
Suppose that $(J_0,\alpha_0,p_0)$ and
$(J_1,\alpha_1,p_1)\in\Cal P\Bbb N\times\Bbb R\times\frak C^+$ are such
that

\Centerline{$\lim_{n\to\Cal F_{p_0}}2^{-n}\#(J_0\cap I_n)=\alpha_0$,
\quad$\lim_{n\to\Cal F_{p_1}}2^{-n}\#(J_1\cap I_n)=\alpha_1$,}

\noindent and that $p\in\frak C^+$, $p\Bsubseteq p_0\Bcap p_1$ and
$p\VVdP\,\dot b_{J_0}=\dot b_{J_1}$.
Then $p\Bcap b_{J_0}=p\Bcap b_{J_1}$ (556Da).
Express $p$ as $c_K$, where $K\subseteq\Bbb N$;  then
$\bigcup_{n\in K}I_n\cap(J_0\symmdiff J_1)\in\Cal Z$, so
$\lim_{n\in K,n\to\infty}2^{-n}\#(I_n\cap(J_0\symmdiff J_1))=0$, that is,
$\lim_{n\to\Cal F_p}2^{-n}\#(I_n\cap(J_0\symmdiff J_1))=0$.   But this
means that

\Centerline{$\alpha_0
=\lim_{n\to\Cal F_p}2^{-n}\#(I_n\cap J_0)
=\lim_{n\to\Cal F_p}2^{-n}\#(I_n\cap J_1)
=\alpha_1$,}

\noindent and surely $p\VVdP\,\check\alpha_0=\check\alpha_1$.   Thus the
condition of 5A3Ea is satisfied and

\Centerline{$\VVdP\,\dot\nu$ is a function.  \Qed}

\medskip

\quad{\bf (iii)} $\VVdP\,\dom\dot\nu=\dot\frak Z$.   \Prf\ Setting

\Centerline{$\dot A
=\{(\dot b_J,p):p\in\frak C^+,\,J\subseteq\Bbb N,\,
  \lim_{n\to\Cal F_p}2^{-n}\#(J\cap I_n)$ is defined$\}$,}

\noindent 5A3E tells us that $\VVdP\,\dom\dot\nu=\dot A$.   Of course
$\VVdP\,\dot A\subseteq\dot\frak Z$ just because
$\VVdP\,\dot b_J\in\dot\frak Z$ for every
$J\subseteq\Bbb N$.   In the other direction, if $p\in\frak C^+$ and
$\dot x$ is a $\Bbb P$-name such that $p\VVdP\,\dot x\in\dot\frak Z$, there
are a $q$ stronger than $p$ and a $b\in\frak Z$ such that
$q\VVdP\,\dot x=\dot b$.   Express $q$ as
$c_K$ and $b$ as $b_J$ where
$K\subseteq\Bbb N$ is infinite and $J\subseteq\Bbb N$.   Then there is an
infinite $L\subseteq K$ such that
$\lim_{n\in L,n\to\infty}2^{-n}\#(J\cap I_n)$ is defined, that is,
$(\dot b,r)\in\dot A$, where $r=c_L$.  So
$r\VVdP\,\dot x=\dot b\in\dot A$.   As $p$ and $\dot x$ are arbitrary,

\Centerline{$\VVdP\,\dot\frak Z\subseteq\dot A$ and
$\dom\dot\nu=\dot\frak Z$. \Qed}

\medskip

\quad{\bf (iv)} Of course $\lim_{n\to\Cal F_p}2^{-n}\#(J\cap I_n)$, if it
is defined, must belong to $[0,1]$.   So

\Centerline{$\VVdP\,\dot\nu$ is a function from $\dot\frak Z$ to $[0,1]$.}

\noindent Next, $((\dot 1,\check 1),1)\in\dot\nu$
(if you can work out how to interpret
each $1$ in this formula), so $\VVdP\,\dot\nu\dot 1=\check 1=1$.

\medskip

\quad{\bf (v)} $\VVdP\,\dot\nu$ is additive.   \Prf\ Suppose that
$p\in\frak C^+$ and that $\dot x_0$, $\dot x_1$
are $\Bbb P$-names such that

\Centerline{$p\VVdP\,\dot x_0$, $\dot x_1\in\dot\frak Z$ are disjoint.}

\noindent If $p_1$ is stronger than $p$ there are $q_0$, $q'_0$, $q_1$,
$r\in\Bbb P$,
$J_0$, $J_1\subseteq\Bbb N$ and $\alpha_0$, $\alpha_1\in\Bbb R$ such that

\Centerline{$((\dot b_{J_0},\check\alpha_0),q_0)
\in\dot\nu$,
\quad$q'_0$ is stronger than both $q_0$ and $p_1$,
\quad$q'_0\VVdP\,\dot b_{J_0}=\dot x_0$,}

\Centerline{$((\dot b_{J_1},\check\alpha_1),q_1)
\in\dot\nu$,
\quad$r$ is stronger than both $q_1$ and $q'_0$,
\quad$r\VVdP\,\dot b_{J_1}=\dot x_1$.}

\noindent As $r\VVdP\,(b_{J_0}\Bcap b_{J_1})^{\centerdot}=0$,
$r\Bcap b_{J_0}\Bcap b_{J_1}=0$.   Express $r$ as
$c_K$, where $K\in[\Bbb N]^{\omega}$.
Then $J_0\cap J_1\cap\bigcup_{n\in K}I_n\in\Cal Z$, so
$\lim_{n\to\Cal F_{r}}2^{-n}\#(J_0\cap J_1\cap I_n)=0$.   At the same
time,

\Centerline{$\lim_{n\to\Cal F_{r}}2^{-n}\#(J_0\cap I_n)
=\lim_{n\to\Cal F_{q_0}}2^{-n}\#(J_0\cap I_n)=\alpha_0$,}

\Centerline{$\lim_{n\to\Cal F_{r}}2^{-n}\#(J_1\cap I_n)
=\lim_{n\to\Cal F_{q_1}}2^{-n}\#(J_1\cap I_n)=\alpha_1$,}

\noindent so

$$\eqalign{\lim_{n\to\Cal F_{r}}&2^{-n}\#((J_0\cup J_1)\cap I_n)\cr
&=\lim_{n\to\Cal F_{r}}2^{-n}\#(J_0\cap I_n)
  +\lim_{n\to\Cal F_{r}}2^{-n}\#(J_1\cap I_n)
  -\lim_{n\to\Cal F_{r}}2^{-n}\#(J_0\cap J_1\cap I_n)\cr
&=\alpha_0+\alpha_1\cr}$$

\noindent and
$(((\dot b_{J_0\cup J_1},(\alpha_0+\alpha_1)\varBspcheck),r)\in\dot\nu$.
Accordingly

\Centerline{$r\VVdP\,\dot\nu(\dot x_0\dot{\Bcup}\dot x_1)
=\dot\nu(\dot b_{J_0\cup J_1})
=(\alpha_0+\alpha_1)\varBspcheck
=\check\alpha_0+\check\alpha_1
=\dot\nu(\dot x_0)+\dot\nu(\dot x_1)$,}

\noindent while $r\Bsubseteq p_1$.   As $p_1$ is arbitrary,

\Centerline{$p\VVdP\,\dot\nu(\dot x_0\dot{\Bcup}\dot x_1)
 =\dot\nu(\dot x_0)+\dot\nu(\dot x_1)$.}

\noindent As $p$, $\dot x_0$ and $\dot x_1$ are arbitrary,

\Centerline{$\VVdP\,\dot\nu$ is additive.  \Qed}

\medskip

\quad{\bf (vi)} $\VVdP\,\dot\nu$ is strictly positive.   \Prf\ Let
$p\in\frak C^+$ and a $\Bbb P$-name $\dot x$ be such that
$p\VVdP\,\dot x\in\dot\frak Z$ and $\dot x\ne\dot 0$.
If $q$ is stronger than $p$ there are a $q'$ stronger than $q$ and a
$J\subseteq\Bbb N$ such that
$q'\VVdP\,\dot x=\dot b_J$.   Express $q'$ as
$ c_K$ where $K\in[\Bbb N]^{\omega}$.
As $q'\VVdP\,\dot b_J\ne\dot 0$,
$q'\Bcap b_J\ne 0$ and $\bigcup_{n\in K}I_n\cap J\notin\Cal Z$.
Accordingly $\limsup_{n\in K,n\to\infty}2^{-n}\#(I_n\cap J)>0$ and there is
an infinite $L\subseteq K$ such that
$\alpha=\lim_{n\in L,n\to\infty}2^{-n}\#(I_n\cap J)$ is defined and greater
than $0$.   Set $r=c_L$;  then
$((\dot b_J,\check\alpha),r)\in\dot\nu$, so

\Centerline{$r\VVdP\,\dot\nu(\dot x)=\dot\nu(\dot b_J)
=\check\alpha>0$,}

\noindent while $r$ is stronger than $q$.   As $q$ is arbitrary,
$p\VVdP\,\dot\nu(\dot x)>0$;  as $p$ and $\dot x$ are arbitrary,
$\VVdP\,\dot\nu$ is strictly positive.\ \Qed

\medskip

\quad{\bf (vii)}

$$\eqalign{\VVdP\,\,&\text{if }\sequence{k}{x_k}
\text{ is a non-increasing sequence in }\dot\frak Z\cr
&\mskip20mu\text{ there is an }x\in\dot\frak Z
\text{ such that }x\dot{\Bsubseteq}x_k\text{ for every }k
\text{ and }\dot\nu(x)=\inf_{k\in\Bbb N}\dot\nu(x_k).\cr}$$

\noindent\Prf\ Let $p\in\frak C^+$ and a sequence $\sequence{k}{\dot x_k}$
of $\Bbb P$-names be such that

\Centerline{$p\VVdP\,\sequencen{\dot x_n}$ is a
non-increasing sequence in $\dot\frak Z$.}

\noindent Let $q$ be stronger than $p$.
Then we can choose $\sequence{k}{q_k}$,
$\sequence{k}{q'_k}$, $\sequence{k}{q''_k}$,
$\sequence{k}{J_k}$ and $\sequence{k}{\alpha_k}$
inductively so that $q'_0=q$ and

\Centerline{$q''_k$ is stronger than $q'_k$, $J_k\subseteq\Bbb N$ and
$q''_k\VVdP\,\dot x_k=\dot b_{J_k}$,}

\Centerline{$q_k$ is stronger than $q''_k$, $\alpha_k\in[0,1]$,
$\lim_{n\to\Cal F_{q_k}}2^{-n}\#(J_k\cap I_n)=\alpha_k$}

\noindent (compare (iii) above),

\Centerline{$q'_{k+1}=q_k$}

\noindent for every $k\in\Bbb N$.

Because $\Bbb P$ is countably closed,
there is an $r\in\frak C^+$ stronger than every $q_k$.   In this case,
$((\dot b_{J_k},\check\alpha_k),r)\in\dot\nu$ for every
$k$, so

\Centerline{$r\VVdP\,\inf_{k\in\Bbb N}\dot\nu(\dot x_k)
=\inf_{k\in\Bbb N}\check\alpha_k=\check\alpha$}

\noindent where $\alpha=\inf_{k\in\Bbb N}\alpha_k$.
Express $r$ as $c_K$ where
$K\subseteq\Bbb N$ is infinite.   For each $k\in\Bbb N$,

\Centerline{$r\VVdP\,\dot b_{J_{k+1}}=\dot x_{k+1}\dot{\Bsubseteq}\dot x_k
   =\dot b_{J_k}$,}

\noindent so $r\Bcap b_{J_{k+1}}\Bsetminus b_{J_k}=0$ and

\Centerline{$\lim_{n\in K,n\to\infty}2^{-n}\#(I_n\cap J_{k+1}\setminus J_k)
=0$,}

\noindent while

\Centerline{$\lim_{n\in K,n\to\infty}2^{-n}\#(I_n\cap J_k)
=\lim_{n\to\Cal F_{q_k}}2^{-n}\#(I_n\cap J_k)
=\alpha_k\ge\alpha$.}

\noindent We can therefore find a strictly increasing sequence
$\sequence{k}{n_k}$ in $K$ such that

\Centerline{$2^{-n_k}\#(I_{n_k}\cap J'_k)\ge\alpha-2^{-k}$}

\noindent for every $k$, where $J'_k=\bigcap_{j\le k}J_j$.   Set
$r'=(\bigcup_{k\in\Bbb N}I_{n_k})^{\ssbullet}$ and
$J=\bigcup_{k\in\Bbb N}I_{n_k}\cap J'_k$.
Then $J\setminus J_k$ is finite, so
$r'\VVdP\,\dot b_J\dot{\Bsubseteq}\dot x_k$ for every
$k$.   Also $((\dot b_J,\check\alpha),r')\in\dot\nu$, so

\Centerline{$r'\VVdP\,\dot\nu(\dot b_J)
=\check\alpha=\inf_{k\in\Bbb N}\dot\nu(\dot x_k)$.}

Thus

\Centerline{$r'\VVdP\,\text{ there is a lower bound }x\text{ for }
\{\dot x_k:k\in\Bbb N\}\text{ such that }
\dot\nu(x)=\inf_{k\in\Bbb N}\dot\nu(\dot x_k)$.}

\noindent As $q$ is arbitrary,

\Centerline{$p\VVdP\,\text{ there is a lower bound }x\text{ for }
\{\dot x_k:k\in\Bbb N\}\text{ such that }
\dot\nu(x)=\inf_{k\in\Bbb N}\dot\nu(\dot x_k)$.}

\noindent As $p$ and $\sequence{k}{\dot x_k}$ are arbitrary,

\doubleinset{$\VVdP$ if $\sequence{k}{x_k}$ is a non-increasing sequence
in $\dot\frak Z$ there is an $x\in\dot\frak Z$ such that
$x\dot{\Bsubseteq}x_k$ for every $k$ and
$\dot\nu(x)=\inf_{k\in\Bbb N}\dot\nu(x_k)$.  \Qed}

\medskip

\quad{\bf (viii)}

\doubleinset{$\VVdP$ there is a family $\family{L}{\Cal P\Bbb N}{x_L}$
in $\dot\frak A$ such that $\dot\nu(\inf_{L\in\Cal L}x_L)=2^{-\#(\Cal L)}$
for every finite set $\Cal L\subseteq\Cal P\Bbb N$.}

\noindent\Prf\ Let $\family{L}{\Cal P\Bbb N}{M_L}$ be an almost disjoint
family of infinite subsets of $\Bbb N$ (5A1Ga).   For each $n\in\Bbb N$,
let $\ofamily{i}{n}{K_{ni}}$ be a family of subsets of $I_n$ such that
$\#(\bigcap_{i\in J}K_{ni})=2^{n-\#(J)}$ for every non-empty set
$J\subseteq n$;  such a family exists because $\#(I_n)=2^n$.   For
$L\subseteq\Bbb N$, set

\Centerline{$A_L=\bigcup_{n\in\Bbb N,n>\min M_L}K_{n,\max(n\cap M_L)}$,
\quad$a_L=b_{A_L}\in\frak Z$.}

\noindent If $\Cal L\subseteq\Cal P\Bbb N$ is finite and not empty, let
$n_0\in\Bbb N$ be such that $M_L\cap M_{L'}\subseteq n_0$ whenever $L$,
$L'\in\Cal L$ are distinct, and $n_1\ge n_0$ such that
$M_L\cap n_1\setminus n_0\ne\emptyset$ for every $L\in\Cal L$.
Then $\max(n\cap M_L)\ne\max(n\cap M_{L'})$ whenever $L$, $L'\in\Cal L$ are
distinct and $n\ge n_1$.   So

$$\eqalign{\lim_{n\to\infty}2^{-n}\#(I_n\cap\bigcap_{L\in\Cal L}A_L)
&=\lim_{n\to\infty}
  2^{-n}\#(I_n\cap\bigcap_{L\in\Cal L}K_{n,\max(n\cap M_L)})\cr
&=\lim_{n\to\infty}2^{-n}2^{n-\#(\Cal L)}
=2^{-\#(\Cal L)}.\cr}$$

\noindent Of course the same formula is valid when $\Cal L=\emptyset$.

It follows that

\Centerline{$\VVdP\,\dot\nu(\inf_{L\in\check\Cal L}\dot a_L)
=2^{-\#(\check\Cal L)}$}

\noindent for every finite $\Cal L\subseteq\Cal P\Bbb N$.   Accordingly

\Centerline{$\VVdP\,\dot\nu(\inf_{L\in\Cal L}\dot a_L)=2^{-\#(\Cal L)}$
for every finite set $\Cal L\subseteq(\Cal P\Bbb N)\varBspcheck$.}

\noindent But we know also that

\Centerline{$\VVdP\,\Cal P\Bbb N=(\Cal P\Bbb N)\varBspcheck$}

\noindent (5A3Qb).   So the family $\family{L}{\Cal P\Bbb N}{\dot a_L}$ of
$\Bbb P$-names, when interpreted as a $\Bbb P$-name
$\family{L}{(\Cal P\Bbb N)\varBspcheck}{\dot a_L}$ as in 5A3Fb, can also be
regarded as a $\Bbb P$-name for a function defined on the whole power
set of the set of natural numbers.   If we do this, we get

\Centerline{$\VVdP\,\dot\nu(\inf_{L\in\Cal L}\dot a_L)=2^{-\#(\Cal L)}$
for every finite set $\Cal L\subseteq\Cal P\Bbb N$,}

\noindent witnessing the truth of the result we seek.\ \Qed

\medskip

\quad{\bf (ix)} $\VVdP\,\#(\dot\frak A)\le\frak c$.   \Prf\ Since

\Centerline{$\dot\frak Z=\{(\dot a,1):a\in\frak Z\}
=\{(\dot b_J,1):J\in\Cal P\Bbb N\}$}

\noindent (556Ab), we get

\Centerline{$\VVdP\,\dot\frak Z
=\{\dot b_J:J\in(\Cal P\Bbb N)\varBspcheck\}$,
so $\#(\dot\frak Z)\le\#((\Cal P\Bbb N)\varBspcheck)
\le\#(\Cal P\Bbb N)=\frak c$.  \Qed}

\medskip

{\bf (e)} Assembling the facts in (d), we see that

\Centerline{$\VVdP\,(\dot\frak Z,\dot\nu)$ satisfies the conditions of
556Q with $\kappa=\frak c$, so $\dot\frak Z\cong\frak B_{\frakc}$.}

\noindent But we also have

\Centerline{$\VVdP\,\frak B_{\frakc}$ is isomorphic to
$\frak B_{\Cal P\Bbb N}=\frak B_{(\Cal P\Bbb N)\varBspcheck}
\cong(\frak B_{\Cal P\Bbb N})\varBspcheck$}

\noindent by 556R.   As $\frak C$ is regularly embedded in $\frak Z$,
we can apply 556Fc to see that $\widehat{\frak Z}$
is isomorphic to the Dedekind completed free product
$\frak C\tensorhat\frak B_{\Cal P\Bbb N}$ and therefore to
$(\Cal P\Bbb N/[\Bbb N]^{<\omega})\tensorhat\frak B_{\frakc}$, by (a).

This completes the proof.
}%end of proof of 556S

\exercises{\leader{556X}{Basic exercises (a)}
%\spheader 556Xa
Let $\frak A$ be a Boolean algebra, not $\{0\}$, and
$\frak C$ a Boolean subalgebra of $\frak A$ which is not regularly
embedded;  let $\Bbb P$ be the forcing notion $\frak C^+$, active
downwards, and let $\dot\frak A$ be the forcing
name for $\frak A$ over $\frak C$.   Show that there is an
$a\in\frak A\setminus\{0\}$ such that $\VVdP\,\dot a=0$, where $\dot a$ is
the forcing name for $a$ over $\frak C$.
%556D

\spheader 556Xb Let $\Bbb P$ be a countably closed forcing notion.
(i) Show that $\VVdP\,\omega_1=\check\omega_1$.
(ii) Show that
$\VVdP\,[\check I]^{\le\omega}=([I]^{\le\omega})\varBspcheck$
for every set $I$.
(iii) Let $\frak A$ be a Dedekind $\sigma$-complete Boolean algebra.
Show that $\VVdP\,\check\frak A$ is Dedekind $\sigma$-complete.
(iv) Let $(X,\rho)$ be a complete metric space.   Show that
$\VVdP\,(\check X,\check\rho)$ is a complete metric space.
%556R

\spheader 556Xc Show that the Dedekind completion $\widehat{\frak Z}$ of
the asymptotic density algebra is a homogeneous Boolean algebra.
\Hint{316Q, 316P.}
%556S   316P:  completion of homog is homog.  316Q:  free product of
%homog is homog

\leader{556Y}{Further exercises (a)}
%\spheader 556Ya
Let $\Bbb P$ be a forcing notion, and $\dot\Bbb Q_1$, $\dot\Bbb Q_2$ two
$\Bbb P$-names for forcing notions such that

\Centerline{$\VVdP\,\RO(\dot\Bbb Q_1)\cong\RO(\dot\Bbb Q_2)$.}

\noindent Show that
$\RO(\Bbb P*\dot\Bbb Q_1)\cong\RO(\Bbb P*\dot\Bbb Q_2)$.
%556F

\spheader 556Yb Let $\Bbb P$ and $\Bbb Q$ be forcing notions.   Show that
$\RO(\Bbb P*\check\Bbb Q)\cong\RO(\Bbb P)\tensorhat\RO(\Bbb Q)$.
%556F

\spheader 556Yc
Give an example of a Dedekind $\sigma$-complete Boolean
algebra $\frak A$ with an order-closed subalgebra $\frak C$ such that

\Centerline{$\VVdP\,\dot\frak A$ is not Dedekind $\sigma$-complete,}

\noindent where
$\Bbb P$ is the forcing notion $\frak C^+$, active downwards,
and $\dot\frak A$ is the forcing name for $\frak A$ over $\frak C$.
%556G mt55bits

\spheader 556Yd Show that the argument of 556Q is sufficient to take us
from ($\dagger$) there to Theorem 395N, as well as to 395P.
%556Q

\spheader 556Ye Show that if the Proper Forcing Axiom is true then
the asymptotic density algebra $\frak Z$ is not homogeneous.   \Hint{5A6H.}
%556S

\spheader 556Yf\dvAnew{2015} Let $(\frak A,\bar\mu)$ be a probability
algebra, $\frak C$ a closed subalgebra and $\Bbb P$ the forcing notion
$\frak C^+$ active downwards.   Set $q(t)=-t\ln t$ for $t>0$,
$0$ for $t\le 0$ (cf.\ 385A).
Let $A$ be a finite partition of unity in $\frak A$, and $\dot A$ the
$\Bbb P$-name $\{(\dot a,1):a\in A\}$.
(i) Confirm that the definition of
$q$ can be interpreted in the forcing universe $V^{\Bbb P}$.
(ii) Show
that if $u\in L^0(\frak C)$ then $\VVdP\,(\bar q(u))\sspvec=q(\vec u)$.
(iii) Set $v=\sum_{a\in A}\bar q(P\chi a)$ where $P$ is the conditional
expectation associated with $\frak C$ (cf.\ 385D).   Show that

\Centerline{$\VVdP\,\dot A$ is a partition of unity
in $(\dot{\frak A},\dot{\bar\mu})$ and its entropy is $\vec v$.}

\noindent(iv) Re-examine Lemma 385Ga in the light of this.
}%end of exercises

\endnotes{
\Notesheader{556} I did not positively instruct you to do so in the
introduction to this section, but I expect that most readers will
have passed rather quickly over the nineteen %\query check
$^{\centerdot}$-infested pages up to
556L, and looked at the target theorems from 556M on.   In each of the
first three we have a pair ($\ddagger$), ($\dagger$) of propositions,
($\dagger$)
being the special case of ($\ddagger$) in which an algebra $\Tau$ or
$\frak C$ is trivial.   If, as I hope, you are already acquainted with at
least one of the assertions ($\ddagger$), you will know that it can be
proved by essentially the same methods as the corresponding ($\dagger$),
but with some non-trivial technical changes.   These technical changes,
already incorporated in the proofs of 388L/556N and 395P/556P in
Volume 3, and indicated in \S458 for 458Ye/556M,
certainly do not amount to nineteen pages of mathematics in total;
moreover, they explore ideas which are of independent interest.
So I cannot on this evidence claim that the approach gives quick proofs of
otherwise inaccessible results.

What I do claim is that the general method gives a way of
looking at a recurrent phenomenon.   Throughout the theory of
measure-preserving transformations, ergodic transformations have a special
place;  and one comes to expect that once one has answered a question for
ergodic transformations, the general case will be easy to determine.
Similarly, every theorem about independent random variables ought to have a
form applying to relatively independent variables.   Indeed there are
standard techniques for developing such extensions, based on
disintegrations, as in \S\S458-459.   What I have tried to do here is to
develop a completely different approach, and in the process to indicate a
new aspect of the theory of forcing.   I note that the method demands
preliminary translations into the language of measure algebras,
which suits my prejudices as already expressed at length in Volume 3.

The message is that everything works.   There are no royal roads in
mathematics, and to use this one you will have to master some non-trivial
machinery.   But perhaps just knowing that a machine exists will give
you the confidence to attack similar problems in your own way.   I offer an
example in 556Yf.   Note that this depends on the fact that the ordinary
functions of elementary calculus have definitions which can be interpreted
in any forcing universe.

The great bulk of the work of this section consists of routine checks that
natural formulae are in fact valid.   You will see that some simple ideas
recur repeatedly, but the details demand a certain amount of attention.
At the very beginning, in finding a forcing name $\dot a$ for an
element of a Boolean algebra, we have to take care that we are exactly
following our preferred formulation of what a name `is'.   (If my preferred
formulation is not yours, you have some work to do, but it should not be
difficult, and might be enlightening.)   It is not
surprising that regularly embedded subalgebras have a special status
(556D);  it is worth taking a moment to think about why it matters so much
(556Xa).   In 556H, I do not think it is obvious that $\frak A$ must be
Dedekind complete, rather than just Dedekind $\sigma$-complete,
to make the ideas work in the straightforward way that they do (556Yc).
When we come to measure algebras (556K), we need to be sure that we have a
description of forcing names for real numbers which is compatible with the
apparatus here.   Again and again, we have sentences with clauses in both
the forcing language and in ordinary language, and we must keep the pieces
properly segregated in our minds.

The last fifth of the section (556Q-556S) %556Q 556R 556S
is quite hard work for the result we get, but I think it is particularly
instructive, in that it cannot be regarded as a technical extension of a
simpler and more important
result.   It is a good example of a theorem proved by a method
unavoidably dependent on the Forcing Theorem (5A3D), and for which it is
not at all clear that a proof avoiding forcing can be made manageably
simple.   Such a proof must exist, but the obvious route to it involves
teasing out the requisite parts of the proof of Maharam's theorem, and
translating them into properties of the set

\Centerline{$\{(J,\alpha,K):K\in[\Bbb N]^{\omega}$,
  $J\subseteq\Bbb N$,
  $\lim_{n\in K,n\to\infty}2^{-n}\#(J\cap I_n)=\alpha\}$}

\noindent as in part (d) of the proof of 556S, but going very much farther.
My own experience is that facing up to such challenges is often profitable,
but for the moment I am happy to present an adaptation of Farah's
original proof.

An easy corollary of Theorem 556S is that $\widehat{\frak Z}$ is
homogeneous (556Xc).   This is striking in view of the fact that $\frak Z$
itself may or may not be homogeneous.   If the continuum hypothesis is
true, then $\frak Z$ is indeed homogeneous ({\smc Farah 03}); %8.2
but if
the Proper Forcing Axiom is true, then $\frak Z$ is {\it not} homogeneous
(556Ye), even though its completion is.
}%end of notes

\discrpage



%}}}}}

\references{
\frfilename{mt5r.tex}
\versiondate{28.4.23}
\copyrightdate{2008}

% except for references in ...X., ...Y., ...Z. give locations in form
%  342L  rather than  342Lb
%  \S511 {\it notes\/}, \S512 {\it intro.}]

\gdef\topparagraph{}
\gdef\bottomparagraph{}
\gdef\newparagraph{}
\gdef\chaptername{References}
\gdef\sectionname{References}
\gdef\headlinesectionname{References}

\centerline{\bf References for Volume 5}

%use subparagraph details (5A1Aa) only for X. Y. Z.
%  \S511 {\it notes\/}, \S512 {\it intro.}]

\medskip%A

%Antonovskii M.Ja.\ \& Chudnovsky D.V.\ [76] `Some questions of general
%topology and Tikhonov semifields II', Russian Math.\ Surveys 31 (1976)
%69-128.

Argyros S.A.\ \& Kalamidas N.D.\ [82] `The $K_{\alpha n}$ property on
spaces with strictly positive measure', Canadian J.\ Math.\ 34 (1982)
1047-1058.
\cmmnt{[525T.]}

Argyros S.A.\ \& Tsarpalias A.\ [82] `Calibers of compact spaces',
Trans.\ Amer.\ Math.\ Soc.\ 270 (1982) 149-162.
\cmmnt{[525N.]}

\medskip%B

Babiker A.G.\ [76] `On uniformly regular topological measure spaces',
Duke Math.\ J.\ 43 (1976) 775-789.
\cmmnt{[533Xd.]}

Balcar B.\ \& Fran\v{e}k F.\ [82] `Independent families in complete Boolean
algebras', Trans.\ Amer.\ Math.\ Soc.\ 274 (1982) 607-618.
\cmmnt{[515H.]}

Balcar B., Jech T.\ \& Paz\'ak T.\ [05]
`Complete ccc Boolean algebras, the order sequential topology,
and a problem of von Neumann',
Bull.\ London Math.\ Soc.\ 37 (2005) 885-898.
\cmmnt{[\S539 {\it intro.}, 539E, 539L, 539N, \S539 {\it notes\/}, 555K.]}
 % arXiv.math.LO/0312473

Balcar B., Jech T.\ \& Zapletal J.\ [97] `Semi-Cohen Boolean algebras',
Ann.\ Pure \& Applied Logic 87 (1997) 187-208.
\cmmnt{[\S515 {\it notes\/}, \S547 {\it notes\/}.]}

Balcar B.\ \& Vojt\'a\v{s} P.\ [77] `Refining systems on Boolean algebras',
pp.\ 45-58 in {\smc Lachlan Srebny \& Zarach 77}.
\cmmnt{[515E.]}

Bar-Hillel Y.\ [70] {\it Mathematical Logic and Foundations of Set Theory.}
North-Holland, 1970.

Bartoszy\'nski T.\ [84] `Additivity of measure implies additivity of
category', Trans.\ Amer.\ Math.\ Soc.\ 281 (1984) 209-213.
\cmmnt{[Chap.\ 52 {\it intro.}, 522Q, \S524 {\it notes\/}.]}

Bartoszy\'nski T.\ [87] `Combinatorial aspects of measure and category',
Fund.\ Math.\ 127 (1987) 225-239.
\cmmnt{[522S.]}
% http://matwbn.icm.edu.pl/ksiazki/fm/fm127/fm127118.pdf

% Bartoszy\'nski T.\ [88] `On covering of real line by null sets',
% Pacific J.\ Math.\ 131 (1988) 1-12.

% Bartoszy\'nski T.\ [p97] `Invariants of measure and category', in
% preparation for {\smc Foreman Kanamori \& Magidor ??}

% Bartoszy\'nski T.\ [00] `A note on duality between measure and category',
% Proc.\ Amer.\ Math.\ Soc.\ 128 (2000) 2745-2748.
   % (2^N,+,meager) not iso to (2^N,+,negligible)

Bartoszy\'nski T.\ \& Judah H.\ [89] `On the cofinality of the smallest
covering of the real line by meager sets', J.\ Symbolic Logic 54 (1989) 828-832.
\cmmnt{[522V.]}

Bartoszy\'nski T.\ \& Judah H.\ [95] {\it Set Theory:  on the structure
of the real line}.  A.K.Peters, 1995.
\cmmnt{[522R, \S522 {\it notes\/},
526M, \S528 {\it notes\/}, 534Q, \S534 {\it notes\/}, 546I,
553C, 554Ya, 5A1Q.]}

%Bartoszy\'nski T., Judah H.\ \& Shelah S.\ [93] `The Cicho\'n diagram',
%J.\ Symbolic Logic 58 (1993) 401-423.

%Bartoszy\'nski T.\ \& Scheepers M.\ [96] (eds.) {\it Set Theory (BEST, 1992-1994).}
%Amer.\ Math.\ Soc., 1996 (Contemporary Mathematics 192).
% ISBN 0821803069

Bartoszy\'nski T.\ \& Shelah S.\ [92] `Closed measure zero sets', Ann.\
Pure \& Applied Logic 58 (1992) 93-110.
\cmmnt{[526M.]}

Baumgartner J.E., Taylor A.D.\ \& Wagon S.\ [77] `On splitting
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\cmmnt{[\S541 {\it notes\/}, \S544 {\it notes\/}.]}

Bell M.G.\ [81] `On the combinatorial principle $P(c)$',
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\cmmnt{[\S517 {\it notes\/}.]}

Benedikt M.\ [98] `Ultrafilters which extend measures',
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\cmmnt{[538M, 538Xm, \S538 {\it notes\/}.]}

Benedikt M.\ [99] `Hierarchies of measure-theoretic ultrafilters',
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Blair C.E.\ [77] `The Baire category theorem implies the principle of
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\cmmnt{[\S566 {\it notes\/}.]}

%Blass A.\ [88] `Selective ultrafilters and homogeneity', Ann.\ Pure and
%Applied Logic 38 (1988) 252-285.

%Blass A.\ [p01b] `Combinatorial cardinal characteristics of the continuum',
%pp. ?? in {\smc Foreman Kanamori \& Magidor ??}.

Blass A.\ \& Laflamme C.\ [89] `Consistency results about filters and the
number of inequivalent growth types', J.\ Symbolic Logic 54 (1989) 50-56.
% u<g  filter dichotomy
\cmmnt{[5A6Ib, 5A6J.]}

Booth D.\ [70] `Ultrafilters on a countable set',
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\cmmnt{[517R.]}

Borel E.\ [1919] `Sur la classification des ensembles de mesure nulle', Bull.\ Soc.\ Math.\ France 47 (1919) 97-125.
\cmmnt{[\S534 {\it intro.}]}

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Research report, March 2000.
\cmmnt{[528N, 529F-529H.]} %529F 529G 529H

%Brendle J.\ [01] `Cardinal invariants of the continuum --- a survey',
%J.\ Research Institute for Mathematical Sciences, Kyoto University
%(JRIMS Kokyuroku) 1202 (2001) 7-32.

Brendle J.\ [06] `Cardinal invariants of the continuum and combinatorics
on uncountable cardinals',
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\cmmnt{[529F, 529H.]}

Burke M.R.\ [93] `Liftings for Lebesgue measure', pp.\ 119-150 in
{\smc Judah 93}.
\cmmnt{[\S535 {\it notes\/}.]}

%Burke M.R.\ [n95] `Consistent liftings', privately circulated, 1995.

%Burke M.R.\ [n96] `A theorem of Gitik and Shelah on disjoint refinements
%of sequences of subsets of the real line', note of 23.8.96.
%\cmmnt{[5x46J, \S547 {\it intro.}]}

Burke M.R.\ [n05] `Non-null sets of minimal cofinality', note of 2.3.05.
\cmmnt{[523K.]}

Burke M.R.\ \& Magidor M.\ [90] `Shelah's pcf theory and
its applications', Ann.\ Pure and Applied Logic 50 (1990) 207-254.
\cmmnt{[513J, \S5A2 {\it intro.}.]}

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\cmmnt{[\S535 {\it notes\/}.]}

\medskip%C

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\cmmnt{[552N, \S552 {\it notes.}]}

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\cmmnt{[534R.]}

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\cmmnt{[532I.]}

%Cicho\'n J., Kamburelis T.\ \& Pawlikowski J.\ [85] `On dense subsets of
%the measure algebra', Proc.\ Amer.\ Math.\ Soc.\ 94 (1985) 142-146.

Cicho\'n J.\ \& Pawlikowski J.\ [86]
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\cmmnt{[527F.]}

%Cohen P.\ [63] `The independence of the continuum hypothesis',
% Proc.\ Nat.\ Acad.\ Sci.\ U.S.A.\ 50 (1963) 1143-1148.

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\cmmnt{[538F, 538Yb.]}

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\cmmnt{[516 {\it notes\/}.]}

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\medskip%D

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\cmmnt{[5A6D.]}

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%preprint, 1998

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%Drake F.\ [74] {\it Set Theory.}  North-Holland, 1974.

%Drake F.R.\ \& Singh D.\ [96] {\it Intermediate Set Theory.}
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% ISBN 0 471 96496 4

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\medskip%E

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\medskip%F

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\cmmnt{[561A.]}

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\cmmnt{[555N.]}

% Fleissner W.G., Hansell R.W.\ \& Junnila H.J.K.\ [82] `PMEA implies
% Proposition P', Topology and its Appl.\ 13 (1982) 255-262.

% Foreman M., Kanamori A.\ \& Magidor M.\ [??] {\it Handbook of Set
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Foreman M.\ \& Kanamori A.\  [10] {\it Handbook of Set Theory.}
Springer, 2010.

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\cmmnt{[563A.]}

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fail everywhere', Annals of Math.\ 133 (1991) 1-35.
\cmmnt{[525Z.]}

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notions of compactness', J.\ London Math.\ Soc. (2) 57 (1998) 1-19.
\cmmnt{[\S566 {\it notes\/}.]}

% Frankiewicz R.\ \& Plebanek G.\ [94] `Nonaccessible filters in
% measure algebras and functionals on $L^{\infty}(\lambda)^*$', Studia
% Math.\ 108 (1994) 191-200.

%Frankiewicz R.\ \& Plebanek G.\ [96] `On asymptotic density and
%uniformly distributed sequences', Studia Math.\ 119 (1996) 17-26.

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\cmmnt{[\S518 {\it notes\/}.]}

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\cmmnt{[537K.]}

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\cmmnt{[564Xc.]}

Fremlin D.H.\ [77]
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\cmmnt{[533J.]}

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\cmmnt{[\S524 {\it notes\/}.]}

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\cmmnt{[551A.]}

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\cmmnt{[525S.]}

Fremlin D.H.\ [91] `The partially ordered sets of measure
theory and Tukey's ordering,' Note di Matematica  11 (1991) 177-214.
[\S524 {\it notes\/}, 526B, 526I, 527J, 529C, 529D, 534L.]

Fremlin D.H.\ [93] `Real-valued-measurable cardinals', pp.\ 151-304
in {\smc Judah 93}.
\cmmnt{[\S537 {\it notes\/}, \S541 {\it notes\/}, \S544 {\it notes\/},
\S545 {\it notes\/}, \S5A2 {\it intro.}.]}

Fremlin D.H.\ [97] `On compact spaces carrying Radon measures of
uncountable Maharam type', Fundamenta Math.\  154 (1997) 295-304.
\cmmnt{[531T.]}

%Fremlin D.H.\ [00] `Weakly $\alpha$-favourable measure spaces',
%Fundamenta Math.\ 165 (2000) 67-94.

Fremlin D.H.\ [03] `Skew products of ideals',
J.\ Applied Analysis 9 (2003) 1-18.
\cmmnt{[\S527 {\it notes\/}.]}
%n01910.tex

%Fremlin D.H.\ [n97] `A theorem of Gitik and Shelah on disjoint
%refinements of sequences of subsets of the real line', note of 4.4.97.
%n96j01.tex

%Fremlin D.H.\ [n99] `Embedding the Lebesgue measure algebra into
%$\Cal P\Bbb N/\Cal Z$', note of 25.5.99.
%n99525.tex

%Fremlin D.H.\ [n99] `Notes on {\smc Farah p99}', note of 26.7.99.
%n99612.tex

%Fremlin D.H.\ [n99] `Notes on {\smc Solecki p96}', note of 17.6.99.
%n99617.tex

%Fremlin D.H.\ [n02] `Games and the Tukey classification', note of 29.9.02.
%n02929.tex

Fremlin D.H.\ \& Grekas S.\ [95] `Products of completion regular
measures', Fundamenta Math.\ 147 (1995) 27-37.
\cmmnt{[532D.]}

Fremlin D.H.\ \& Miller A.W.\ [88]
`On some properties of Hurewicz, Menger and Rothberger',
Fund.\ Math.\ 129 (1988) 17-33.
\cmmnt{[534Q.]}

Fremlin D.H., Natkaniec T.\ \& Rec{\l}aw I.\ [00]
`Universally Kuratowski-Ulam spaces', Fundamenta Math.\ 165 (2000) 239-247.
\cmmnt{[\S527 {\it notes\/}.]}

%Friedman H.\ [80]
%`A consistent Fubini-Tonelli theorem for nonmeasurable functions',
% Illinois J.\ Math.\ 24 (1980) 390-395.

Friedman S.D. \& Koepke P.\ [97] `An elementary approach to the fine
structure of L', Bull.\ Symbolic Logic 3 (1997) 453-468.
\cmmnt{[5A6D.]}

Fuchino S.,  Geschke S., Shelah S.\ \& Soukup L.\ [01]
`On the weak Freese-Nation property of complete Boolean algebras',
Ann.\ Pure Appl.\ Logic 110 (2001) 89-105.
\cmmnt{[518K.]}

Fuchino S., Geschke S.\ \& Soukup L.\ [01] `On the weak Freese-Nation
property of $\Cal P(\omega)$', Arch.\ Math.\ Logic 40 (2001) 425-435.
\cmmnt{[522U, \S522 {\it notes\/}.]}

Fuchino S., Koppelberg S.\ \& Shelah S.\ [96] `Partial orderings with
the weak Freese-Nation property',
Ann.\ Pure and Applied Logic 80 (1996) 35-54.
\cmmnt{[518A, 518G, 518Yb, \S518 {\it notes\/}, 522U.]}

Fuchino S.\ \& Soukup L.\ [97] `More set theory around the weak
Freese-Nation property', Fundamenta Math.\ 154 (1997) 159-176.
\cmmnt{[518I, 518K.]}

\medskip%G

Galvin F.\ [80] `Chain conditions and products',
Fundamenta Math.\ 108 (1980) 33-48.
\cmmnt{[\S553 {\it notes\/}.]}

Galvin F., Mycielski J.\ \& Solovay R.M.\ [79] `Strong measure zero sets',
Notices Amer.\ Math.\ Soc.\ 26 (1979) A280.
%(abstract 79T-E25);  often assigned to year 1973
\cmmnt{[534K.]}

Gandy R.O.\ \& Hyland J.M.E.\ [77](eds.) {\it Logic Colloquium '76.}
North-Holland, 1977.

Geschke S.\ [02] `On tightly $\kappa$-filtered Boolean algebras',
Algebra Universalis 47 (2002) 69-93.
\cmmnt{[518P, 518S, \S535 {\it notes\/}.]}

Gitik M.\ \& Shelah S.\ [89] `Forcings with ideals and
simple forcing notions', Israel J.\ Math.\ 68 (1989) 129-160.
\cmmnt{[543E, 547F.]}

Gitik M.\ \& Shelah S.\ [93]
`More on simple forcing notions and forcings with ideals',
Annals of Pure and Applied Logic 59 (1993) 219-238.
\cmmnt{[542E, 543E, 547F.]}

Gitik M.\ \& Shelah S.\ [01]
`More on real-valued measurable cardinals and forcing with ideals',
Israel J.\ Math. 124 (2001) 221-242.
\cmmnt{[548E.]}
%GiSh582

G\l\'owczy\'nski W.\ [91] `Measures on Boolean algebras',
Proc.\ Amer.\ Math.\ Soc.\ 111 (1991) 845-849.
\cmmnt{[555K.]}

G\l\'owczy\'nski W.\ [08] `Outer measure on Boolean algebras', Acta Univ.\
Carolinae (Math.\ et Phys.) 49 (2008) 3-8.
\cmmnt{[555K.]}

Goldstern M., Judah H.\ \& Shelah S.\ [93] `Strong measure zero sets
without Cohen reals',
J.\ Symbolic Logic 58 (1993) 1323-1341.
\cmmnt{[\S534 {\it notes\/}.]}

\medskip%H

% Hart J.\ \& Kunen K.\ [98] `Weak Measure Extension Axioms',
% Topology and its Applications 85 (1998) 219-246.
% http://www.math.wisc.edu/~kunen/me.ps

Haydon R.G.\ [77] `On Banach spaces which contain $\ell^1(\tau)$ and
types of measures on compact spaces', Israel J.\ Math.\ 28 (1977) 313-324.
\cmmnt{[531E, 531L.]}

%Haydon R.G.\ [78] `On dual $L^1$-spaces and injective bidual Banach
%spaces,' Israel J.\ Math.\ 31 (1978) 142-152.

Hodges W., Hyland M., Steinhorn C.\ \& Truss J.\ [96] (eds.)
{\it Logic: from Foundations to Applications,
European Logic Colloquium, 1993.}
Clarendon, 1996.
%[ISBN 0-19-853862-6]

Humke P.D.\ \& Laczkovich M.\ [05] `Symmetrically approximately
continuous functions, consistent density theorems, and Fubini type
inequalities', Trans.\ Amer.\ Math.\ Soc.\ 357 (2005) 31-44.
\cmmnt{[537Q, \S537 {\it notes\/}.]}

\medskip%I

Ihoda J.I.\ [88] `Strong measure zero sets and rapid filters',
J.\ Symbolic Logic 53 (1988) 393-402.
%model of Borel conjecture + Ramsey filter
\cmmnt{[534Q.]}

\medskip%J

Jech T.\ [73] {\it The Axiom of Choice.}  North-Holland, 1973.
\cmmnt{[561A, 561Xc, 561Yc, 561Yi, 564Yd.]}

Jech T.\ [78] {\it Set Theory.}  Academic, 1978 (ISBN 0123819504).
\cmmnt{[{\it Intro.}, 553H,
555O, \S555 {\it notes\/}, \S562 {\it notes\/}, \S5A1, 5A6B.]}
%5A1B 5A1D 5A1E 5A1F B
%ASIN B0095LMZ9O

Jech T.\ [03] {\it Set Theory, Millennium Edition.}  Springer, 2003
(ISBN 3540440852).
\cmmnt{[\S521 {\it notes\/}, \S555 {\it notes\/},
\S562 {\it notes\/}, \S567 {\it notes\/}, \S5A1, 5A3N, 5A3P, 5A6B.]}
%5A1B 5A1D 5A1E 5A1F B

%Jin R.\ \& Shelah S.\ [p96] `Compactness of Loeb spaces', preprint, 1996.

%Jovanovi\'c A.\ [p01]
%`Real valued measurability, some set-theoretic aspects',
%pp.\ ?? in {\smc Pap 0?}

Judah H.\ [93] (ed.) {\it Proceedings of the Bar-Ilan Conference on Set
Theory and the Reals, 1991.} Amer.\ Math.\ Soc.\ (Israel Mathematical
Conference Proceedings 6), 1993.

Judah H.\ \& Repick\'y M.\ [95] `Amoeba reals', J.\ Symbolic Logic 60 (1995) 1168-1185.
\cmmnt{[528N.]}

%Juh\'asz I.\ [71]  {\it Cardinal Functions in Topology.}  North-Holland,
%1975 (Math.\ Centre Tracts 34).

%Juh\'asz I.\ [80]  {\it Cardinal Functions in Topology -- Ten Years Later.}  Mathematisch Centrum, Amsterdam, 1980 (Math.\ Centre Tracts 123).   ISBN 9061961963.

Just W.\ [92] `A modification of Shelah's oracle-c.c., with
applications', Trans.\ Amer.\ Math.\ Soc.\ 329 (1992) 325-356.
\cmmnt{[527M.]}

%Just W.\ \& Krawczyk A.\ [84] `On certain Boolean algebras
%$\Cal P(\omega)/I$', Trans.\ Amer.\ Math.\ Soc.\ 285 (1984) 411-429.

Just W.\ \& Weese M.\ [96] {\it Discovering Modern Set Theory I.}
Amer.\ Math.\ Soc., 1996 (Graduate Studies in Mathematics 8).
\cmmnt{[{\it Intro.}.]}

Just W.\ \& Weese M.\ [97] {\it Discovering Modern Set Theory II.}
Amer.\ Math.\ Soc., 1997 (Graduate Studies in Mathematics 18).
\cmmnt{[5A1B, 5A1F, 5A1H.]}

\medskip%K

%Kada M. \& Yuasa Y.\ [96] `Cardinal invariants about shrinkability of unbounded
%sets', Topology and its Applications 74 (1996) 215-223.

%Kamburelis A.\ [89] `Iterations of Boolean algebras with measure',
%Arch.\ Math.\ Logic 29 (1989) 21-28.
% chargeable algebras

%Kamburelis A.\ [90] `A new proof of the Gitik-Shelah theorem', Israel
%J.\ Math.\ 72 (1990) 373-380.

Kanamori A.\ [03] {\it The Higher Infinite.}   Springer, 2003.
\cmmnt{[\S541 {\it notes\/}, \S555 {\it notes\/}, \S567 {\it notes\/},
5A1H, 5A6B, 5A6F.]}

Kechris A.S.\ [95] {\it Classical Descriptive Set Theory.}  Springer,
1995.
\cmmnt{[5A1D, \S567 {\it notes\/}.]}
%QA248.K4

Keisler H.J.\ \& Tarski A.\ [64] `From accessible to
inaccessible cardinals', Fundamenta Math.\ 53 (1964) 225-308;  errata
Fundamenta Math.\ 57 (1965) 119.
\cmmnt{[\S541 {\it intro.\/}, \S541 {\it notes\/}.]}

Kelley J.L.\ [50] `The Tychonoff product theorem implies the axiom of
choice', Fundamenta Math.\ 37 (1950) 75-76.
\cmmnt{[561D.]}

Koppelberg S.\ [75]
`Homomorphic images of $\sigma$-complete Boolean algebras',
Proc.\ Amer.\ Math.\ Soc.\ 51 (1975) 171-175.
\cmmnt{[515L.]}

Koppelberg S.\ [89] {\it General Theory of Boolean
Algebras\/}, vol.\ 1 of {\smc Monk 89}.
\cmmnt{[\S515 {\it notes\/}.]}

Koppelberg S.\ \& Shelah S.\ [96]
`Subalgebras of Cohen algebras need not be Cohen',
pp.\ 261-275 in {\smc Hodges Hyland Steinhorn \& Truss 96}.
\cmmnt{[\S515 {\it notes\/}, \S547 {\it notes\/}.]}

Kraszewski J.\ [01] `Properties of ideals on the generalized Cantor spaces',
J.\ Symbolic Logic 66 (2001) 1303-1320.
\cmmnt{[523G, 523H, 523J, \S523 {\it notes\/}.]}

%Krawczyk A.\ [n85] `Dominating reals add random reals', 1985.

Kumar A.\ [13] `Avoiding rational distances', Real Analysis Exchange 38
(2012/13), 493-498.
%{\tt http://www.math.wisc.edu/\~{}akumar/ARD.pdf}.
%{\tt
%https:/{\bsp}projecteuclid.org/journals/real-analysis-exchange/volume-38%
%/issue-2/Avoiding-Rational-{\vthsp}Distances/rae/1403894908.full}
\cmmnt{[548Xb.]}

%Kumar A.\ [n16] `On a theorem of Gitik and Shelah', note of January 2016
%({\tt http:/{\bsp}www.math.huji.ac.il/\~{}akumar/otgs.pdf}).

Kumar A.\ \& Shelah S.\ [17] `A transversal of full outer measure',
Advances in Math.\ 321 (2017) 475-485.
\cmmnt{[547P, 547Q, \S548 {\it intro.\/}, 548C.]}
%http://home.iitk.ac.in/~krashu/tfom.pdf;  printed version not seen

%Kunen K.\ [75] `Some points in $\beta N$',
% Math.\ Proc.\ Cambridge Phil.\ Soc.\ 80 (1975) 385-398.

Kunen K.\ [80] {\it Set Theory.}  North-Holland, 1980.
\cmmnt{[{\it Intro.}, 511A, \S521 {\it notes\/}, \S522 {\it notes\/},
\S541 {\it notes\/}, 551Q, \S551 {\it notes\/}, 556F,
\S562 {\it notes}, Appendix {\it intro.}, \S5A1, \S5A3, 5A6B.]}

Kunen K.\ [81] `A compact $L$-space under CH', Topology and its Appl.\
12 (1981) 283-287.
\cmmnt{[\S531 {\it notes\/}.]}

Kunen K.\ [84] `Random and Cohen reals', pp.\ 887-911 in
{\smc Kunen \& Vaughan 84}.
\cmmnt{[\S554 {\it notes}.]}

Kunen K.\ [n70] `$\Pi^1_1$ reflection at the continuum', note of January
1970.
\cmmnt{[543C, 544C, 544E, 544F.]}

Kunen K.\ \& Mill J.van [95] `Measures on Corson compact spaces',
Fundamenta Math.\ 147 (1995) 61-72.
\cmmnt{[531O.]}
%http://www.math.wisc.edu/~kunen/corson.ps

Kunen K.\ \& Vaughan J.E.\ [84] (eds.) {\it Handbook of Set-Theoretic
Topology.}   North-Holland, 1984.

Kuratowski K.\ [66] {\it Topology,} vol.\ I.   Academic, 1966.
\cmmnt{[562G.]}

\medskip%L

Lachlan A., Srebny M.\ \& Zarach A.\ [77] (eds.) {\it Set Theory and Hierarchy Theory.}  Springer, 1977 (Lecture Notes in Math.\ 619).

%Laczkovich M.\ \& Miller A.W.\ [95] `Measurability of functions with
%approximately continuous vertical sections and measurable horizontal
%sections', Colloquium Math.\ 69 (1995) 299-308.

Larson P.\ [09] `The filter dichotomy and medial limits',
J.\ Math.\ Logic 9 (2009) 159-165.
\cmmnt{[538S.]}

Larson P., Neeman I.\ \& Shelah S.\ [10] `Universally measurable sets in
generic extensions', Fundamenta Math.\ 208 (2010) 173-192.
\cmmnt{[553O.]}
	
%Larson P.\ \& Shelah S.\ [p09]
%`There may be just continuum many universally measurable sets', 2009.
%LrSh:947

Laver R.\ [76] `On the consistency of Borel's conjecture', Acta Math.\
137 (1976) 151-169.
\cmmnt{[534Q.]}

Laver R.\ [87] `Random reals and Souslin trees',
Proc.\ Amer.\ Math.\ Soc.\ 100 (1987) 531-534.
\cmmnt{[553M.]}

Levinski J.-P., Magidor M.\ \& Shelah S.\ [90] `Chang's conjecture for
$\aleph_{\omega}$', Israel J.\ Math.\ 69 (1990) 161-172.
\cmmnt{[5A6F.]}

Levy A.\ [71] `The sizes of the indescribable cardinals',
pp.\ 205-218 in {\smc Scott 71}.
\cmmnt{[\S541 {\it notes\/}, \S544 {\it notes\/}.]}

Lipecki Z.\ [09]
`Semivariations of an additive function on a Boolean ring',
Colloquium Math.\ 117 (2009) 267-279.
\cmmnt{[552Xe.]}

%Loeb P.A.\ \& Talvila E.\ [p00]
%`Covering theorems and Lebesgue integration', preprint, June 2000.

%Losada M. \& Todor\v{c}evi\'c S. [p98] `Chains and antichains in Boolean
%algebras', note of 5.12.98.

%Louveau A.\ \& Veli\v{c}kovi\'c B.\ [99] `Analytic ideals and cofinal
%types', Ann.\ Pure and Applied Logic 99 (1999) 171-195.
%\ell^1\not\prT\CalNwd

\medskip%M

Maharam D.\ [1947] `An algebraic characterization of measure algebras',
Ann.\ Math.\ 48 (1947) 154-167.
\cmmnt{[539P.]}

Makowsky J.A.\ \& Ravve E.V.\ (eds.) [98]
{\it Logic Colloquium '95.}  Springer, 1998 (Lecture Notes in Logic 11).
%ISBN 3-540-63994-2

%Malliaris M.\ \& Shelah S.\ [16] `Cofinality spectrum theorems in
%model theory, set theory and general topology',
%J.\ Amer.\ Math.\ Soc.\ 29 (2016) 237-297.

Martin D.A.\ [70] `Measurable cardinals and analytic games',
Fundamenta Math.\ 66 (1970) 287-291.
\cmmnt{[567N.]}

Martin D.A.\ [75] `Borel determinacy', Ann.\ of Math.\ (2) 102 (1975)
363-371.
\cmmnt{[\S567 {\it notes\/}.]}

Martin D.A.\ \& Solovay R.M.\ [70] `Internal Cohen
extensions', Ann.\ Math.\ Logic 2 (1970) 143-178.
\cmmnt{[Chap.\ 52 {\it intro.}, \S528 {\it intro.}, \S528 {\it notes\/}.]}

%Mathias A.R.D.\ [78] `0\# and the $p$-point problem', pp.\ 375-384 in
%{\smc M\"uller \& Scott 78}.

M\'atrai T.\ [p09] `More cofinal types of definable directed orders', 2009
({\tt https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.414.3938}).
\cmmnt{[526L, \S526 {\it notes\/}.]}
% books/matrai.p09.pdf

%Mercourakis S.\ [96] `Some remarks on countably determined measures and
%uniform distribution of sequences', Monatschefte f\"ur Math.\ 121 (1996)
%79-101.

Meyer P.-A.\ [73] `Limites m\'ediales d'apr\`es Mokobodzki',
pp.\ 198-204 in {\smc Dellacherie Meyer \& Weil 73}.
\cmmnt{[538Yl.]}

Meyer P.A.\ [75] (ed.) {\it S\'eminaire de Probabilit\'es IX.}
Springer, 1975 (Lecture Notes in Mathematics 465).

Miller A.W.\ [80] `There are no Q-points in Laver's model for the Borel
conjecture', Proc.\ Amer.\ Math.\ Soc.\ 78 (1980) 103-106.
\cmmnt{[\S538 {\it notes\/}.]}
%actually, no rapid ultrafilters, therefore no rapid filters

Miller A.W.\ [81] `Some properties of measure and category',
Trans.\ Amer.\ Math.\ Soc.\ 266 (1981) 93-114.
\cmmnt{[522J.]}

Miller A.W.\ [82] `The Baire category theorem and cardinals of countable
cofinality', J.\ Symbolic Logic 47 (1982) 275-288.
\cmmnt{[552G, 552Xb.]}

%Miller A.W.\ [83] `Mapping a set of reals onto the reals',
%J.\ Symbolic Logic 48 (1983) 575-584.

Mokobodzki G.\ [75] `R\'el\`evement bor\'elien compatible avec une
classe d'ensembles n\'egligeables.  Application \grv{a} la
d\'esint\'egration des mesures', pp.\ 437-442 in {\smc Meyer 75}.
\cmmnt{[535I.]}

Mokobodzki G.\ [7?] `D\'esintegration des mesures et rel\`evements
Boreliens de sous-espaces de $L^{\infty}(X,\Cal B,\mu)$',
Lab.\ d'Analyse Fonctionnelle, Univ.\ Paris VI (?).
\cmmnt{[535E, \S535 {\it notes\/}.]\footnote{I have not been able to
locate this paper;  I believe it was a seminar report.   I took notes from
it in 1977.}}
% "On two theorems of Mokobodzki", 23.6.77.

Monk J.D.\ [89] (ed.) {\it Handbook of Boolean Algebra.}
North-Holland, 1989.

%Monk J.D.\ [96] {\it Cardinal Invariants on Boolean Algebras.}
%Birkhauser, 1996.
% QA266

Moore J.T.\ [05] `Set mapping reflection', J.\ Math.\ Logic 5 (2005) 87-98.
\cmmnt{[517O.]}

Moschovakis Y.N.\ [70] `Determinacy and prewellorderings of the continuum',
pp.\ 24-62 in {\smc Bar-Hillel 70}.
\cmmnt{[567M.]}

%M\"uller G.H.\ \& Scott D.S.\ [78] (eds.) {\it Higher Set Theory.}
%Springer, 1978 (Lecture Notes in Mathematics 669).

Mycielski J.\ [64] `On the axiom of determinateness',
Fund.\ Math.\ 53 (1964) 205-224.
\cmmnt{[567D.]}

%Mycielski J.\ \& Steinhaus H.\ [62]
%`A mathematical axiom contradicting the axiom of choice',
%Bull.\ Acad.\ Polon.\ Sci.\ (Math., Astron., Phys.) 10 (1962) 1-3.

Mycielski J.\ \& \'Swierczkowski S.\ [64]
`On the Lebesgue measurability and the axiom of determinateness',
Fund.\ Math.\ 54 (1964) 67-71.
\cmmnt{[567F.]}

\medskip%N

Naimark M.A. [70] {\it Normed Rings.}  Wolters-Noordhoff, 1970.
\cmmnt{[\S561 {\it notes\/}.]}

Neumann J.von [1931] `Algebraische Repr\"asentanten der Funktionen ``bis
auf eine Menge vom Masse Null''\nobreak', Crelle's J.\ Math.\ 165 (1931)
109-115.
\cmmnt{[535G.]}

Neumann J.von \& Stone M.H.\ [1935] `The determination of representative
elements in the residual classes of a Boolean algebra', Fundamenta
Math.\ 25 (1935) 353-378.
\cmmnt{[\S535 {\it notes\/}.]}

%Normann D.\ [76] `Martin's axiom and medial functions', Math.\
%Scandinavica 38 (1976) 167-176.

%Nyikos P.J.\ [80] `A provisional solution to the normal Moore space
%problem', Proc.\ Amer.\ Math.\ Soc.\ 78 (1980) 429-435.

\medskip%O\medskip%Pa

%Pap E.\ [0?] (ed.) {\it Handbook of Measure Theory.}  North-Holland, 200?.

Pawlikowski J.\ [86] `Why Solovay real produces Cohen real',
J.\ Symbolic Logic 51 (1986) 957-968.
\cmmnt{[552G, 552H.]}

%Pb%Pc%Pd%Pe

Pe{\l}czy'nski A.\ [68] `On Banach spaces containing $L^1(\mu)$, Studia
Math.\ 30 (1968) 231-246.
\cmmnt{[531Ye.]}

Perovi\'c \v{Z}.\ \& Veli\v{c}kovi\'c B.\ [18] `Ranks of Maharam algebras',
Advances in Math.\ 330 (2018) 253-279.
\cmmnt{[539V.]}

%Pf%Pg%Ph%Pu%Pi

% Pierce R.S.\ [58] `A note on complete Boolean algebras',
% Proc.\ Amer.\ Math.\ Soc.\ 9 (1958) 892-896.
% the Dedekind complete case of 515L

%Pj%Pk%Pl

%Plebanek G.\ [93] `Remarks on measurable Boolean algebras and
%sequential cardinals', Fundamenta Math.\ 143 (1993) 11-22.

Plebanek G.\ [95] `On Radon measures on first-countable compact spaces',
Fundamenta Math.\ 148 (1995) 159-164.
\cmmnt{[531O.]}

Plebanek G.\ [97] `Non-separable Radon measures and small compact
spaces', Fundamenta Math.\ 153 (1997) 25-40.
\cmmnt{[531L, 531M, 531V.]}

Plebanek G.\ [00]
`Approximating Radon measures on first-countable compact spaces',
Colloquium Math.\ 86 (2000) 15-23.
\cmmnt{[533H, 533Yc.]}

Plebanek G.\ [02]
`On compact spaces carrying Radon measures of large Maharam type',
Acta Univ.\ Carolinae 43 (2002) 87-99.
\cmmnt{[531U.]}

Plebanek G.\ \& Sobota D.\ [15] `Countable tightness in the spaces of
regular probability measures', Fund.\ Math.\ 229 (2015) 159-169.
\cmmnt{[531U.]}

%Pm%Pn%Po

Pol R.\ [82] `Note on the spaces $P(S)$ of regular probability measures
whose topology is determined by countable subsets', Pacific J.\ Math.\
100 (1982) 185-201.
\cmmnt{[533Ya.]}

%Pp%Pq%Pr

Prikry K.\ [75] `Ideals and powers of cardinals', Bull.\
Amer.\ Math.\ Soc.\ 81 (1975) 907-909.
\cmmnt{[555N.]}

%Ps%Pt%Pu%Pv%Pw%Px%Py%Pz
\medskip%Q

Quickert S.\ [02] `CH and the Saks property', Fundamenta Math.\ 171
(2002) 93-100.
\cmmnt{[539L.]}

\medskip%R

Raisonnier J.\ \& Stern J.\ [85] `The strength of measurability hypotheses', Israel J.\ Math.\ 50 (1985) 337-349.
\cmmnt{[Chap.\ 52 {\it intro.}, 522Q.]}

%Roitman J.\ [79] `Adding a random or a Cohen real;  topological
%consequences and the effect on Martin's axiom', Fundamenta Math.\ 103
%(1979) 47-60.

%Ros{\l}anowski A.\ \& Shelah S.\ [p00] `Measured creatures', preprint, 7.10.00

Rothberger F.\ [1938a] `Eine \"Aquivalenz zwischen der kontinuumhypothese
unter der Existenz der Lusinschen und Sierpinschischen Mengen',
Fundamenta Math.\ 30 (1938) 215-217.
\cmmnt{[522G.]}

Rothberger F.\ [1938b] `Eine Versch\"arfung der Eigenschaft C',
Fundamenta Math.\ 30 (1938) 50-55.
\cmmnt{[\S534 {\it notes\/}.]}

Rothberger F.\ [1941] `Sur les familles ind\'enombrables de suites de nombres naturels et les probl\`emes concernant la propri\'et\'e {\bf C}', Proc.\ Cambridge Phil.\ Soc.\ 37 (1941) 109-126.
\cmmnt{[534R.]}

\medskip%S

Scott D.S.\ [71] (ed.) {\it Axiomatic Set Theory.}  Amer.\
Math.\ Soc., 1971
(Proceedings of Symposia in Pure Mathematics XIII, vol.\ 1).

Shelah S.\ [82] {\it Proper Forcing.}  Springer, 1982
(Lecture Notes in Mathematics 940).
\cmmnt{[\S538 {\it notes\/}.]}

Shelah S.\ [83] `Lifting problem of the measure algebra', Israel J.\
Math.\ 45 (1983) 90-96.   % Sh185
\cmmnt{[\S535 {\it notes\/}.]}

Shelah S.\ [84] `Can you take Solovay inaccessible away?', Israel J.\ Math.\ 48 (1984) 1-47.
\cmmnt{[\S522 {\it notes\/}.]}

Shelah S.\ [92] `Cardinal arithmetic for skeptics',
Bull.\ Amer.\ Math.\ Soc.\ 26 (1992) 197-210.
% Sh400a
\cmmnt{[5A2D.]}

Shelah S.\ [94] {\it Cardinal Arithmetic.}  Oxford U.P., 1994.
% Sh400
\cmmnt{[5A2D, 5A2G.]}

Shelah S.\ [96] `Further cardinal arithmetic', Israel J.\ Math.\ 95
(1996) 61-114.
\cmmnt{[542I.]}
%Sh 430

Shelah S.\ [98a] {\it Proper and Improper Forcing.}  Springer, 1998.
\cmmnt{[\S538 {\it notes\/}.]}
%http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.pl/1235419814

Shelah S.\ [98b] `There may be no nowhere dense ultrafilter', pp.\ 305-324
in {\smc Makovsky \& Ravve 98}.
\cmmnt{[538H, \S538 {\it notes\/}.]}
% Sh594;  arXiv:math.LO/9611221.

% Shelah S.\ [99] `Special subsets of $^{\cf(\mu)}\mu$, Boolean
% algebras and Maharam measure algebras',
% Topology and its Applications 99 (1999) 135-235.
 % Sh620

Shelah S.\ [00] `Covering of the null ideal may have countable
cofinality', Fundamenta Math.\ 166 (2000) 109-136.
\cmmnt{[\S522 {\it notes\/}.]}
% Sh592

Shelah S.\ [03] `The null ideal restricted to some non-null set may be
$\aleph_1$-saturated',  Fundamenta Math.\ 179 (2003) 97-129.
\cmmnt{[ \S548 {\it notes\/}.]}
% Sh619

%Shelah S.\ \& Fremlin D.H.\ [93] `Pointwise compact and stable sets of
%measurable functions', J.\ Symbolic Logic 58 (1993) 435-455.
  %FrSh 406

Shelah S.\ \& Stepr\=ans J.\ [05]
`Comparing the uniformity invariants of null sets for different measures',
Advances in Math.\ 192 (2005) 403-426.
\cmmnt{[534Za.]}
% Con(\non\Cal N(\mu_H)<\non\Cal N)

%Shelah S.\ \& Woodin H.\ [90] `Large cardinals imply that every
%reasonably definable set of reals is Lebesgue measurable', Israel J.\
%Math.\ 70 (1990) 381-394.

Shipman J.\ [90] `Cardinal conditions for strong Fubini
theorems', Trans.\ Amer.\ Math.\ Soc.\ 321 (1990) 465-481.
\cmmnt{[537K.]}

%Solecki S.\ [06] `Amenability, free subgroups, and Haar null sets in
%non-locally-compact groups',
%Proc.\ London Math.\ Soc.\ 93 (2006) 693-722.

Solecki S.\ \& Todor\v{c}evi\'c S.\ [04]
`Cofinal types of topological directed orders',
Ann.\ Inst.\ Fourier 54 (2004) 1877-1911.
\cmmnt{[513K, 513O, 513Yi.]}

Solecki S.\ \& Todor\v{c}evi\'c S.\ [10]
`Avoiding families and Tukey functions on the nowhere-dense ideal',
J.\ Inst.\ Math.\ Jussieu 9 (2010) 1-31.
\cmmnt{[526L.]}

Solovay R.M.\ [66] `New proof of a theorem of Gaifman and Hales',
Bull.\ Amer.\ Math.\ Soc.\ 72 (1966) 282-284.
\cmmnt{[514Xi.]}

Solovay R.M.\ [70] `A model of set theory in which every set of reals
is Lebesgue measurable', Annals of Math.\ 92 (1970) 1-56.
\cmmnt{[\S522 {\it notes\/}.]}

Solovay R.M.\ [71] `Real-valued measurable cardinals', pp.\ 397-428 in
{\smc Scott 71}.
\cmmnt{[541J, 541P, 541Ya, \S541 {\it notes\/}, 555D, 555O.]}

%Solovay R.M., Reinhardt W.N.\ \& Kanamori A.\ [78]
%`Strong axioms of infinity and elementary embeddings',
%Annals of Mathematical Logic 13 (1978) 73-116.

%Solovay R.M.\ \& Tennenbaum S.\ [71]
%`Iterated Cohen extensions and Souslin's problem',
%Ann.\ Math.\ (2) 94 (1971) 201-245.

%Stepr\=ans J.\ [96] `Cardinal invariants associated with Hausdorff
%capacities', pp.\ 147-184 in {\smc Bartoszy\'nski \& Scheepers 96}.

%Szyma\'nski A.\ [88] `Some remarks on real-valued
%measurable cardinals', Proc.\ Amer.\ Math.\ Soc.\ 104 (1988)
%596-602.

\medskip%T\medskip

Talagrand M.\ [84] {\it Pettis Integral and Measure Theory.}
Mem.\ Amer.\ Math.\ Soc.\  307 (1984).
\cmmnt{[536C.]}

Talagrand M.\ [08] `Maharam's problem', Annals of Math.\ 168 (2008)
981-1009.
\cmmnt{[539A.]}

Tarski A.\ [1945] `Ideale in volst\"andigen Mengenk\"orpen II', Fundamenta Math.\ 33 (1945) 51-65.
\cmmnt{[541P.]}

Todor\v{c}evi\'c S.\ [85] `Remarks on chain conditions in products',
Compositio Math.\ 55 (1985) 295-302.
\cmmnt{[537G.]}

Todor\v{c}evi\'c S.\ [87] `Partitioning pairs of countable ordinals',
Acta Math.\ 159 (1987) 261-294.
\cmmnt{[554Yc.]}

Todor\v{c}evi\'c S.\ [00] `A dichotomy for $P$-ideals of countable sets',
Fundamenta Math.\ 166 (2000) 251-267.
\cmmnt{[5A6G.]}

T\"ornquist A.\ [11] `On the pointwise implementation of near-actions',
Trans.\ Amer.\ Math.\ Soc.\ 363 (2011) 4929-4944.
%{\tt http://arxiv.org/abs/0909.3869}
\cmmnt{[535Yd.]}

Truss J.K.\ [77] `Sets having calibre $\aleph_1$',
pp.\ 595-612 in {\smc Gandy \& Hyland 77}.
\cmmnt{[522J.]}

Truss J.K.\ [88] `Connections between different amoeba algebras',
Fund.\ Math.\ 130 (1988) 137-155.
\cmmnt{[Chap.\ 52 {\it intro.}, 528A, 528Da, 528K, \S528 {\it notes\/}.]}

Tukey J.W.\ [1940] {\it Convergence and Uniformity in Topology.}
Princeton U.P., 1940 (Ann.\ Math.\ Studies 1).
\cmmnt{[513F, \S513 {\it notes}.]}

%U
\medskip

Ulam S.\ [1930] `Zur Masstheorie in der allgemeinen
Mengenlehre', Fundamenta Math.\ 16 (1930) 140-150.
\cmmnt{[\S541 {\it intro.\/}, \S541 {\it notes\/}.]}

%V
\medskip

Veli\v{c}kovi\'c B.\ [92] `Forcing axioms and stationary sets', Advances in
Math.\ 94 (1992) 256-284.
\cmmnt{[517O.]}

Veli\v{c}kovi\'c B.\ [05] `CCC forcing and splitting reals',
Israel J.\ Math.\ 147 (2005) 209-221.
\cmmnt{[\S539 {\it intro.}, 539E, 539N.]}

Vojt\'a\v{s} P.\ [93] `Generalized Galois-Tukey connections between
explicit relations on classical objects of real analysis', pp.\ 619-643
in {\smc Judah 93}.
\cmmnt{[512A.]}

%W
\medskip

Wimmers E.\ [82] `The Shelah $P$-point independence theorem',
Israel J.\ Math.\ 43 (1982) 28-48.
\cmmnt{[\S538 {\it notes\/}.]}

\medskip%X\medskip%Y\medskip%Z

Zakrzewski P.\ [92] `Strong Fubini theorems from measure extension
axioms', Comm.\ Math.\ Univ.\ Carolinae 33 (1992) 291-297.
\cmmnt{[544J.]}

%Zindulka O.\ [00] `Residual measures in locally compact spaces', Topology Appl.\ 108 (2000) 253-265.

%Zindulka O.\ [p98] `Killing residual measures', preprint, 1998.

\discrpage

}%end of references

%---------------------------------------------------------------%

\end



