% 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\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\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{mt33.tex} 
\versiondate{17.11.10} 
\copyrightdate{1995} 
 
\def\chaptername{Maharam's theorem} 
\def\sectionname{Introduction} 
 
\newchapter{33} 
 
We are now ready for the astonishing central fact about measure 
algebras:  there are very few of them.   Any localizable measure algebra 
has a canonical expression as a simple product of measure algebras of 
easily described types.   This complete classification necessarily 
dominates all further discussion of measure algebras;  to the point that 
all the results of Chapter 32 have to be regarded as `elementary', since 
however complex their formulation they have been proved by techniques 
not involving, nor providing, any particular insight into the special 
nature of measure algebras.   The proof depends, of course, on 
developing methods of defining measure-preserving homomorphisms and 
isomorphisms;  I give a number of results, progressively more elaborate, 
but all based on the same idea.   These techniques are of great power, 
leading, for instance, to an effective classification of closed 
subalgebras and their embeddings. 
 
`Maharam's theorem' itself, the classification of localizable measure 
algebras, is in \S332.   I devote \S331 to the definition and 
description of `homogeneous' probability algebras.   In \S333 I turn to 
the problem of describing pairs $(\frak A,\frak C)$ where $\frak A$ is a 
probability algebra and $\frak C$ is a closed subalgebra.   Finally, in 
\S334, I give some straightforward results on the classification of free 
products of probability algebras. 
 
 
 
 
\discrpage 
 
 
\frfilename{mt331.tex}
\versiondate{1.2.05}
\copyrightdate{1995}

\def\chaptername{Maharam's theorem}
\def\sectionname{Maharam types and homogeneous measure algebras}

\newsection{331}

I embark directly on the principal theorem of this chapter (331I), split
between 331B, 331D and 331I;  331B and 331D will be the basis of many of
the results in later sections of this chapter.   In 331E-331H I
introduce the concepts of `Maharam type' and `Maharam-type-homogeneity'.
I discuss the measure algebras of products $\{0,1\}^{\kappa}$, showing
that these provide a complete set of examples of \Mth\
probability algebras (331J-331L).

\leader{331A}{Definition}\cmmnt{ The following idea is almost the key
to the whole chapter.}   Let $\frak A$ be a Boolean algebra and
$\frak B$ an order-closed subalgebra of $\frak A$.
A non-zero element $a$ of $\frak A$ is
a {\bf relative atom} over $\frak B$ if every $c\Bsubseteq a$ is of the
form $a\Bcap b$ for some $b\in\frak B$\cmmnt{;  that is,
$\{a\Bcap b:b\in\frak B\}$ is the principal ideal generated by $a$}.
\cmmnt{We say that} $\frak A$ is {\bf relatively atomless} over $\frak B$
if there are no relative atoms in $\frak A$ over $\frak B$.

\cmmnt{(I'm afraid the phrases `relative atom', `relatively atomless'
are bound to seem opaque at this stage.   I hope that after the
structure theory of \S333 they will seem more natural.   For the moment,
note only that $a$ is an atom in $\frak A$ iff it is a relative atom
over the smallest subalgebra $\{0,1\}$, and every element of $\frak A$
is a relative atom over the largest subalgebra $\frak A$.   In a way,
$a$ is a relative atom over $\frak B$ if its image is an atom in a kind
of quotient $\frak A/\frak B$.   But we are two volumes away from any
prospect of making sense of this kind of quotient.)}

At some point I ought to remark that if $a$ is an atom of $\frak A$ then it is
\cmmnt{surely} a relative atom over $\frak B$;  
\cmmnt{so} if $\frak A$ is relatively atomless
over $\frak B$, then it is atomless\cmmnt{ in the sense of 316Kb}.

\leader{331B}{}\dvArevised{2010}\cmmnt{ The first lemma is the heart of
Maharam's theorem.

\medskip

\noindent}{\bf Lemma} Let $(\frak A,\bar\mu)$ be a totally finite
measure algebra, $\frak B$ a closed subalgebra of $\frak A$ such that
$\frak A$ is relatively atomless over $\frak B$, and $a_0\in\frak A$.
Let $\nu:\frak B\to\Bbb R$ be an additive functional such
that $0\le\nu b\le\bar\mu(b\Bcap a_0)$ for every $b\in\frak B$.
Then there is a $c\in\frak A$ such that
$c\Bsubseteq a_0$ and $\nu b=\bar\mu(b\Bcap c)$ for every $b\in\frak B$.

\cmmnt{\medskip

\noindent{\bf Remark} Recall that by 323H we need not distinguish
between `order-closed' and `topologically closed' subalgebras.
}

\proof{{\bf (a)} It is worth noting straight away that $\nu$ is
necessarily countably additive.   This is easy to check from first
principles, but if you want to trace the underlying ideas they are in
313O (the identity map from $\frak B$ to $\frak A$ is
order-continuous), 326Jf (so $\mu\restrp\frak B:\frak B\to\Bbb R$ is
countably additive) and 326Kb (therefore $\nu$ is countably additive).

\medskip

{\bf (b)} For each $a\in\frak A$ set
$\nu_ab=\bar\mu(b\Bcap a)$ for every $b\in\frak B$;  
then $\nu_a:\frak B\to\Bbb R$ is
countably additive (326Jd).   Note that $\nu_{c\Bcup d}=\nu_c+\nu_d$ whenever
$c$. $d\in\frak A$ are disjoint.   The key idea is the
following fact:  for every non-zero $a\in\frak A$ there is a non-zero
$d\Bsubseteq a$ such that $\nu_d\le\bover12\nu_a$.   \Prf\ Because
$\frak A$ is relatively atomless over $\frak B$, there is an
$e\Bsubseteq a$ such that $e\ne a\Bcap b$ for any $b\in\frak B$.
Consider the countably additive
functional $\lambda=\nu_a-2\nu_e:\frak B\to\Bbb R$.   By 326M, there is
a $b_0\in\frak B$ such that $\lambda b\ge 0$ whenever $b\in\frak B$ and
$b\Bsubseteq b_0$, while $\lambda b\le 0$ whenever $b\in\frak B$ and
$b\Bcap b_0=0$.

If $e\Bcap b_0\ne 0$, try $d=e\Bcap b_0$.   Then $0\ne d\Bsubseteq a$,
and for every $b\in\frak B$

\Centerline{$\nu_db=\nu_e(b\Bcap b_0)
=\Bover12(\nu_a(b\Bcap b_0)-\lambda(b\Bcap b_0))\le\Bover12\nu_ab$}


\noindent (because $\lambda(b\Bcap b_0)\ge 0$) so
$\nu_d\le\bover12\nu_a$.

If $e\Bcap b_0=0$, then (by the choice of $e$) $e\ne a\Bcap(1\Bsetminus
b_0)$, so $d=a\Bsetminus(e\Bcup b_0)\ne 0$, and of course $d\Bsubseteq
a$.   In this case, for every $b\in\frak B$,

\Centerline{$\nu_db=\nu_a(b\Bsetminus b_0)-\nu_e(b\Bsetminus b_0)
=\Bover12(\lambda(b\Bsetminus b_0)+\nu_a(b\Bsetminus b_0))
\le\Bover12\nu_ab$}


\noindent (because $\lambda(b\Bsetminus b_0)\le 0$), so once again
$\nu_d\le\bover12\nu_a$.


Thus in either case we have a suitable $d$.\   \Qed

\medskip

{\bf (c)} It follows at once, by induction on $n$, that if $a$ is any
non-zero element of $\frak A$ and $n\in\Bbb N$ then there is a non-zero
$d\Bsubseteq a$ such that $\nu_d\le 2^{-n}\nu_a$.

\medskip

{\bf (d)}
Now suppose that $a\in\frak A$ and that $\lambda:\frak B\to\coint{0,\infty}$ is a
non-zero countably additive functional such that $\lambda\le\nu_a$.   Then there is a
non-zero $d\Bsubseteq a$ such that $\nu_d\le\lambda$.   \Prf\ Let $b^*\in\frak B$ be
such that $\lambda b^*>0$;  then

\Centerline{$\lambda(b^*\setminus a)\le\nu_a(b^*\setminus a)=0$,}

\noindent so $\lambda b^*=\lambda(b^*\Bcap a)$.   Take $n\in\Bbb N$ such that
$2^{-n}\nu_ab^*<\lambda b^*$;  set $\lambda_1=\lambda-2^{-n}\nu_a$.   
By 326M (for the
second time), there is a $b_1\in\frak B$ such that $\lambda_1b\ge 0$ if
$b\Bsubseteq b_1$ and $\lambda_1b\le 0$ if $b\Bcap b_1=0$.   Set $c=a\Bcap b_1$.
Now

\Centerline{$2^{-n}\nu_a(a\Bcap b^*)
=2^{-n}\nu_ab^*<\lambda b^*=\lambda(a\Bcap b^*)$,}

\noindent so $\lambda_1(a\Bcap b^*)>0$ and $a\Bcap b^*\Bcap b_1\ne 0$ and $c\ne 0$.
By (c), we have a non-zero $d\Bsubseteq c$ such that $\nu_d\le 2^{-n}\nu_c$.
If $b\in\frak B$ then

\Centerline{$\nu_db\le 2^{-n}\nu_cb=2^{-n}\nu_a(b\Bcap b_1)
=\lambda(b\Bcap b_1)-\lambda_1(b\Bcap b_1)\le\lambda(b\Bcap b_1)\le\lambda b$,}

\noindent so $\nu_d\le\lambda$, as required.\ \Qed

\medskip

{\bf (e)} Let $C$ be the set

\Centerline{$\{a:a\in\frak A$, $a\Bsubseteq a_0$, $\nu_a\le\nu\}$.}

\noindent Then $0\in C$, so $C\ne\emptyset$.    If $D\subseteq C$ is
upwards-directed and not empty, then $a=\sup D$ is defined in $\frak A$ and
included in $a_0$, and

\Centerline{$\nu_{\sup D}b=\bar\mu(b\Bcap\sup D)
=\bar\mu(\sup_{d\in D}b\Bcap d)=\sup_{d\in D}\bar\mu(b\Bcap d)
=\sup_{d\in D}\nu_db\le\nu b$}

\noindent  using 313Ba and 321C.   So $a\in C$ and is an upper bound for
$D$ in $C$.   In particular, any non-empty totally ordered subset of $C$
has an upper bound in $C$.   By Zorn's Lemma, $C$ has a maximal element
$c$ say.   Of course $c\Bsubseteq a_0$.

\medskip

{\bf (e)} \Quer\ Suppose, if possible, that $\nu_c\ne\nu$.
Set $\lambda=\nu-\nu_c$;  note that

\Centerline{$\lambda b
=\nu b-\nu_cb\le\bar\mu(b\Bcap a_0)-\bar\mu(b\Bcap c)
=\nu_{a_0\Bsetminus c}b$}

\noindent for every $b\in\frak B$, so $\lambda\le\nu_{a_0\Bsetminus c}$.
Because $c\in C$, $\lambda\ge 0$, and we are supposing that $\lambda\ne 0$.
By (d), there is a non-zero $d\Bsubseteq a_0\Bsetminus c$ such that
$\nu_d\le\lambda$.   But now $c\Bcup d\Bsubseteq a_0$,

\Centerline{$\nu_{c\Bcup d}=\nu_c+\nu_d\le\nu_c+\lambda=\nu$}

\noindent and $c\Bcup d\in C$, so $c$ is not maximal in $C$.\ \Bang

Thus $c$ is an element of $\frak A$, included in $a_0$,
giving a representation of $\nu$.
}%end of proof of 331B

\leader{331C}{Corollary} Let $(\frak A,\bar\mu)$ be an atomless
semi-finite measure algebra, and $a\in\frak A$.   Suppose that
$0\le\gamma\le\bar\mu a$.   Then there is a $c\in\frak A$ such that
$c\Bsubseteq a$ and $\bar\mu c=\gamma$.

\proof{ If $\gamma=\bar\mu a$, take $c=a$.    If $\gamma<\bar\mu a$,
there is a
$d\in\frak A$ such that $d\Bsubseteq a$ and $\gamma\le\bar\mu d<\infty$
(322Eb).   Apply 331B to
the principal ideal $\frak A_d$ generated by $d$, with $a_0=d$,
$\frak B=\{0,d\}$
and $\nu d=\gamma$.   (The point is that because $\frak A$ is atomless,
no non-trivial principal ideal of $\frak A_d$ can be of the form
$\{c\Bcap b:b\in\frak B\}=\{0,c\}$, so $\frak A_d$ is relatively atomless
over $\{0,d\}$.)
}%end of proof of 331C

\cmmnt{\medskip

\noindent{\bf Remark} Of course this is also an easy consequence of
either 215D or the one-dimensional case of 326H.
}%end of comment

\leader{331D}{Lemma} Let $(\frak A,\bar\mu)$, $(\frak B,\bar\nu)$ be
totally finite measure algebras and $\frak C\subseteq\frak A$ a closed
subalgebra.   Suppose that $\pi:\frak C\to\frak B$ is a
measure-preserving Boolean homomorphism such that $\frak B$ is
relatively atomless over $\pi[\frak C]$.
Take any $a\in\frak A$, and let $\frak C_1$ be the subalgebra
of $\frak A$ generated by $\frak C\cup\{a\}$.   Then  there is a
measure-preserving homomorphism from $\frak C_1$ to $\frak B$ extending
$\pi$.

\proof{ We know that $\pi[\frak C]$ is a closed subalgebra of $\frak B$
(324Kb), and that $\pi$ is a Boolean isomorphism between $\frak C$ and
$\pi[\frak C]$.   Consequently the countably additive functional
$c\mapsto\bar\mu(c\Bcap a):\frak C\to\Bbb R$ is transferred to a
countably additive functional $\lambda:\pi[\frak C]\to\Bbb R$, writing
$\lambda(\pi c)=\bar\mu(c\Bcap a)$ for every $c\in\frak C$.   Of course
$\lambda(\pi c)\le\bar\mu c=\bar\nu(\pi c)$ for every $c\in\frak C$.
So by 331B there is a $b\in\frak B$ such that
$\lambda(\pi c)=\bar\nu(b\Bcap\pi c)$ for every $c\in\frak C$.

If $c\in\frak C$ and $c\Bsubseteq a$ then

\Centerline{$\bar\nu(b\Bcap\pi c)=\lambda(\pi c)
=\bar\mu(a\Bcap c)=\bar\mu c=\bar\nu(\pi c)$,}

\noindent so $\pi c\Bsubseteq b$.   Similarly, if $a\Bsubseteq c\in\frak C$, then

\Centerline{$\bar\nu(b\Bcap\pi c)=\bar\mu(a\Bcap c)=\bar\mu(a\Bcap 1)
=\bar\nu(b\Bcap\pi 1)=\bar\nu b$,}

\noindent so $b\Bsubseteq\pi c$.   It follows from 312O that there is
a Boolean homomorphism $\pi_1:\frak C_1\to\frak B$, extending $\pi$,
such that $\pi_1a=b$.

To see that $\pi_1$ is measure-preserving, take any member of $\frak C_1$.
By 312N, this is expressible as $e=(c_1\Bcap a)\Bcup(c_2\Bsetminus a)$,
where $c_1$, $c_2\in\frak C$.   Now

$$\eqalign{\bar\nu(\pi_1e)
&=\bar\nu((\pi c_1\Bcap b)\Bcup(\pi c_2\Bsetminus b))
=\bar\nu(\pi c_1\Bcap b)+\bar\nu(\pi c_2)-\bar\nu(\pi c_2\Bcap b)\cr
&=\bar\mu(c_1\Bcap a)+\bar\mu c_2-\bar\mu(c_2\Bcap a)
=\bar\mu e.\cr}$$

\noindent As $e$ is arbitrary, $\pi_1$ is measure-preserving.
}%end of proof of 331D

\leader{331E}{Generating sets}\ifresultsonly{ If}
\else{ For the sake of the next definition, we
need a language a little more precise than I have felt the need to use
so far.   The point is that if }\fi
$\frak A$ is a Boolean algebra and $B$ is
a subset of $\frak A$\cmmnt{, there is more than one subalgebra of $\frak A$
which can be said to be `generated' by $B$, because} we can look at any
of the three algebras

\quad-- $\frak B$, the smallest subalgebra of $\frak A$ including $B$;

\quad-- $\frak B_{\sigma}$, the smallest $\sigma$-subalgebra of
$\frak A$ including $B$;

\quad-- $\frak B_{\tau}$, the smallest order-closed subalgebra of
$\frak A$ including $B$.

\noindent\cmmnt{(See 313Fb.)  Now} I will say henceforth\cmmnt{, in this context,} that

\quad-- $\frak B$ is the subalgebra of $\frak A$ generated by $B$, and
$B$ {\bf generates} $\frak A$ if $\frak A=\frak B$;

\quad-- $\frak B_{\sigma}$ is the $\sigma$-subalgebra
of $\frak A$ generated by $B$, and $B$ {\bf $\sigma$-generates}
$\frak A$ if $\frak A=\frak B_{\sigma}$;

\quad-- $\frak B_{\tau}$ is the order-closed subalgebra of $\frak A$
generated by $B$, and $B$ {\bf $\tau$-generates} or {\bf completely generates} %Monk
$\frak A$ if $\frak A=\frak B_{\tau}$.

\cmmnt{There is a danger inherent in these phrases, because if we have
$B\subseteq\frak A'$, where $\frak A'$ is a subalgebra of $\frak A$, it
is possible that the smallest order-closed subalgebra of $\frak A'$
including $B$ might not be recoverable from the smallest order-closed
subalgebra of $\frak A$ including $B$.   (See
331Yb-331Yc.)   This problem will not seriously
interfere with the ideas below;  but for definiteness let me say that
the phrases `$B\,\,\sigma$-generates $\frak A$',
`$B\,\,\tau$-generates $\frak A$' will always refer to suprema and
infima taken in $\frak A$ itself, not in any larger algebra in which it
may be embedded.
}

\leader{331F}{Maharam types (a)}
\ifresultsonly{ If $\frak A$ is a Boolean algebra, its {\bf Maharam
type} $\tau(\frak A)$}
\else{ With the language of 331E established,
I can now define the {\bf Maharam type} or
{\bf complete generation number}
$\tau(\frak A)$ of any Boolean algebra $\frak A$;  it }\fi
is the smallest cardinal of any subset of $\frak A$ which $\tau$-generates
$\frak A$.

\cmmnt{(I think that this is the first `cardinal function' which I have
mentioned in this treatise.   All you need to know, to confirm that the
definition is well-conceived, is that there is {\it some} set which
$\tau$-generates $\frak A$;  and obviously $\frak A\,\,\tau$-generates
itself.   For this means that (assuming the axiom of choice) the set
$A=\{\#(B):B\subseteq\frak A\,\,\tau$-generates $\frak A\}$ is a non-empty
class of cardinals, and therefore has a least member (2A1Lf).
In 331Ye-331Yf I mention a further function, the `density' of a
topological space, which is closely related to
Maharam type.)
}

\header{331Fb}{\bf (b)} A Boolean algebra $\frak A$ is {\bf\Mth} if
$\tau(\frak A_a)=\tau(\frak A)$ for every non-zero
$a\in\frak A$, writing $\frak A_a$ for the principal ideal of $\frak A$
generated by $a$.

\header{331Fc}{\bf (c)} Let $(X,\Sigma,\mu)$ be a measure space, with
measure algebra $(\frak A,\bar\mu)$.   Then the {\bf Maharam type} of
$(X,\Sigma,\mu)$, or of $\mu$, is the Maharam type of $\frak A$;  and
$(X,\Sigma,\mu)$, or $\mu$, is {\bf\Mth} if $\frak A$ is.

\cmmnt{
\medskip

\noindent{\bf Remark} I should perhaps remark that the phrases `Maharam
type' and `\Mth', while well established in the context
of probability algebras, are not in common use for general Boolean
algebras.   But the cardinal $\tau(\frak A)$ is important in the general
context, and is such an obvious extension of Maharam's idea 
({\smc Maharam 1942}) that I am happy to propose this extension of terminology.
}

\leader{331G}{}\cmmnt{ For the sake of those who have not mixed set
theory and
algebra before, I had better spell out some basic facts.

\medskip

\noindent}{\bf Proposition} Let $\frak A$ be a Boolean algebra, $B$ a
subset of $\frak A$.   Let $\frak B$ be the subalgebra of $\frak A$
generated by $B$, $\frak B_{\sigma}$ the
$\sigma$-subalgebra of $\frak A$ generated by $B$, and $\frak B_{\tau}$
the order-closed subalgebra of $\frak A$ generated by $B$.

(a) $\frak B\subseteq\frak B_{\sigma}\subseteq\frak B_{\tau}$.

(b) If $B$ is finite, so is $\frak B$, and in this case
$\frak B=\frak B_{\sigma}=\frak B_{\tau}$.

(c) For every $a\in\frak B$, there is a finite $B'\subseteq B$ such that
$a$ belongs to the subalgebra of $\frak A$ generated by $B'$.
Consequently $\#(\frak B)\le\max(\omega,\#(B))$.

(d) For every $a\in\frak B_{\sigma}$, there is a countable
$B'\subseteq B$ such that $a$ belongs to the $\sigma$-subalgebra of
$\frak A$ generated by $B'$.

(e) If $\frak A$ is ccc, then $\frak B_{\sigma}=\frak B_{\tau}$.

\proof{{\bf (a)} All we need to know is that $\frak B_{\sigma}$ is a
subalgebra of $\frak A$ including $B$, and that $\frak B_{\tau}$ is a
$\sigma$-subalgebra of $\frak A$ including $B$.

\medskip

{\bf (b)} Induce on $\#(B)$, using 312N for the inductive step, to see
that $\frak B$ is finite.   In this case it must be order-closed, so is
equal to $\frak B_{\tau}$.

\medskip

{\bf (c)(i)} For $I\subseteq B$, let $\frak C_I$ be the subalgebra of
$\frak A$ generated by $I$.   If $I$, $J\subseteq B$ then $\frak
C_I\cup\frak C_J\subseteq\frak C_{I\cup J}$.   So $\bigcup\{\frak
C_I:I\subseteq B$ is finite$\}$ is a subalgebra of $\frak A$, and must
be equal to $\frak B$, as claimed.

\medskip

\quad{\bf (ii)} To estimate the size of $\frak B$, recall that the set
$[B]^{<\omega}$ of all finite subsets of $B$ has cardinal at most
$\max(\omega,\#(B))$ (3A1Cd).    For each $I\in[B]^{<\omega}$,
$\frak C_I$ is finite, so

\Centerline{$\#(\frak B)=\#(\bigcup_{I\in[B]^{<\omega}}\frak C_I)
\le\max(\omega,\#(I),\sup_{I\in [B]^{<\omega}}\#(\frak C_I))
\le\max(\omega,\#(B))$}

\noindent by 3A1Cc.

\medskip

{\bf (d)} For $I\subseteq B$, let $\frak D_I\subseteq \frak B_{\sigma}$
be the $\sigma$-subalgebra of $\frak A$ generated by $I$.   If $I$,
$J\subseteq B$ then $\frak D_I\cup\frak D_J\subseteq\frak D_{I\cup J}$,
so $\frak B'_{\sigma}=\bigcup\{\frak D_I:I\subseteq B$ is countable$\}$
is a subalgebra of $\frak A$.   But also it is sequentially
order-closed in $\frak A$.   \Prf\ Let $\sequencen{a_n}$ be a
non-decreasing sequence in $\frak B'_{\sigma}$ with supremum $a$ in
$\frak
A$.   For each $n\in\Bbb N$ there is a countable $I(n)\subseteq B$ such
that $a_n\in\frak C_{I(n)}$.   Set $K=\bigcup_{n\in\Bbb N}I(n)$;  then
$K$ is a countable subset of $B$ and every $a_n$ belongs to $\frak D_K$,
so $a\in\frak D_K\subseteq\frak B'_{\sigma}$.\  \Qed\   So $\frak
B'_{\sigma}$ is a $\sigma$-subalgebra of $\frak A$
including $B$ and must be the whole of $\frak B_{\sigma}$.

\medskip

{\bf (e)} By 316Fb, $\frak B_{\sigma}$ is order-closed in $\frak A$, so
must be equal to $\frak B_{\tau}$.
}%end of proof of 331G

\leader{331H}{Proposition} Let $\frak A$ be a Boolean algebra.

(a)(i) $\tau(\frak A)=0$ iff $\frak A$ is either $\{0\}$ or $\{0,1\}$.

\quad(ii) $\tau(\frak A)$ is finite iff $\frak A$ is finite.

(b) If $\frak B$ is another Boolean algebra and $\pi:\frak A\to\frak B$
is a surjective order-continuous Boolean homomorphism, then 
$\tau(\frak B)\le\tau(\frak A)$.

(c) If $a\in\frak A$ then $\tau(\frak A_a)\le\tau(\frak A)$, where
$\frak A_a$ is the principal ideal of $\frak A$ generated by $a$.

(d) If $\frak A$ has an atom and is \Mth, then $\frak A=\{0,1\}$.

\proof{{\bf (a)(i)} $\tau(\frak A)=0$ iff $\frak A$ has no proper
subalgebras.

\medskip

\quad{\bf (ii)} If $\frak A$ is finite, then
$\tau(\frak A)\le\#(\frak A)$ is
finite.   If $\tau(\frak A)$ is finite, then there
is a finite set $B\subseteq\frak A$ which $\tau$-generates $\frak A$;
by 331Gb, $\frak A$ is finite.

\medskip

{\bf (b)} We know that there is a set $A\subseteq\frak A$,
$\tau$-generating $\frak A$, with $\#(A)=\tau(\frak A)$.   Now $\pi[A]$
$\tau$-generates $\pi[\frak A]=\frak B$ (313Mb), so

\Centerline{$\tau(\frak B)\le\#(\pi[A])\le\#(A)=\tau(\frak A)$.}

\medskip

{\bf (c)} Apply (b) to the map $b\mapsto a\Bcap b:\frak A\to\frak A_a$.

\medskip

{\bf (d)} If $a\in\frak A$ is an atom, then $\tau(\frak A_a)=0$, so if
$\frak A$ is \Mth\ then $\tau(\frak A)=0$ and $\frak A=\{0,a\}=\{0,1\}$.
}% end of proof of 331H

\leader{331I}{}\cmmnt{ We are now ready for the theorem.

\medskip

\noindent}{\bf Theorem} Let $(\frak A,\bar\mu)$ and $(\frak B,\bar\nu)$
be \Mth\ measure algebras of the same Maharam type, with
$\bar\mu 1=\bar\nu 1<\infty$.
Then they are isomorphic as measure algebras.

\proof{{\bf (a)} If $\tau(\frak A)=\tau(\frak B)=0$, this is trivial.
So let us take $\kappa=\tau(\frak A)=\tau(\frak B)>0$.   In this case,
because $\frak A$ and $\frak B$ are \Mth, they can have
no atoms and must be infinite, so $\kappa$ is
infinite (331H).   Let $\langle a_{\xi}\rangle_{\xi<\kappa}$ and
$\langle b_{\xi}\rangle_{\xi<\kappa}$ enumerate $\tau$-generating
subsets of $\frak A$, $\frak B$ respectively.

The strategy of the proof is to define a measure-preserving isomorphism
$\pi:\frak A\to\frak B$ as the last of an increasing family
$\langle\pi_{\xi}\rangle_{\xi\le\kappa}$ of isomorphisms between closed
subalgebras $\frak C_{\xi}$, $\frak D_{\xi}$ of $\frak A$ and $\frak B$.
The inductive hypothesis will be that, for some families $\langle
a'_{\xi}\rangle_{\xi<\kappa}$, $\langle b'_{\xi}\rangle_{\xi<\kappa}$ to
be determined,

\inset{$\frak C_{\xi}$ is the closed subalgebra of $\frak A$
generated by $\{a_{\eta}:\eta<\xi\}\cup\{a'_{\eta}:\eta<\xi\}$,

$\frak D_{\xi}$ is the closed subalgebra of $\frak B$
generated by $\{b_{\eta}:\eta<\xi\}\cup\{b'_{\eta}:\eta<\xi\}$,

$\pi_{\xi}:\frak C_{\xi}\to\frak D_{\xi}$ is a measure-preserving
isomorphism,

$\pi_{\xi}$ extends $\pi_{\eta}$ whenever $\eta<\xi$.}

\noindent (Formally speaking, this will be a transfinite recursion,
defining a function
$\xi\mapsto f(\xi)=(\frak C_{\xi},\frak D_{\xi},\pi_{\xi},a'_{\xi},b'_{\xi})$
on the ordinal $\kappa+1$ by a
rule which chooses $f(\xi)$ in terms of $f\restr\xi$, as described in
2A1B.   The construction of an actual function $F$ for which
$f(\xi)=F(f\restr\xi)$ will necessitate the axiom of choice.)

\medskip

{\bf (b)} The induction starts with $\frak C_0=\{0,1\}$,
$\frak D_0=\{0,1\}$, $\pi_0(0)=0$, $\pi_0(1)=1$.   (The hypothesis
$\bar\mu 1=\bar\nu 1$ is what we need to ensure that $\pi_0$ is
measure-preserving.)

\medskip

{\bf (c)} For the inductive step to a successor ordinal $\xi+1$, where
$\xi<\kappa$, suppose that $\frak C_{\xi}$, $\frak D_{\xi}$ and
$\pi_{\xi}$ have been defined.

\medskip

\quad{\bf (i)} For any non-zero $b\in\frak B$, the principal ideal
$\frak B_b$ of $\frak B$ generated by $b$ has Maharam type $\kappa$,
because $\frak B$ is \Mth.   On the other hand, the
Maharam type of $\frak D_{\xi}$ is at most

\Centerline{$\#(\{b_{\eta}:\eta<\xi\}\cup\{b'_{\eta}:\eta<\xi\})
\le\#(\xi\times\{0,1\})<\kappa$,}

\noindent because if $\xi$ is finite so is $\xi\times\{0,1\}$, while if
$\xi$ is infinite then $\#(\xi\times\{0,1\})=\#(\xi)\le\xi<\kappa$.
Consequently $\frak B_b$ cannot be an order-continuous image of
$\frak D_{\xi}$ (331Hb).   Now the map
$c\mapsto c\Bcap b:\frak D_{\xi}\to\frak B_b$ is order-continuous, because
$\frak D_{\xi}$ is closed, so that the
embedding $\frak D_{\xi}\embedsinto\frak B$ is order-continuous.
It therefore cannot be surjective, and

\Centerline{$\{b\Bcap\pi_{\xi}a:a\in\frak C_{\xi}\}
=\{b\Bcap d:d\in\frak D_{\xi}\}\ne\frak B_b$.}

This means that $\pi_{\xi}:\frak C_{\xi}\to\frak D_{\xi}$ satisfies the
conditions of 331D, and must have an extension $\phi_{\xi}$ to a
measure-preserving homomorphism from the subalgebra $\frak C'_{\xi}$ of
$\frak A$ generated by $\frak C_{\xi}\cup\{a_{\xi}\}$ to $\frak B$.
We know that $\frak C'_{\xi}$ is a closed subalgebra of $\frak A$
(314Ja), so it must be the closed subalgebra of $\frak A$ generated by
$\{a_{\eta}:\eta\le\xi\}\cup\{a'_{\eta}:\eta<\xi\}$.   Also $\frak
D'_{\xi}=\phi_{\xi}[\frak C'_{\xi}]$ will be the subalgebra of $\frak B$
generated by $\frak D_{\xi}\cup\{b'_{\xi}\}$, where
$b'_{\xi}=\phi_{\xi}(a_{\xi})$, so is closed in $\frak B$, and is
the closed subalgebra of $\frak B$ generated by
$\{b_{\eta}:\eta<\xi\}\cup\{b'_{\eta}:\eta\le\xi\}$.

\medskip

\quad{\bf (ii)} The next step is to repeat the whole of the argument
above, but applying it to 
$\phi_{\xi}^{-1}:\frak D'_{\xi}\to\frak C_{\xi}$, $b_{\xi}$ in place of 
$\pi_{\xi}:\frak C_{\xi}\to\frak D_{\xi}$ and $a_{\xi}$.   Once again, we 
have $\tau(\frak D'_{\xi})<\kappa=\tau(\frak A_a)$ for every 
$a\in\frak A$, so we can use
Lemma 331D to find a measure-preserving isomorphism 
$\psi_{\xi}:\frak D_{\xi+1}\to\frak C_{\xi+1}$ extending 
$\phi_{\xi}^{-1}$, where $\frak D_{\xi+1}$ is the subalgebra of $\frak B$ 
generated by 
$\frak D'_{\xi}\cup\{b_{\xi}\}$, and $\frak C_{\xi+1}$ is the subalgebra of
$\frak A$ generated by $\frak C'_{\xi}\cup\{a'_{\xi}\}$, setting
$a'_{\xi}=\psi_{\xi}(b_{\xi})$.   As in (i), we find that 
$\frak C_{\xi+1}$ is the closed subalgebra of $\frak A$ generated by
$\{a_{\eta}:\eta\le\xi\}\cup\{a'_{\eta}:\eta\le\xi\}$, while 
$\frak D_{\xi+1}$ is the closed subalgebra of $\frak B$ generated by
$\{b_{\eta}:\eta\le\xi\}\cup\{b'_{\eta}:\eta\le\xi\}$.

\medskip

\quad{\bf (iii)} We can therefore take $\pi_{\xi+1}
=\psi_{\xi}^{-1}:\frak C_{\xi+1}\to\frak D_{\xi+1}$, and see that
$\pi_{\xi+1}$ is a measure-preserving isomorphism, extending
$\pi_{\xi}$, such that $\pi_{\xi+1}(a_{\xi})=b'_{\xi}$ and
$\pi_{\xi+1}(a'_{\xi})=b_{\xi}$.   Evidently $\pi_{\xi+1}$ extends
$\pi_{\eta}$ for every $\eta\le\xi$ because it extends $\pi_{\xi}$ and
(by the inductive hypothesis) $\pi_{\xi}$ extends $\pi_{\eta}$ for every
$\eta<\xi$.

\medskip

{\bf (d)} For the inductive step to a limit ordinal $\xi$, where
$0<\xi\le\kappa$, suppose that $\frak C_{\eta}$, $\frak D_{\eta}$,
$a'_{\eta}$, $b'_{\eta}$, $\pi_{\eta}$ have been defined for $\eta<\xi$.
Set $\frak C^*_{\xi}=\bigcup_{\eta<\xi}\frak C_{\xi}$.   Then $\frak
C^*_{\xi}$ is a subalgebra of $\frak A$, because it is the union of an
upwards-directed family of subalgebras;  similarly, $\frak
D^*_{\xi}=\bigcup_{\eta<\xi}\frak D_{\xi}$ is a subalgebra of $\frak B$.
Next, we have a function $\pi^*_{\xi}:\frak C^*_{\xi}\to\frak D^*_{\xi}$
defined by setting $\pi^*_{\xi}a=\pi_{\eta}a$ whenever $\eta<\xi$ and
$a\in\frak C_{\eta}$;  for if $\eta$, $\zeta<\xi$ and $a\in\frak
C_{\eta}\cap\frak C_{\zeta}$, then
$\pi_{\eta}a=\pi_{\max(\eta,\zeta)}a=\pi_{\zeta}a$.   Clearly

\Centerline{$\pi^*_{\xi}[\frak C^*_{\xi}]
=\bigcup_{\eta<\xi}\pi_{\eta}[\frak C_{\eta}]
=\frak D^*_{\xi}$.}

\noindent Moreover, $\bar\nu(\pi^*_{\xi}a)=\bar\mu a$ for every $a\in\frak C^*_{\xi}$,
since $\bar\nu(\pi_{\eta}a)=\bar\mu a$ whenever $\eta<\xi$ and
$a\in\frak C_{\eta}$.

Now let $\frak C_{\xi}$ be the smallest closed subalgebra of
$\frak A$ including $\frak C^*_{\xi}$, that is, the topological closure of
$\frak C^*_{\xi}$ in $\frak A$ (323J).   Since $\frak C_{\xi}$ is the
smallest closed subalgebra of $\frak A$ including $\frak C_{\eta}$ for every
$\eta<\xi$, it must be the closed subalgebra of $\frak A$ generated by
$\{a_{\eta}:\eta<\xi\}\cup\{a'_{\eta}:\eta<\xi\}$.
By 324O, $\pi^*_{\xi}$ has an extension to a measure-preserving
homomorphism $\pi_{\xi}:\frak C_{\xi}\to\frak B$.   Set
$\frak D_{\xi}=\pi_{\xi}[\frak C_{\xi}]$;  by 324Kb again, $\frak D_{\xi}$ is a
closed subalgebra of $\frak B$.   Because
$\pi_{\xi}:\frak C_{\xi}\to\frak B$ is continuous (also noted in 324Kb),

\Centerline{$\frak D^*_{\xi}=\pi^*_{\xi}[\frak C^*_{\xi}]
=\pi_{\xi}[\frak C^*_{\xi}]$}

\noindent is topologically dense in $\frak D_{\xi}$ (3A3Eb), and
$\frak D_{\xi}=\overline{\frak D^*_{\xi}}$ is the closed subalgebra of
$\frak B$-generated by
$\{b_{\eta}:\eta<\xi\}\cup\{b'_{\eta}:\eta<\xi\}$.   Finally, if
$\eta<\xi$, $\pi_{\xi}$ extends $\pi_{\eta}$ because $\pi^*_{\xi}$
extends $\pi_{\eta}$.   Thus the induction continues.

\medskip

{\bf (e)} The induction ends with $\xi=\kappa$, $\frak C_{\kappa}=\frak A$,
$\frak D_{\kappa}=\frak B$ and $\pi=\pi_{\kappa}:\frak A\to\frak B$
the required measure algebra isomorphism.
}%end of proof of 331I

\leader{331J}{Lemma} Let $(\frak A,\bar\mu)$ be a totally finite measure
algebra, and $\kappa$ an infinite cardinal.

(a)\dvAnew{2008}
If there is a family $\ofamily{\xi}{\kappa}{a_{\xi}}$ in $\frak A$ such
that $\inf_{\xi\in I}a_{\xi}=0$ and $\sup_{\xi\in I}a_{\xi}=1$ for every
infinite $I\subseteq\kappa$, then $\tau(\frak A_d)\ge\kappa$ for every
non-zero $d\in\frak A$.

(b)\discrversionA{\footnote{Elaborated 2008.}}{}
Let $\nu_{\kappa}$ be the usual measure on
$\{0,1\}^{\kappa}$\cmmnt{ (254J)} and
$(\frak B_{\kappa},\bar\nu_{\kappa})$ its measure algebra.   If there is an
order-continuous Boolean homomorphism
from $\frak B_{\kappa}$ to $\frak A$, $\tau(\frak A_d)\ge\kappa$ for every
non-zero $d\in\frak A$.

\proof{{\bf (a)(i)} To begin with (down to the end of (iii)), let us take
it that $d=1$.   For $a\in\frak A$, $\delta>0$ set
$U(a,\delta)=\{a':\bar\mu(a'\Bsymmdiff a)<\delta\}$, the ordinary open
$\delta$-neighbourhood of $a$.   If $a\in\frak A$, then there is a
$\delta>0$ such that $\{\xi:\xi<\kappa,\,a_{\xi}\in U(a,\delta)\}$ is
finite.    \Prf\Quer\ Suppose, if possible, otherwise.   Then there is a
sequence $\sequencen{\xi_n}$ of distinct elements of $\kappa$ such that
$\bar\mu(a\Bsymmdiff a_{\xi_n})\le 2^{-n-2}\bar\mu 1$ for every
$n\in\Bbb N$.    Now $\inf_{n\in\Bbb N}a_{\xi_n}=0$, so

\Centerline{$\bar\mu a
=\bar\mu(a\Bsetminus\inf_{n\in\Bbb N}a_{\xi_n})
\le\sum_{n=0}^{\infty}\bar\mu(a\Bsetminus a_{\xi_n})$.}

\noindent Similarly

\Centerline{$\bar\mu(1\Bsetminus a)
=\bar\mu(\sup_{n\in\Bbb N}a_{\xi_n}\Bsetminus a)
\le\sum_{n=0}^{\infty}\bar\mu( a_{\xi_n}\Bsetminus a)$.}

\noindent Putting these together,

$$\eqalign{\bar\mu 1
&=\bar\mu a+\bar\mu(1\Bsetminus a)
\le\sum_{n=0}^{\infty}\bar\mu(a\Bsetminus a_{\xi_n})
  +\sum_{n=0}^{\infty}\bar\mu( a_{\xi_n}\Bsetminus a)\cr
&=\sum_{n=0}^{\infty}\bar\mu(a\Bsymmdiff a_{\xi_n})
\le\sum_{n=0}^{\infty}2^{-n-2}\bar\mu 1
<\bar\mu 1,\cr}$$

\noindent which is impossible.\ \Bang\Qed

\medskip

\quad{\bf (ii)} Note that $\frak A$ is infinite;  for if $a\in\frak A$
the set
$\{\xi: a_{\xi}=a\}$ must be finite, and $\kappa$ is supposed to be
infinite.   So $\tau(\frak A)$ must be infinite.

\medskip

\quad{\bf (iii)} Now take a set $C\subseteq\frak A$, with cardinal
$\tau(\frak A)$, which $\tau$-generates $\frak A$.   By (ii), $C$ is
infinite.   Let $\frak C$ be the subalgebra of $\frak A$ generated by
$C$;  then $\#(\frak C)=\#(C)=\tau(\frak A)$, by 331Gc, and $\frak C$ is
topologically dense in $\frak A$ (323J again).
If $a\in\frak A$, there are
$c\in\frak C$ and $k\in\Bbb N$ such that $a\in U(c,2^{-k})$ and
$\{\xi: a_{\xi}\in U(c,2^{-k})\}$ is finite.   \Prf\ By (b), there is a
$\delta>0$ such that $\{\xi: a_{\xi}\in U(a,\delta)\}$ is finite.
Take $k\in\Bbb N$ such that $2\cdot 2^{-k}\le\delta$, and
$c\in\frak C\cap U(a,2^{-k})$;  then
$U(c,2^{-k})\subseteq U(a,\delta)$ can contain only finitely many
$ a_{\xi}$, so these $c$, $k$ serve.\   \Qed

Consider

\Centerline{$\Cal U
=\{U(c,2^{-k}):c\in\frak C,\,k\in\Bbb N,\,\{\xi: a_{\xi}\in U(c,2^{-k})\}$
is finite$\}$.}

\noindent Then $\#(\Cal U)\le\max(\#(\frak C),\omega)=\tau(\frak A)$.
Also $\Cal U$ is a cover of $\frak A$.   In particular,
$\kappa=\bigcup_{U\in\Cal U}J_U$, where $J_U=\{\xi: a_{\xi}\in U\}$.
But this means that

\Centerline{$\kappa
=\#(\kappa)\le\max(\omega,\#(\Cal U),\sup_{U\in\Cal U}\#(J_U))
=\tau(\frak A)$.}

\noindent This proves the result when $d=1$.

\medskip

\quad{\bf (iv)} For the general case, given $d\in\frak A\setminus\{0\}$,
set $a'_{\xi}=a_{\xi}\Bcap d$ for each $\xi$.   Since
$\inf_{\xi\in I}a_{\xi}\Bcap d=0$ and $\sup_{\xi\in I}a_{\xi}\Bcap d=d$ for
every infinite $I\subseteq\kappa$, we can apply (i)-(iii) to
$(\frak A_d,\bar\mu\restrp\frak A_d,\ofamily{\xi}{\kappa}{a'_{\xi}})$ to
see that $\tau(\frak A_d)\ge\kappa$, as required.

\medskip

{\bf (b)} Let $\pi:\frak B_{\kappa}\to\frak A$ be an order-continuous
Boolean homomorphism.
Set $E_{\xi}=\{x:x\in\{0,1\}^{\kappa},\,x(\xi)=1\}$,
$e_{\xi}=E_{\xi}^{\ssbullet}\in\frak B_{\kappa}$ and
$a_{\xi}=\pi e_{\xi}\in\frak A$ for each $\xi<\kappa$.
If $\sequencen{\xi_n}$ is any sequence of distinct elements of $\kappa$,

\Centerline{$\nu_{\kappa}(\bigcap_{n\in\Bbb N}E_{\xi_n})
=\lim_{n\to\infty}\nu_{\kappa}(\bigcap_{i\le n}E_{\xi_n})
=\lim_{n\to\infty}2^{-n-1}=0$,}

\noindent so that $\bar\nu_{\kappa}(\inf_{n\in\Bbb N}e_{\xi_n})=0$ and
$\inf_{n\in\Bbb N}e_{\xi_n}=0$.
Because $\pi$ is order-continuous,
$\inf_{n\in\Bbb N}a_{\xi_n}=0$ in $\frak A$.    Similarly,
$\nu_{\kappa}(\bigcup_{n\in\Bbb N}E_{\xi_n})=1$ and
$\sup_{n\in\Bbb N}a_{\xi_n}=1$.   As $\sequencen{\xi_n}$ is arbitrary,
$\inf_{\xi\in I}a_{\xi}=0$ and $\sup_{\xi\in I}a_{\xi}=1$ for every
infinite $I\subseteq\kappa$.   So we can apply (a) to get the result.
}%end of proof of 331J

\leader{331K}{Theorem}  Let $\kappa$ be any infinite cardinal.   Let
$\nu_{\kappa}$ be the usual measure on $\{0,1\}^{\kappa}$ and
$(\frak B_{\kappa},\bar\nu_{\kappa})$
its measure algebra.   Then $\frak B_{\kappa}$ is \Mth, with Maharam type
$\kappa$.

\proof{ Set $X=\{0,1\}^{\kappa}$ and write $\Sigma$ for the domain of
$\nu_{\kappa}$.

\medskip

{\bf (a)} To see that $\tau(\frak B_{\kappa})\le\kappa$, set
$E_{\xi}=\{x:x\in X,\,x(\xi)=1\}$ and
$e_{\xi}=E_{\xi}^{\ssbullet}$ for each
$\xi<\kappa$.    Writing $\Cal E$ for the algebra of subsets of $X$
generated by
$\{E_{\xi}:\xi<\kappa\}$, we see that every measurable cylinder in $X$,
as defined in 254Aa, belongs to $\Cal E$, so that every member of
$\Sigma$ is approximated, in measure, by members of $\Cal E$ (254Fe),
that is, $\{E^{\ssbullet}:E\in\Cal E\}$ is topologically dense in
$\frak A$.   But this means just that the subalgebra
$\frak E$ of $\frak B_{\kappa}$ generated
by $\{e_{\xi}:\xi<\kappa\}$ is topologically dense in $\frak B_{\kappa}$,
so that
$\{e_{\xi}:\xi<\kappa\}\,\,\tau$-generates $\frak B_{\kappa}$, and
$\tau(\frak B_{\kappa})\le\kappa$.

\medskip

{\bf (b)} Next, if $c\in\frak B_{\kappa}\setminus\{0\}$ and
$(\frak B_{\kappa})_c$
is the principal ideal of $\frak B_{\kappa}$ generated by $c$, the map
$b\mapsto b\Bcap c$ is an order-continuous Boolean homomorphism from
$\frak B_{\kappa}$ to $(\frak B_{\kappa})_c$, so by 331Jb we must have
$\tau((\frak B_{\kappa})_c)\ge\kappa$.   Thus

\Centerline{$\kappa\le\tau((\frak B_{\kappa})_c)
\le\tau(\frak B_{\kappa})\le\kappa$.}

\noindent As $c$ is arbitrary, $\frak B_{\kappa}$ is \Mth\ of
Maharam type $\kappa$.
}%end of proof of 331K

\leader{331L}{Theorem} Let $(\frak A,\bar\mu)$ be a \Mth\
probability algebra.   Then there is exactly one $\kappa$, either $0$ or
an infinite cardinal, such that $(\frak A,\bar\mu)$ is isomorphic, as
measure algebra, to the measure algebra $(\frak B_{\kappa},\bar\nu_{\kappa})$
of the usual measure on $\{0,1\}^{\kappa}$.

\proof{ If $\tau(\frak A)$ is finite, it is zero, and $\frak A=\{0,1\}$
(331Ha, 331Hd) so that (interpreting $\{0,1\}^0$ as $\{\emptyset\}$) we
have the case $\kappa=0$.   If $\kappa=\tau(\frak A)$ is infinite, then
by 331K we know that $(\frak B_{\kappa},\bar\nu_{\kappa})$ also is
\Mth\ with Maharam type $\kappa$, so 331I gives the required
isomorphism.   Of course $\kappa$ is uniquely defined by $\frak A$.
}%end of proof of 331L

\leader{331M}{Homogeneous Boolean algebras} Of course a homogeneous Boolean
algebra\cmmnt{ (definition:  316N)} must be
\Mth, since $\tau(\frak A)=\tau(\frak A_c)$ whenever
$\frak A$ is isomorphic to $\frak A_c$.
In general, a Boolean algebra can be \Mth\ without being
homogeneous
(331Xj, 331Yg).   But for $\sigma$-finite measure algebras this doesn't
happen.

\leader{331N}{Proposition} Let $(\frak A,\bar\mu)$ be a \Mth\
$\sigma$-finite measure algebra.   Then it is homogeneous as
a Boolean algebra.

\proof{ If $\frak A=\{0\}$ this is trivial;  so suppose that
$\frak A\ne\{0\}$.
By 322G, there is a measure $\bar\nu$ on $\frak A$ such
that $(\frak A,\bar\nu)$ is a probability algebra.   Now let $c$ be any
non-zero member of $\frak A$, and set $\gamma=\bar\nu c$,
$\bar\nuprime_c=\gamma^{-1}\bar\nu_c$, where $\bar\nu_c$ is the restriction
of $\bar\nu$ to the principal ideal $\frak A_c$ of $\frak A$ generated
by $c$.   Then $(\frak A,\bar\nu)$ and $(\frak A_c,\bar\nuprime_c)$ are
\Mth\ probability algebras of the same Maharam type, so
are isomorphic as measure algebras, and {\it a fortiori} as Boolean
algebras.
}%end of proof of 331N

\leader{331O}{}\dvAnew{2011}
I will wait until Chapter 52 of Volume 5 for a systematic
discussion of properties of measure algebras
which depend on their Maharam
types.   There are however a couple of results which are easy, 
useful and expressible in terms already introduced.

\medskip

\noindent{\bf Proposition} Let $(\frak A,\bar\mu)$ be a measure algebra
with countable Maharam type.   Then $\frak A$ is separable in its
measure-algebra topology.

\proof Let $B\subseteq\frak A$ be a countable set which $\tau$-generates
$\frak A$.   Then the subalgebra $\frak B$ of $\frak A$ generated by $B$ is
countable (331Gc).   Now $\frak B$ is dense for the measure-algebra
topology.   \Prf\ Let $G$ be a non-empty open subset of $\frak A$, and $c$
any element of $G$.   Let $\Rho=\{\rho_a:a\in\frak A^f\}$ be the
upwards-directed family of pseudometrics defining the topology of
$\frak A$, as described in 323A.   Then there must be an $a\in\frak A^f$
and an $\epsilon>0$ such that $\{b:\rho_a(b,c)\le\epsilon\}\subseteq G$.
Let $\frak C$ be the order-closed subalgebra of the principal ideal
$\frak A_a$ generated by $\frak B_a=\{b\Bcap a:b\in\frak B\}$.
Because $b\mapsto b\Bcap a:\frak A\to\frak A_a$ is an order-continuous
Boolean homomorphism, $\{b:b\in\frak A$, $b\Bcap a\in\frak C\}$ is
an order-closed subalgebra of $\frak A$, and must be the whole of
$\frak A$, because it includes $B$.   So $\frak C=\frak A_a$.   By 323J,
$\frak C$ is the topological closure of $\frak B_a$ in $\frak A_a$, and
there must be a $b\in\frak B_a$ such that
$\bar\mu(b\Bsymmdiff(c\Bcap a))\le\epsilon$;  that is, there is a
$b\in\frak B$ such that $\bar\mu(a\Bcap(b\Bsymmdiff c))\le\epsilon$ and
$b\in G$.   Thus $\frak B$ meets $G$;  as $G$ is arbitrary, $\frak B$ is
dense.\ \Qed

So $\frak B$ is a countable dense subset of $\frak A$ and $\frak A$ is
separable.

\leader{331P}{Proposition}\dvAnew{2016} 
Let $(\frak A,\bar\mu)$ be an atomless probability
algebra of countable Maharam type.   Then it is isomorphic to the measure
algebras of the usual measure on $\{0,1\}^{\Bbb N}$ and of Lebesgue measure on
$[0,1]$.

\proof{ The point is that $\frak A$ is Maharam-type-homogeneous.   \Prf\ For
any non-zero $a\in\frak A$, 
$\frak A_a$ is atomless, so must be infinite, and
$\tau(\frak A_a)\ge\omega$ (331H(a-ii));  as also 
$\tau(\frak A_a)\le\tau(\frak A)$ (331Hc), and we are supposing that 
$\tau(\frak A)\le\omega$, we have $\tau(\frak A_a)=\tau(\frak A)$.\ \QeD\
So $(\frak A,\bar\mu)\cong(\frak B_{\kappa},\nu_{\kappa})$ for some $\kappa$
(331L);  but as $\tau(\frak B_{\kappa})=\kappa$ (331K), $\kappa=\omega$ and
$(\frak A,\bar\mu)$ is isomorphic to the measure algebra of the usual measure on
$\{0,1\}^{\Bbb N}$.

Since Lebesgue measure on $[0,1]$ is isomorphic to the usual measure on
$\{0,1\}^{\Bbb N}$ (254K), they surely have isomorphic measure algebras.
}%end of proof of 331P

\exercises{
\leader{331X}{Basic exercises (a)}
%\spheader 331Xa
Let $(X,\Sigma,\mu)$ be a probability space and $\Tau$ a
$\sigma$-subalgebra of $\Sigma$ such that for any
non-negligible $E\in\Sigma$
there is an $F\in\Sigma$ such that $F\subseteq E$ and
$\mu(F\symmdiff(E\cap H))>0$ for every $H\in\Tau$.   Suppose that
$f:X\to[0,1]$ is a measurable function.   Show that there is an
$F\in\Sigma$ such that $\int_Hf=\mu(H\cap F)$ for every $H\in\Tau$.
%331B

\sqheader 331Xb Write out a direct proof of 331C not relying on
331B or 321J.
%331C

\spheader 331Xc Let $\frak A$ be a finite Boolean algebra with $n$
atoms.   Show that $\tau(\frak A)$ is the least $k$ such that $n\le 2^k$.
%331F

\sqheader 331Xd Show that the measure algebra of Lebesgue measure on
$\Bbb R$ is \Mth\ with Maharam type $\omega$.
\Hint{show that it is $\tau$-generated by
$\{\ocint{-\infty,q}^{\ssbullet}:q\in\Bbb Q\}$.}
%331H

\spheader 331Xe Show that the measure algebra of Lebesgue measure on
$\BbbR^r$ is \Mth\ with Maharam type $\omega$, for any
$r\ge 1$. \Hint{show that it is $\tau$-generated by
$\{\ocint{-\infty,q}^{\ssbullet}:q\in\Bbb Q^r\}$.}
%331H, 331Xd

\spheader 331Xf Show that the measure algebra of any Radon measure on
$\BbbR^r$ (256Ad) has countable Maharam type.    \Hint{show that
it is $\tau$-generated by
$\{\ocint{-\infty,q}^{\ssbullet}:q\in\Bbb Q^r\}$.}
%331H, 331Xd, 331Xe

\sqheader 331Xg Show that $\Cal P\Bbb R$ has Maharam type $\omega$.
\Hint{show that it is $\tau$-generated by
$\{\ocint{-\infty,q}:q\in\Bbb Q\}$.}
%331H

\sqheader 331Xh Show that the regular open algebra of $\Bbb R$ is
\Mth\ with Maharam type $\omega$.   \Hint{show that it is
$\tau$-generated by $\{\ocint{-\infty,q}^{\ssbullet}:q\in\Bbb Q\}$.}
%331H

\spheader 331Xi Let $(\frak A,\bar\mu)$ be a totally finite measure
algebra, and $\kappa$ an infinite cardinal.   Suppose that there is a
family $\langle a_{\xi}\rangle_{\xi<\kappa}$ in $\frak A$ such that
$\inf_{\xi\in I}a_{\xi}=0$, $\sup_{\xi\in I}a_{\xi}=1$ for every
infinite $I\subseteq\kappa$.   Show that $\tau(\frak A_a)\ge\kappa$ for
every non-zero principal ideal $\frak A_a$ of $\frak A$.
%331K

\spheader 331Xj Let $\frak A$ be the measure algebra of Lebesgue measure
on $\Bbb R$, and $\frak G$ the regular open algebra of $\Bbb R$.   Show
that the simple product $\frak A\times\frak G$ is \Mth\ of
Maharam type $\omega$, but is not homogeneous.   \Hint{$\frak A$ is
weakly
$(\sigma,\infty)$-distributive, but $\frak G$ is not, so they are not
isomorphic.}
%316N

\spheader 331Xk Show that a homogeneous semi-finite measure algebra is
$\sigma$-finite.
%316N

\spheader 331Xl Let $(X,\Sigma,\mu)$ be a measure space, and $A$ a subset
of $X$ which has a measurable envelope.
Show that the Maharam type of the subspace measure on $A$ is less
than or equal to the Maharam type of $\mu$.
%331H  useful in 552M (i) => (iii)

\spheader 331Xm Let $\frak A$ be a Boolean algebra, and $\frak B$ an
order-dense subalgebra of $\frak A$.   Show that
$\tau(\frak A)\le\tau(\frak B)$.
%331H

\spheader 331Xn Let $\mu$ be a semi-finite measure, and
$\tilde\mu$ the c.l.d.\ version of $\mu$.   Show that the Maharam type
of $\tilde\mu$ is at most the Maharam type of $\mu$.   \Hint{322Db.}
%331Xm 331H

\spheader 331Xo Let $\familyiI{\mu_i}$ be a non-empty
countable family of $\sigma$-finite
measures all with the same domain;  let $\mu$ be the sum measure
$\sum_{i\in I}\mu_i$.   Writing $\tau(\mu)$, $\tau(\mu_i)$ for the Maharam
types of the measures, show that
$\sup_{i\in I}\tau(\mu_i)
\le\tau(\mu)\le\max(\omega,\sup_{i\in I}\tau(\mu_i))$.
%331H

\leader{331Y}{Further exercises (a)}
%\spheader 331Ya
Suppose that $\frak A$ is a Dedekind complete Boolean algebra,
$\frak B$ is an order-closed subalgebra of $\frak A$ and $\frak C$ is an
order-closed
subalgebra of $\frak B$.   Show that if $a\in\frak A$ is a relative atom
in $\frak A$ over $\frak C$, then $\upr(a,\frak B)$ is a relative atom
in $\frak B$ over $\frak C$.   So if $\frak B$ is relatively atomless
over $\frak C$, then $\frak A$ is relatively atomless over $\frak C$.
%331B

\spheader 331Yb Give an example of a Boolean algebra $\frak A$
with a subalgebra $\frak A'$ and a proper subalgebra $\frak B$ of $\frak
A'$ which is order-closed in $\frak A'$, but $\tau$-generates $\frak A$.
\Hint{take $\frak A$ to be the measure algebra $\frak A_L$ of
Lebesgue measure on $\Bbb R$ and $\frak B$ the subalgebra $\frak B_{\Bbb
Q}$ of $\frak A$ generated by $\{[a,b]^{\ssbullet}:a,\,b\in\Bbb Q\}$.
Take $E\subseteq\Bbb R$ such that $I\cap E$, $I\setminus E$ have
non-zero measure for every
non-trivial interval $I\subseteq\Bbb R$ (134Jb), and let $\frak A'$ be
the subalgebra of $\frak A$ generated by
$\frak B\cup\{E^{\ssbullet}\}$.}
%331E

\spheader 331Yc Give an example of a Boolean algebra $\frak A$
with a subalgebra $\frak A'$ and a proper subalgebra $\frak B$ of
$\frak A'$ which is order-closed in $\frak A$, but $\tau$-generates $\frak A'$.
\Hint{in the notation of 331Yb, take $Z$ to be the Stone space of
$\frak A_L$, and set $\frak A'=\{\widehat{a}:a\in\frak A_L\}$,
$\frak B=\{\widehat{a}:a\in\frak B_{\Bbb Q}\}$;  let $\frak A$ be the
subalgebra of $\Cal PZ$ generated by $\frak A'\cup\{\{z\}:z\in Z\}$.}
%331E

\spheader 331Yd Let $\frak A$ be a Dedekind complete purely atomic
Boolean algebra, and $A$ the set of its atoms.   Show that
$\tau(\frak A)$ is the least
cardinal $\kappa$ such that $\#(A)\le 2^{\kappa}$.
%331F, 331Xc

\spheader 331Ye Let $(\frak A,\bar\mu)$ be a measure algebra.   Write
$d(\frak A)$ for the smallest cardinal of any subset of $\frak A$ which
is dense for the measure-algebra topology.   Show that
$d(\frak A)\le\max(\omega,\tau(\frak A))$.   Show that if
$(\frak A,\bar\mu)$ is semi-finite, then $\tau(\frak A)\le d(\frak A)$.
%331F mt33bits

\spheader 331Yf Let $(X,\rho)$ be a metric space.   Write $d(X)$ for the
{\bf density} of $X$, the smallest cardinal of any dense subset of $X$.
(i) Show that if $\Cal G$ is any family of open subsets of $X$, there is
a family $\Cal H\subseteq\Cal G$ such that $\bigcup\Cal H=\bigcup\Cal G$
and $\#(\Cal H)\le\max(\omega,d(X))$.   (ii) Show that if
$\kappa>\max(\omega,d(X))$ and $\langle x_{\xi}\rangle_{\xi<\kappa}$ is
any family in $X$, then there is an $x\in X$ such that
$\#(\{\xi:x_{\xi}\in G\})>\max(\omega,d(X))$ for every open set $G$
containing $x$, and that there is a strictly increasing
sequence $\sequencen{\xi_n}$ in $\kappa$ such that $x=\lim_{n\to\infty}x_{\xi_n}$.
%331J

\spheader 331Yg Let $(\frak A,\bar\mu)$ be the simple product (322L) of
$\omega_1$ copies of the measure algebra of the usual measure on
$\{0,1\}^{\omega_1}$.   Show that $\frak A$ is \Mth\ but not
homogeneous.
%316N

\spheader 331Yh Let $\kappa$ be an infinite cardinal,
$\nu_{\kappa}$ the usual measure on $\{0,1\}^{\kappa}$ and
$(\frak B_{\kappa},\bar\nu_{\kappa})$ its measure algebra.   Suppose that
$(\frak A,\bar\mu)$ is a totally finite measure algebra and such that
$\tau(\frak A)<\kappa$, and $\pi:\frak B_{\kappa}\to\frak A$
a Boolean homomorphism.   Show that (i) for every $\epsilon>0$ there is a
$b\in\frak B_{\kappa}$ such that $\bar\nu_{\kappa}b\ge 1-\epsilon$ and
$\bar\mu(\pi b)\le\epsilon$ (ii) $\pi$ is not injective.
%331J out of order query

\spheader 331Yi Give an example of a semi-finite measure space
$(X,\Sigma,\mu)$ such that the Maharam type of $\mu$ is greater than the
Maharam type of its c.l.d.\ version.
%331Xn 331H

\spheader 331Yj\dvAnew{2011}
Let $(\frak A,\bar\mu)$ be a semi-finite measure algebra
which is separable when given
its measure-algebra topology.   Show that it has countable Maharam type.
%331O

\spheader 331Yk\dvAnew{2014}
Let $\familyiI{(\frak A_i,\bar\mu_i)}$ be a non-empty family of homogeneous
probability algebras, and $\Cal F$ an ultrafilter on $I$.   Show that the
probability algebra reduced product 
$\prod_{i\in I}(\frak A_i,\bar\mu_i)|\Cal F$ (328C) is homogeneous.
%331N out of order query
}%end of exercises

\endnotes{
\Notesheader{331}
Maharam's theorem belongs with the Radon-Nikod\'ym theorem, Fubini's
theorem and the strong law of large numbers as one of the theorems which
make measure theory what it is.   Once you have this theorem and its
consequences in the next section properly absorbed, you will never again
look at a measure space without trying to classify its measure algebra in
terms of the Maharam types of its homogeneous principal ideals.
As one might expect,
a very large proportion of the important measure algebras of analysis are
homogeneous, and indeed a great many are homogeneous with Maharam type
$\omega$.

In this section I have contented myself with the basic statement of
Theorem 331I on the isomorphism of \Mth\ measure algebras
and the identification of representative
homogeneous probability algebras (331K).   The same techniques lead to
an enormous number of further facts, some of which I will describe in
the rest of the chapter.
For the moment, it gives us a complete description of \Mth\ probability algebras (331L).   There is the atomic algebra
$\{0,1\}$, with Maharam type $0$, and for each infinite cardinal
$\kappa$ there is the measure algebra of $\{0,1\}^{\kappa}$, with
Maharam type $\kappa$;  these are all non-isomorphic, and
every \Mth\ probability algebra is isomorphic to exactly
one of them.   The isomorphisms here are not unique;  indeed, it is
characteristic of measure algebras that they have very large
automorphism groups (see Chapter 38 below), and there are
correspondingly large numbers of isomorphisms between any isomorphic
pair.   The proof of 331I already suggests this, since we have such a
vast amount of choice concerning the lists
$\langle a_{\xi}\rangle_{\xi<\kappa}$ and
$\langle b_{\xi}\rangle_{\xi<\kappa}$, and even with these fixed there
remains a good deal of scope in the choice of
$\langle a'_{\xi}\rangle_{\xi<\kappa}$ and
$\langle b'_{\xi}\rangle_{\xi<\kappa}$.

The isomorphisms described in Theorem 331I are measure algebra
isomorphisms, that is, measure-preserving Boolean isomorphisms.
Obvious questions arise concerning Boolean isomorphisms which are not
necessarily measure-preserving;  the theorem also helps us to settle
many of these (see 331N).   But we can observe straight
away the remarkable fact that two homogeneous probability
algebras which are isomorphic as Boolean algebras are also isomorphic as
probability algebras, since they must have the same Maharam type.

I have already mentioned certain measure space isomorphisms (254K,
255A).   Of course any isomorphism between measure spaces must induce an
isomorphism between their measure algebras (see 324M), and any
isomorphism between measure algebras corresponds to an isomorphism
between their Stone spaces (see 324N).   But there are many important
examples of isomorphisms between measure algebras which do not
correspond to isomorphisms between the measure spaces most naturally
involved.   (I describe one in 343J.)   Maharam's theorem really is a
theorem about measure {\it algebras} rather than measure {\it spaces}.

The particular method I use to show that the measure algebra of the
usual measure on $\{0,1\}^{\kappa}$ is homogeneous for infinite $\kappa$
(331J-331K) is chosen with a view to a question in the next section
(332O).   There are other ways of doing it.   But I recommend study of
this particular one because of the way in which it involves the
topological, algebraic and order properties of the algebra $\frak B$.
I have extracted some of the elements of the argument in 331Xi and
331Ye-331Yf.   These use the concept of `density' of a topological
space.   This does not seem the moment to go farther along this road,
but I hope you can see that there are likely to be many further
`cardinal functions' to provide useful measures of complexity in both
algebraic and topological structures.
}%end of comment

\discrpage





\frfilename{mt332.tex}
\versiondate{19.3.05}
\copyrightdate{1996}

\def\chaptername{Maharam's theorem}
\def\sectionname{Classification of localizable measure algebras}

\newsection{332}

In this section I present what I call `Maharam's theorem', that every
localizable measure algebra is expressible as a weighted simple product
of measure algebras of spaces of the form $\{0,1\}^{\kappa}$ (332B).
Among its many
consequences is a complete description of the isomorphism classes of
localizable measure algebras (332J).   This description needs the
concepts of `cellularity' of a Boolean algebra (332D) and its
refinement, the `magnitude' of a measure algebra (332G).   I end this
section with a discussion of those pairs of measure algebras for which
there is a measure-preserving homomorphism from one to the other
(332P-332Q), and a general formula for the Maharam type of a localizable
measure algebra (332S).

\leader{332A}{Lemma} Let $\frak A$ be any Boolean algebra.   Writing
$\frak A_a$ for the principal ideal generated by $a\in\frak A$,
\cmmnt{the set }
$\{a:a\in\frak A,\,\frak A_a$ is \Mth$\}$ is order-dense in $\frak A$.

\proof{ Take any $a\in\frak A\setminus\{0\}$.   Then
$A=\{\tau(\frak A_b):0\ne b\Bsubseteq a\}$ has a least member;
take $c\Bsubseteq a$ such that $c\ne 0$ and $\tau(\frak A_c)=\min A$.
If $0\ne b\Bsubseteq c$, then $\tau(\frak A_b)\le\tau(\frak A_c)$, by 331Hc,
while $\tau(\frak A_b)\in A$, so
$\tau(\frak A_c)\le\tau(\frak A_b)$.   Thus
$\tau(\frak A_b)=\tau(\frak A_c)$ for every non-zero $b\Bsubseteq c$, and
$\frak A_c$ is \Mth.
}%end of proof of 332A


\leader{332B}{Maharam's theorem} Let $(\frak A,\bar\mu)$ be a localizable
measure algebra.   Then it is isomorphic to the simple product of a
family $\langle(\frak A_i,\bar\mu_i)\rangle_{i\in I}$ of measure
algebras, where for each $i\in I\,\,(\frak A_i,\bar\mu_i)$ is isomorphic,
up to a re-normalization of
the measure, to the measure algebra of the usual measure on
$\{0,1\}^{\kappa_i}$, where $\kappa_i$ is either $0$ or an infinite
cardinal.

\proof{{\bf (a)} For $a\in\frak A$, let $\frak A_a$ be the principal
ideal of $\frak A$ generated by $a$.   Then

\Centerline{$D=\{a:a\in\frak A,\,0<\bar\mu a<\infty,\,\frak A_a$ is
\Mth$\}$}

\noindent is order-dense in $\frak A$.    \Prf\  If
$a\in\frak A\setminus\{0\}$, then (because $(\frak A,\bar\mu)$ is
semi-finite)
there is a $b\Bsubseteq a$ such that $0<\bar\mu b<\infty$;  now by 332A
there is
a non-zero $d\Bsubseteq b$ such that $\frak A_d$ is \Mth.\   \Qed

\medskip

{\bf (b)} By 313K, there is a partition of unity
$\langle e_i\rangle_{i\in I}$ consisting of members of $D$;
by 322L(d-i), $(\frak A,\bar\mu)$ is isomorphic, as measure algebra,
to the simple product of the principal ideals $\frak A_i=\frak A_{e_i}$.

\medskip

{\bf (c)} For each $i\in I$, $(\frak A_i,\bar\mu_i)$ is a non-trivial
totally finite \Mth\ measure algebra, writing
$\bar\mu_i=\bar\mu\restrp\frak A_i$.   Take
$\gamma_i=\bar\mu_i(1_{\frak A_i})=\bar\mu e_i$, and set
$\bar\mu_i'=\gamma_i^{-1}\bar\mu_i$.   Then $(\frak A_i,\bar\mu'_i)$
is a \Mth\ probability algebra, so by 331L is isomorphic
to the measure algebra $(\frak B_{\kappa_i},\bar\nu_{\kappa_i})$ of the
usual measure on $\{0,1\}^{\kappa_i}$, where $\kappa_i$ is either $0$
or an infinite cardinal.   Thus $(\frak A_i,\bar\mu_i)$ is isomorphic,
up to a scalar multiple of the measure, to
$(\frak B_{\kappa_i},\bar\nu_{\kappa_i})$.
}%end of proof of 332B

\cmmnt{\medskip

\noindent{\bf Remark} For the case of totally finite measure algebras,
this is Theorem 2 of {\smc Maharam 1942}.
}

\leader{332C}{Corollary} Let $(\frak A,\bar\mu)$ be a localizable
measure algebra.   For any cardinal $\kappa$, write $\nu_{\kappa}$ for
the usual
measure on $\{0,1\}^{\kappa}$, and $\Tau_{\kappa}$ for its domain.
Then we can find families $\langle\kappa_i\rangle_{i\in I}$, $\langle
\gamma_i\rangle_{i\in I}$ such that every $\kappa_i$ is either $0$ or
an infinite cardinal, every $\gamma_i$ is a strictly positive real
number, and $(\frak A,\bar\mu)$ is isomorphic to the measure algebra of
$(X,\Sigma,\nu)$, where

\Centerline{$X=\{(x,i):i\in I,\,x\in\{0,1\}^{\kappa_i}\}$,}

\Centerline{$\Sigma=\{E:E\subseteq X,\,
\{x:(x,i)\in E\}\in\Tau_{\kappa_i}$ for every $i\in I\}$,}

\Centerline{$\nu E
=\sum_{i\in I}\gamma_i\nu_{\kappa_i}\{x:(x,i)\in E\}$}

\noindent for every $E\in\Sigma$.

\proof{ Take the family $\langle\kappa_i\rangle_{i\in I}$ from the last
theorem, take the $\gamma_i=\bar\mu e_i$ to be the normalizing factors
of the proof there, and apply 322Lb to identify the simple product of
the measure algebras of
$(\{0,1\}^{\kappa_i},\Tau_{\kappa_i},\gamma_i\nu_{\kappa_i})$ with
the measure algebra of their direct sum $(X,\Sigma,\nu)$.
}%end of proof of 332C


\vleader{72pt}{332D}{The cellularity of a Boolean algebra}\cmmnt{ In
order to  properly describe
non-sigma-finite measure algebras, we need the following concept.}   If
$\frak A$ is any Boolean algebra, write

\Centerline{$c(\frak A)=\sup\{\#(C):C\subseteq\frak A\setminus\{0\}$ is
disjoint$\}$,}

\noindent the {\bf cellularity} of $\frak A$.
\cmmnt{(If $\frak  A=\{0\}$, take $c(\frak A)=0$.)   Thus $\frak A$ is
ccc\cmmnt{ (316A)} iff $c(\frak A)\le\omega$.}

\leader{332E}{Proposition} Let $(\frak A,\bar\mu)$ be any semi-finite
measure algebra, and $C$ any partition of unity in $\frak A$ consisting
of elements of finite measure.   Then
$\max(\omega,\#(C))=\max(\omega,c(\frak A))$.

\proof{ Of course $\#(C\setminus\{0\})\le c(\frak A)$, because
$C\setminus\{0\}$ is disjoint, so

\Centerline{$\max(\omega,\#(C))
=\max(\omega,\#(C\setminus\{0\})\le\max(\omega,c(\frak A))$.}

Now suppose that $D$ is any disjoint set in $\frak A\setminus\{0\}$.
For $c\in C$, $\{d\Bcap c:d\in D\}$ is a disjoint set in the
principal ideal $\frak A_c$ generated by $c$.   But $\frak A_c$ is ccc
(322G), so $\{d\Bcap c:d\in D\}$ must be countable, and
$D_c=\{d:d\in D,\,d\Bcap c\ne 0\}$ is countable.   Because $\sup C=1$,
$D=\bigcup_{c\in C}D_c$, so

\Centerline{$\#(D)\le\max(\omega,\#(C),\sup_{c\in C}\#(D_c))=\max(\omega,\#(C))$.}

\noindent As $D$ is arbitrary, $c(\frak A)\le\max(\omega,\#(C))$ and
$\max(\omega,c(\frak A))=\max(\omega,\#(C))$.
}%end of proof of 332E

\leader{332F}{Corollary} Let $(\frak A,\bar\mu)$ be any semi-finite
measure
algebra.   Then there is a disjoint set in $\frak A\setminus\{0\}$ of
cardinal $c(\frak A)$.

\proof{ Start by taking any partition of unity $C$ consisting of
non-zero elements of finite measure.   If $\#(C)=c(\frak A)$ we can
stop, because $C$ is a disjoint set in $\frak A\setminus\{0\}$.
Otherwise, by 332E, we must have $C$ finite and $c(\frak
A)=\omega$.   Let $A$ be the set of atoms in $\frak A$.   If $A$ is
infinite, it is a disjoint set with cardinal $\omega$, so we can stop.
Otherwise, since there is certainly a disjoint
set $D\subseteq\frak A\setminus\{0\}$ with cardinal greater than $\#(A)$,
and since each member of $A$ can meet at most one member of $D$, there
must be a member $d$ of $D$ which does not include any atom.
Accordingly we can choose inductively a sequence $\sequencen{d_n}$ such
that $d_0=d$, $0\ne d_{n+1}\Bsubset d_n$ for every $n$.   Now
$\{d_n\Bsetminus d_{n+1}:n\in\Bbb N\}$ is a disjoint set in
$\frak A\setminus\{0\}$ with cardinal $\omega=c(\frak A)$.
}%end of proof of 332F

\leader{332G}{Definitions}\cmmnt{ For the next theorem, it will be
convenient to have some special terminology.

\medskip

}{\bf (a)}\cmmnt{ The first word I wish to introduce is
a variant of the idea of
`cellularity', adapted to measure algebras.}   If $(\frak A,\bar\mu)$ is
a semi-finite measure algebra, let us say that the {\bf magnitude} of an
$a\in\frak A$ is $\bar\mu a$ if $\bar\mu a$ is finite, and otherwise is
the cellularity of the principal ideal $\frak A_a$ generated by $a$.
\cmmnt{(This
is necessarily infinite, since any partition of $a$ into sets of finite
measure must be infinite.)}
If we take it that any real number is less
than any infinite cardinal, then the class of possible magnitudes is
totally ordered.

I shall sometimes speak of the {\bf magnitude} of the measure algebra
$(\frak A,\bar\mu)$ itself, meaning the magnitude of $1_{\frak A}$.
Similarly, if $(X,\Sigma,\mu)$ is a semi-finite measure space, the {\bf
magnitude} of $(X,\Sigma,\mu)$, or of $\mu$, is the magnitude of its
measure algebra.

\header{332Gb}{\bf (b)} Next, for any Dedekind complete Boolean algebra
$\frak A$, and any cardinal $\kappa$, we can look at the element

\Centerline{$e_{\kappa}=\sup\{a:a\in\frak A\setminus\{0\},\,\frak A_a$
is \Mth\ with Maharam type $\kappa\}$,}

\noindent writing $\frak A_a$ for the principal ideal of $\frak A$
generated by $a$, as usual.   I will call this the
{\bf Maharam-type-$\kappa$ component} of $\frak A$.   \cmmnt{Of course
$e_{\kappa}\Bcap e_{\lambda}=0$ whenever $\lambda$, $\kappa$ are
distinct cardinals.
\prooflet{\Prf\ $a\Bcap b=0$ whenever $\frak A_a$, $\frak A_b$
are \Mth\ of different Maharam types, since
$\tau(\frak A_{a\Bcap b})$
cannot be equal simultaneously to $\tau(\frak A_a)$ and
$\tau(\frak A_b)$.\  \Qed}
}%end of comment

\cmmnt{Also} $\{e_{\kappa}:\kappa$ is a cardinal$\}$ is a partition of
unity in $\frak A$\cmmnt{, because

\Centerline{$\sup\{e_{\kappa}:\kappa$ is a cardinal$\}
=\sup\{a:\frak A_a$ is \Mth$\}=1$}

\noindent by 332A}.   \cmmnt{Note that there is no claim that
$\frak A_{e_{\kappa}}$ itself is homogeneous;  but we do have a useful
result in this direction.}

\leader{332H}{Lemma} Let $\frak A$ be a Dedekind complete Boolean
algebra and $\kappa$ an
infinite cardinal.   Let $e$ be the Maharam-type-$\kappa$ component of
$\frak A$.   If $0\ne d\Bsubseteq e$ and the principal ideal $\frak A_d$
generated by $d$ is ccc, then it is \Mth\ with Maharam
type $\kappa$.

\proof{{\bf (a)} The point is that $\tau(\frak A_d)\le\kappa$.   \Prf\
Set

\Centerline{$A=\{a:a\in\frak A\setminus\{0\},\,\frak A_a$ is \Mth\ of
Maharam type $\kappa\}$.}

\noindent Then $d=\sup\{a\Bcap d:a\in A\}$.   Because $\frak A_d$ is
ccc, there is a sequence $\sequencen{a_n}$ in $A$ such that
$d=\sup_{n\in\Bbb N}d\Bcap a_n$ (316E);  set $b_n=d\Bcap a_n$.   We have
$\tau(\frak A_{b_n})\le\tau(\frak A_{a_n})=\kappa$ for each $n$;  let
$D_n$ be a subset of $\frak A_{b_n}$, with cardinal at most $\kappa$,
which $\tau$-generates $\frak A_{b_n}$.   Set

\Centerline{$D=\bigcup_{n\in\Bbb N}D_n\cup\{b_n:n\in\Bbb
N\}\subseteq\frak A_d$.}

\noindent   If $\frak C$ is the
order-closed subalgebra of $\frak A_d$ generated by $D$, then
$\frak C\cap\frak A_{b_n}$ is an order-closed subalgebra of
$\frak A_{b_n}$
including $D_n$, so is equal to $\frak A_{b_n}$, for every $n$.   But
$a=\sup_{n\in\Bbb N}a\Bcap b_n$ for every $a\in\frak A_d$, so $\frak
C=\frak A_d$.   Thus $D\,\,\tau$-generates $\frak A_d$, and


\Centerline{$\tau(\frak A_d)\le\#(D)\le\max(\omega,\sup_{n\in\Bbb
N}\#(D_n))=\kappa$.   \Qed}

\medskip

{\bf (b)} If now $b$ is any non-zero member of $\frak A_d$, there is
some $a\in A$ such that $b\Bcap a\ne 0$, so that

\Centerline{$\kappa=\tau(\frak A_{b\Bcap a})
\le\tau(\frak A_b)\le\tau(\frak A_d)\le\kappa$.}

\noindent Thus we must have $\tau(\frak A_b)=\kappa$ for every non-zero
$b\in\frak A_d$, and $\frak A_d$ is \Mth\ with type
$\kappa$, as claimed.
}%end of proof of 332H

\leader{332I}{Lemma} Let $(\frak A,\bar\mu)$ be an atomless semi-finite
measure algebra which is not totally finite.   Then it has a partition
of unity consisting of elements of measure $1$.

\proof{ Let $A$ be the set $\{a:\bar\mu a=1\}$, and $\Cal C$ the family
of disjoint subsets of $A$.   By Zorn's lemma, $\Cal C$ has a maximal
member $C_0$ (compare the proof of 313K).
Set $D=\{d:d\in\frak A,\,d\Bcap c=0$ for every $c\in C_0\}$.   Then $D$
is upwards-directed.   If $d\in D$, then $\bar\mu a\ne 1$ for every
$a\Bsubseteq d$, so $\bar\mu d<1$, by 331C.   So $d_0=\sup D$ is defined
in $\frak A$ (321C);  of course $d_0\in D$, so $\bar\mu d_0<1$.
Observe that $\sup C_0=1\Bsetminus d_0$.

Because $\bar\mu 1=\infty$, $C_0$ must be
infinite;  let $\sequencen{a_n}$ be any sequence of distinct elements of
$C_0$.   For each $n\in\Bbb N$, use 331C again to choose an
$a'_n\Bsubseteq a_n$ such that $\bar\mu a'_n=\bar\mu d_0$.   Set

\Centerline{$b_0=d_0\Bcup(a_0\Bsetminus a'_0)$,
\quad $b_n=a'_{n-1}\Bcup(a_n\Bsetminus a'_n)$}

\noindent for every $n\ge 1$.   Then $\sequencen{b_n}$ is a disjoint
sequence of elements of measure $1$ and $\sup_{n\in\Bbb
N}b_n=\sup_{n\in\Bbb N}a_n\Bcup d_0$.    Now

\Centerline{$(C_0\setminus\{a_n:n\in\Bbb N\})\cup\{b_n:n\in\Bbb N\}$}

\noindent is a partition of unity consisting of elements of measure $1$.
}%end of proof of 332I

\leader{332J}{}\cmmnt{ Now I can formulate a complete classification
theorem for localizable measure algebras, refining the expression in
332B.

\medskip

\noindent}{\bf Theorem} Let $(\frak A,\bar\mu)$ and $(\frak B,\bar\nu)$
be localizable measure algebras.   For each cardinal $\kappa$, let
$e_{\kappa}$, $f_{\kappa}$ be the Maharam-type-$\kappa$ components of
$\frak A$, $\frak B$ respectively.   Then $(\frak A,\bar\mu)$ and
$(\frak B,\bar\nu)$ are isomorphic, as measure algebras, iff (i)
$e_{\kappa}$ and
$f_{\kappa}$ have the same magnitude for every infinite cardinal
$\kappa$ (ii) for every $\gamma\in\ooint{0,\infty}$, $(\frak A,\bar\mu)$
and $(\frak B,\bar\nu)$ have the same number of atoms of measure
$\gamma$.

\proof{ Throughout the proof, write $\frak A_a$ for the principal ideal
of $\frak A$ generated by $a$, and $\bar\mu_a$ for the restriction of
$\bar\mu$ to $\frak A_a$;  and define $\frak B_b$, $\bar\nu_b$ similarly
for $b\in\frak B$.

\medskip

{\bf (a)} If $(\frak A,\bar\mu)$ and $(\frak B,\bar\nu)$ are isomorphic,
then of course the isomorphism matches their Maharam-type components
together and retains their magnitudes, and matches atoms of the same
measure together;  so the conditions are surely satisfied.


\medskip

{\bf (b)} Now suppose that the conditions are satisfied.   Set

\Centerline{$K
=\{\kappa:\kappa$ is an infinite cardinal, $e_{\kappa}\ne 0\}
=\{\kappa:\kappa$ is an infinite cardinal, $f_{\kappa}\ne 0\}$.}

\noindent For $\gamma\in\ooint{0,\infty}$, let $A_{\gamma}$ be the set
of atoms of measure $\gamma$ in $\frak A$, and set $e_{\gamma}=\sup
A_{\gamma}$.   Write $I=K\cup\ooint{0,\infty}$.   Then $\langle
e_i\rangle_{i\in I}$ is a partition of unity in $\frak A$, so $(\frak
A,\bar\mu)$ is isomorphic to the simple product of $\langle(\frak
A_{e_i},\bar\mu_{e_i})\rangle_{i\in I}$, writing $\frak A_{e_i}$ for the
principal ideal generated by $e_i$ and $\bar\mu_{e_i}$ for the
restriction $\bar\mu\restrp\frak A_{e_i}$.

In the same way, writing
$B_{\gamma}$ for the set of atoms of measure $\gamma$ in $\frak B$,
$f_{\gamma}$ for $\sup B_{\gamma}$,  $\frak B_{f_i}$ for the principal
ideal generated by $f_i$ and $\bar\nu_{f_i}$ for the restriction of
$\bar\nu$
fo $\frak B_{f_i}$, we have $(\frak B,\bar\nu)$ isomorphic to the simple
product of $\langle(\frak B_{f_i},\bar\nu_{f_i})\rangle_{i\in I}$.

\medskip

{\bf (c)} It will therefore be enough if I can show that
$(\frak A_{e_i},\bar\mu_{e_i})\cong(\frak B_{f_i},\bar\nu_{f_i})$
for every $i\in I$.

\medskip

\quad{\bf (i)} For $\kappa\in K$, the hypothesis is that $e_{\kappa}$
and $f_{\kappa}$ have the same magnitude.   If they are both of finite
magnitude, that is, $\bar\mu e_{\kappa}=\bar\nu f_{\kappa}<\infty$, then
both $(\frak A_{e_{\kappa}},\bar\mu_{e_{\kappa}})$ and
$(\frak B_{f_{\kappa}},\bar\nu_{f_{\kappa}})$ are homogeneous and of
Maharam type $\kappa$, by 332H.   So 331I tells us that they are
isomorphic.   If they are both of infinite magnitude $\lambda$, then
332I tells us that both $\frak A_{e_{\kappa}}$, $\frak B_{f_{\kappa}}$
have partitions of unity $C$, $D$ consisting of sets of measure $1$.
So $(\frak A_{e_{\kappa}},\bar\mu_{e_{\kappa}})$ is isomorphic to the
simple product of $\langle(\frak A_c,\bar\mu_c)\rangle_{c\in C}$, while
$(\frak B_{f_{\kappa}},\bar\nu_{f_{\kappa}})$ is isomorphic to the
simple product of
$\langle(\frak B_d,\bar\nu_d)\rangle_{d\in D}$.  But we know also that
every $(\frak A_c,\bar\mu_c)$, $(\frak B_d,\bar\nu_d)$ is a homogeneous
probability
algebra with Maharam type $\kappa$, by 332H again, so by Maharam's theorem
again they are all isomorphic.   Since $C$, $D$ and $\lambda$ are all
infinite,

\Centerline{$\#(C)=c(\frak A_{e_{\kappa}})=\lambda=c(\frak
B_{f_{\kappa}})=\#(D)$}

\noindent by 332E.   So we are taking the same number of factors in each
product and $(\frak A_{e_{\kappa}},\bar\mu_{e_{\kappa}})$ must be
isomorphic
to $(\frak B_{f_{\kappa}},\bar\nu_{f_{\kappa}})$.

\medskip

\quad{\bf (ii)} For $\gamma\in\ooint{0,\infty}$, our hypothesis is that
$\#(A_{\gamma})=\#(B_{\gamma})$.   Now $A_{\gamma}$ is a partition of
unity in $\frak A_{e_{\gamma}}$, so
$(\frak A_{e_{\gamma}},\bar\mu_{e_{\gamma}})$ is isomorphic to the
simple product of
$\langle(\frak A_a,\bar\mu_a)\rangle_{a\in A_{\gamma}}$.   Similarly,
$(\frak B_{f_{\gamma}},\bar\nu_{f_{\gamma}})$ is isomorphic to the
simple product of
$\langle(\frak B_b,\bar\nu_b)\rangle_{b\in B_{\gamma}}$.
Since every $(\frak A_a,\bar\mu_a)$, $(\frak B_b,\bar\nu_b)$ is just a
simple atom of
measure $\gamma$, these are all isomorphic;  since we are taking the
same number of factors in each product,
$(\frak A_{e_{\gamma}},\bar\mu_{e_{\gamma}})$ must be isomorphic to
$(\frak B_{f_{\gamma}},\bar\nu_{f_{\gamma}})$.

\medskip

\quad{\bf (iii)} Thus we have the full set of required isomorphisms, and
$(\frak A,\bar\mu)$ is isomorphic to $(\frak B,\bar\nu)$.
}%end of proof of 332J

\cmmnt{
\leader{332K}{Remarks (a)} The partition of unity
$\{e_i:i\in I\}$ of
$\frak A$ used in the above theorem is in some sense canonical.   (You
might feel it more economical to replace $I$ by
$K\cup\{\gamma:A_{\gamma}\ne\emptyset\}$.)   The further partition of
the atomic part into individual atoms (part (c-ii) of the proof) is also
canonical.   But of course the partition of the $e_{\kappa}$ of infinite
magnitude into elements of measure $1$ requires a degree of arbitrary
choice.

The value of the expressions in 332C is that the parameters $\kappa_i$,
$\gamma_i$ there are sufficient to identify the measure algebra up to
isomorphism.   For, amalgamating the language of 332C and 332J, we see
that the magnitude of $e_{\kappa}$ in 332J is just
$\sum_{\kappa_i=\kappa}\gamma_i$ if this is finite,
$\#(\{i:\kappa_i=\kappa\})$ otherwise (using 332E, as usual);  while
the number of atoms of measure $\gamma$ is
$\#(\{i:\kappa_i=0,\,\gamma_i=\gamma\})$.

\header{332Kb}{\bf (b)} The classification which Maharam's theorem gives
us is not merely a listing.   It involves a real insight into the nature
of the algebras, enabling us to answer a very wide variety of natural
questions.   I give the next couple of results as a sample of what we
can expect these methods to do for us.
}%end of comment

\leader{332L}{Proposition} Let $(\frak A,\bar\mu)$ be a measure algebra,
and $a$, $b\in\frak A$ two elements of finite measure.   Suppose that
$\pi:\frak A_a\to\frak A_b$ is a measure-preserving isomorphism, where
$\frak A_a$, $\frak A_b$ are the principal ideals generated by $a$ and
$b$.   Then there is a measure-preserving automorphism
$\phi:\frak A\to\frak A$ which extends $\pi$.

\proof{ The point is that $\frak A_{b\Bsetminus a}$ is isomorphic, as
measure algebra, to $\frak A_{a\Bsetminus b}$.   \Prf\ Set $c=a\Bcup b$.
For each infinite cardinal $\kappa$, let $e_{\kappa}$ be the
Maharam-type-$\kappa$ component of $\frak A_c$.   Then
$e_{\kappa}\Bcap a$ is the Maharam-type-$\kappa$ component of
$\frak A_a$, because if $d\Bsubseteq c$ and $\frak A_d$ is Maharam
homogeneous with Maharam type $\kappa$, then $\frak A_{d\Bcap a}$ is
either $\{0\}$ or again Maharam-type-homogeneous with Maharam type $\kappa$.
Similarly, $e_{\kappa}\Bsetminus a$ is the Maharam-type-$\kappa$
component of $\frak A_{c\setminus a}=\frak A_{b\setminus a}$, $e_{\kappa}\Bcap b$ is the
Maharam-type-$\kappa$ component of $\frak A_b$ and
$e_{\kappa}\Bsetminus b$ is the Maharam-type-$\kappa$ component of
$\frak A_{a\setminus b}$.   Now $\pi:\frak A_a\to\frak A_b$ is an
isomorphism, so $\pi(e_{\kappa}\cap a)$ must be $e_{\kappa}\cap b$, and

$$\eqalign{\bar\mu(e_{\kappa}\setminus a)
&=\bar\mu e_{\kappa}-\bar\mu(e_{\kappa}\cap a)
=\bar\mu e_{\kappa}-\bar\mu\pi(e_{\kappa}\cap a)\cr
&=\bar\mu e_{\kappa}-\bar\mu(e_{\kappa}\cap b)
=\bar\mu(e_{\kappa}\setminus b).\cr}$$

In the same way, if we write $n_{\gamma}(d)$ for the number of atoms of
measure $\gamma$ in $\frak A_d$, then

\Centerline{$n_{\gamma}(b\Bsetminus a)
=n_{\gamma}(c)-n_{\gamma}(a)
=n_{\gamma}(c)-n_{\gamma}(b)
=n_{\gamma}(a\Bsetminus b)$}

\noindent for every $\gamma\in\ooint{0,\infty}$.   By 332J, there is a
measure-preserving isomorphism
$\pi_1:\frak A_{b\Bsetminus a}\to\frak A_{a\setminus b}$.\ \Qed

If we now set

\Centerline{$\phi d
=\pi(d\Bcap a)\Bcup\pi_1(d\Bcap b\Bsetminus a)\Bcup(d\Bsetminus c)$}

\noindent for every $d\in\frak A$, $\phi:\frak A\to\frak A$ is a
measure-preserving isomorphism which agrees with $\pi$ on $\frak A_a$.
}%end of proof of 332L

\cmmnt{\medskip

\noindent{\bf Remark} There is an elementary proof of this result
not relying on Maharam's theorem;  see 381Ye(ii).
}

\leader{332M}{Lemma} Suppose that $(\frak A,\bar\mu)$ and
$(\frak B,\bar\nu)$
are homogeneous measure algebras, with  $\tau(\frak A)\le\tau(\frak B)$
and $\bar\mu 1=\bar\nu 1<\infty$.   Then there is a
measure-preserving Boolean homomorphism from $\frak A$ to $\frak B$.

\proof{ The case $\tau(\frak A)=0$ is trivial.   Otherwise, considering
normalized versions of the measures, we are reduced to the case
$\bar\mu 1=\bar\nu 1=1$, $\tau(\frak A)=\kappa\ge\omega$,
$\tau(\frak B)=\lambda\ge\kappa$, so that $(\frak A,\bar\mu)$ is isomorphic to the
measure algebra $(\frak B_{\kappa},\bar\nu_{\kappa})$ of the usual
measure $\bar\nu_{\kappa}$ on $\{0,1\}^{\kappa}$;   and similarly
$(\frak B,\bar\nu)$ is isomorphic to the measure algebra
$(\frak B_{\lambda},\bar\nu_{\lambda})$ of the usual measure on
$\{0,1\}^{\lambda}$.   Now (identifying the cardinals $\kappa$, $\lambda$ with von
Neumann ordinals, as usual),
$\kappa\subseteq\lambda$, so we have an inverse-measure-preserving map
$x\mapsto x\restr\kappa:\{0,1\}^{\lambda}\to\{0,1\}^{\kappa}$ (254Oa),
which induces a measure-preserving Boolean homomorphism from
$\frak B_{\kappa}$ to $\frak B_{\lambda}$ (324M), and hence a
measure-preserving homomorphism from $\frak A$ to $\frak B$.
}%end of proof of 332M

\leader{332N}{Lemma} If $(\frak A,\bar\mu)$ is a probability algebra and
$\kappa\ge\max(\omega,\tau(\frak A))$, then there is a
measure-preserving Boolean
homomorphism from $(\frak A,\bar\mu)$ to the measure
algebra $(\frak B_{\kappa},\bar\nu_{\kappa})$ of the usual measure $\nu$ on
$\{0,1\}^{\kappa}$;  that is, $(\frak A,\bar\mu)$ is isomorphic to a
closed subalgebra of $(\frak B_{\kappa},\bar\nu_{\kappa})$.

\proof{ Let $\familyiI{c_i}$ be a partition of unity in $\frak A$ such
that every principal ideal $\frak A_{c_i}$ is homogeneous and no $c_i$
is zero.   Then $I$ is countable and $\sum_{i\in I}\bar\mu c_i=1$.
Accordingly there is a partition of unity $\familyiI{d_i}$ in
$\frak B_{\kappa}$ such that $\bar\nu d_i=\bar\mu c_i$ for every $i$.
\Prf\ Because $I$ is countable, we may suppose that it is either
$\Bbb N$ or an initial segment of $\Bbb N$.   In this case, choose
$\familyiI{d_i}$
inductively such that $d_i\Bsubseteq 1\Bsetminus\sup_{j<i}d_j$ and
$\bar\nu d_i=\bar\mu d_i$ for each $i\in I$, using 331C.\ \Qed

If $i\in I$, then $\tau(\frak A_{c_i})\le\kappa=\tau((\frak B_{\kappa})_{d_i})$, so
there is a measure-preserving Boolean homomorphism
$\pi_i:\frak A_{c_i}\to(\frak B_{\kappa})_{d_i}$.   Setting
$\pi a=\sup_{i\in I}\pi_i(a\Bcap c_i)$ for $a\in\frak A$, we have a
measure-preserving Boolean homomorphism $\pi:\frak A\to\frak B_{\kappa}$.   By
324Kb, $\pi[\frak A]$ is a closed subalgebra of $\frak B_{\kappa}$, and of course
$(\pi[\frak A],\bar\nu_{\kappa}\restr\pi[\frak A])$ is isomorphic to
$(\frak A,\bar\mu)$.
}%end of proof of 332N

\leader{332O}{Lemma} Let $(\frak A,\bar\mu)$ and $(\frak B,\bar\nu)$ be
localizable measure algebras.   For each infinite cardinal $\kappa$ let
$e_{\kappa}$,
$f_{\kappa}$ be their Maharam-type-$\kappa$ components, and for
$\gamma\in\ooint{0,\infty}$ let $e_{\gamma}$, $f_{\gamma}$ be the
suprema of the atoms of measure $\gamma$ in $\frak A$, $\frak B$
respectively.   If there is a measure-preserving Boolean homomorphism
from $\frak A$ to $\frak B$, then the magnitude of
$\sup_{\kappa\ge\lambda}e_{\kappa}$ is not greater than the magnitude of
$\sup_{\kappa\ge \lambda}f_{\kappa}$ whenever $\lambda$ is an infinite
cardinal, while the magnitude of
$\sup_{\kappa\ge\omega}e_{\kappa}\Bcup\sup_{\gamma\le\delta}e_{\gamma}$
is not greater than the magnitude of
$\sup_{\kappa\ge\omega}f_{\kappa}\Bcup\sup_{\gamma\le\delta}f_{\gamma}$
for any $\delta\in\ooint{0,\infty}$.

\proof{ Suppose that $\pi:\frak A\to\frak B$ is a measure-preserving
Boolean homomorphism.   For infinite cardinals $\lambda$, set
$e^*_{\lambda}=\sup_{\kappa\ge\lambda}e_{\kappa}$,
$f^*_{\lambda}=\sup_{\kappa\ge\lambda}f_{\kappa}$, while for
$\delta\in\ooint{0,\infty}$ set $e^*_{\delta}
=\sup_{\kappa\ge\omega}e_{\kappa}\Bcup\sup_{\gamma\le\delta}e_{\gamma}$,
$f^*_{\delta}
=\sup_{\kappa\ge\omega}f_{\kappa}\Bcup\sup_{\gamma\le\delta}f_{\gamma}$.
Let $\langle c_i\rangle_{i\in I}$ be a partition of unity in
$\frak A$ such that all the principal ideals $\frak A_{c_i}$ are totally
finite and homogeneous, as in 332B.   Then $c_i\Bsubseteq e_{\kappa}$
whenever $\kappa=\tau(\frak A_{c_i})$ is infinite, and $c_i\Bsubseteq
e_{\gamma}$ if
$c_i$ is an atom of measure $\gamma$.   Take $v$ to be either an
infinite cardinal or a strictly positive real
number.   Set

\Centerline{$J=\{i:i\in I,\,c_i\Bsubseteq e^*_v\}$;}

\noindent then $e^*_v=\sup_{i\in J}c_i$.

Now the point is that if $i\in J$ then $\pi c_i\Bsubseteq f^*_{{v}}$.
\Prf\   We need to consider two cases.   (i) If $c_i$ is an atom, then
$v\in\ooint{0,\infty}$ and $\bar\mu c_i\le v$.   So we need only observe
that $1\Bsetminus f^*_v$ is just the supremum in $\frak B$ of the atoms
of measure greater than $v$, none of which can meet $\pi c_i$, since
this has measure at most $v$.   (ii)   Now suppose that $\frak A_{c_i}$
is atomless, with $\tau(\frak A_{c_i})=\kappa\ge v$.   If $0\ne
b\Bsubseteq\pi c_i$, then
$a\mapsto b\Bcap\pi a:\frak A_{c_i}\to\frak B_b$ is an order-continuous
Boolean homomorphism, while $\frak A_{c_i}$ is isomorphic (as Boolean
algebra) to the measure algebra of $\{0,1\}^{\kappa}$, so 331J tells us
that $\tau(\frak B_b)\ge\kappa$.   This means, first, that $b$ cannot be
an atom, so that $\pi c_i$ cannot meet
$\sup_{\gamma\in\ooint{0,\infty}}f_{\gamma}$;  and also that $b$ cannot
be included in $f_{\kappa'}$ for any infinite $\kappa'<\kappa$, so that
$\pi c_i$ cannot meet $\sup_{\omega\le\kappa'<\kappa}f_{\kappa}$.   Thus
$\pi c_i$ must be included in
$\sup_{\kappa'\ge\kappa}f_{\kappa}\Bsubseteq f^*_v$.\ \Qed

Of course $\langle\pi c_i\rangle_{i\in J}$ is disjoint.   So if
$e^*_{{v}}$ has finite magnitude, the magnitude of $f^*_{{v}}$ is at
least

\Centerline{$\sum_{i\in J}\bar\nu\pi c_i=\sum_{i\in J}\bar\mu
c_i=\bar\mu
e^*_{{v}}$,}

\noindent the magnitude of $e^*_{{v}}$.   While if $e^*_{{v}}$ has
infinite magnitude, this is $\#(J)$, by 332E, which is not greater than
the magnitude of $f^*_{{v}}$.
}%end of proof of 332O

\leader{332P}{Proposition} Let $(\frak A,\bar\mu)$ and $(\frak B,\bar\nu)$
be atomless totally finite measure algebras.   For each infinite
cardinal
$\kappa$ let $e_{\kappa}$, $f_{\kappa}$ be their Maharam-type-$\kappa$
components.   Then the following are equiveridical:

(i) $(\frak A,\bar\mu)$ is isomorphic to a closed subalgebra of a
principal ideal of $(\frak B,\bar\nu)$;

(ii) for every cardinal $\lambda$,

\Centerline{$\bar\mu(\sup_{\kappa\ge\lambda}e_{\kappa})
\le\bar\nu(\sup_{\kappa\ge\lambda}f_{\kappa})$.}

\proof{{\bf (a)(i)$\Rightarrow$(ii)} Suppose that $\pi:\frak A\to\frak
B_d$ is a measure-preserving isomorphism between $\frak A$ and a closed
subalgebra of a principal ideal $\frak B_d$ of $\frak B$.   The
Maharam-type-$\kappa$ component of $\frak B_d$ is just $d\Bcap
f_{\kappa}$, so 332O tells us that

\Centerline{$\bar\mu(\sup_{\kappa\ge\lambda}e_{\kappa})
\le\bar\nu(\sup_{\kappa\ge\lambda}d\Bcap f_{\kappa})
\le\bar\nu(\sup_{\kappa\ge\lambda}f_{\kappa})$}


\noindent for every $\lambda$.

\medskip

{\bf (b)(ii)$\Rightarrow$(i)} Now suppose that the condition is
satisfied.

\medskip

\quad\grheada\ Let $P$ be the set of all measure-preserving Boolean
homomorphisms $\pi$ from principal ideals  $\frak A_{c_{\pi}}$ of
$\frak A$ to principal ideals $\frak B_{d_{\pi}}$ of $\frak B$ such that

\Centerline{$\bar\mu(\sup_{\kappa\ge\lambda}
  e_{\kappa}\Bsetminus c_{\pi})
\le\bar\nu(\sup_{\kappa\ge\lambda}
  \bar\nu f_{\kappa}\Bsetminus d_{\pi})$}

\noindent for every cardinal $\lambda\ge\omega$.
Then the trivial homomorphism from $\frak A_0$ to $\frak B_0$ belongs to
$P$, so $P$ is not empty.   Order $P$ by saying that $\pi\le\pi'$ if
$\pi'$ extends $\pi$, that is, if $c_{\pi}\Bsubseteq c_{\pi'}$ and
$\pi'a=\pi a$ for every $a\in\frak A_{c_{\pi}}$.   Then $P$ is a
partially ordered set.

\medskip

\quad\grheadb\
If $Q\subseteq P$ is non-empty and totally ordered, it is bounded above
in $P$.   \Prf\ Set $c^*=\sup_{\pi\in Q}c_{\pi}$,
$d^*=\sup_{\pi\in Q}d_{\pi}$.   For $a\Bsubseteq c^*$ set
$\pi^*a=\sup_{\pi\in Q}\pi(a\Bcap c_{\pi})$.   Because $Q$ is totally
ordered, $\pi^*$ extends all the
functions in $Q$.   It is also easy to check that $\pi^*0=0$,
$\pi^*(a\Bcap a')=\pi^*a\Bcap\pi^*a'$ and
$\pi^*(a\Bcup a')=\pi^*a\Bcup\pi^*a'$ for all $a$, $a'\in\frak A_{c^*}$,
$\pi^*c^*=d^*$ and that $\bar\nu\pi^*a=\bar\mu a$ for every
$a\in\frak A_{c^*}$; so that $\pi^*$ is a measure-preserving Boolean
homomorphism from $\frak A_{c^*}$ to $\frak B_{d^*}$.

Now suppose that $\lambda$ is any cardinal;  then

$$\eqalign{\bar\mu(\sup_{\kappa\ge\lambda}e_{\kappa}\Bsetminus c^*)
&=\inf_{\pi\in Q}
   \bar\mu(\sup_{\kappa\ge\lambda}e_{\kappa}\Bsetminus c_{\pi})
\le\inf_{\pi\in Q}
   \bar\nu(\sup_{\kappa\ge\lambda}f_{\kappa}\Bsetminus d_{\pi})
=\bar\nu(\sup_{\kappa\ge\lambda}f_{\kappa}\Bsetminus d^*).\cr}$$

\noindent So $\pi^*\in P$ and is the required upper bound of $Q$.\
\Qed

\medskip

\quad\grheadc\  By Zorn's Lemma, $P$ has a maximal element $\tilde\pi$
say.   Now $c_{\tilde\pi}=1$.   \Prf\Quer\ If not, then let $\kappa_0$
be the least cardinal such that
$e_{\kappa_0}\Bsetminus c_{\tilde\pi}\ne 0$.  Then

\Centerline{$0
<\bar\mu(\sup_{\kappa\ge\kappa_0}e_{\kappa}\Bsetminus c_{\tilde\pi})
\le\bar\nu(\sup_{\kappa\ge\kappa_0}\bar\nu f_{\kappa}\Bsetminus d_{\tilde\pi})$,}

\noindent so there is a least $\kappa_1\ge\kappa_0$ such that
$f_{\kappa_1}\Bsetminus d_{\tilde\pi}\ne 0$.   Set
$\delta=\min(\bar\mu(e_{\kappa_0}\Bsetminus c_{\tilde\pi}),
\bar\nu(f_{\kappa_1}\Bsetminus d_{\tilde\pi}))>0$.   Because $\frak A$
and $\frak B$ are atomless, there are
$a\Bsubseteq e_{\kappa_0}\Bsetminus c_{\tilde\pi}$ and
$b\Bsubseteq f_{\kappa_1}\Bsetminus d_{\tilde\pi}$ such
that $\bar\mu a=\bar\nu b=\delta$ (331C).   Now $\frak A_a$ is
homogeneous with
Maharam type $\kappa_0$, while $\frak B_b$ is homogeneous with Maharam
type $\kappa_1$ (332H), so there is a
measure-preserving Boolean homomorphism $\phi:\frak A_a\to\frak B_b$
(332M).   Set

\Centerline{$c^*=c_{\tilde\pi}\Bcup a$, \quad $d^*=d_{\tilde\pi}\Bcup
b$,}

\noindent and define $\pi^*:\frak A_{c^*}\to\frak B_{d^*}$ by setting
$\pi^*(g)=\tilde\pi(g\Bcap c_{\tilde\pi})\Bcup\phi(g\Bcap a)$ for every
$g\Bsubseteq c^*$.  It is easy to check that $\pi^*$ is a
measure-preserving Boolean homomorphism.

If $\lambda$ is a cardinal and $\lambda\le\kappa_0$,

$$\eqalign{\bar\mu(\sup_{\kappa\ge\lambda}e_{\kappa}\Bsetminus c^*)
&=\bar\mu(\sup_{\kappa\ge\lambda}e_{\kappa}\Bsetminus c_{\tilde\pi})
-\delta
\le\bar\nu(\sup_{\kappa\ge\lambda}f_{\kappa}\Bsetminus d_{\tilde\pi})
-\delta
=\bar\nu(\sup_{\kappa\ge\lambda}\bar\nu f_{\kappa}\Bsetminus d^*).\cr}$$

\noindent If $\kappa_0<\lambda\le\kappa_1$,

$$\eqalignno{\bar\mu(\sup_{\kappa\ge\lambda}e_{\kappa}\Bsetminus c^*)
&=\bar\mu(\sup_{\kappa\ge\lambda}e_{\kappa}\Bsetminus c_{\tilde\pi})
\le\bar\mu(\sup_{\kappa\ge\kappa_0}e_{\kappa}\Bsetminus c_{\tilde\pi})
-\bar\mu(e_{\kappa_0}\Bsetminus c_{\tilde\pi})\cr
&\le\bar\mu(\sup_{\kappa\ge\kappa_0}e_{\kappa}\Bsetminus c_{\tilde\pi})
-\delta
\le\bar\nu(\sup_{\kappa\ge\kappa_0}f_{\kappa}\Bsetminus d_{\tilde\pi})
-\delta\cr
&=\bar\nu(\sup_{\kappa\ge\kappa_1}f_{\kappa}\Bsetminus d_{\tilde\pi})
-\delta\cr
\noalign{\noindent (by the choice of $\kappa_1$)}
&=\bar\nu(\sup_{\kappa\ge\kappa_1}f_{\kappa}\Bsetminus d^*)
\le\bar\nu(\sup_{\kappa\ge\lambda}f_{\kappa}\Bsetminus d^*).\cr}$$

\noindent If $\lambda>\kappa_1$,

$$\eqalign{\bar\mu(\sup_{\kappa\ge\lambda}e_{\kappa}\Bsetminus c^*)
&=\bar\mu(\sup_{\kappa\ge\lambda}e_{\kappa}\Bsetminus c_{\tilde\pi})
\le\bar\nu(\sup_{\kappa\ge\lambda}f_{\kappa}\Bsetminus d_{\tilde\pi})
=\bar\nu(\sup_{\kappa\ge\lambda}f_{\kappa}\Bsetminus d^*).\cr}$$

\noindent But this means that $\pi^*\in P$, and evidently it is a proper
extension of $\tilde\pi$, which is supposed to be impossible.\
\Bang\Qed

\medskip

\quad\grheadd\ Thus $\tilde\pi$ has domain $\frak A$ and is the required
measure-preserving homomorphism from $\frak A$ to the principal ideal
$\frak B_{d_{\tilde\pi}}$ of $\frak B$.
}%end of proof of 332P

\leader{332Q}{Proposition} Let $(\frak A,\bar\mu)$ and
$(\frak B,\bar\nu)$ be totally finite
measure algebras, and suppose that there are
measure-preserving Boolean homomorphisms $\pi_1:\frak A\to\frak B$ and
$\pi_2:\frak B\to\frak A$.   Then $(\frak A,\bar\mu)$ and
$(\frak B,\bar\nu)$ are isomorphic.

\proof{ Writing $e_{\kappa}$, $f_{\kappa}$ for their
Maharam-type-$\kappa$ components, 332O (applied to both $\pi_1$ and
$\pi_2$) tells us that

\Centerline{$\bar\mu(\sup_{\kappa\ge\lambda}e_{\kappa})
=\bar\nu(\sup_{\kappa\ge\lambda}f_{\kappa})$}

\noindent for every $\lambda$.   Because all these measures are finite,

$$\eqalign{\bar\mu e_{\lambda}
&=\bar\mu(\sup_{\kappa\ge\lambda}e_{\kappa})
  -\bar\mu(\sup_{\kappa>\lambda}e_{\kappa})\cr
&=\bar\nu(\sup_{\kappa\ge\lambda}f_{\kappa})
  -\bar\nu(\sup_{\kappa>\lambda}f_{\kappa})
=\bar\nu f_{\lambda}\cr}$$

\noindent for every $\lambda$.

Similarly, writing $e_{\gamma}$, $f_{\gamma}$ for the suprema in $\frak
A$, $\frak B$ of the atoms of measure $\gamma$, 332O tells us that

\Centerline{$\bar\mu(\sup_{\gamma\le\delta}e_{\gamma})
=\bar\nu(\sup_{\gamma\le\delta}f_{\gamma})$}

\noindent for every $\delta\in\ooint{0,\infty}$, and hence that $\bar\mu
e_{\gamma}=\bar\nu f_{\gamma}$ for every $\gamma$, that is, that $\frak
A$
and $\frak B$ have the same number of atoms of measure $\gamma$.

So $(\frak A,\bar\mu)$ and $(\frak B,\bar\nu)$ are isomorphic, by 332J.
}%end of proof of 332Q

\leader{332R}{}\cmmnt{ 332J tells us that if we know the magnitudes of
the Maharam-type-$\kappa$ components of a localizable measure algebra,
we shall have specified the algebra completely, so that all its
properties are determined.   The calculation of its Maharam type is
straightforward and useful, so I give the details.

\medskip

\noindent}{\bf Lemma} Let $(\frak A,\bar\mu)$ be a semi-finite measure
algebra.   Then $c(\frak A)\le 2^{\tau(\frak A)}$.

\proof{ Let $C\subseteq\frak A\setminus\{0\}$ be a disjoint set, and
$B\subseteq\frak A$ a $\tau$-generating set with cardinal $\tau(\frak A)$.

\medskip

{\bf (a)} If $\frak A$ is purely atomic, then for each $c\in C$ choose
an atom $c'\Bsubseteq c$, and set $f(c)=\{b:b\in B,\,c'\Bsubseteq b\}$.
If $c_1$, $c_2$ are distinct members of $C$, the set

\Centerline{$\{a:a\in\frak A,\,c'_1\Bsubseteq a\iff c'_2\Bsubseteq a\}$}

\noindent is an order-closed subalgebra of $\frak A$ not containing
either $c'_1$ or $c'_2$, so cannot include $B$, and $f(c_1)\ne f(c_2)$.
Thus $f$ is injective, and

\Centerline{$\#(C)\le\#(\Cal PB)=2^{\tau(\frak A)}$.}

\medskip

{\bf (b)} Now suppose that $\frak A$ is not purely atomic;  in this case
$\tau(\frak A)$ is infinite.    For each $c\in C$ choose an element
$c'\Bsubseteq c$ of non-zero finite measure.   Let $\frak B$ be the
subalgebra of $\frak A$ generated by $B$.   Then the topological closure
of $\frak B$ is $\frak A$ itself (323J), and $\#(\frak B)=\tau(\frak A)$
(331Gc).   For $c\in C$ set

\Centerline{$f(c)=\{b:b\in\frak B,
  \,\bar\mu(b\Bcap c')\ge\Bover12\bar\mu c'\}$.}

\noindent Then $f:C\to\Cal P\frak B$ is injective.   \Prf\ If $c_1$,
$c_2$ are distinct members of $C$, then (because $\frak B$ is
topologically dense in $\frak A$) there is a $b\in\frak B$ such that

\Centerline{$\bar\mu((c'_1\Bcup c'_2)\Bcap(c'_1\Bsymmdiff b))
\le\Bover13\min(\bar\mu c'_1,\bar\mu c'_2)$.}

\noindent But in this case

\Centerline{$\bar\mu(c'_1\Bsetminus b)\le\Bover13\bar\mu c'_1$,
\quad$\bar\mu(c'_2\Bcap b)\le\Bover13\bar\mu c'_2$,}

\noindent and $b\in f(c_1)\symmdiff f(c_2)$, so $f(c_1)\ne f(c_2)$.\
\QeD\
Accordingly $\#(C)\le 2^{\#(\frak B)}=2^{\tau(\frak A)}$ in this case
also.

As $C$ is arbitrary, $c(\frak A)\le 2^{\tau(\frak A)}$.
}%end of proof of 332R

\leader{332S}{Theorem} Let $(\frak A,\bar\mu)$ be a localizable measure
algebra.   Then $\tau(\frak A)$ is the least cardinal $\lambda$ such
that ($\alpha$) $c(\frak A)\le 2^{\lambda}$ ($\beta$)
$\tau(\frak A_a)\le\lambda$ for every \Mth\ principal ideal
$\frak A_a$ of $\frak A$.

\proof{ Fix $\lambda$ as the least cardinal satisfying ($\alpha$) and
($\beta$).

\medskip

{\bf (a)} By 331Hc, $\tau(\frak A_a)\le\tau(\frak A)$ for every
$a\in\frak A$, while $c(\frak A)\le 2^{\tau(\frak A)}$ by 332R;  so
$\lambda\le\tau(\frak A)$.

\medskip

{\bf (b)}  Let $C$ be a partition of unity in $\frak A$ consisting of
elements of non-zero finite measure generating \Mth\
principal ideals (as in the proof of 332B);  then
$\#(C)\le c(\frak A)\le 2^{\lambda}$, and there is an injective function
$f:C\to\Cal P\lambda$.   For each $c\in C$, let $B_c\subseteq\frak A_c$ be a
$\tau$-generating set with cardinal $\tau(\frak A_c)$, and
$f_c:B_c\to\lambda$ an injection.   Set

\Centerline{$b_{\xi}=\sup\{c:c\in C,\,\xi\in f(c)\}$,}

\Centerline{$b'_{\xi}=\sup\{b:$ there is some $c\in C$ such that
$b\in B_c$ and $f_c(b)=\xi\}$}

\noindent for $\xi<\lambda$.   Set
$B=\{b_{\xi}:\xi<\lambda\}\cup\{b'_{\xi}:\xi<\lambda\}$ if $\lambda$ is
infinite, $\{b_{\xi}:\xi<\lambda\}$ if $\lambda$ is finite;  then
$\#(B)\le\lambda$.   Note that if $c\in C$ and $b\in B_c$ there is a
$b'\in B$ such that $b=b'\Bcap c$.   \Prf\ Since $B_c\ne\emptyset$,
$\tau(\frak A_c)>0$;  but this means that $\tau(\frak A_c)$ is infinite
(see 331H) so $\lambda$ is infinite and $b'_{\xi}\in B$, where
$\xi=f_c(b)$;  now $b=b'_{\xi}\Bcap c$.\   \Qed

Let $\frak B$ be the
closed subalgebra of $\frak A$ generated by $B$.   Then
$C\subseteq\frak B$.   \Prf\ For $c\in C$, we surely have
$c\Bsubseteq b_{\xi}$ if
$\xi\in f(c)$;  but also, because $C$ is disjoint, $c\Bcap b_{\xi}=0$ if
$\xi\in\lambda\setminus f(c)$.   Consequently

\Centerline{$c^*
=\inf_{\xi\in f(c)}b_{\xi}
  \Bcap\inf_{\xi\in\lambda\setminus f(c)}(1\Bsetminus b_{\xi})$}

\noindent includes $c$.   On the other hand, if $d$ is any other member
of $C$, there is some $\xi\in f(c)\symmdiff f(d)$, so that

\Centerline{$d^*\Bcap c^*\Bsubseteq b_{\xi}\Bcap(1\Bsetminus
b_{\xi})=0$.}

\noindent Since $\sup C=1$, it follows that $c=c^*$;  but $c^*\in\frak
B$, so $c\in\frak B$.\  \Qed

For any $c\in C$, look at $\{b\Bcap c:b\in\frak B\}\subseteq\frak B$.
This is a closed subalgebra of $\frak A_c$
(314F(a-i)) including $B_c$, so must be the whole of $\frak A_c$.   Thus
$\frak A_c\subseteq\frak B$ for every $c\in C$.   But $\sup C=1$, so
$a=\sup_{c\in C}a\Bcap c\in\frak B$ for every $a\in\frak A$, and $\frak
A=\frak B$.   Consequently $\tau(\frak A)\le\#(B)\le\lambda$, and
$\tau(\frak A)=\lambda$.
}%end of proof of 332S

\leader{332T}{Proposition} Let $(\frak A,\bar\mu)$ be a localizable
measure algebra and $\frak B$ a closed subalgebra of $\frak A$.   Then

(a) there is a function $\bar\nu:\frak B\to[0,\infty]$ such that
$(\frak B,\bar\nu)$ is a localizable measure algebra;

(b) $\tau(\frak B)\le\tau(\frak A)$.

\proof{{\bf (a)} Let $D$ be the set of those $b\in\frak B$ such that the
principal ideal $\frak B_b$ has Maharam type at most $\tau(\frak A)$ and
is a totally finite measure algebra when endowed with an appropriate
measure.   Then $D$ is order-dense in $\frak B$.   \Prf\ Take any
non-zero $b_0\in\frak B$.   Then there is an $a\in\frak A$ such that
$a\Bsubseteq b_0$ and $0<\bar\mu a<\infty$.   Set
$c=\upr(a,\frak B)=\min\{b:b\in\frak B,\,a\Bsubseteq b\}$;
then $c\in\frak B$ and $a\Bsubseteq c\Bsubseteq b_0$.
If $0\ne b\in\frak B_c$, then $c\Bsetminus b$ belongs to $\frak B$ and is properly
included in $c$, so cannot include $a$;  accordingly
$a\Bcap b\ne 0$.    For $b\in\frak B_c$, set
$\bar\nu b=\bar\mu(a\Bcap b)$.   Because the map $b\mapsto a\Bcap b$ is an injective
order-continuous Boolean homomorphism, $\bar\nu$ is countably additive
and strictly positive, that is, $(\frak B_c,\bar\nu)$ is a measure
algebra.  It is totally finite because $\bar\nu c=\bar\mu a<\infty$.

Let $d\in\frak B_c\setminus\{0\}$ be such that $\frak B_d$ is \Mth;
suppose that its Maharam type is $\kappa$.   The map
$b\mapsto b\Bcap a$ is a measure-preserving Boolean homomorphism from
$\frak B_d$ to $\frak A_{a\Bcap d}$, so by 332O $\frak A_{a\Bcap d}$
must have a non-zero Maharam-type-$\kappa'$ component for some
$\kappa'\ge\kappa$;  but this means that

\Centerline{$\tau(\frak B_d)\le\kappa\le\kappa'
\le\tau(\frak A_{a\Bcap d})\le\tau(\frak A)$.}

\noindent Thus $d\in D$, while $0\ne d\Bsubseteq c\Bsubseteq b_0$.   As
$b_0$ is arbitrary, $D$ is order-dense.\  \Qed

Accordingly there is a partition of unity $C$ in $\frak B$ such that
$C\subseteq D$.   For each $c\in C$ we have a functional
$\bar\nu_c$ such that $(\frak B_c,\bar\nu_c)$ is a totally finite
measure algebra with Maharam type at most $\tau(\frak A)$;  define
$\bar\nu:\frak B\to[0,\infty]$ by setting
$\bar\nu b=\sum_{c\in C}\bar\nu_c(b\Bcap c)$ for every $b\in\frak B$.
It is easy to check
that $(\frak B,\bar\nu)$ is a measure algebra (compare 322La);  it is
localizable because $\frak B$ (being
order-closed in a Dedekind complete partially ordered set) is Dedekind
complete.

\medskip

{\bf (b)} The construction above ensures that every homogeneous
principal ideal of $\frak B$ can have Maharam type at most
$\tau(\frak A)$, since it must share a principal ideal with some $\frak B_c$ for
$c\in C$.   Moreover, any disjoint set in $\frak B$ is also a disjoint
set in $\frak A$, so $c(\frak B)\le c(\frak A)$.   So 332S tells us that
$\tau(\frak B)\le\tau(\frak A)$.
}%end of proof of 332T

\cmmnt{\medskip

\noindent{\bf Remark} I think the only direct appeal I shall make to
this result will be when $(\frak A,\bar\mu)$ is a probability algebra,
in which case (a) above becomes trivial, and the proof of (b) can be
shortened to some extent, though I think we still need some of the ideas
of 332S.
}%end of comment

%332L out of order

\exercises{
\leader{332X}{Basic exercises (a)} Let $\frak A$ be a Dedekind complete
Boolean algebra.   Show that it is isomorphic to a simple
product of \Mth\ Boolean algebras.
%332A

\header{332Xb}{\bf (b)} Let $\frak A$ be a Boolean algebra of finite
cellularity.   Show that $\frak A$ is purely atomic.
%332D

\header{332Xc}{\bf (c)} Let $\frak A$ be a purely atomic Boolean
algebra.   Show that $c(\frak A)$ is the number of atoms in $\frak A$.
%332D

\header{332Xd}{\bf (d)} Let $\frak A$ be any Boolean algebra, and $Z$
its Stone space.   Show that $c(\frak A)$ is equal to

\Centerline{$c(Z)=\sup\{\#(\Cal G):G$ is a disjoint family of non-empty
open subsets of $Z\}$,}

\noindent the {\bf cellularity} of the topological space $Z$.
%332D

\header{332Xe}{\bf (e)} Let $X$ be a topological space, and $\RO(X)$
its regular open algebra.   Show that $c(\RO(X))=c(X)$ as defined in
332Xd.
%332Xe, 332D

\header{332Xf}{\bf (f)} Let $\frak A$ be a Boolean algebra, and
$\frak B$ any subalgebra of $\frak A$.   Show that
$c(\frak B)\le c(\frak A)$.
%332D

\spheader 332Xg Let $\langle\frak A_i\rangle_{i\in I}$ be any
family of Boolean algebras, with simple product $\frak A$.   Show that
the cellularity of $\frak A$ is at most
$\max(\omega,\#(I),\sup_{i\in I}c(\frak A_i))$.   Devise an
elegant expression of a necessary and sufficient condition for equality.
%332D

\spheader 332Xh Let $\frak A$ be any Boolean algebra, and
$a\in\frak A$;  let $\frak A_a$ be the principal ideal generated by $a$.
Show that $c(\frak A_a)\le c(\frak A)$.
%332D

\header{332Xi}{\bf (i)} Let $(\frak A,\bar\mu)$ be a semi-finite measure
algebra.   Show that it has a partition of unity with cardinal $c(\frak
A)$.
%332F

\spheader 332Xj Let $(\frak A,\bar\mu)$ and $(\frak B,\bar\nu)$ be
localizable measure algebras.   For each cardinal $\kappa$ let
$e_{\kappa}$, $f_{\kappa}$ be their Maharam-type-$\kappa$ components,
and $\frak A_{e_{\kappa}}$, $\frak B_{f_{\kappa}}$ the corresponding
principal ideals.   Show that $\frak A$ and $\frak B$ are isomorphic, as
Boolean algebras, iff $c(\frak A_{e_{\kappa}})=c(\frak B_{f_{\kappa}})$
for every $\kappa$.
%332J

\spheader 332Xk Let $\zeta$ be an ordinal, and
$\langle\alpha_{\xi}\rangle_{\xi<\zeta}$,
$\langle\beta_{\xi}\rangle_{\xi<\zeta}$ two families of non-negative
real numbers such that $\sum_{\theta\le\xi<\zeta}\alpha_{\xi}
\le\sum_{\theta\le\eta<\zeta}\beta_{\eta}<\infty$ for every
$\theta\le\zeta$.   Show that there is a family
$\langle\gamma_{\xi\eta}\rangle_{\xi\le\eta<\zeta}$ of non-negative real
numbers such that
$\alpha_{\xi}=\sum_{\xi\le\eta<\zeta}\gamma_{\xi\eta}$ for every
$\xi<\zeta$ and
$\beta_{\eta}\ge\sum_{\xi\le\eta}\gamma_{\xi\eta}$ for every
$\eta<\zeta$.   (If only finitely many of the $\alpha_{\xi}$,
$\beta_{\xi}$ are non-zero, this is an easy special case of the max-flow
min-cut theorem;  see {\smc Bollob\'as 79}, {\S}III.1 or 
{\smc Anderson 87}, 12.3.1;
there is a statement of the theorem in 4A4N in the next volume.)
Show that the $\gamma_{\xi\eta}$ can be chosen in such a way that if
$\xi<\xi'\le\eta'<\eta$ then at least one of $\gamma_{\xi\eta}$,
$\gamma_{\xi'\eta'}$ is zero.
%for 332Xl

\spheader 332Xl Use 332Xk and 332M to give another proof of 332P.
%332Xk, 332P

\spheader 332Xm For each cardinal $\kappa$, write
$(\frak B_{\kappa},\bar\nu_{\kappa})$ for the measure algebra of the usual
measure on $\{0,1\}^{\kappa}$.   Let $(\frak A,\bar\mu)$ be the simple
product of $\sequencen{(\frak B_{\omega_n},\bar\nu_{\omega_n})}$ and
$(\frak B,\bar\nu)$ the simple product of $(\frak A,\bar\mu)$ with
$(\frak B_{\omega_{\omega}},\bar\nu_{\omega_{\omega}})$.   (See 3A1E if
you are puzzled by the names $\omega_n$, $\omega_{\omega}$.)   Show that there is a
measure-preserving Boolean homomorphism from $\frak A$ to $\frak B$, but
that no such homomorphism can be order-continuous.
%332P

\spheader 332Xn For each cardinal $\kappa$, write
$(\frak B_{\kappa},\bar\nu_{\kappa})$ for the measure algebra of the usual
measure on $\{0,1\}^{\kappa}$.  Let $(\frak A,\bar\mu)$ be the simple product of
$\sequencen{(\frak B_{\kappa_{n}},\bar\nu_{\kappa_{n}})}$ and
$(\frak B,\bar\nu)$ the simple product of
$\sequencen{(\frak B_{\lambda_{n}},\bar\nu_{\lambda_{n}})}$, where
$\kappa_n=\omega$ for
even $n$, $\omega_n$ for odd $n$, while $\lambda_n=\omega$ for odd $n$,
$\omega_n$ for even $n$.   Show that there are
order-continuous measure-preserving Boolean homomorphisms from $\frak A$
to $\frak B$ and from $\frak B$ to $\frak A$, but that these two measure
algebras are not isomorphic.
%332Q, 332Xm

\spheader 332Xo Let $\frak C$ be a Boolean algebra.    Show that
the following are equiveridical:  (i) $\frak C$ is isomorphic (as Boolean
algebra) to a closed subalgebra of a localizable measure algebra;  (ii)
there is a $\bar\mu$ such that $(\frak C,\bar\mu)$ is itself a
localizable measure algebra;   (iii) $\frak C$ is Dedekind complete and
for every non-zero $c\in\frak C$ there is a completely additive
real-valued functional
$\nu$ on $\frak C$ such that $\nu c\ne 0$.   ({\it Hint for
(iii)$\Rightarrow$(ii)\/}:  show that the set of supports of
non-negative
completely additive functionals is order-dense in $\frak C$, so includes
a partition of unity.)
%332T

\leader{332Y}{Further exercises (a)}
%\spheader 332Ya
Let $(\frak A,\bar\mu)$ and
$(\frak B,\bar\nu)$ be atomless localizable measure algebras.   For each
infinite cardinal $\kappa$ let $e_{\kappa}$, $f_{\kappa}$ be their
Maharam-type-$\kappa$ components.   Show that the following are
equiveridical:  (i) $(\frak A,\bar\mu)$ is isomorphic to a closed
subalgebra of a principal ideal of $(\frak B,\bar\nu)$;
(ii) for every cardinal $\lambda$,
the magnitude of $\sup_{\kappa\ge\lambda}e_{\kappa}$ is not greater than
the magnitude of $\sup_{\kappa\ge\lambda}f_{\kappa}$.
%332P

\header{332Yb}{\bf (b)} Let $(\frak A,\bar\mu)$ and $(\frak B,\bar\nu)$
be any semi-finite measure algebras, and $(\widehat{\frak A},\hat\mu)$,
$(\widehat{\frak B},\hat\nu)$ their localizations (322P-322Q).   Let
$\langle e_i\rangle_{i\in I}$, $\langle f_j\rangle_{j\in J}$ be
partitions of
unity in $\frak A$, $\frak B$ respectively into elements of finite
measure generating homogeneous principal ideals $\frak A_{e_i}$,
$\frak B_{f_j}$.   For each infinite cardinal $\kappa$ set
$I_{\kappa}=\{i:\tau(\frak A_{e_i})=\kappa\}$,
$J_{\kappa}=\{j:\tau(\frak B_{f_j})=\kappa\}$;  for
$\gamma\in\ooint{0,\infty}$, set $I_{\gamma}=\{i:e_i$ is an atom,
$\bar\mu e_i=\gamma\}$, $J_{\gamma}=\{j:f_j$ is an atom,
$\bar\nu f_j=\gamma\}$.   Show that $(\widehat{\frak A},\hat\mu)$ and
$(\widehat{\frak B},\hat\nu)$ are isomorphic iff for each $u$, either
$\sum_{i\in I_u}\bar\mu e_i=\sum_{j\in J_u}\bar\nu f_j<\infty$
or $\sum_{i\in I_u}\bar\mu e_i=\sum_{j\in J_u}\bar\nu f_j=\infty$ and
$\#(I_u)=\#(J_u)$.

\spheader 332Yc Let $(\frak A,\bar\mu)$ and $(\frak B,\bar\nu)$ be
non-zero localizable measure algebras;  let $e_{\kappa}$, $f_{\kappa}$
be their Maharam-type-$\kappa$ components.   Show that the following are
equiveridical:  (i) $\frak A$ is isomorphic, as Boolean algebra, to an
order-closed subalgebra of a principal ideal of $\frak B$;   (ii)
$c(\frak A^*_{\lambda})\le c(\frak B^*_{\lambda})$ for every cardinal
$\lambda$, where $\frak A^*_{\lambda}$, $\frak B^*_{\lambda}$ are the
principal ideals generated by $\sup_{\kappa\ge\lambda}e_{\kappa}$ and
$\sup_{\kappa\ge\lambda}f_{\kappa}$ respectively.

}%end of exercises

\cmmnt{
\Notesheader{332}
Maharam's theorem tells us that all localizable measure algebras -- in
particular, all $\sigma$-finite measure algebras -- can be obtained from
the basic algebra $\frak A=\{0,a,1\Bsetminus a,1\}$, with $\bar\mu
a=\bar\mu(1\Bsetminus a)=\bover12$, by combining the constructions of
probability algebra free products, scalar multiples of measures and
simple products.   But what is much more important is the fact that we
get a description of our measure algebras in terms sufficiently explicit
to make a very wide variety of questions resolvable.   The description I
offer in 332J hinges on the complementary concepts of `Maharam type' and
`magnitude'.   If you like, the magnitude of a measure algebra is a
measure of its width, while its Maharam type is a measure of its depth.
The latter is more important just because, for localizable algebras, we
have such a simple decomposition into algebras of finite magnitude.   Of
course there is a good deal of scope for further complications if we
seek to consider non-localizable semi-finite algebras.   For these, the
natural starting point is a proper description of their localizations,
which is not difficult (332Yb).

Observe that 332C gives a representation of a localizable measure
algebra as the measure algebra of a measure space which is completely
different from the Stone representation in 321K.   It is less canonical
(since there is a degree of choice about the partition
$\langle e_i\rangle_{i\in I}$) but very much more informative, since the
$\kappa_i$, $\gamma_i$ carry enough information to identify the measure
algebra up to isomorphism (332K).

`Cellularity' is the second cardinal function I have introduced in this
chapter.   It refers as much to topological spaces as to Boolean
algebras (see 332Xd-332Xe).   There is an interesting question in this
context.   If $\frak A$ is an arbitrary Boolean algebra, is there
necessarily a disjoint set in $\frak A$ with cardinal $c(\frak A)$?   This
is believed to be undecidable from the ordinary axioms of set theory
(including the axiom of choice);  see the `Erd\H{o}s-Tarski theorem' in
Volume 5.
%511Db 513B 513Ya 514Nc
But for semi-finite measure algebras we have a definite answer (332F).

Maharam's classification not only describes the isomorphism classes of
localizable measure algebras, but also tells us when to expect Boolean
homomorphisms between them (332P, 332Yc).   I have given 332P only for
atomless totally finite measure algebras because the non-totally-finite
case (332Ya, 332Yc) seems to require a new idea, while atoms introduce
combinatorial complications.

I offer 332T as an example of the kind of result which these methods
make very simple.
It fails for general Boolean algebras;  in fact, there is for any $\kappa$ a
countably $\tau$-generated Dedekind complete Boolean algebra $\frak A$ with
cellularity $\kappa$ (514Xi in Volume 5, or {\smc Koppelberg 89}, 13.1),
so that $\Cal P\kappa$ is isomorphic to an order-closed subalgebra of
$\frak A$,
and if $\kappa>\frak c$ then $\tau(\Cal P\kappa)>\omega$ (332R).

For totally finite measure algebras we have a kind of weak
Schr\"oder-Bernstein theorem:  if we have two of them, each isomorphic
to a closed subalgebra of the other, they are isomorphic (332Q).   This
fails for $\sigma$-finite algebras (332Xn).   I call it a `weak'
Schr\"oder-Bernstein theorem because it is not clear how to build the
isomorphism from the two injections;  `strong'
Schr\"oder-Bernstein theorems include definite recipes for constructing
the isomorphisms declared to exist (see, for instance, 344D below).
}%end of notes

\discrpage


\frfilename{mt333.tex}
\versiondate{27.6.08}
\copyrightdate{1996}

\def\chaptername{Maharam's theorem}
\def\sectionname{Closed subalgebras}

\newsection{333}

Proposition 332P tells us, in effect, which totally finite measure
algebras can be
embedded as closed subalgebras of each other.   Similar techniques make
it possible to describe the possible forms of such embeddings.   In this
section I give the fundamental theorems on extension of
measure-preserving homomorphisms from closed subalgebras (333C, 333D);
these rely on the concept of `relative Maharam type' (333A).   I go on
to describe possible canonical forms for structures
$(\frak A,\bar\mu,\frak C)$, where $(\frak A,\bar\mu)$ is a totally
finite measure algebra and
$\frak C$ is a closed subalgebra of $\frak A$ (333K, 333N).   I end the
section with a description of fixed-point subalgebras (333R).

\leader{333A}{Definitions (a)} Let $\frak A$ be a Boolean algebra and
$\frak C$ a subalgebra of $\frak A$.   The {\bf relative Maharam type of
$\frak A$ over $\frak C$}, $\tau_{\frak C}(\frak A)$, is the smallest
cardinal of any set $A\subseteq\frak A$ such that
$A\cup\frak C\,\,\tau$-generates $\frak A$.

\spheader 333Ab In this section, I will regularly use the following
notation:
if $\frak A$ is a Boolean algebra, $\frak C$ is a subalgebra of
$\frak A$, and $a\in\frak A$, then I will write $\frak C_a$ for
$\{c\Bcap a:c\in\frak C\}$.   Observe that $\frak C_a$ is a subalgebra of
the principal ideal $\frak A_a$\prooflet{ (because
$c\mapsto c\Bcap a:\frak C\to\frak A_a$ is a Boolean homomorphism);
it is included in $\frak C$ iff $a\in\frak C$}.

\spheader 333Ac\cmmnt{ Still taking $\frak A$ to be a Boolean algebra and
$\frak C$
to be a subalgebra of $\frak A$,} I will say that an element $a$ of
$\frak A$ is {\bf relatively \Mth\ over $\frak C$} if
$\tau_{\frak C_b}(\frak A_b)=\tau_{\frak C_a}(\frak A_a)$ for every
non-zero $b\Bsubseteq a$.

\spheader 333Ad If $\kappa$ is a cardinal\cmmnt{ which
is either infinite or zero},
I will write $(\frak B_{\kappa},\bar\nu_{\kappa})$ for the measure algebra
of the usual measure $\nu_{\kappa}$ on $\{0,1\}^{\kappa}$.   \cmmnt{I
hope that there will be no confusion between this notation and the use, in
333C-333F, %333C 333D 333E 333F
of the formula $\frak B_b$ for the principal ideal generated by $b$
in an arbitrary Boolean algebra $\frak B$.}

\leader{333B}{}\cmmnt{ Evidently this is a generalization of the
ordinary concept of Maharam type as used in \S\S331-332;  if
$\frak C=\{0,1\}$ then $\tau_{\frak C}(\frak A)=\tau(\frak A)$.   The
first step is naturally to check the results corresponding to 331H.

\medskip

\noindent}{\bf Lemma} Let $\frak A$ be a Boolean algebra and $\frak C$ a
subalgebra of $\frak A$.

(a) If $a\Bsubseteq b$ in $\frak A$, then
$\tau_{\frak C_a}(\frak A_a)\le\tau_{\frak C_b}(\frak A_b)$.
In particular, $\tau_{\frak C_a}(\frak A_a)\le\tau_{\frak C}(\frak A)$
for every $a\in\frak A$.

(b) The set $\{a:a\in\frak A$ is relatively \Mth\ over
$\frak C\}$ is order-dense in $\frak A$.

(c) If $\frak A$ is Dedekind complete and $\frak C$ is order-closed in
$\frak A$, then $\frak C_a$ is order-closed in $\frak A_a$.

(d) If $a\in\frak A$ is relatively \Mth\ over $\frak C$
then either $\frak A_a=\frak C_a$, so that
$\tau_{\frak C_a}(\frak A_a)=0$ and $a$ is a relative atom of $\frak A$
over $\frak C$\cmmnt{ (definition:  331A)}, or
$\tau_{\frak C_a}(\frak A_a)\ge\omega$.

(e) If $\frak D$ is another subalgebra of $\frak A$ and
$\frak D\subseteq\frak C$, then

\Centerline{$\tau(\frak A_a)=\tau_{\{0,a\}}(\frak A_a)
\ge\tau_{\frak D_a}(\frak A_a)
\ge\tau_{\frak C_a}(\frak A_a)
\ge\tau_{\frak A_a}(\frak A_a)=0$}

\noindent for every $a\in\frak A$.

\proof{{\bf (a)} Let $D\subseteq\frak A_b$ be a set with cardinal
$\tau_{\frak C_b}(\frak A_b)$ such that
$D\cup\frak C_b\,\,\tau$-generates $\frak A_b$.   Set
$D'=\{d\Bcap a:d\in D\}$.   Then $D'\cup\frak C_a\,\,\tau$-generates
$\frak A_a$.   \Prf\ Apply 313Mc to the map
$d\mapsto d\Bcap a:\frak A_b\to\frak A_a$, as in 331Hc.\ \QeD\
Consequently

\Centerline{$\tau_{\frak C_a}(\frak A_a)\le\#(D')\le\#(D)
=\tau_{\frak C_b}(\frak A_b)$,}

\noindent as claimed.    Setting $b=1$ we get
$\tau_{\frak C_a}(\frak A_a)\le\tau_{\frak C}(\frak A)$.

\medskip

{\bf (b)} Just as in the proof of 332A, given
$b\in\frak A\setminus\{0\}$, there is an $a\in\frak A_b\setminus\{0\}$
minimising $\tau_{\frak C_a}(\frak A_a)$, and this $a$ must be
relatively \Mth\ over $\frak C$.

\medskip

{\bf (c)} $\frak C_a$ is the image of the Dedekind complete Boolean
algebra $\frak C$ under the
order-continuous Boolean homomorphism $c\mapsto c\Bcap a$, so must be
order-closed (314F(a-i)).

\medskip

{\bf (d)} Suppose that $\tau_{\frak C_a}(\frak A_a)$ is finite.   Let
$D\subseteq\frak A_a$ be a finite set such that $D\cup\frak
C_a\,\,\tau$-generates $\frak A_a$.   Then there is a non-zero
$b\in\frak A_a$ such that $b\Bcap d$ is either $0$ or $b$ for every
$d\in D$.   But this means that $\frak C_b=\{d\Bcap b:d\in D\cup\frak
C_a\}$, which $\tau$-generates $\frak A_b$;  so that $\tau_{\frak
C_b}(\frak A_b)=0$.   Since $a$ is relatively \Mth\ over
$\frak C$, $\tau_{\frak C_a}(\frak A_a)$ must be zero, that is, $\frak
A_a=\frak C_a$.

\medskip

{\bf (e)} The middle inequality is true just because $\frak A_a$ will be
$\tau$-generated by $D\cup\frak C_a$ whenever it is $\tau$-generated by
$D\cup\frak D_a$.   The neighbouring inequalities are special cases of
the middle one, and the outer equalities are elementary.
}%end of proof of 333B

\leader{333C}{Theorem} Let $(\frak A,\bar\mu)$ and $(\frak B,\bar\nu)$
be totally finite measure algebras, and $\frak C$ a closed subalgebra of
$\frak A$.   Let $\phi:\frak C\to\frak B$ be a measure-preserving
Boolean homomorphism.

(a) If\cmmnt{, in the notation of 333A,}
$\tau_{\frak C}(\frak A)\le\tau_{\phi[\frak C]_b}(\frak B_b)$
for every non-zero
$b\in\frak B$, there is a measure-preserving Boolean homomorphism
$\pi:\frak A\to\frak B$ extending $\phi$.

(b) If $\tau_{\frak C_a}(\frak A_a)=\tau_{\phi[\frak C]_b}(\frak B_b)$
for every non-zero $a\in\frak A$, $b\in\frak B$, then there is a measure
algebra isomorphism $\pi:\frak A\to\frak B$ extending $\phi$.

\proof{ In both parts, the idea is to use the technique of the proof of
331I to construct $\pi$ as the last of an increasing family
$\langle\pi_{\xi}\rangle_{\xi\le\kappa}$ of measure-preserving
homomorphisms from closed subalgebras $\frak C_{\xi}$ of $\frak A$,
where $\kappa=\tau_{\frak C}(\frak A)$.   Let
$\langle a_{\xi}\rangle_{\xi<\kappa}$ be a family in $\frak A$ such that
$\frak C\cup\{a_{\xi}:\xi<\kappa\}\,\,\tau$-generates $\frak A$.   Write
$\frak D$ for $\phi[\frak C]$;  remember that $\frak D$ is a closed
subalgebra of $\frak B$ (324L).

\medskip

{\bf (a)(i)} In this case, we can describe the $\frak C_{\xi}$
immediately;   $\frak C_{\xi}$ will be the closed subalgebra of
$\frak A$ generated by $\frak C\cup\{a_{\eta}:\eta<\xi\}$.   The
induction starts with $\frak C_0=\frak C$, $\pi_0=\phi$.

\medskip

\quad{\bf (ii)} For the inductive step to a successor ordinal $\xi+1$,
where $\xi<\kappa$, suppose that $\frak C_{\xi}$ and $\pi_{\xi}$ have
been defined.   Take any non-zero $b\in\frak B$.   We are supposing that
$\tau_{\frak D_b}(\frak B_b)\ge\kappa>\#(\xi)$, so $\frak B_b$ cannot be
$\tau$-generated by

\Centerline{$D=\frak D_b\cup\{b\Bcap\pi_{\xi}a_{\eta}:\eta<\xi\}
=\pi_{\xi}[\frak C]_b\cup\{b\Bcap\pi_{\xi}a_{\eta}:\eta<\xi\}
=\psi[\frak C\cup\{a_{\eta}:\eta<\xi\}]$,}

\noindent writing $\psi c=b\Bcap\pi_{\xi}c$ for $c\in\frak C_{\xi}$.
As $\psi$ is order-continuous, $\psi[\frak C_{\xi}]$ is precisely the
closed subalgebra of $\frak B_b$ generated by $D$ (314H), and is
therefore not the whole of $\frak B_b$.

But this means that
$\frak B_b\ne\{b\Bcap \pi_{\xi}c:c\in\frak C_{\xi}\}$.   As $b$ is
arbitrary, $\pi_{\xi}$ satisfies the conditions
of 331D, and has an extension to a measure-preserving Boolean
homomorphism $\pi_{\xi+1}:\frak C_{\xi+1}\to\frak B$, since
$\frak C_{\xi+1}$ is just the closed subalgebra of $\frak A$ generated
by $\frak C\cup\{a_{\xi}\}$.

\medskip

\quad{\bf (iii)} For the inductive step to a non-zero limit ordinal
$\xi\le\kappa$, we can argue exactly as in part (d) of the proof of
331I;  $\frak C_{\xi}$ will be the metric closure of
$\frak C^*_{\xi}=\bigcup_{\eta<\xi}\frak C_{\eta}$, so we can take
$\pi_{\xi}:\frak C_{\xi}\to\frak B$ to be the unique measure-preserving
homomorphism extending $\pi^*_{\xi}=\bigcup_{\eta<\xi}\pi_{\eta}$.

Thus the induction proceeds, and evidently $\pi=\pi_{\kappa}$ will be a
measure-preserving homomorphism from $\frak A$ to $\frak B$ extending
$\phi$.

\medskip

{\bf (b)} (This is rather closer to the proof of 331I, being indeed a
direct generalization of it.)   Observe that the hypothesis (b) implies
that $1_{\frak A}$ is relatively \Mth\ over $\frak C$;  so
either $\kappa=0$, in which case $\frak A=\frak C$, $\frak B=\phi[\frak C]$
and the result is trivial, or $\kappa\ge\omega$, by 333Bd.   Let us
therefore take it that $\kappa$ is infinite.

We are supposing, among other things,
that $\tau_{\frak D}(\frak B)=\kappa$;  let
$\langle b_{\xi}\rangle_{\xi<\kappa}$ be a family in $\frak B$ such that
$\frak B$ is $\tau$-generated by $\frak D\cup\{b_{\xi}:\xi<\kappa\}$.
This time, as in 331I, we shall have to choose further families
$\langle a'_{\xi}\rangle_{\xi<\kappa}$ and
$\langle b'_{\xi}\rangle_{\xi<\kappa}$, and

\inset{$\frak C_{\xi}$ will be the closed subalgebra of $\frak A$
generated by

\Centerline{$\frak
C\cup\{a_{\eta}:\eta<\xi\}\cup\{a'_{\eta}:\eta<\xi\}$,}
}

\inset{$\frak D_{\xi}$ will be the closed subalgebra of $\frak B$
generated by

\Centerline{$\frak
D\cup\{b_{\eta}:\eta<\xi\}\cup\{b'_{\eta}:\eta<\xi\}$,}
}

\inset{$\pi_{\xi}:\frak C_{\xi}\to\frak D_{\xi}$ will be a
measure-preserving homomorphism.
}

\noindent The induction will start with $\frak C_0=\frak C$, $\frak
D_0=\frak D$ and $\pi_0=\phi$, as in (a).

\medskip

\quad{\bf (i)} For the inductive step to a successor ordinal $\xi+1$,
where $\xi<\kappa$, suppose that $\frak C_{\xi}$, $\frak D_{\xi}$ and
$\pi_{\xi}$ have been defined.

\medskip

\qquad\grheada\ Let $b\in\frak B\setminus\{0\}$.  Because

\Centerline{$\tau_{\frak D_b}(\frak B_b)
=\kappa>\#(\{b_{\eta}:\eta<\xi\}\cup\{b'_{\eta}:\eta<\xi\})$,}

\noindent $\frak B_b$ cannot be $\tau$-generated by
$\frak D_b\cup\{b\Bcap b_{\eta}:\eta<\xi\}
  \cup\{b\Bcap b'_{\eta}:\eta<\xi\}$, and
cannot be equal to $\{b\Bcap d:d\in\frak D_{\xi}\}$.   As $b$ is
arbitrary, there is an extension of $\pi_{\xi}$ to a
measure-preserving homomorphism $\phi_{\xi}$ from $\frak C'_{\xi}$ to
$\frak B$,
where $\frak C'_{\xi}$ is the closed subalgebra of $\frak A$ generated
by $\frak C\cup\{a_{\eta}:\eta\le\xi\}\cup\{a'_{\eta}:\eta<\xi\}$.
Setting $b'_{\xi}=\phi_{\xi}(a_{\xi})$, the image
$\frak D'_{\xi}=\phi_{\xi}[\frak C'_{\xi}]$ will be the closed subalgebra
of $\frak B$ generated by
$\frak D\cup\{b_{\eta}:\eta<\xi\}\cup\{b'_{\eta}:\eta\le \xi\}$.

\medskip

\qquad\grheadb\ Next, as in 331I, we must repeat the argument of
($\alpha$), applying it now to
$\phi_{\xi}^{-1}:\frak D_{\xi}\to\frak A$.   If
$a\in\frak A\setminus\{0\}$,

\Centerline{$\tau_{\frak C_a}(\frak A_a)
=\kappa>\#(\{a_{\eta}:\eta\le\xi\}\cup\{a'_{\eta}:\eta<\xi\})$,}

\noindent so that $\frak A_a$ cannot be
$\{a\Bcap c:c\in\frak C'_{\xi}\}$.   As $a$ is arbitrary,
$\phi_{\xi}^{-1}$ has an extension
to a measure-preserving homomorphism
$\psi_{\xi}:\frak D_{\xi+1}\to\frak C_{\xi+1}$, where $\frak D_{\xi+1}$
is the subalgebra of $\frak B$
generated by $\frak D'_{\xi}\cup\{b_{\xi}\}$, that is, the closed
subalgebra of $\frak B$ generated by
$\frak D\cup\{b_{\eta}:\eta\le\xi\}\cup\{b'_{\eta}:\eta\le\xi\}$, and
$\frak C_{\xi+1}$ is the subalgebra of $\frak A$ generated by
$\frak C'_{\xi}\cup\{a'_{\xi}\}$, setting
$a'_{\xi}=\psi_{\xi}(b_{\xi})$.

We can therefore take
$\pi_{\xi+1}=\psi_{\xi}^{-1}:\frak C_{\xi+1}\to\frak D_{\xi+1}$, as in
331I.

\medskip

\quad{\bf (ii)} The inductive step to a non-zero limit ordinal
$\xi\le\kappa$ is exactly the same as in (a) above or in 331I;
$\frak C_{\xi}$ is the metric closure of
$\frak C^*_{\xi}=\bigcup_{\eta<\xi}\frak C_{\eta}$,
$\frak D_{\xi}$ is the metric closure of $\frak
D^*_{\xi}=\bigcup_{\eta<\xi}\frak D_{\eta}$, and $\pi_{\xi}$ is the
unique measure-preserving homomorphism from $\frak C_{\xi}$ to $\frak
D_{\xi}$ extending every $\pi_{\eta}$ for $\eta<\xi$.

\medskip

\quad{\bf (iii)} The induction stops, as before, with
$\pi=\pi_{\kappa}:\frak C_{\kappa}\to\frak D_{\kappa}$, where $\frak
C_{\kappa}=\frak A$, $\frak D_{\kappa}=\frak B$.
}%end of proof of 333C

\leader{333D}{Corollary} Let $(\frak A,\bar\mu)$ and $(\frak B,\bar\nu)$
be
totally finite measure algebras and $\frak C$ a closed subalgebra of
$\frak A$.   Suppose that

\Centerline{$\tau(\frak C)<\max(\omega,\tau(\frak A))
\le\min\{\tau(\frak B_b):b\in\frak B\setminus\{0\}\}$.}

\noindent Then any measure-preserving Boolean homomorphism
$\phi:\frak C\to\frak B$ can be extended to a measure-preserving Boolean
homomorphism $\pi:\frak A\to\frak B$.

\proof{ Set $\kappa=\min\{\tau(\frak B_b):b\in\frak B\setminus\{0\}\}$.
Then for any non-zero $b\in\frak B$,

\Centerline{$\tau_{\phi[\frak C]_b}(\frak B_b)\ge\kappa$.}

\noindent\Prf\ There is a set $C\subseteq\frak C$, with cardinal
$\tau(\frak C)$, which $\tau$-generates $\frak C$, so that
$C'=\{b\Bcap\phi c:c\in C\}\,\,\tau$-generates $\phi[\frak C]_b$.   Now
there is a set $D\subseteq \frak B_b$, with cardinal $\tau_{\phi[\frak
C]_b}(\frak B_b)$, such that $\phi[\frak C]_b\cup D\,\,\tau$-generates
$\frak B_b$.   In this case $C'\cup D$ must $\tau$-generate $\frak B_b$,
so
$\kappa\le\#(C'\cup D)$.   But $\#(C')\le\#(C)<\kappa$ and $\kappa$ is
infinite, so we must have $\#(D)\ge\kappa$, as claimed.\ \Qed

On the other hand, $\tau_{\frak C}(\frak A)\le\tau(\frak A)\le\kappa$.
So we can apply 333Ca to give the result.
}%end of proof of 333D

\vleader{60pt}{333E}{Theorem} Let $(\frak C,\bar\mu)$ be a totally finite
measure algebra and $\kappa$ an infinite cardinal.
Let $(\frak A,\bar\lambda)$ be the localizable measure algebra free
product of $(\frak C,\bar\mu)$ and
$(\frak B_{\kappa},\bar\nu_{\kappa})$\cmmnt{ (notation:  333Ad)}, and
$\varepsilon:\frak C\to\frak A$ the corresponding homomorphism.   Then for any
non-zero $a\in\frak A$,

\Centerline{$\tau_{\varepsilon[\frak C]_a}(\frak A_a)=\kappa$\dvro{.}{,}}

\cmmnt{\noindent in the notation of 333A above.}

\proof{ Recall from 325Dd that $\varepsilon[\frak C]$ is a closed
subalgebra of $\frak A$.

\medskip

{\bf (a)} Let $\langle e_{\xi}\rangle_{\xi<\kappa}$ be the standard
generating family in $\frak B_{\kappa}$, corresponding to the sets
$\{x:x\in\{0,1\}^{\kappa}$, $x(\xi)=1\}$.
Let $\varepsilon':\frak B_{\kappa}\to\frak A$ be the
canonical map, and set $e'_{\xi}=\varepsilon'e_{\xi}$ for each $\xi$.

We know that $\{e_{\xi}:\xi<\kappa\}\,\,\tau$-generates
$\frak B_{\kappa}$ (see part (a) of the proof of 331K).   Consequently
$\varepsilon[\frak C]\cup\{e'_{\xi}:\xi<\kappa\}\,\,\tau$-generates
$\frak A$.
\Prf\ Let $\frak A_1$ be the closed subalgebra of $\frak A$ generated by
$\varepsilon[\frak C]\cup\{e'_{\xi}:\xi<\kappa\}$.   Because
$\varepsilon':\frak B_{\kappa}\to\frak A$ is
order-continuous (325Da),
$\varepsilon'[\frak B_{\kappa}]\subseteq\frak A_1$
(313Mb).   But this means that $\frak A_1$ includes
$\varepsilon[\frak  C]\cup\varepsilon'[\frak B_{\kappa}]$
and therefore includes the image of
$\frak C\otimes\frak B_{\kappa}$ in $\frak A$;  because this is
topologically dense in $\frak A$ (325Dc), $\frak A_1=\frak A$, as
claimed.\ \Qed

\medskip

{\bf (b)} It follows that

\Centerline{$\tau_{\varepsilon[\frak C]_a}(\frak A_a)
\le\tau_{\varepsilon[\frak C]}(\frak A)\le\kappa$}

\noindent (333Ba).

\medskip

{\bf (c)} We need to know that if $\xi<\kappa$ and $a$ belongs to the
closed subalgebra $\frak E_{\xi}$ of $\frak A$ generated by
$\varepsilon[\frak C]\cup\{e'_{\eta}:\eta\ne\xi\}$, then
$\bar\lambda(a\Bcap e'_{\xi})=\bover12\bar\lambda a$.   \Prf\ Set

\Centerline{$E=\varepsilon[\frak C]\cup\{e'_{\eta}:\eta\ne\xi\}$, \quad
$F=\{a_0\Bcap\ldots\Bcap a_n:a_0,\ldots,a_n\in E\}$.}

\noindent Then every member of $F$ is expressible in the form

\Centerline{$a=\varepsilon c\Bcap\inf_{\eta\in J}e'_{\eta}$,}

\noindent where $c\in\frak C$ and $J\subseteq\kappa\setminus\{\xi\}$ is
finite.   Now

\Centerline{$\bar\lambda a
=\bar\mu c\cdot\bar\nu(\inf_{\eta\in J}e_{\eta})
=2^{-\#(J)}\bar\mu c$,}

\Centerline{$\bar\lambda(e'_{\xi}\Bcap a)
=\bar\mu c\cdot\bar\nu(e_{\xi}\Bcap\inf_{\eta\in J}e_{\eta})
=2^{-\#(J\cup\{\xi\})}\bar\mu c=\Bover12\bar\lambda a$.}

\noindent Now consider the set

\Centerline{$G=\{a:a\in\frak A,\,\bar\lambda(e_{\xi}\Bcap a)
=\Bover12\bar\lambda a\}$.}

\noindent We have $1_{\frak A}\in F\subseteq G$, and $F$ is closed under
$\Bcap$.   Secondly, if $a$, $a'\in G$ and $a\Bsubseteq a'$, then

\Centerline{$\bar\lambda(e_{\xi}\Bcap(a'\Bsetminus a))
=\bar\lambda(e_{\xi}\Bcap a')-\bar\lambda(e_{\xi}\Bcap a)
=\Bover12\bar\lambda a'-\Bover12\bar\lambda a
=\Bover12\bar\lambda(a'\Bsetminus a)$,}

\noindent so $a'\Bsetminus a\in G$.   Also, if $H\subseteq G$ is
non-empty and upwards-directed,

\Centerline{$\bar\lambda(e_{\xi}\Bcap\sup H)
=\bar\lambda(\sup_{a\in H}e_{\xi}\Bcap a)
=\sup_{a\in H}\bar\lambda(e_{\xi}\Bcap a)
=\sup_{a\in H}\Bover12\bar\lambda a
=\Bover12\bar\lambda(\sup H)$,}

\noindent so $\sup H\in G$.   By the Monotone Class Theorem (313Gc), $G$
includes the order-closed
subalgebra of $\frak D$ generated by $F$.   But this is just
$\frak E_{\xi}$.\ \Qed

\medskip

{\bf (d)}  The next step is to see that
$\tau_{\varepsilon[\frak C]_a}(\frak A_a)>0$.   \Prf\ By (a) and 323J,
$\frak A$ is the metric closure of
the subalgebra $\frak A_0$ generated by
$\varepsilon[\frak C]\cup\{e'_{\eta}:\eta<\kappa\}$, so there must be an
$a_0\in\frak A_0$
such that $\bar\lambda(a_0\Bsymmdiff a)\le\bover14\bar\lambda a$.   Now
there is
a finite $J\subseteq\kappa$ such that $a_0$ belongs to the subalgebra
$\frak A_1$ generated by $\varepsilon[\frak C]\cup\{e'_{\eta}:\eta\in J\}$.
Take any $\xi\in\kappa\setminus J$ (this is where I use the hypothesis
that $\kappa$ is infinite).   If $c\in\frak C$, then by (c) we have

$$\eqalignno{\bar\lambda((a\Bcap\varepsilon c)\Bsymmdiff(a\Bcap e'_{\xi}))
&=\bar\lambda(a\Bcap(\varepsilon c\Bsymmdiff e'_{\xi}))
\ge\bar\lambda(a_0\Bcap(\varepsilon c \Bsymmdiff e'_{\xi}))
 -\bar\lambda(a\Bsymmdiff a_0)\cr
&=\bar\lambda(a_0\Bcap e'_{\xi})+\bar\lambda(a_0\Bcap \varepsilon c)
  -2\bar\lambda(a_0\Bcap \varepsilon c\Bcap e'_{\xi})
  -\bar\lambda(a\Bsymmdiff a_0)\cr
&=\Bover12\bar\lambda a_0-\bar\lambda(a\Bsymmdiff a_0)\cr
\noalign{\noindent (because both $a_0$ and $a_0\Bcap\varepsilon c$
belong to $\frak E_{\xi}$)}
&\ge\Bover12\bar\lambda a-\Bover32\bar\lambda(a\Bsymmdiff a_0)
>0.\cr}$$

\noindent Thus $a\Bcap e'_{\xi}$ is not of the form $a\Bcap\varepsilon c$
for any $c\in\frak C$, and $\frak A_a\ne\varepsilon[\frak C]_a$, so that
$\tau_{\varepsilon[\frak C]_a}(\frak A_a)>0$.\ \Qed

\medskip

{\bf (e)} It follows that $\tau_{\varepsilon[\frak C]_a}(\frak A_a)$ is
infinite.   \Prf\ There is a non-zero $d\Bsubseteq a$ which is
relatively \Mth\ over $\varepsilon[\frak C]$.   By (d), applied
to $d$, $\tau_{\varepsilon[\frak C]_d}(\frak A_d)>0$;  but now 333Bd tells us
that $\tau_{\varepsilon[\frak C]_d}(\frak A_d)$ must be infinite, so
$\tau_{\varepsilon[\frak C]_a}(\frak A_a)$ is infinite.\ \Qed

\medskip

{\bf (f)} If $\kappa=\omega$, we can stop here.   If $\kappa>\omega$, we
continue, as follows.   Let $D\subseteq\frak A_a$ be any set with cardinal
less than $\kappa$.   Each $d\in D\cup\{a\}$ belongs to the closed
subalgebra of $\frak A$ generated by
$C=\varepsilon[\frak C]\cup\{e'_{\xi}:\xi<\kappa\}$.
But because $\frak A$ is
ccc, this is just the $\sigma$-subalgebra of $\frak A$ generated by
$C$ (331Ge).   So $d$ belongs to the closed subalgebra of $\frak A$
generated by some countable subset $C_d$ of $C$, by 331Gd.   Now
$J_d=\{\eta:e'_{\eta}\in C_d\}$ is countable.   Set
$J=\bigcup_{d\in D\cup\{a\}}J_d$;  then

\Centerline{$\#(J)\le\max(\omega,\#(D\cup\{a\}))
=\max(\omega,\#(D))<\kappa$,}

\noindent so $J\ne\kappa$, and there is a $\xi\in\kappa\setminus J$.
Accordingly $\varepsilon[\frak C]\cup D\cup\{a\}$ is included in
$\frak E_{\xi}$, as defined in (c) above, and
$\varepsilon[\frak C]_a\cup D\subseteq\frak E_{\xi}$.   As
$\frak A_a\cap\frak E_{\xi}$ is a closed
subalgebra of $\frak A_a$, it includes the closed subalgebra
generated by $\varepsilon[\frak C]_a\cup D$.   But $a\Bcap e'_{\xi}$ surely
does not belong to $\frak E_{\xi}$, since

\Centerline{$\bar\lambda(a\Bcap e'_{\xi}\Bcap e'_{\xi})
=\bar\lambda(a\Bcap e'_{\xi})=\Bover12\bar\lambda a>0$,}

\noindent and $\bar\lambda(a\Bcap e'_{\xi}\Bcap
e'_{\xi})\ne\bover12\bar\lambda(a\Bcap e'_{\xi})$.   Thus
$a\Bcap e'_{\xi}$
cannot belong to the closed subalgebra of $\frak A_a$ generated by
$\varepsilon[\frak C]_a\cup D$, and $\varepsilon[\frak C]_a\cup D$ does not
$\tau$-generate $\frak A_a$.   As $D$ is arbitrary,
$\tau_{\phi[\frak C]_a}(\frak A_a)\ge\kappa$.

This completes the proof.
}%end of proof of 333E

\leader{333F}{Corollary} Let $(\frak A,\bar\mu)$ be a totally finite
measure algebra, $\frak C$ a closed subalgebra of $\frak A$ and $\kappa$
an infinite cardinal.

(a) Suppose that $\kappa\ge\tau_{\frak C}(\frak A)$.   Let
$(\frak C\widehat{\otimes}\frak B_{\kappa},\bar\lambda)$ be the localizable
measure algebra free product of $(\frak C,\bar\mu\restrp\frak C)$ and
$(\frak B_{\kappa},\bar\nu_{\kappa})$, and
$\varepsilon:\frak C\to\frak C\widehat{\otimes}\frak B_{\kappa}$ the
corresponding homomorphism.
Then there is a measure-preserving Boolean homomorphism
$\pi:\frak A\to\frak C\widehat{\otimes}\frak B_{\kappa}$ extending
$\varepsilon$.

(b) Suppose further that $\kappa=\tau_{\frak C_a}(\frak A_a)$
for every non-zero $a\in\frak A$.   Then $\pi$ can be
taken to be an isomorphism.

\proof{ All we have to do is apply 333C with
$\frak B=\frak C\widehat{\otimes}\frak B_{\kappa}$, using 333E to see
that the hypothesis

\Centerline{$\tau_{\varepsilon[\frak C]_b}(\frak B_b)=\kappa$ for every
non-zero $b\in\frak B$}

\noindent is satisfied.
}%end of proof of 333F

\leader{333G}{Corollary} Let $(\frak C,\bar\mu)$ be a
totally finite measure algebra.
Suppose that $\kappa\ge\max(\omega,\tau(\frak C))$ is
a cardinal.   Let
$(\frak C\widehat{\otimes}\frak B_{\kappa},\bar\lambda)$ be the
localizable measure algebra free product of $(\frak C,\bar\mu)$ and
$(\frak B_{\kappa},\bar\nu_{\kappa})$.   Then

(a) $\frak C\widehat{\otimes}\frak B_{\kappa}$ is \Mth,
with Maharam type $\kappa$ if $\frak C\ne\{0\}$;

(b) for every measure-preserving Boolean homomorphism
$\phi:\frak C\to\frak C$ there is a measure-preserving automorphism
$\pi:\frak C\widehat{\otimes}\frak B_{\kappa}
\to\frak C\widehat{\otimes}\frak B_{\kappa}$ such that
$\pi(c\otimes 1)=\phi c\otimes 1$ for every
$c\in\frak C$, writing $c\otimes 1$ for the canonical image in
$\frak C\widehat{\otimes}\frak B_{\kappa}$ of any $c\in\frak C$.

\proof{ Write $\frak A$ for $\frak C\widehat{\otimes}\frak B_{\kappa}$,
as in 333E, and $\frak D$ for $\{c\otimes 1:c\in\frak C\}\subseteq\frak A$.

\medskip

{\bf (a)} If $C\subseteq\frak C$ is a set with cardinal $\tau(\frak C)$
which $\tau$-generates $\frak C$, and $B\subseteq\frak B_{\kappa}$ a set
with cardinal $\kappa$ which $\tau$-generates $\frak B_{\kappa}$ (331K),
then $\{c\otimes b:c\in C,\,b\in B\}$ is a set with cardinal at most
$\max(\omega,\tau(\frak C),\kappa)=\kappa$ which $\tau$-generates
$\frak A$ (because the subalgebra it generates is topologically dense in
$\frak A$, by 325Dc).   So $\tau(\frak A)\le\kappa$.   On the other
hand, if $a\in\frak A$ is non-zero, then
$\tau(\frak A_a)\ge\tau_{\frak D_a}(\frak A_a)\ge\kappa$, by 333E;
so $\frak A$ is \Mth, with Maharam type $\kappa$ unless $\frak C=\{0\}$.

\medskip

{\bf (b)} We have a
measure-preserving automorphism $\phi_1:\frak D\to\frak D$ defined by
setting $\phi_1(c\otimes 1)=\phi c\otimes 1$ for every $c\in\frak C$.
Because $\phi_1[\frak D]\subseteq\frak D$, 333Be and 333E tell us that

\Centerline{$\kappa=\tau(\frak A_a)\ge\tau_{\phi_1[\frak D]_a}(\frak
A_a)\ge\tau_{\frak D_a}(\frak A_a)=\kappa$}

\noindent for every non-zero $a\in\frak A$, so we can use 333Cb, with
$\frak B=\frak A$, to see that $\phi_1$ can be extended to a
measure-preserving automorphism on $\frak A$.
}%end of proof of 333G

\vleader{72pt}{333H}{}\cmmnt{ I turn now to the classification of closed
subalgebras.

\medskip

\noindent}{\bf Theorem} Let $(\frak A,\bar\mu)$ be a localizable measure
algebra and $\frak C$ a closed subalgebra of $\frak A$.   Then there are
$\langle\mu_i\rangle_{i\in I}$, $\langle c_i\rangle_{i\in I}$,
$\langle\kappa_i\rangle_{i\in I}$ such that

\inset{for each $i\in I$, $\mu_i$ is a non-negative completely additive
functional on $\frak C$,

\inset{$c_i=\Bvalue{\mu_i>0}\in\frak C$,}

\inset{$\kappa_i$ is $0$ or an infinite cardinal,}

\inset{$(\frak C_{c_i},\mu_i\restrp\frak C_{c_i})$ is
a totally finite measure algebra, writing $\frak C_{c_i}$ for the
principal ideal of $\frak C$ generated by $c_i$,}}

\inset{$\sum_{i\in I}\mu_ic=\bar\mu c$ for every $c\in\frak C$,}

\inset{there is a measure-preserving isomorphism $\pi$ from $\frak A$ to
the simple product
$\prod_{i\in I}\frak C_{c_i}\widehat{\otimes}\frak B_{\kappa_i}$ of the
localizable measure algebra free products
$\frak C_{c_i}\widehat{\otimes}\frak B_{\kappa_i}$ of
$(\frak C_{c_i},\mu_i\restrp\frak C_{c_i})$ and
$(\frak B_{\kappa_i},\bar\nu_{\kappa_i})$.}

\noindent Moreover, $\pi$ may be taken such that

\inset{for every $c\in\frak C$, $\pi c=\langle (c\Bcap c_i)\otimes
1\rangle_{i\in I}$, writing $c\otimes 1$ for the image in $\frak
C_{c_i}\widehat{\otimes}\frak B_{\kappa_i}$ of $c\in\frak C_{c_i}$.}

\cmmnt{\medskip

\noindent{\bf Remark} Recall that $\Bvalue{\mu_i>0}$ is that element of
$\frak C$ such that $\mu_ic>0$ whenever $c\in\frak C$ and
$0\ne c\Bsubseteq\Bvalue{\mu_i>0}$, $\mu_ic\le 0$ whenever $c\in\frak C$
and $c\Bcap\Bvalue{\mu_i>0}=0$ (326S).
}%end of comment

\proof{{\bf (a)} Let $A$ be the set of those elements of $\frak A$ which
are relatively \Mth\ over $\frak C$ (see 333Ac).      By
333Bb, $A$ is order-dense in $\frak A$ (compare part (a) of the proof of
332B), and consequently $A'=\{a:a\in A,\,\bar\mu a<\infty\}$ is
order-dense in $\frak A$.   So there is a partition of unity
$\langle a_i\rangle_{i\in I}$ in $\frak A$ consisting of members of $A'$
(313K).   For
each $i\in I$, set $\mu_ic=\bar\mu(a_i\Bcap c)$ for every $c\in\frak C$;
then $\mu_i$ is
non-negative, and it is completely additive by 327E.   Because
$\langle a_i\rangle_{i\in I}$ is a partition of unity in
$\frak A$,

\Centerline{$\bar\mu c=\sum_{i\in I}\bar\mu(c\Bcap a_i)=\sum_{i\in
I}\mu_ic$}

\noindent for every $c\in\frak C$.   Next, $(\frak
C_{c_i},\mu_i\restrp\frak C_{c_i})$
is a totally finite measure algebra.   \Prf\ $\frak C_{c_i}$ is a
Dedekind $\sigma$-complete Boolean algebra because $\frak C$ is.
$\mu_i\restrp\frak C_{c_i}$ is a non-negative countably additive
functional because $\mu_i$ is.   If $c\in\frak C_{c_i}$ and $\mu_ic=0$,
then $c=0$ by the choice of $c_i$.\ \Qed\   Note also that

\Centerline{$\bar\mu(a_i\Bsetminus c_i)=\mu_i(1\Bsetminus c_i)=0$,}

\noindent so that $a_i\Bsubseteq c_i$.

\medskip

{\bf (b)} By 333Bd, any finite $\kappa_i$ must actually be zero.   The
next element we need is the fact that, for each $i\in I$,
we have a measure-preserving isomorphism $c\mapsto c\Bcap a_i$ from
$(\frak C_{c_i},\mu_i\restrp\frak C_{c_i})$ to $(\frak
C_{a_i},\bar\mu\restrp\frak C_{a_i})$.   \Prf\ Of course this is a ring
homomorphism.   Because $a_i\Bsubseteq c_i$, it is a surjective Boolean
homomorphism.   It is
measure-preserving by the definition of $\mu_i$, and therefore
injective.\ \Qed

\medskip

{\bf (c)} Still focusing on a particular $i\in I$, let $\frak A_{a_i}$
be the principal ideal of $\frak A$ generated by $a_i$.   Then we have a
measure-preserving isomorphism
$\tilde\pi_i:\frak A_{a_i}
\to\frak C_{a_i}\widehat{\otimes}\frak B_{\kappa_i}$,
extending the canonical homomorphism
$c\mapsto c\otimes 1:
 \frak C_{a_i}\to\frak C_{a_i}\widehat{\otimes}\frak B_{\kappa_i}$.
\Prf\ When $\kappa_i$ is
infinite, this is just 333Fb.   But the only other case is
when $\kappa_i=0$, that is, $\frak C_{a_i}=\frak A_{a_i}$, while
$\frak B_{\kappa_i}=\{0,1\}$ and
$\frak C_{a_i}\widehat{\otimes}\frak B_{\kappa_i}\cong\frak C_{c_i}$.\
\Qed

The isomorphism between $(\frak C_{c_i},\mu_i\restrp\frak C_{c_i})$ and
$(\frak C_{a_i},\bar\mu\restrp\frak C_{a_i})$ induces an isomorphism
between
$\frak C_{c_i}\widehat{\otimes}\frak B_{\kappa_i}$ and
$\frak C_{a_i}\widehat{\otimes}\frak B_{\kappa_i}$.   So we have a
measure-preserving isomorphism
$\pi_i:\frak A_{a_i}\to\frak C_{c_i}\widehat{\otimes}\frak B_{\kappa_i}$
such that $\pi_i(c\Bcap a_i)=c\otimes 1$ for every $c\in\frak C_{c_i}$.

\medskip

{\bf (d)} By 322Le, we have a measure-preserving isomorphism
$a\mapsto\langle a\Bcap a_i\rangle_{i\in I}:
\frak A\to\prod_{i\in I}\frak A_{a_i}$.
Putting this together with the isomorphisms of (c), we have a
measure-preserving isomorphism $\pi$ from $\frak A$ to
$\prod_{i\in I}\frak C_{c_i}\widehat{\otimes}\frak B_{\kappa_i}$,
setting $\pi a=\langle\pi_i(a\Bcap a_i)\rangle_{i\in I}$ for
$a\in\frak A$.   Observe that, for $c\in\frak C$,

\Centerline{$\pi c=\langle\pi_i(c\Bcap a_i)\rangle_{i\in I}
=\langle(c\Bcap c_i)\otimes 1\rangle_{i\in I}$,}

\noindent as required.
}%end of proof of 333H


\vleader{36pt}{333I}{Remarks\cmmnt{ (a)}}\dvro{ Whenever}{ I hope it is
clear that whenever} $(\frak C,\bar\mu)$ is a Dedekind complete measure
algebra, $\langle\mu_i\rangle_{i\in I}$ is a family of non-negative
completely additive functionals on $\frak C$ such that
$\sum_{i\in I}\mu_i=\bar\mu$, and $\langle\kappa_i\rangle_{i\in I}$ is a
family of
cardinals all infinite or zero, then the construction above can be
applied to give a measure algebra $(\frak A,\bar\lambda)$, the product
of the family
$\langle\frak C_{c_i}\widehat{\otimes}\frak B_{\kappa_i}\rangle_{i\in I}$,
together with an order-continuous
measure-preserving
homomorphism $\pi:\frak C\to\frak A$;  and\cmmnt{ that} the partition
of unity
$\langle a_i\rangle_{i\in I}$ in $\frak A$ corresponding to this
product\cmmnt{ (315E)} has $\mu_ic=\bar\lambda(a_i\Bcap\pi c)$ for
every $c\in\frak C$ and $i\in I$, while each principal ideal
$\frak A_{a_i}$ can be
identified with $\frak C_{c_i}\widehat{\otimes}\frak B_{\kappa_i}$, so
that $a_i$ is relatively \Mth\ over $\pi[\frak C]$.   Thus
any structure $(\frak C,\bar\mu,\langle\mu_i\rangle_{i\in
I},\langle\kappa_i\rangle_{i\in I})$ of the type described here
corresponds to an embedding of $\frak C$ as a closed subalgebra of
a localizable measure algebra.

\cmmnt{\medskip

{\bf (b)} The obvious next step is to seek a complete classification of
objects $(\frak A,\bar\mu,\frak C)$, where $(\frak A,\bar\mu)$ is a
localizable
measure algebra and $\frak C$ is a closed subalgebra, corresponding to
the classification of localizable measure algebras in terms of the
magnitudes of their Maharam-type-$\kappa$ components in 332J.   The
general case seems to be complex.   But I can deal with the special case
in which $(\frak A,\bar\mu)$ is totally finite.   In this case, we have
the following facts.
}

\vleader{48pt}{333J}{Lemma} Let $(\frak A,\bar\mu)$ be a totally finite measure
algebra, and $\frak C$ a closed subalgebra.   Let $A$ be the set of
relative atoms of $\frak A$ over $\frak C$.   Then there is
a unique sequence $\sequencen{\mu_n}$ of additive functionals on
$\frak C$ such that (i) $\mu_{n+1}\le\mu_n$ for every $n$ (ii) there is
a disjoint sequence $\sequencen{a_n}$ in $A$ such that
$\sup_{n\in\Bbb N}a_n=\sup A$ and $\mu_nc=\bar\mu(a_n\Bcap c)$ for every
$n\in\Bbb N$ and $c\in\frak C$.

\cmmnt{\medskip

\noindent{\bf Remark} I hope it is plain from my wording that it is the
${\mu_n}$ which are unique, not the $a_n$.
}

\proof{{\bf (a)} For each $a\in\frak A$ set $\theta_a(c)=\bar\mu(c\Bcap a)$
for $c\in\frak C$.   Then $\theta_a$ is a non-negative completely additive
real-valued functional on $\frak C$ (see 326Od).

The key step is I suppose in (c) below;  I approach by a two-stage
argument.   For each $b\in\frak A$ write $A^{\perp}_b$ for
$\{a:a\in A,\,a\Bcap b=0\}$.

\medskip

{\bf (b)} For every $b\in\frak A$ and non-zero $c\in\frak C$ there are
$a\in A^{\perp}_b$, $c'\in\frak C$ such that $0\ne c'\Bsubseteq c$ and
$\theta_a(d)\ge\theta_e(d)$ whenever $d\in\frak C$, $e\in A^{\perp}_b$ and
$d\Bsubseteq c'$.   \Prf\Quer\ Otherwise, choose $\sequencen{a_n}$ and
$\sequencen{c_n}$ as follows.   Since $0$, $c$ won't serve for $a$,
$c'$, there must be an $a_0\in A^{\perp}_b$ such that $\theta_{a_0}(c)>0$.   Let
$\delta>0$ be such that $\theta_{a_0}(c)>\delta\bar\mu c$ and set
$c_0=c\Bcap\Bvalue{\theta_{a_0}>\delta\bar\mu\restrp\frak C}$;  then
$c_0\in\frak C$ and $0\ne c_0\Bsubseteq c$.   Given that $a_n\in A^{\perp}_b$,
$c_n\in\frak C$ and $0\ne c_n\Bsubseteq c$, then there must be
$a_{n+1}\in A^{\perp}_b$, $d_n\in \frak C$ such that $d_n\Bsubseteq c_n$ and
$\theta_{a_{n+1}}(d_n)>\theta_{a_n}(d_n)$.   Set
$c_{n+1}=d_n\Bcap\Bvalue{\theta_{a_{n+1}}>\theta_{a_n}}$, so that
$c_{n+1}\in\frak C$ and $0\ne c_{n+1}\Bsubseteq c_n$, and continue.

There is some $n\in\Bbb N$ such that $n\delta\ge 1$.   For any $i<n$,
the construction ensures that

\Centerline{$0\ne c_{n+1}\Bsubseteq
c_{i+1}\Bsubseteq\Bvalue{\theta_{a_{i+1}}>\theta_{a_i}}$,}

\noindent so $\theta_{a_i}(c_{n+1})<\theta_{a_{i+1}}(c_{n+1})$;  also
$c_{n+1}\Bsubseteq c_0$ so

\Centerline{$\bar\mu(a_i\Bcap c_{n+1})
=\theta_{a_i}(c_{n+1})\ge\theta_{a_0}(c_{n+1})>\delta\bar\mu c_{n+1}$.}

\noindent But this means that
$\sum_{i=0}^{n-1}\bar\mu(a_i\Bcap c_{n+1})>\bar\mu c_{n+1}$ and there
must be distinct $j$, $k<n$ such that $a_j\Bcap a_k\Bcap c_{n+1}$ is non-zero.
Because $a_j$, $a_k\in A$ there are $d'$, $d''\in\frak C$ such that
$a_j\Bcap a_k=a_j\Bcap d'=a_k\Bcap d''$;  set $d=c_{n+1}\Bcap d'\Bcap
d''$, so that $d\in\frak C$ and

\Centerline{$a_j\Bcap d=a_j\Bcap a_k\Bcap c_{n+1}=a_k\Bcap d$,
\quad$\theta_{a_j}(d)=\bar\mu(a_j\Bcap a_k\Bcap c_{n+1})=\theta_{a_k}(d)$.}

\noindent  But as $0\ne d\Bsubseteq\Bvalue{\theta_{a_{i+1}}>\theta_{a_i}}$ for
every $i<n$, $\theta_{a_0}(d)<\theta_{a_1}(d)<\ldots<\theta_{a_n}(d)$, so this is
impossible.   \Bang\Qed

\medskip

{\bf (c)} Now for a global, rather than local, version of the same idea.
For every $b\in\frak A$ there is an $a\in A^{\perp}_b$ such that and
$\theta_a\ge\theta_e$ whenever $e\in A^{\perp}_b$.   \Prf\ (i) By (b), the set $C$
of those $c\in\frak C$ such that there is an $a\in A^{\perp}_b$ such that
$\theta_a\restrp\frak C_c\ge\theta_e\restrp\frak C_c$ for every $e\in A^{\perp}_b$ is
order-dense in $\frak C$.   Let $\langle c_i\rangle_{i\in I}$ be a
partition of unity in $\frak C$ consisting of members of $C$, and for
each $i\in I$ choose $a_i\in A^{\perp}_b$ such that $\theta_{a_i}\restrp\frak
C_{c_i}\ge\theta_e\restrp\frak C_{c_i}$ for every $e\in A^{\perp}_b$.   Consider
$a=\sup_{i\in I}a_i\Bcap c_i$.   (ii) If $a'\in\frak A$ and
$a'\Bsubseteq a$, then for each $i\in I$ there is a $d_i\in\frak C$ such
that $a_i\Bcap a'=a_i\Bcap d_i$.   Set $d'=\sup_{i\in I}c_i\Bcap d_i$;
then (because $\langle c_i\rangle_{i\in I}$ is disjoint)

\Centerline{$a\Bcap d'=\sup_{i\in I}a_i\Bcap c_i\Bcap d_i=\sup_{i\in
I}a_i\Bcap c_i\Bcap a'=a\Bcap a'=a'$.}

\noindent As $a'$ is arbitrary, this shows that $a\in A$.   (iii) Of
course $a\Bcap b=0$, so $a\in A^{\perp}_b$.   Now take any $e\in A^{\perp}_b$ and
$d\in\frak C$.   Then

\Centerline{$\theta_a(d)=\sum_{i\in I}\theta_{a_i}(c_i\Bcap d)\ge\sum_{i\in
I}\theta_e(c_i\Bcap d)=\theta_e(d)$.}

\noindent So this $a$ has the required property.\  \Qed

\medskip

{\bf (d)} Choose $\sequencen{a_n}$ inductively in $A$ so that, for each
$n$, $a_n\Bcap\sup_{i<n}a_i=0$ and $\theta_{a_n}\ge \theta_e$ whenever
$e\in A$ and $e\Bcap\sup_{i<n}a_i=0$.   Set $\mu_n=\theta_{a_n}$.   Because
$a_{n+1}\Bcap\sup_{i<n}a_i=0$, $\mu_{n+1}\le\mu_n$ for each $n$.   Also
$\sup_{n\in\Bbb N}a_n=\sup A$.   \Prf\  Take any $a\in A$ and set
$e=a\Bsetminus\sup_{n\in\Bbb N}a_n$.   Then $e\in A$ and, for any
$n\in\Bbb N$, $e\Bcap\sup_{i<n}a_i=0$, so $\theta_e\le\theta_{a_n}$ and

\Centerline{$\bar\mu e=\theta_e(1)\le\theta_{a_n}(1)=\bar\mu a_n$.}

\noindent   But as $\sequencen{a_n}$ is disjoint, this means that $e=0$,
that is, $a\Bsubseteq\sup_{n\in\Bbb N}a_n$.    As $a$ is arbitrary,
$\sup A\Bsubseteq\sup_{n\in\Bbb N}a_n$.\   \Qed

\medskip

{\bf (e)} Thus we have a sequence $\sequencen{\mu_n}$ of the required
type, witnessed by $\sequencen{a_n}$.   To see that it is unique,
suppose that $\sequencen{\mu'_n}$, $\sequencen{a'_n}$ are another pair
of sequences with the same properties.    Note first that if $c\in\frak
C$ and $0\ne c\Bsubseteq\Bvalue{\mu'_i>0}$ there is some $k\in\Bbb N$
such that $c\Bcap a'_i\Bcap a_k\ne 0$;  this is because
$\bar\mu(a'_i\Bcap c)=\mu'_i(c)>0$, so that $a'_i\Bcap c\ne 0$, while
$a'_i\Bsubseteq\sup A=\sup_{k\in\Bbb N}a_k$.    \Quer\  Suppose, if
possible, that there is some $n$ such that $\mu_n\ne\mu'_n$;  since the
situation is symmetric, there is no loss of generality in supposing that
$\mu'_n\not\le\mu_n$, that is, that $c=\Bvalue{\mu'_n>\mu_n}\ne 0$.
For any $i\le n$, $\mu'_i\ge\mu'_n$ so $c\Bsubseteq\Bvalue{\mu'_i>0}$.
We may therefore choose $c_0,\ldots, c_{n+1}\in\frak C_c\setminus\{0\}$
and $k(0),\ldots,k(n)\in\Bbb N$ such that $c_0=c$ and, for $i\le n$,

\Centerline{$c_i\Bcap a'_i\Bcap a_{k(i)}\ne 0$}

\noindent (choosing $k(i)$, recalling that $0\ne c_i\Bsubseteq
c\Bsubseteq\Bvalue{\mu'_i>0}$),

\Centerline{$c_{i+1}\in\frak C$,
\quad$c_{i+1}\Bsubseteq c_i$,
\quad$c_{i+1}\Bcap a'_i=c_{i+1}\Bcap a_{k(i)}=c_i\Bcap a'_i\Bcap a_{k(i)}$}

\noindent (choosing $c_{i+1}$, using the fact that $a'_i$ and $a_{k(i)}$
both belong to $A$ -- see the penultimate sentence in part (b) of the
proof).   On reaching $c_{n+1}$, we have $0\ne c_{n+1}\Bsubseteq c$ so
$\mu_n(c_{n+1})<\mu'_n(c_{n+1})$.   On the other hand, for each $i\le
n$,

\Centerline{$c_{n+1}\Bcap a'_i\Bcap a_{k(i)}
=c_{n+1}\Bcap c_{i+1}\Bcap a'_i\Bcap a_{k(i)}
=c_{n+1}\Bcap a'_i=c_{n+1}\Bcap a_{k(i)}$,}

so

\Centerline{$\mu_n(c_{n+1})<\mu'_n(c_{n+1})
\le\mu'_i(c_{n+1})=\bar\mu(c_{n+1}\Bcap a'_i)=\bar\mu(c_{n+1}\Bcap
a_{k(i)})=\mu_{k(i)}(c_{n+1})$,}

\noindent and $k(i)$ must be less than $n$.   There are therefore
distinct $i$, $j\le n$ such that $k(i)=k(j)$.   But in this case

\Centerline{$c_{n+1}\Bcap a'_i=c_{n+1}\Bcap a_{k(i)}=c_{n+1}\Bcap
a_{k(j)}=c_{n+1}\Bcap a'_j\ne 0$}

\noindent because $0\ne c_{n+1}\Bsubseteq\Bvalue{\mu'_j>0}$.   So
$a'_i$, $a'_j$ cannot be disjoint, breaking one of the rules of the
construction.\ \BanG\  Thus $\mu_n=\mu'_n$ for every $n\in\Bbb N$.

This completes the proof.
}%end of proof of 333J

\leader{333K}{Theorem} Let $(\frak A,\bar\mu)$ be a totally finite
measure algebra and $\frak C$ a closed subalgebra of $\frak A$.   Then
there are unique families $\sequencen{\mu_n}$,
$\langle\mu_{\kappa}\rangle_{\kappa\in K}$ such that

\inset{$K$ is a countable set of infinite cardinals,}

\inset{for $ i\in\Bbb N\cup K$, $\mu_i$ is a non-negative countably
additive functional on $\frak C$, and
$\sum_{i\in\Bbb N\cup K}\mu_ic=\bar\mu c$ for every $c\in\frak C$,}

\inset{$\mu_{n+1}\le\mu_n$ for every $n\in\Bbb N$, and
$\mu_{\kappa}\ne 0$ for $\kappa\in K$,}

\inset{setting $e_i=\Bvalue{\mu_i>0}\in\frak C$, and
giving the principal ideal $\frak C_{e_i}$ generated by $e_i$ the
measure $\mu_i\restrp\frak C_{e_i}$ for each $ i\in\Bbb N\cup K$,
we have a measure algebra isomorphism

\Centerline{$\pi:\frak A\to\prod_{n\in\Bbb N}\frak C_{e_n}
  \times\prod_{\kappa\in K}
    \frak C_{e_{\kappa}}\widehat{\otimes}\frak B_{\kappa}$}

\noindent such that

\Centerline{$\pi c
=(\sequencen{c\Bcap e_n},
  \langle(c\Bcap e_{\kappa})\otimes 1\rangle_{\kappa\in K})$}

\noindent for each $c\in C$, writing $c\otimes 1$ for the canonical
image in $C_{e_{\kappa}}\widehat{\otimes}\frak B_{\kappa}$ of
$c\in\frak C_{e_{\kappa}}$.}

\proof{{\bf (a)} I aim to use the construction of 333H, but taking much
more care over the choice of $\langle a_i\rangle_{i\in I}$ in part (a)
of the proof there.   We start by taking $\sequencen{a_n}$ as in 333J,
and setting $\mu_nc=\bar\mu(a_n\Bcap c)$ for every $n\in\Bbb N$,
$c\in\frak C$;  then these $a_n$ will deal with the relative atoms over
$\frak C$.

\medskip

{\bf (b)} The further idea required here concerns the treatment of
infinite $\kappa$.   Let $\langle b_i\rangle_{i\in I}$ be any partition
of unity in $\frak A$ consisting of non-zero members of $\frak A$ which
are relatively \Mth\ over $\frak C$, and
$\langle\kappa_i\rangle_{i\in I}$ the corresponding cardinals, so that
$\kappa_i=0$ iff $b_i$ is a relative atom.   Set
$I_1=\{i:i\in I,\,\kappa_i\ge\omega\}$.   Set $K=\{\kappa_i:i\in I_1\}$,
so that $K$
is a countable set of infinite cardinals, and for $\kappa\in K$ set
$J_{\kappa}=\{i:\kappa_i=\kappa\}$,
$a_{\kappa}=\sup_{i\in J_{\kappa}}b_i$ for $\kappa\in K$.   Now every
$a_{\kappa}$ is relatively \Mth\ over $\frak C$.   \Prf\ (Compare 332H.)
$J_{\kappa}$ must be countable, because $\frak A$ is ccc.   If
$0\ne a\Bsubseteq a_{\kappa}$, there is some $i\in J_{\kappa}$ such that
$a\Bcap b_i\ne 0$;  now

\Centerline{$\tau_{\frak C_a}(\frak A_a)
\ge\tau_{\frak C_{a\Bcap b_i}}(\frak A_{a\Bcap b_i})
=\kappa_i=\kappa$}

\noindent (333Ba).
At the same time, for each $i\in J_{\kappa}$, there is a set
$D_i\subseteq\frak A_{b_i}$ such that $\#(D_i)=\kappa$ and
$\frak C_{b_i}\cup D_i\,\,\tau$-generates $\frak A_{b_i}$.   Set
$D=\bigcup_{i\in J_{\kappa}}D_i\cup\{b_i:i\in J_{\kappa}\}$;  then

\Centerline{$\#(D)\le\max(\omega,\#(J_{\kappa}),\sup_{i\in
K}\#(D_i))=\kappa$.}

\noindent   Let $\frak B$ be the closed subalgebra of
$\frak A_{a_{\kappa}}$ generated by $\frak C_{a_{\kappa}}\cup D$.   Then

\Centerline{$\frak C_{b_i}\cup D_i\subseteq\{b\cap b_i:b\in \frak B\}
=\frak B\cap\frak A_{b_i}$}

\noindent so $\frak B\supseteq\frak A_{b_i}$ for each $i\in J_{\kappa}$,
and $\frak B=\frak A_{a_{\kappa}}$.   Thus $\frak C_{a_{\kappa}}\cup
D\,\,\tau$-generates $\frak A_{a_{\kappa}}$, and

\Centerline{$\tau_{\frak C_{a_{\kappa}}}(\frak A_{a_{\kappa}})
\le\kappa\le\min_{0\ne a\Bsubseteq a_{\kappa}}\tau_{\frak C_a}(\frak
A_a)$.}

\noindent This shows that $a_{\kappa}$ is relatively \Mth\
over $\frak C$, with
$\tau_{\frak C_{a_{\kappa}}}(\frak A_{a_{\kappa}})=\kappa$.\ \Qed

Since evidently $\langle J_{\kappa}\rangle_{\kappa\in K}$ and
$\langle a_{\kappa}\rangle_{\kappa\in K}$ are disjoint, and
$\sup_{\kappa\in K}a_{\kappa}=\sup_{i\in I_1}b_i$,
this process yields a partition
$\langle a_i\rangle_{i\in\Bbb N\cup K}$ of unity in $\frak A$.
Now the arguments of 333H show that we get an isomorphism $\pi$ of the
kind described.

\medskip

{\bf (c)} To see that the families $\sequencen{\mu_n}$,
$\langle\mu_{\kappa}\rangle_{\kappa\in K}$ (and therefore the $e_i$ and
the $(\frak C_{e_i},\mu_i\restrp\frak C_{e_i})$, but not $\pi$) are
uniquely defined, argue as follows.   Let $A$ be the set of
those $a\in\frak A$ which are relatively \Mth\ over $\frak C$.
Take  families
$\sequencen{\tilde\mu_n}$,
$\langle\tilde\mu_{\kappa}\rangle_{\kappa\in\tilde K}$
which correspond to an isomorphism

\Centerline{$\tilde\pi:\frak A\to\frak D
=\prod_{n\in\Bbb N}\frak C_{\tilde e_n}
  \times\prod_{\kappa\in\tilde K}
  \frak C_{\tilde e_{\kappa}}\widehat{\otimes}\frak B_{\kappa}$,}

\noindent writing $\tilde e_i=\Bvalue{\tilde\mu_i>0}$ for
$i\in\Bbb N\cup\tilde K$.   In the simple product
$\prod_{n\in\Bbb N}\frak C_{\tilde e_n}
  \times\prod_{\kappa\in \tilde K}
  \frak C_{\tilde e_{\kappa}}\widehat{\otimes}\frak B_{\kappa}$,
we have a partition of unity
$\langle e_i^*\rangle_{i\in\Bbb N\cup\tilde K}$ corresponding to the
product structure.   Now for $d\Bsubseteq e_i^*$, we have

$$\eqalign{\tau_{\tilde\pi[\frak C]_d}(\frak D_d)
&=0\text{ if } i\in\Bbb N,\cr
&=\kappa\text{ if } i=\kappa\in\tilde K.\cr}$$

\noindent So $\tilde K$ must be

\Centerline{$\{\kappa:\kappa\ge\omega,\,\Exists a\in A,
\,\tau_{\frak C_a}(\frak A_a)=\kappa\}=K$,}

\noindent and for $\kappa\in\tilde K$,

\Centerline{$\tilde\pi^{-1} e_{\kappa}^*
=\sup\{a:a\in A,\,\tau_{\frak C_a}(\frak A_a)=\kappa\}=a_{\kappa}$,}

\noindent so that $\tilde\mu_{\kappa}=\mu_{\kappa}$.   On the other hand,
$\sequencen{\tilde\pi^{-1}e_n^*}$ must be a disjoint sequence with
supremum $\sup A$, and the corresponding functionals $\tilde\mu_n$ are
supposed to form a non-increasing sequence, so must be equal to the
$\mu_n$ by 333J.
}%end of proof of 333K

\leader{333L}{Remark} Thus for the classification of structures
$(\frak A,\bar\mu,\frak C)$, where $(\frak A,\bar\mu)$ is a totally finite
measure algebra and $\frak C$ is a closed subalgebra, it will be enough
to classify objects $(\frak C,\bar\mu,\sequencen{\mu_n},
\langle\mu_{\kappa}\rangle_{\kappa\in K})$, where

\inset{$(\frak C,\bar\mu)$ is a totally finite measure algebra,}

\inset{$\sequencen{\mu_n}$ is a non-increasing sequence of
non-negative countably
additive functionals on $\frak C$,}

\inset{$K$ is a countable set of infinite cardinals\cmmnt{ (possibly
empty)},}

\inset{$\langle\mu_{\kappa}\rangle_{\kappa\in K}$ is a family of
non-zero non-negative countably additive functionals on $\frak C$,}

\inset{$\sum_{n=0}^{\infty}\mu_n+\sum_{\kappa\in K}\mu_{\kappa}=\bar\mu$.}

\cmmnt{\noindent To do this we need the concept of `standard
extension' of a
countably additive functional on a closed subalgebra of a measure
algebra, treated in 327F-327G, together with the following idea.
}%end of comment

\vleader{48pt}{333M}{Lemma} Let $(\frak C,\bar\mu)$ be a
totally finite measure
algebra and $\langle\mu_i\rangle_{i\in I}$ a family of countably
additive functionals on $\frak C$.   For $i\in I$, $\alpha\in\Bbb R$
set $e_{i\alpha}=\Bvalue{\mu_i>\alpha\bar\mu}$\cmmnt{ (326T)}, and
let $\frak C_0$ be the closed subalgebra of $\frak C$
generated by $\{e_{i\alpha}:i\in I,\,\alpha\in\Bbb R\}$.   Write
$\Sigma$ for the
$\sigma$-algebra of subsets of $\BbbR^I$ generated by sets of the form
$E_{i\alpha}=\{x:x(i)>\alpha\}$ as $i$ runs over $I$ and $\alpha$ runs
over $\Bbb R$.   Then

(a) there is a measure $\mu$, with domain $\Sigma$, such that there
is a measure-preserving isomorphism
$\pi:\Sigma/\Cal N_{\mu}\to\frak C_0$ for which
$\pi E_{i\alpha}^{\ssbullet}=e_{i\alpha}$ for every $i\in I$ and
$\alpha\in\Bbb R$, writing $\Cal N_{\mu}$ for $\mu^{-1}[\{0\}]$;

(b) this formula determines both $\mu$ and $\pi$;

(c) for every $E\in\Sigma$ and $i\in I$, we have

\Centerline{$\mu_i\pi E^{\ssbullet}=\int_Ex(i)\mu(dx)$;}

(d) for every $i\in I$, $\mu_i$ is the standard extension of
$\mu_i\restrp\frak C_0$ to $\frak C$;

(e) for every $i\in I$, $\mu_i\ge 0$ iff $x(i)\ge 0$ for
$\mu$-almost every $x$;

(f) for every $i$, $j\in I$, $\mu_i\ge\mu_j$ iff $x(i)\ge x(j)$ for
$\mu$-almost every $x$;

(g) for every $i\in I$, $\mu_i=0$ iff $x(i)=0$ for $\mu$-almost
every $x$.

\proof{{\bf (a)} Express $(\frak C,\bar\mu)$ as the measure algebra of a
measure space $(Y,\Tau,\nu)$;  write $\phi:\Tau\to\frak C$ for the
corresponding homomorphism.   For each $i\in I$ let $f_i:Y\to\Bbb R$ be
a $\Tau$-measurable, $\nu$-integrable function such that
$\int_Hf_i=\mu_i\phi H$ for every $H\in\Tau$.   Define
$\psi:Y\to\BbbR^I$ by setting $\psi(y)=\langle f_i(y)\rangle_{i\in I}$;
then $\psi^{-1}[E_{i\alpha}]\in\Sigma$, and
$e_{i\alpha}=\phi(\psi^{-1}[E_{i\alpha}])$ for
every $i\in I$ and $\alpha\in\Bbb R$.   (See part (a) of the proof of
327F.)   So
$\{E:E\subseteq\BbbR^I,\,\psi^{-1}[E]\in\Tau\}$, which is a
$\sigma$-algebra of subsets of $\BbbR^I$, contains every $E_{i\alpha}$,
and therefore includes $\Sigma$;  that is, $\psi^{-1}[E]\in\Tau$ for
every $E\in\Sigma$.   Accordingly we may define $\mu$ by setting
$\mu E=\nu\psi^{-1}[E]$ for every $E\in\Sigma$, and $\mu$ will be a
measure on $\BbbR^I$ with domain $\Sigma$.   The Boolean homomorphism
$E\mapsto\phi\psi^{-1}[E]:\Sigma\to\frak C$ has kernel $\Cal N_{\mu}$,
so descends to a homomorphism $\pi:\Sigma/\Cal N_{\mu}\to\frak C$,
which is measure-preserving.   To see that
$\pi[\Sigma/\Cal N_{\mu}]=\frak C_0$, observe that because $\Sigma$ is
the $\sigma$-algebra generated by
$\{E_{i\alpha}:i\in I,\,\alpha\in\Bbb R\}$,
$\pi[\Sigma/\Cal N_{\mu}]$ must be the closed subalgebra of
$\frak C$ generated by
$\{\pi E_{i\alpha}^{\ssbullet}:i\in I,\,\alpha\in\Bbb R\}
=\{e_{i\alpha}:i\in I,\,\alpha\in\Bbb R\}$, which is $\frak C_0$.

\medskip

{\bf (b)} Now suppose that $\mu'$, $\pi'$ have the same properties.
Consider

\Centerline{$\Cal A=\{E:E\in\Sigma,\,\pi E^{\ssbullet}=\pi'
E^{\smallcirc}\}$,}

\noindent where I write $E^{\ssbullet}$ for the equivalence class of $E$
in $\Sigma/\Cal N_{\mu}$, and $E^{\smallcirc}$ for the equivalence
class of $E$ in $\Sigma/\Cal N_{\mu'}$.   Then $\Cal A$ is a
$\sigma$-subalgebra of $\Sigma$, because $E\mapsto\pi E^{\ssbullet}$,
$E\mapsto \pi'E^{\smallcirc}$ are both sequentially order-continuous
Boolean homomorphisms, and contains every $E_{i\alpha}$, so must be the
whole
of $\Sigma$.   Consequently

\Centerline{$\mu E=\bar\mu\pi
E^{\ssbullet}=\bar\mu\pi'E^{\smallcirc}=\mu'E$}

\noindent for every $E\in\Sigma$, and $\mu'=\mu$;  it follows at
once that $\pi'=\pi$.   So $\mu$ and $\pi$ are uniquely determined.

\medskip

{\bf (c)} If $E\in\Sigma$ and $i\in I$,

$$\eqalignno{\int_Ex(i)\mu(dx)
&=\int x(i)\chi E(x)\mu(dx)
=\int\psi(y)(i)\chi E(\psi(y))\nu(dy)\cr
\noalign{\noindent (applying 235G\formerly{2{}35I}
to the inverse-measure-preserving function $\psi:Y\to\BbbR^I$)}
&=\int_{\psi^{-1}[E]}f_i(y)\nu(dy)\cr
\noalign{\noindent (by the definition of $\psi$)}
&=\mu_i\phi(\psi^{-1}[E])\cr
\noalign{\noindent (by the choice of $f_i$)}
&=\mu_i\pi E^{\ssbullet}\cr}$$

\noindent by the definition of $\pi$.

\medskip

{\bf (d)} For every $\alpha\in\Bbb R$,
$\Bvalue{\mu_i>\alpha\bar\mu}$ belongs to $\frak C_0$, so must be
equal to $\Bvalue{\mu_i\restrp\frak C_0>\alpha\bar\mu\restrp\frak C_0}$.
Thus $\mu_i$ is the standard extension of $\mu_i\restrp\frak C_0$ (327G).

\medskip

{\bf (e)-(g)} The point is that, because the standard-extension operator
is order-preserving (327F(b-ii)),

$$\eqalignno{\mu_i\ge 0
&\iff\mu_i\restrp\frak C_0\ge 0\cr
&\iff\int_Ex(i)\mu(dx)\ge 0\text{ for every }E\in\Sigma\cr
&\iff x(i)\ge 0\,\,\mu\text{-a.e.},\cr
\mu_i\ge\mu_j
&\iff\mu_i\restrp\frak C_0\ge \mu_j\restrp\frak C_0\cr
&\iff\int_Ex(i)\mu(dx)\ge\int_Ex(j)\mu(dx)\text{ for every
}E\in\Sigma\cr
&\iff x(i)\ge x(j)\,\,\mu\text{-a.e.},\cr
\mu_i= 0
&\iff\mu_i\restrp\frak C_0= 0\cr
&\iff\int_Ex(i)\mu(dx)= 0\text{ for every }E\in\Sigma\cr
&\iff x(i)= 0\,\,\mu\text{-a.e.}.\cr}$$
}%end of proof of 333M


\leader{333N}{A canonical form for closed subalgebras} We now have all
the elements required to describe a canonical form for structures

\Centerline{$(\frak A,\bar\mu,\frak C)$,}

\noindent where $(\frak A,\bar\mu)$ is a totally finite measure algebra
and
$\frak C$ is a closed subalgebra of $\frak A$.   The first step is the
matching of such structures with structures

\Centerline{$(\frak
C,\bar\mu,\sequencen{\mu_n},\langle\mu_{\kappa}\rangle_{\kappa\in K})$,}

\noindent where $(\frak C,\bar\mu)$ is a totally finite measure algebra,
$\sequencen{\mu_n}$ is a non-increasing sequence of
non-negative countably additive
functionals on $\frak C$, $K$ is a countable set of infinite cardinals,
$\langle\mu_{\kappa}\rangle_{\kappa\in K}$ is a family of non-zero
non-negative countably additive functionals on $\frak C$, and
$\sum_{n=0}^{\infty}\mu_n+\sum_{\kappa\in K}\mu_{\kappa}
=\bar\mu$\cmmnt{;  this is
the burden of 333K}.

Next, given any structure of this second kind, we have a corresponding
closed subalgebra $\frak C_0$ of $\frak C$, a measure $\mu$ on $\Bbb
R^I$, where $I=\Bbb N\cup K$, and an isomorphism $\pi$ from the measure
algebra $\frak C_0^*$ of $\mu$ to $\frak C_0$, all uniquely
defined from the family $\langle\mu_i\rangle_{i\in I}$\cmmnt{ by the
process of
333M}.   \cmmnt{For any $E$ belonging to the domain $\Sigma$ of $\mu$,
and
$i\in I$, we have

\Centerline{$\mu_i\pi E^{\ssbullet}=\int_Ex(i)\mu(dx)$}

\noindent (333Mc), so that} $\mu_i\restrp\frak C_0$ is fixed by $\pi$ and
$\mu$.   Moreover, the functionals $\mu_i$ can be recovered from their
restrictions to $\frak C_0$\cmmnt{ by the formulae of 327F
(333Md)}.   Thus
from $(\frak C,\bar\mu,\langle\mu_i\rangle_{i\in I})$ we are led, by a
canonical and reversible process, to the structure

\Centerline{$(\frak C,\bar\mu,\frak C_0,I,\mu,\pi)$.}

But the extension $\frak C$ of $\frak C_0=\pi[\frak C_0^*]$ can
be described, up to isomorphism, by the same process as before;  that
is, it corresponds to a sequence $\sequencen{\theta'_n}$ and a family
$\langle\theta'_{\kappa}\rangle_{\kappa\in L}$ of countably additive
functionals on $\frak C_0$ satisfying the conditions of 333K.   We can
transfer these to $\frak C_0^*$, where they correspond to
families $\sequencen{\theta_n}$,
$\langle\theta_{\kappa}\rangle_{\kappa\in L}$ of absolutely continuous
countably additive functionals defined on $\Sigma$, setting

\Centerline{$\theta_jE=\theta'_j\pi E^{\ssbullet}$}

\noindent for $E\in\Sigma$, $j\in\Bbb N\cup L$.   This process too is
reversible;  every absolutely continuous countably additive functional
$\nu$ on $\Sigma$ corresponds to countably additive functionals on
$\frak C_0^*$ and $\frak C_0$.   \cmmnt{Let me repeat that the
results of 327F mean that the whole structure
$(\frak C,\bar\mu,\langle\mu_i\rangle_{i\in I})$ can be recovered from
$(\frak C_0,\bar\mu\restrp\frak C_0,
\langle\mu_i\restrp\frak C_0\rangle_{i\in I})$ if we can get the
description of $(\frak C,\bar\mu)$ right, and that
the requirements $\mu_i\ge 0$, $\mu_n\ge\mu_{n+1}$, $\mu_{\kappa}\ne 0$,
$\sum_{i\in I}\mu_i=\bar\mu$ imposed in 333K will survive the process
(327F(b-iv)).
}

Putting all this together, a structure $(\frak A,\bar\mu,\frak C)$
leads, in
a canonical and (up to isomorphism) reversible way, to a structure

\Centerline{$(K,\mu,L,\langle\theta_j\rangle_{j\in\Bbb N\cup L})$}

\noindent such that

\inset{$K$ and $L$ are countable sets of infinite cardinals,}

\inset{$\mu$ is a totally finite measure on $\BbbR^I$, where
$I=\Bbb N\cup K$, and its domain $\Sigma$ is precisely the
$\sigma$-algebra of subsets of $\BbbR^I$ defined by the coordinate
functionals,}

\inset{for $\mu$-almost every $x\in\BbbR^I$ we have $x(i)\ge 0$ for
every $i\in I$, $x(n)\ge x(n+1)$ for every $n\in\Bbb N$ and $\sum_{i\in
I}x(i)=1$,}

\inset{for $\kappa\in K$, $\mu\{x:x(\kappa)>0\}>0$,}

\cmmnt{\noindent (these two clauses corresponding to the requirements
$\mu_i\ge 0$, $\mu_n\ge\mu_{n+1}$, $\sum_{i\in I}\mu_i=\bar\mu$,
$\mu_{\kappa}\ne 0$ -- see 333M(e)-(g))}

\inset{for $j\in J=\Bbb N\cup L$, $\theta_j$ is a non-negative
countably additive functional on $\Sigma$,}

\inset{$\theta_n\ge\theta_{n+1}$ for every $n\in\Bbb N$,
$\theta_{\kappa}\ne 0$ for every $\kappa\in L$, $\sum_{j\in
J}\theta_j=\mu$.}

\cmmnt{
\leader{333O}{Remark} I do not envisage quoting the result above very
often.   Indeed I do not claim that its final form adds anything to the
constituent results 333K, 327F and 333M.   I have taken the trouble
to spell it out, however, because it does not seem to me obvious that
the trail is going to end quite as quickly as it does.   We need to use
333K twice, but only twice.   The most important use of the ideas
expressed here, I suppose, is in constructing examples to strengthen our
intuition for the structures $(\frak A,\bar\mu,\frak C)$ under
consideration, and
I hope that you will experiment in this direction.}

\leader{333P}{}\cmmnt{ At the risk of trespassing on the province of
Chapter 38, I turn now to a special type of closed subalgebra, in which
there is a particularly elegant alternative form for a canonical
description.
The first step is an important result concerning automorphisms of
homogeneous probability algebras.

\medskip

\noindent}{\bf Proposition} Let $(\frak B,\bar\nu)$ be a homogeneous
probability algebra.   Then there is a measure-preserving automorphism
$\phi:\frak B\to\frak B$ such that

\Centerline{$\lim_{n\to\infty}\bar\nu(c\Bcap\phi^n(b))
=\bar\nu c\cdot\bar\nu b$}

\noindent for all $b$, $c\in\frak B$.

\proof{{\bf (a)} The case $\frak B=\{0,1\}$ is trivial ($\phi$ is, and
must be, the identity map) so we may take it that
$(\frak B,\bar\nu)=(\frak B_{\kappa},\bar\nu_{\kappa})$ for some infinite
cardinal $\kappa$ is an infinite cardinal.   Because
$\#(\kappa\times\Bbb Z)=\max(\omega,\kappa)=\kappa$, there must be a
permutation $\theta:\kappa\to\kappa$ such that every orbit of $\theta$ in
$\kappa$ is infinite (take $\theta$ to correspond to the bijection
$(\xi,n)\mapsto(\xi,n+1):\kappa\times\Bbb Z\to\kappa\times\Bbb Z$).
This induces a permutation
$\hat\theta:\{0,1\}^{\kappa}\to\{0,1\}^{\kappa}$ through the formula
$\hat\theta(x)=x\theta$ for every $x\in\{0,1\}^{\kappa}$, and of course
$\hat\theta$ is an automorphism of the measure space
$(\{0,1\}^{\kappa},\nu_{\kappa})$.   We therefore have a
corresponding automorphism $\phi$ of $\frak B$, setting
$\phi E^{\ssbullet}=(\hat\theta^{-1}[E])^{\ssbullet}$ for every $E$ in the
domain $\Tau_{\kappa}$ of $\nu_{\kappa}$.

\medskip


{\bf (b)} Let $\Cal E$ be the family of subsets $E$ of
$\{0,1\}^{\kappa}$ which are determined by coordinates in finite sets,
that is, are expressible in the form $E=\{x:x\restr J\in \tilde E\}$ for
some finite set $J\subseteq\kappa$ and some
$\tilde E\subseteq\{0,1\}^J$;  equivalently, expressible as a finite union of
basic cylinder sets $\{x:x\restr J=y\}$.   Then $\Cal E$ is a
subalgebra of $\Tau_{\kappa}$, so $\frak C=\{E^{\ssbullet}:E\in\Cal E\}$ is a
subalgebra of $\frak B$.

\medskip


{\bf (c)} Now if $b$, $c\in\frak C$, there is an $n\in\Bbb N$ such that
$\bar\nu(c\Bcap\phi^m(b))=\bar\nu c\cdot\bar\nu b$ for every $m\ge n$.
\Prf\ Express $b$, $c$ as $E^{\ssbullet}$, $F^{\ssbullet}$ where
$E=\{x:x\restr J\in\tilde E\}$, $F=\{x:x\restr K\in\tilde F\}$ and $J$,
$K$ are finite
subsets of $\kappa$.   For $\xi\in K$, all the $\theta^n(\xi)$ are
distinct, so only finitely many of them can belong to $J$;  as $K$ is
also finite, there is an $n$ such that $\theta^m[J]\cap K=\emptyset$ for
every $m\ge n$.   Fix $m\ge n$.   Then $\phi^m(b)=H^{\ssbullet}$ where

\Centerline{$H=\{x:x\theta^m\in E\}=\{x:x\theta^m\restr J\in\tilde E\}
=\{x:x\restr L\in\tilde H\}$,}

\noindent where $L=\theta^m[J]$ and
$\tilde H=\{z\theta^{-m}:z\in\tilde E\}$.   So
$\bar\nu(c\Bcap\phi^m(b))=\nu(F\cap H)$.   But $L$ and $K$ are disjoint,
because $m\ge n$, so $F$ and $H$ must be independent (cf.\ 272K), and

\Centerline{$\bar\nu(c\Bcap\phi^m(b))=\nu F\cdot\nu H
=\nu F\cdot\nu E=\bar\nu c\cdot\bar\nu b$,}

\noindent as claimed.\ \Qed

\medskip

{\bf (d)} Now recall that for every $E\in\Tau_{\kappa}$ and
$\epsilon>0$ there is an $E'\in\Cal E$ such that
$\nu(E\symmdiff E')\le\epsilon$ (254Fe).   So, given $b$, $c\in\frak B$
and $\epsilon>0$, we can
find $b'$, $c'\in\frak C$ such that $\bar\nu(b\Bsymmdiff b')\le\epsilon$
and $\bar\nu(c\Bsymmdiff c')\le\epsilon$, and in this case

$$\eqalign{\limsup_{n\to\infty}
&|\bar\nu(c\Bcap\phi^n(b))-\bar\nu c\cdot\bar\nu b|\cr
&\le\limsup_{n\to\infty}|
\bar\nu(c\Bcap\phi^n(b))-\bar\nu(c'\Bcap\phi^n(b'))|\cr
&\qquad\qquad\qquad\qquad
  +|\bar\nu(c'\Bcap\phi^n(b'))-\bar\nu c'\cdot\bar\nu b'|
  +|\bar\nu c\cdot\bar\nu b-\bar\nu c'\cdot\bar\nu b'|\cr
&=\limsup_{n\to\infty}|
\bar\nu(c\Bcap\phi^n(b))-\bar\nu(c'\Bcap\phi^n(b'))|
  +|\bar\nu c\cdot\bar\nu b-\bar\nu c'\cdot\bar\nu b'|\cr
&\le\limsup_{n\to\infty}\bar\nu(c\Bsymmdiff c')
  +\bar\nu(\phi^n(b)\Bsymmdiff\phi^n(b'))\cr
&\qquad\qquad\qquad\qquad
  +\bar\nu c|\bar\nu b-\bar\nu b'|+|\bar\nu c-\bar\nu c'|\bar\nu b'\cr
&\le\bar\nu(c\Bsymmdiff c')+\bar\nu(b\Bsymmdiff b')
  +\bar\nu c\cdot\bar\nu(b\Bsymmdiff b')
  +\bar\nu(c\Bsymmdiff c')\cdot\bar\nu b'
\le 4\epsilon.\cr}$$

\noindent As $\epsilon$ is arbitrary,

\Centerline{$\lim_{n\to\infty}\bar\nu(c\Bcap\phi^n(b))
=\bar\nu c\cdot\bar\nu b$,}

\noindent as required.
}%end of proof of 333P

\cmmnt{\medskip

\noindent{\bf Remark} Automorphisms of this type are called {\bf
mixing} (see 372O below).
}

\leader{333Q}{Corollary} Let $(\frak C,\bar\mu_0)$ be a totally finite
measure algebra and $(\frak B,\bar\nu)$ a probability algebra which is
{\it either} homogeneous {\it or} purely atomic with finitely many atoms
all of the same measure.   Let $(\frak A,\bar\mu)$ be the localizable
measure algebra free product of $(\frak C,\bar\mu_0)$ and
$(\frak B,\bar\nu)$. Then
there is a measure-preserving automorphism $\pi:\frak A\to\frak A$ such
that

\Centerline{$\{a:a\in\frak A,\,\pi a=a\}=\{c\otimes 1:c\in\frak C\}$.}

\medskip

\cmmnt{
\noindent{\bf Remark} I am following 315N in using the notation
$c\otimes b$ for the intersection in $\frak A$ of the canonical images
of $c\in\frak C$ and $b\in\frak B$.   By 325D(c-i) I need not distinguish
between the free product $\frak C\otimes\frak B$ and its image in
$\frak A$.
}

\proof{ Set $\gamma=\bar\mu 1=\bar\mu_01$.

\medskip

{\bf (a)} Let me deal with the case of atomic $\frak B$ first.
In this case, if $\frak B$ has $n+1$ atoms $b_0,\ldots,b_n$, let
$\phi:\frak B\to\frak B$ be the measure-preserving homomorphism
cyclically permuting these atoms, so that
$\phi b_0=b_1,\ldots,\phi b_n=b_0$.   Because $\phi$ is an
automorphism of $(\frak B,\bar\nu)$, it
induces an automorphism $\pi$ of $(\frak A,\bar\mu)$;  any member of
$\frak A$ is uniquely expressible as $a=\sup_{i\le n}c_i\otimes b_i$, and
now $\pi a=\sup_{i\le n}c_i\otimes b_{i+1}$, if we set $b_{n+1}=b_0$.   So
$\pi a=a$ iff
$c_i=c_{i+1}$ for $i<n$ and $c_n=c_0$, that is, iff all the $c_i$
are the same and $a=\sup_{i\le n}c\otimes b_i=c\otimes 1$ for some
$c\in\frak C$.

\medskip

{\bf (b)} If $\frak B$ is homogeneous, then take a mixing
measure-preserving
automorphism $\phi:\frak B\to\frak B$ as described in 333P.   As in (a),
this corresponds to an automorphism $\pi$ of $\frak A$, defined by
saying that $\pi(c\otimes b)=c\otimes\phi(b)$ for every $c\in\frak C$,
$b\in\frak A$.   Of course $\pi(c\otimes 1)=c\otimes 1$ for every
$c\in\frak C$.

Now suppose that $a\in\frak A$ and $\pi a=a$;  I need to show that
$a\in\frak C_1=\{c\otimes 1:c\in\frak C\}$.
Take any $\epsilon\in\ocint{0,\bover14}$.
We know that $\frak C\otimes\frak B$ is topologically dense in $\frak A$
(325Dc), so there is an $a'\in\frak C\otimes\frak B$ such that
$\bar\mu(a\Bsymmdiff a')\le\epsilon^2$.   Express $a'$ as
$\sup_{i\in I}c_i\otimes b_i$, where $\langle c_i\rangle_{i\in I}$ is a
finite partition of unity in $\frak C$ (315Oa).   Then

\Centerline{$\pi a'=\sup_{i\in I}c_i\otimes\phi(b_i)$,
\quad$\pi^n(a')=\sup_{i\in I}c_i\otimes\phi^n(b_i)$ for every
$n\in\Bbb N$.}

\noindent So we can get a formula for

$$\eqalign{\lim_{n\to\infty}\bar\mu(a'\Bcap\pi^n(a'))
&=\lim_{n\to\infty}\bar\mu(\sup_{i\in
I}c_i\otimes(b_i\Bcap\phi^n(b_i)))\cr
&=\lim_{n\to\infty}\sum_{i\in
I}\bar\mu_0c_i\cdot\bar\nu(b_i\Bcap\phi^n(b_i))
=\sum_{i\in I}\bar\mu_0c_i\cdot(\bar\nu b_i)^2.\cr}$$

\noindent It follows that

$$\eqalign{\sum_{i\in I}\bar\mu_0c_i\cdot(\bar\nu b_i)^2
&=\lim_{n\to\infty}\bar\mu(a'\Bcap\pi^n(a'))\cr
&\ge\limsup_{n\to\infty}\bar\mu(a\Bcap\pi^n(a))
  -\bar\mu(a\Bsymmdiff a')-\bar\mu(\pi^n(a)\Bsymmdiff\pi^n(a'))\cr
&=\bar\mu a-2\bar\mu(a\Bsymmdiff a')
\ge\bar\mu a'-3\bar\mu(a\Bsymmdiff a')
\ge\sum_{i\in I}\bar\mu_0c_i\cdot\bar\nu b_i-3\epsilon^2,\cr}$$

\noindent that is,

\Centerline{$\sum_{i\in I}
 \bar\mu_0c_i\cdot\bar\nu b_i\cdot(1-\bar\nu b_i)
\le 3\epsilon^2$.}

\noindent But this means that, setting
$J=\{i:i\in I,\,\bar\nu b_i\cdot(1-\bar\nu b_i)\ge\epsilon\}$, we must have
$\sum_{i\in J}\bar\mu_0c_i\le 3\epsilon$.   Set

\Centerline{$K=\{i:i\in I,\,\bar\nu b_i\ge 1-2\epsilon\}$,
\quad$L=\{i:i\in I\setminus K,\,\bar\nu b_i\le 2\epsilon\}$,
\quad$c=\sup_{i\in K}c_i$.}

\noindent   Then $I\setminus(K\cup L)\subseteq J$, so

$$\eqalign{\bar\mu(a'\Bsymmdiff(c\otimes 1))
&=\sum_{i\in I\setminus K}\bar\mu_0c_i\cdot\bar\nu b_i+\sum_{i\in
K}\bar\mu_0c_i\cdot(1-\bar\nu
b_i)\cr
&\le\sum_{i\in J}\bar\mu_0c_i\cdot\bar\nu b_i+\sum_{i\in
L}\bar\mu_0c_i\cdot\bar\nu b_i+\sum_{i\in
K}\bar\mu_0c_i\cdot(1-\bar\nu b_i)\cr
&\le\sum_{i\in J}\bar\mu_0c_i+2\epsilon\sum_{i\in
L}\bar\mu_0c_i+2\epsilon\sum_{i\in K}\bar\mu_0c_i
\le 3\epsilon+2\epsilon\gamma,\cr}$$

\noindent and

\Centerline{$\bar\mu(a\Bsymmdiff(c\otimes 1))
\le\epsilon^2+3\epsilon+2\epsilon\gamma$.}

As $\epsilon$ is arbitrary, $a$ belongs to the topological closure of
$\frak C_1$.   But of course $\frak C_1$ is a closed subalgebra of
$\frak A$ (325Dd), so must actually contain $a$.

As $a$ is arbitrary, $\pi$ has the required property.
}%end of proof of 333Q

\leader{333R}{}\cmmnt{ Now for the promised special type of closed
subalgebra.   It will be convenient to have the following temporary
notation.}   For an integer
$n\ge 1$, I will\cmmnt{ (for this paragraph only)}
write $\frak B_n$ for the power set of
$\{0,\ldots,n\}$ and set $\bar\nu_nb=\bover1{n+1}\#(b)$ for
$b\in\frak B_n$.    \cmmnt{(The natural interpretation of
$(\frak B_n,\bar\nu_n)$ as defined in
333Ad corresponds to $(\frak B_{2^{n+1}-1},\bar\nu_{2^{n+1}-1})$ here,
so we have a match if $n=0$.)}

\medskip

\noindent{\bf Theorem} Let $(\frak A,\bar\mu)$ be a totally finite
measure algebra and $\frak C$ a subset of $\frak A$.
Then the following are equiveridical:

(i) there is some set $G$ of measure-preserving automorphisms of
$\frak A$ such that

\Centerline{$\frak C=\{c:c\in\frak A,\,\pi c=c$ for every $\pi\in G\}$;}

(ii) $\frak C$ is a closed subalgebra of $\frak A$ and there is a
partition of unity $\langle e_i\rangle_{i\in I}$ in $\frak C$, where $I$
is a countable set of cardinals, such that $\frak A$ is
isomorphic to $\prod_{i\in I}\frak C_{e_i}\widehat{\otimes}\frak B_i$,
writing $\frak C_{e_i}$ for the principal ideal of $\frak C$ generated
by $e_i$ and endowed with $\bar\mu\restrp\frak C_{e_i}$, and
$\frak C_{e_i}\widehat{\otimes}\frak B_i$ for the localizable measure
algebra
free product of $\frak C_{e_i}$ and $\frak B_i$ -- the isomorphism being
one which takes any $c\in\frak C$ to
$\langle (c\Bcap e_i)\otimes 1\rangle_{i\in I}$\cmmnt{, as in 333H and
333K};

(iii) there is a single measure-preserving automorphism $\pi$ of
$\frak A$ such that

\Centerline{$\frak C=\{c:c\in\frak A,\,\pi c=c\}$.}

\proof{{\bf (a)(i)$\Rightarrow$(ii)($\pmb{\alpha}$)} $\frak C$ is a
subalgebra because every $\pi\in G$ is a Boolean homomorphism, and it is
order-closed because every $\pi$ is order-continuous (324Kb).   (Or,
if you prefer, $\frak C$ is topologically closed because every $\pi$ is
continuous.)

\medskip

\quad\grheadb\ Because $\frak C$ is a closed subalgebra of $\frak A$,
its embedding can be described in terms of families $\sequencen{\mu_n}$,
$\langle\mu_{\kappa}\rangle_{\kappa\in K}$ as in Theorem 333K.   Set
$I=K\cup\Bbb N$.    Recall that each $\mu_i$ is defined by setting
$\mu_ic=\bar\mu(c\Bcap a_i)$, where $\langle a_i\rangle_{i\in I}$ is a
partition of unity in $\frak A$ (see the proofs of 333H and 333K).
For $\kappa\in K$, $a_{\kappa}$ is the maximal element of $\frak A$
which is relatively \Mth\ over $\frak C$ with relative
Maharam type $\kappa$ (part (b) of the proof of 333K).   Consequently we
must have $\pi a_{\kappa}=a_{\kappa}$ for any measure algebra
automorphism of $(\frak A,\bar\mu)$ which leaves $\frak C$ invariant;
in particular, for every $\pi\in G$.   Thus $a_{\kappa}\in \frak C$ for
every $\kappa\in K$.

\medskip

\quad\grheadc\
Now consider the relatively atomic part of $\frak A$.   The elements
$a_n$, for $n\in\Bbb N$, are not uniquely defined.   However, the
functionals $\mu_n$ and their supports $e'_n=\Bvalue{\mu_n>0}$
are uniquely defined from the structure $(\frak A,\bar\mu,\frak C)$
and therefore invariant under $G$.
Since

$$\eqalign{e'_n
&=1\Bsetminus\sup\{c:c\in\frak C,\,\mu_nc\le 0\}\cr
&=1\Bsetminus\sup\{c:c\in\frak C,\,c\Bcap a_n=0\}
=\inf\{c:c\in\frak C,\,c\Bsupseteq a_n\},\cr}$$

\noindent and
$\sup_{n\in\Bbb N}a_n=1\Bsetminus\sup_{\kappa\in K}a_{\kappa}$ belongs
to $\frak C$, while
$e'_n\supseteq e'_{n+1}$ for every $n$, we must have
$e'_0=\sup_{n\in\Bbb N}a_n$.

Let $G^*$ be the set of all those automorphisms $\pi$ of the measure
algebra $(\frak A,\bar\mu)$ such that $\pi c=c$ for every $c\in\frak C$.
Then of course $G^*$ is a group including $G$.   Now $\sup_{\pi\in
G^*}\pi a_n$ must be invariant under every member of $G^*$, so belongs
to $\frak C$;  it includes $a_n$ and is included in any member of $\frak
C$ including $a_n$, so must be $e'_n$.

\medskip

\quad\grheadd\ I claim now that if $n\in\Bbb N$ then
$e'_n\Bcap\Bvalue{\mu_0>\mu_n}=0$.   \Prf\Quer\   Otherwise, set
$c=\Bvalue{\mu_0>\mu_n}\Bcap e'_n$.   Then $\mu_0c>0$ so
$c\Bcap a_0\ne 0$.   By the last remark in ($\gamma$), there is a
$\pi\in G^*$ such
that $c\Bcap a_0\Bcap\pi a_n\ne 0$.   Now there is a $c'\in\frak C$ such
that $c\Bcap a_0\Bcap\pi a_n=c'\Bcap a_0$, and of course we may suppose
that $c'\Bsubseteq c$.   But this means that

\Centerline{$\pi(c'\Bcap a_n)
=c'\Bcap\pi a_n\Bsupseteq c'\Bcap a_0\Bcap\pi a_n= c'\Bcap a_0$,}

\noindent so that

\Centerline{$\mu_nc'=\bar\mu(c'\Bcap a_n)
=\bar\mu\pi(c'\Bcap a_n)
\ge\bar\mu(c'\Bcap a_0)
=\mu_0c'$,}

\noindent which is impossible, because
$0\ne c'\Bsubseteq\Bvalue{\mu_0>\mu_n}$.\ \Bang\Qed

So $\mu_0c\le\mu_nc$ whenever $c\in\frak C$ and $c\Bsubseteq e'_n$.
Because the $\mu_k$ have been chosen to be a non-increasing sequence, we
must have $\mu_0c=\mu_1c=\ldots=\mu_nc$ for every $c\Bsubseteq e'_n$.

\medskip

\quad\grheade\ Recalling now that
$\sum_{i\in I}\mu_i=\bar\mu\restrp\frak C$, we see that
$\mu_0c\le\bover1{n+1}\bar\mu c$ for every $c\Bsubseteq e'_n$.   It
follows that if $e^*=\inf_{n\in\Bbb N}e'_n$, $\mu_0e^*=0$;
but this must mean that $e^*=0$.   Consequently, setting
$e_n=e'_n\Bsetminus e'_{n+1}$ for $n\in\Bbb N$,
$e_{\kappa}=a_{\kappa}$ for $\kappa\in K$,  we find that
$\langle e_i\rangle_{i\in I}$ is a partition of unity in $\frak C$.

Moreover, for $n\in\Bbb N$ and $c\Bsubseteq e_n$, we must have
$\mu_{n+1}c=0$,

\Centerline{$\bar\mu c=\sum_{i\in I}\mu_ic=\sum_{k=0}^n\mu_kc
=(n+1)\mu_0c$,}

\noindent so that $\mu_kc=\bover1{n+1}\mu c$ for every $k\le n$.   But this
means that we have a measure-preserving homomorphism
$\psi_n:\frak A_{e_n}\to\frak C_{e_n}\widehat{\otimes}\frak B_n$ given
by setting

\Centerline{$\psi_n(a_k\Bcap c)=c\otimes\{k\}$}

\noindent whenever $c\in\frak C_{e_n}$ and $k\le n$;  this is well-defined
because $e_n\Bsubseteq e'_k$, so that $a_k\Bcap c\ne a_k\Bcap c'$ if
$c$, $c'$ are distinct members of $\frak C_{e_n}$, and it is
measure-preserving because

\Centerline{$\bar\mu(a_k\Bcap c)=\mu_kc=\Bover1{n+1}\bar\mu c
=\bar\mu c\cdot\bar\nu_n\{k\}$}

\noindent for all relevant $k$ and $c$.    Because $\frak B_n$ is
finite, $\psi_n$ is surjective.

\medskip

\quad\grheadz\ Just as in 333H, we now see that because $\langle
e_i\rangle_{i\in I}$ is a partition of unity in $\frak A$ as well as in
$\frak C$, we can identify $\frak A$ with $\prod_{i\in I}\frak A_{e_i}$
and therefore with $\prod_{i\in I}\frak C_{e_i}\widehat{\otimes}\frak
B_i$.

\medskip

{\bf (b)(ii)$\Rightarrow$(iii)} Let us work in
$\frak D=\prod_{i\in I}\frak C_{e_i}\widehat{\otimes}\frak B_i$, writing
$\psi:\frak A\to\frak D$ for the given isomorphism.
For each $i\in I$, we have a measure-preserving automorphism $\pi_i$ of
$\frak C_{e_i}\widehat{\otimes}\frak B_i$ with fixed-point subalgebra
$\{c\otimes 1:c\in\frak C_{e_i}\}$ (333Q).   For
$d=\langle d_i\rangle_{i\in I}\in\frak D$, set

\Centerline{$\pi d=\langle \pi_id_i\rangle_{i\in I}$.}

\noindent Then $\pi$ is a measure-preserving automorphism because every
$\pi_i$ is.   If $\pi d=d$, then for every $i\in I$ there must be a
$c_i\Bsubseteq e_i$ such that $d_i=c_i\otimes 1$.   But this means that
$d=\psi c$, where $c=\sup_{i\in I}c_i\in\frak C$.   Thus the fixed-point
subalgebra of $\pi$ is just $\psi[\frak C]$.   Transferring the
structure $(\frak D,\psi[\frak C],\pi)$ back to $\frak A$, we obtain a
measure-preserving automorphism $\psi^{-1}\pi\psi$ of $\frak A$ with
fixed-point subalgebra $\frak C$, as required.

\medskip

{\bf (c)(iii)$\Rightarrow$(i)} is trivial.
}%end of proof of 333R

\exercises{
\leader{333X}{Basic exercises $\pmb{>}$(a)}
%\sqheader 333Xa
Show that, in the proof of 333H,
$c_i=\upr(a_i,\frak C)$ (definition: 313S) for every $i\in I$.
%333H

\spheader 333Xb In Lemma 333J, show that every relative
atom in $\frak A$ over $\frak C$ belongs to the closed subalgebra of
$\frak A$ generated by $\frak C\cup\{a_n:n\in\Bbb N\}$.
%333J

\spheader 333Xc In the context of Lemma 333M, show that
if $I$ is countable we have
a one-to-one correspondence between atoms $c$ of $\frak C_0$ and points
$x$ of non-zero mass in $\BbbR^I$, given by the formula
$\pi\{x\}^{\ssbullet}=c$.
%333M

\spheader 333Xd Let $(\frak A,\bar\mu)$ be totally finite measure
algebra and $G$ a set of measure-preserving Boolean homomorphisms from
$\frak A$ to itself such that $\pi\phi\in G$ for all $\pi$, $\phi\in G$.
(i) Show that $a\Bsubseteq\sup_{\pi\in G}\pi a$ for every $a\in\frak A$.
\Hint{if $\pi c\Bsubseteq c$, where $\pi\in G$ and $c\in\frak A$, then
$\pi c=c$;  apply this to $c=\sup_{\pi\in G}\pi a$.}   (ii) Set
$\frak C=\{c:c\in\frak A,\,\pi c=c$ for every $\pi\in G\}$.   Show that
$\sup_{\pi\in G}\pi a=\upr(a,\frak C)$ for every $a\in\frak A$.
%333R

\leader{333Y}{Further exercises (a)} Show that when $I=\Bbb N$ the
algebra $\Sigma$ of
subsets of $\BbbR^I$, used in 333M, is precisely the Borel
$\sigma$-algebra as described in 271Ya.
%333M

\spheader 333Yb Let $(\frak A,\bar\mu)$ be a totally finite measure
algebra, and $\frak B$, $\frak C$ two closed subalgebras of $\frak A$
with $\frak C\subseteq\frak B$.   Show that
$\tau_{\frak C}(\frak B)\le\tau_{\frak C}(\frak A)$.   \Hint{use 333K
and the ideas of 332T.}
%333K

\spheader 333Yc Let $(\frak A,\bar\mu)$ be a probability algebra.
Show that $\frak A$ is homogeneous iff there is a measure-preserving
automorphism of $\frak A$ which is mixing in the sense of 333P.
%333P

\spheader 333Yd Let $(\frak A,\bar\mu)$ be a totally finite measure
algebra, and $G$ a set of measure-preserving Boolean homomorphisms from
$\frak A$ to itself.   Set $\frak C=\{c:c\in\frak A,\,\pi c=c$ for every
$\pi\in G\}$.   Show that $\frak C$ is a closed subalgebra of $\frak A$
of the type described in 333R.   \Hint{in the language of part (a) of
the proof of 333R, show that every $a_{\kappa}$ still belongs to
$\frak C$.}
%333R
}%end of exercises

\cmmnt{
\Notesheader{333} I have done my best, in the first part of this
section, to follow the lines already laid out in \S\S331-332, using
what should (once you have seen them) be natural generalizations of the
former definitions and arguments.   Thus the Maharam type
$\tau(\frak A)$ of an algebra is just the relative Maharam type
$\tau_{\{0,1\}}(\frak A)$, and $\frak A$ is \Mth\ iff it
is relatively \Mth\ over $\{0,1\}$.   To help you trace
through the correspondence, I list the code numbers:  331Fa$\to$333Aa,
331Fb$\to$333Ac, 331Hc$\to$333Ba, 331Hd$\to$333Bd, 332A$\to$333Bb,
331I$\to$333Cb, 331K$\to$333E,
331L$\to$333Fb, 332B$\to$333H, 332J$\to$333K.   333D overlaps
with 332P.   Throughout, the principle is the same:  everything can be
built up from products and free products.

Theorem 333Ca does not generalize any explicitly stated result, but
overlaps with Proposition 332P.   In the proof of 333E I have used a
new idea;  the same method would of course have worked just as well for
331K, but I thought it worth while to give an example of an
alternative
technique, which displays a different facet of homogeneous algebras, and
a different way in which the algebraic, topological and metric
properties of homogeneous algebras interact.   The argument of
331K-331L
relies (without using the term) on the fact that measure algebras of
Maharam type $\kappa$ have topological density at most
$\max(\kappa,\omega)$ (see 331Ye), while the the argument of 333E uses
the rather more sophisticated concept of stochastic independence.

Corollary 333Fa is cruder than the more complicated results which
follow, but I think that it is invaluable as a first step in forming a
picture of the possible embeddings of a given (totally finite) measure
algebra $\frak C$ in a larger algebra $\frak A$.   If we think of
$\frak C$ as the measure algebra of a measure space $(X,\Sigma,\mu)$,
then we can be sure that $\frak A$ is representable as a closed
subalgebra of
the measure algebra of $X\times\{0,1\}^{\kappa}$ for some $\kappa$, that
is, the measure algebra of $\lambda\restrp\Tau$ where $\lambda$ is the
product measure on $X\times\{0,1\}^{\kappa}$ and $\Tau$ is some
$\sigma$-subalgebra of the domain of $\lambda$;  the embedding of
$\frak C$ in $\frak A$ being defined by the formula
$E^{\ssbullet}\to(E\times\{0,1\}^{\kappa})^{\ssbullet}$ for $E\in\Sigma$
(325A, 325D).   Identifying, in our imaginations, both $X$ and
$\{0,1\}^{\kappa}$ with the unit interval, we can try to picture
everything in the unit square -- and these pictures, although
necessarily inadequate for algebras of uncountable Maharam type, already
give a great deal of scope for invention.

I said above that everything can be constructed from simple products and
free products, judiciously combined;  of course some further ideas must
be mixed with these.   The difference between 332B and 333H, for
instance, is partly in the need for the functionals $\mu_i$ in the
latter, whereas in the former the decomposition involves only principal
ideals with the induced measures.   Because the $\mu_i$ are
completely additive, they all have supports $c_i$ (326Xl) and we get
measure
algebras $(\frak C_{c_i},\mu_i\restrp\frak C_{c_i})$ to use in the
products.   (I note that the $c_i$ can be obtained directly from the
$a_i$, without mentioning the functionals $\mu_i$, by the process of
333Xa.)   The fact that the $c_i$ can overlap means that the `relatively
atomic' part of the larger algebra $\frak A$ needs a much more careful
description than before;  this is the burden of 333J, and also the
principal complication in the proof of 333R.   The `relatively atomless'
part is (comparatively) straightforward, since we can use the same kind
of amalgamation as before (part (c-i) of the proof of 332J, part (b)
of the proof of 333K), simplified because I am no longer seeking to deal
with algebras of infinite magnitude.

Theorem 333K gives a canonical form for superalgebras of a given totally
finite measure algebra $(\frak C,\bar\mu)$, taking the structure
$(\frak C,\bar\mu)$ itself for granted.   I hope it is clear that while
the $\mu_i$ and $e_i$ and the algebra
$\widehat{\frak A}
=\prod_{n\in\Bbb N}\frak C_{e_n}
  \times\prod_{\kappa\in K}
  \frak C_{e_{\kappa}}\widehat{\otimes}\frak B_{\kappa}$ and the embedding of
$\frak C$ in $\widehat{\frak A}$ are uniquely defined, the rest of the
isomorphism $\pi:\frak A\to\widehat{\frak A}$ generally is not.   Even
when the $a_{\kappa}$ are uniquely defined the isomorphisms between
$\frak A_{a_{\kappa}}$ and
$\frak C_{e_{\kappa}}\widehat{\otimes}\frak B_{\kappa}$ depend on
choosing generating families in the $\frak A_{a_{\kappa}}$;
see the proof of 333Cb.

To understand the possible structures
$(\frak C,\langle\mu_i\rangle_{i\in I})$ of that theorem, we have to go
rather deeper.   The route I have chosen is to pick out the subalgebra
$\frak C_0$ of $\frak C$ determined by $\langle\mu_i\rangle_{i\in I}$
and identify it with the measure algebra of a particular measure on
$\BbbR^I$.   Perhaps I should apologise for not stating explicitly in
the course of 333N that the measure $\mu$ there is a `Borel
measure' (see 333Ya);  but I am afraid of opening a door to an
invasion of ideas which belong in Volume 4.   Besides, if I were going
to do anything more with these measures than observe that they are
uniquely defined by the construction proposed, I would complete them and
call them Radon measures.   In order to validate this approach, I must
show that the $\mu_i$ can be recovered from their restrictions to
$\frak C_0$;  this is 333Md, and is the motive for the discussion of
`standard
extensions' in \S327.   No doubt there are other ways of doing it.
One temptation which I felt it right to resist was the idea of
decomposing $\frak C$ into its homogeneous principal ideals;  this
seemed merely an additional complication.   Of course the subalgebra
$\frak C_0$ has countable Maharam type (being $\tau$-generated by the
elements $e_{iq}$, for $i\in I$ and $q\in\Bbb Q$, of 333M), so that its
decomposition is relatively simple, being just a matter of picking out
the atoms (333Xc).

Another way of looking at the expression in 333N is to observe that
$\frak A$ is obtained by amalgamating two extensions of the core subalgebra
$\frak C_0$;  one defined by
$\langle\mu_i\restrp\frak C_0\rangle_{i\in\Bbb N\cup K}$, and one by
$\langle\theta'_j\rangle_{j\in\Bbb N\cup L}$.   After using the
second family to represent $(\frak C,\bar\mu\restrp\frak C,\frak C_0)$,
we obtain standard extensions $\mu_i$ from which we can represent
$(\frak A,\bar\mu,\frak C)$.   Put this way, it seems that the process
demands that the steps be performed in the given order.
In fact it can be made symmetric;  but for that I think we need the theory
of `relative free products', which I will come to in \S458
of Volume 4.

In 333P I find myself presenting an important fact about homogeneous
measure algebras, rather out of context;  but I hope that it will help
you to believe that I have by no means finished with the insights which
Maharam's theorem provides.   I give it here for the sake of 333R.   For
the moment, I invite you to think of 333R as just a demonstration of the
power of the techniques I have developed in this chapter, and of the
kind of simplification (in the equivalence of conditions (i) and (iii))
which seems to arise repeatedly in the theory of measure algebras.  But
you will see that the first step to understanding any automorphism will
be a description of its fixed-point subalgebra, so 333R will also be
basic to the theory of automorphisms of measure algebras.   I note that
the hypothesis (i) of 333R can in fact be relaxed (333Yd), but this
seems to need an extra idea.
}%end of notes

\discrpage
\frfilename{mt334.tex}
\versiondate{26.9.08}
\copyrightdate{1995}

\def\chaptername{Maharam's theorem}
\def\sectionname{Products}

\newsection{334}

I devote a short section to results on the Maharam classification of the
measure algebras of product measures, or, if you prefer, of the free
products of measure algebras.   The complete classification, even for
probability algebras, is complex\cmmnt{ (334Xc, 334Ya)}, so I content
myself
with a handful of the most useful results.   I start with upper bounds
for the Maharam type of the c.l.d.\ product of two measure spaces (334A)
and the localizable measure algebra free product of two semi-finite
measure algebras (334B), and go on to the corresponding results for
general
products of probability spaces and algebras (334C-334D).   Finally, I
show that any infinite power of a probability space is \Mth\ (334E).

In this section I will write $\tau(\mu)$ for the Maharam type of a measure
$\mu$, defined as the Maharam type of its measure algebra (331Fc).

\leader{334A}{Theorem} Let $(X,\Sigma,\mu)$ and $(Y,\Tau,\nu)$
be measure spaces, and
$\lambda$ the c.l.d.\ product measure on $X\times Y$.
Then $\tau(\lambda)\le\max(\omega,\tau(\mu),\tau(\nu))$.

\proof{ Let $\frak A$, $\frak B$, $\frak C$ be the measure algebras of
$\mu$, $\nu$ and $\lambda$, respectively.
Recall from 325A that we have order-continuous Boolean
homomorphisms $\varepsilon_1:\frak A\to\frak C$ and
$\varepsilon_2:\frak B\to\frak C$ defined by setting
$\varepsilon_1(E^{\ssbullet})=(E\times Y)^{\ssbullet}$,
$\varepsilon_2(F^{\ssbullet})=(X\times F)^{\ssbullet}$
for $E\in\Sigma$ and $F\in\Tau$.    Let $A\subseteq\frak A$,
$B\subseteq\frak B$ be $\tau$-generating sets with
$\#(A)=\tau(\frak A)=\tau(\mu)$, $\#(B)=\tau(\nu)$;  set
$C=\varepsilon_1[A]\cup\varepsilon_2[B]$.   Then $C\,\,\tau$-generates
$\frak C$.   \Prf\ Let $\frak C_1$ be the order-closed subalgebra of
$\frak C$ generated by $C$.   Because $\varepsilon_1$ is
order-continuous, $\varepsilon_1^{-1}[\frak C_1]$ is an
order-closed subalgebra of $\frak A$, and it includes $A$, so must be
the whole of $\frak A$;  thus $\varepsilon_1a\in\frak C_1$ for every
$a\in\frak A$.   Similarly, $\varepsilon_2b\in\frak C_1$ for every
$b\in\frak B$.

This means that

\Centerline{$\Lambda_1=\{W:W\in\Lambda,\,W^{\ssbullet}\in\frak C_1\}$}

\noindent contains $E\times F$ for every $E\in\Sigma$, $F\in\Tau$.
Also $\Lambda_1$ is a $\sigma$-algebra of subsets of $X\times Y$,
because $\frak C_1$ is (sequentially) order-closed in $\frak C$.   So
$\Lambda_1\supseteq\Sigma\tensorhat\Tau$ (definition:  251D).   But this
means that
if $W\in\Lambda$ there is a $V\in\Lambda_1$ such that $V\subseteq W$ and
$\lambda V=\lambda W$ (251Ib);  that is, if $c\in\frak C$ there
is a $d\in\frak C_1$ such that
$d\Bsubseteq c$ and $\bar\lambda d=\bar\lambda c$.   Thus $\frak C_1$ is
order-dense in $\frak C$, and

\Centerline{$c=\sup\{d:d\in\frak C_1,\,d\Bsubseteq c\}\in\frak C_1$}

\noindent for every $c\in\frak C$.   So $\frak C_1=\frak C$ and
$C\,\,\tau$-generates $\frak C$, as claimed.\ \Qed

Consequently

\Centerline{$\tau(\lambda)=\tau(\frak C)\le\#(C)
\le\max(\omega,\tau(\mu),\tau(\nu))$.}
}%end of proof of 334A

\leader{334B}{Corollary} Let $(\frak A,\bar\mu)$,
$(\frak B,\bar\nu)$ be
semi-finite measure algebras, with localizable measure algebra free
product $(\frak C,\bar\lambda)$\cmmnt{ (325E)}.   Then
$\tau(\frak C)\le\max(\omega,\tau(\frak A),\tau(\frak B))$.

\proof{ By the construction of part (a) of the proof of 325D, $\frak C$
can be regarded as the measure algebra of the c.l.d.\ product of the
Stone representations of $(\frak A,\bar\mu)$ and
$(\frak B,\bar\nu)$;  so the result follows at once from 334A.
}%end of proof of 334B

\vleader{36pt}{334C}{Theorem} Let
$\langle(X_i,\Sigma_i,\mu_i)\rangle_{i\in I}$ be a family of probability
spaces, with product $(X,\Lambda,\lambda)$.   Then

\Centerline{$\tau(\lambda)
\le\max(\omega,\#(I),\sup_{i\in I}\tau(\mu_i))$.}

\proof{ For $i\in I$, let $\frak A_i$ be the measure algebra of $\mu_i$;
let $\frak C$ be the measure algebra of $\lambda$.
Recall from 325I that we have order-continuous Boolean
homomorphisms $\varepsilon_i:\frak A_i\to\frak C$ corresponding to the
inverse-measure-preserving maps $x\mapsto\pi_i(x)=x(i):X\to X_i$.
For each $i\in I$, let $A_i\subseteq\frak A_i$ be a set with cardinal
$\tau(\mu_i)$ which $\tau$-generates $\frak A_i$.   Set
$C=\bigcup_{i\in I}\varepsilon_i[A_i]$.   Then $C\,\,\tau$-generates
$\frak C$.
\Prf\ Let $\frak C_1$ be the order-closed subalgebra of $\frak C$
generated by $C$.   Because $\varepsilon_i$ is
order-continuous, $\varepsilon_i^{-1}[\frak C_1]$ is an
order-closed subalgebra of $\frak A_i$, and it includes $A_i$, so must
be the whole of $\frak A_i$;  thus $\varepsilon_ia\in\frak C_1$ for every
$a\in\frak A_i$, $i\in I$.

This means that

\Centerline{$\Lambda_1=\{W:W\in\Lambda,\,W^{\ssbullet}\in\frak C_1\}$}

\noindent contains $\pi_i^{-1}[E]$ for every $E\in\Sigma_i$, $i\in I$.
Also $\Lambda_1$ is a $\sigma$-algebra of subsets of $X$, because
$\frak C_1$ is (sequentially) order-closed in $\frak C$.   So
$\Lambda_1\supseteq\Tensorhat_{i\in I}\Sigma_i$.   But this means
that if $W\in\Lambda$ there is a $V\in\Lambda_1$ such that
$V^{\ssbullet}=W^{\ssbullet}$ (254Ff), that is, that
$\frak C_1=\frak C$, and $C\,\,\tau$-generates $\frak C$, as claimed.\
\Qed

Consequently

\Centerline{$\tau(\lambda)=\tau(\frak C)\le\#(C)
\le\max(\omega,\#(I),\sup_{i\in I}\tau(\mu_i))$.}
}%end of proof of 334C

\leader{334D}{Corollary} Let
$\langle(\frak A_i,\bar\mu_i)\rangle_{i\in I}$
be a family of probability algebras, with probability algebra free
product $(\frak C,\bar\lambda)$.    Then

\Centerline{$\tau(\frak C)
\le\max(\omega,\#(I),\sup_{i\in I}\tau(\frak A_i))$.}

\proof{ See 325J-325K.}


\leader{334E}{}\cmmnt{ I come now to the question of when a
product of probability spaces is \Mth.   I give just one
result in detail, leaving others to the exercises.

\medskip

\noindent}{\bf Theorem} Let $(X,\Sigma,\mu)$ be a probability space
and $I$ an infinite set;  let $\lambda$ be the 
product measure on $X^I$.   Then $\lambda$ is \Mth.   If
$\tau(\mu)=0$ then $\tau(\lambda)=0$;   otherwise
$\tau(\lambda)=\max(\tau(\mu),\#(I))$.

\proof{{\bf (a)} Let $(\frak A,\bar\mu)$ be the measure algebra of $\mu$
and $(\frak C,\bar\lambda)$ the measure algebra of $\lambda$.
If $\tau(\mu)=0$, that is, $\frak A=\{0,1\}$, then
$\frak C=\{0,1\}$ (by 254Fe, or 325Jc, or otherwise), and in this case
is surely homogeneous, with $\tau(\frak C)=0$;  so that $\lambda$ is \Mth\
and $\tau(\lambda)=0$.   So let us suppose
hencforth that $\tau(\mu)>0$.  We have

\Centerline{$\tau(\frak C)
\le\max(\omega,\#(I),\tau(\frak A))=\max(\#(I),\tau(\frak A)$}

\noindent by 334C.

\medskip

{\bf (b)} Fix on $b\in\frak A\setminus\{0,1\}$.   For each $i\in I$, let
$\varepsilon_i:\frak A\to\frak C$ be the canonical measure-preserving
homomorphism corresponding to the \imp\ function
$x\mapsto x(i):X^I\to X$.   For each $n\in\Bbb N$, there is a set
$J\subseteq I$ with cardinal
$n$, and now the finite subalgebra of $\frak C$ generated by
$\{\varepsilon_ib:i\in J\}$ has atoms of measure at most $\delta^n$, where
$\delta=\max(\bar\mu b,1-\bar\mu b)<1$.   Consequently $\frak C$ can
have no
atom of measure greater than $\delta^n$, for any $n$, and is therefore
atomless.

\medskip

{\bf (c)} Because $I$ is infinite, there is a bijection between $I$ and
$I\times\Bbb N$;  that is, there is a partition 
$\langle J_i\rangle_{i\in I}$ of $I$ into countably infinite sets.   Now
$(X^I,\lambda)$ can be identified with the product of the family
$\langle(X^{J_i},\lambda_i)\rangle_{i\in I}$, where $\lambda_i$
is the product measure on $X^{J_i}$ (254N).   By (b), every $\lambda_i$
is atomless, so there are sets $E_i\subseteq X^{J_i}$ of measure
$\bover12$.   The sets $E'_i=\{x:x\restr J_i\in E_i\}$ are now
stochastically independent in $X$.   Accordingly we have an \imp\
function $f:X\to\{0,1\}^I$, endowed with its usual measure $\nu_I$,
defined by setting $f(x)(i)=1$ if $x\in E'_i$, $0$ otherwise, and
therefore a measure-preserving Boolean homomorphism 
$\pi:\frak B_I\to\frak C$, writing $\frak B_I$ for the measure algebra of
$\nu_I$.

Now if $c\in\frak C\setminus\{0\}$ and $\frak C_c$ is the corresponding
ideal, $b\mapsto c\Bcap\pi b:\frak B_I\to\frak C_c$ is an
order-continuous Boolean homomorphism.   It follows that
$\tau(\frak C_c)\ge\#(I)$ (331Jb).

\medskip

{\bf (d)} Again take any non-zero $c\in\frak C$.   For each $i\in I$,
set $a_i=\inf\{a:\varepsilon_ia\Bsupseteq c\}$.
Writing $\frak A_{a_i}$ for
the corresponding principal ideal of $\frak A$, we have an
order-continuous Boolean homomorphism
$\varepsilon'_i:\frak A_{a_i}\to\frak C_c$, given by the formula

\Centerline{$\varepsilon'_ia=\varepsilon_ia\Bcap c$
for every $a\in\frak A_{a_i}$.}

\noindent Now $\varepsilon'_i$ is injective, so is a Boolean isomorphism
between $\frak A_{a_i}$ and its image $\varepsilon'_i[\frak A_{a_i}]$,
which by 314F(a-i) is a closed subalgebra of $\frak C_c$.   So

\Centerline{$\tau(\frak A_{a_i})
=\tau(\varepsilon'_i[\frak A_{a_i}])\le\tau(\frak C_c)$}

\noindent by 332Tb.

For any finite $J\subseteq I$,

\Centerline{$0<\bar\lambda c\le\bar\lambda(\inf_{i\in J}\varepsilon_ia_i)
=\prod_{i\in J}\bar\lambda(\varepsilon_ia_i)=\prod_{i\in J}\bar\mu a_i$.}

\noindent So for any $\delta<1$, $\{i:\bar\mu a_i\le\delta\}$ must be
finite, and $\sup_{i\in I}\bar\mu a_i=1$.   In particular,
$\sup_{i\in I}a_i=1$ in $\frak A$.   But this means that if $\zeta$ is
any cardinal
such that the Maharam-type-$\zeta$ component $e_{\zeta}$ of $\frak A$ is
non-zero, then $e_{\zeta}\Bcap a_i\ne 0$ for some $i\in I$, so that

\Centerline{$\zeta\le\tau(\frak A_{e_{\zeta}\Bcap a_i})
\le\tau(\frak A_{a_i})\le\tau(\frak C_c)$.}

\noindent As $\zeta$ is arbitrary,
$\tau(\frak A)\le\max(\omega,\tau(\frak C_c))$ (332S).

\medskip

{\bf (e)} Putting (a)-(d) together, we have

\Centerline{$\max(\tau(\frak A),\#(I))\le\max(\omega,\tau(\frak C_c))
=\tau(\frak C_c)\le\tau(\frak C)\le\max(\tau(\frak A),\#(I))$}

\noindent for every non-zero $c\in\frak C$;  so $\frak C$ is
homogeneous, with $\tau(\frak C)=\max(\tau(\frak A),\#(I))$.   
Re-stating this in terms of $\lambda$ and $\mu$, $\lambda$ is \Mth\
and $\tau(\lambda)=\max(\tau(\mu),\#(I))$.
}%end of proof of 334E

\exercises{
\leader{334X}{Basic exercises (a)}
%spheader 334Xa
Let $(X,\Sigma,\mu)$ and
$(Y,\Tau,\nu)$ be complete locally determined measure spaces with c.l.d.\
product $(X\times Y,\Lambda,\lambda)$.   Show that
if $\nu Y>0$ then $\tau(\mu)\le\tau(\lambda)$.
%334A

\sqheader 334Xb Let $\langle(\frak A_i,\bar\mu_i)\rangle_{i\in I}$ be a
family of probability algebras, with probability algebra free product
$(\frak C,\bar\lambda)$.    Show that $\tau(\frak A_i)\le\tau(\frak C)$
for every $i$, and that

\Centerline{$\#(\{i:i\in I,\,\tau(\frak A_i)>0\})\le\tau(\frak C)$.}
%334C

\spheader 334Xc Let $(X,\Sigma,\mu)$ and $(Y,\Tau,\nu)$ be
$\sigma$-finite measure spaces, and $\lambda$ the product measure on
$X\times Y$.   Show that $\lambda$ is \Mth\ iff one of
$\mu$, $\nu$ is \Mth\ with Maharam type at least
as great as the Maharam type of the other.
%334E

\spheader 334Xd Show that the product of any family of \Mth\ probability
spaces is again \Mth.
%334E

\sqheader 334Xe Let $(X,\Sigma,\mu)$ be a probability space of
Maharam type $\kappa$, and $I$ any set with cardinal at least
$\max(\omega,\kappa)$.   Show that the product measure on
$X\times\{0,1\}^I$ is \Mth, with Maharam type $\#(I)$.
%334E

\leader{334Y}{Further exercises (a)} Let
$\langle(X_i,\Sigma_i,\mu_i)\rangle_{i\in I}$ be an infinite family
of probability spaces, with product $(X,\Lambda,\lambda)$.   Let
$\kappa_i$ be the Maharam type of $\mu_i$ for each $i$;  set
$\kappa=\max(\#(I),\sup_{i\in I}\kappa_i)$.  Show that {\it either}
$\lambda$ is \Mth, with Maharam type $\kappa$, {\it
or} there are $\kappa'<\kappa$, $X'_i\in\Sigma_i$ such that
$\sum_{i\in I}\mu_i(X_i\setminus X'_i)<\infty$, the Maharam type of
the subspace measure on $X'_i$ is at most $\kappa'$ for every $i\in I$
and $\#(\{i:\kappa_i\ne 0\})\le\kappa'$.
}%end of exercises

\cmmnt{
\Notesheader{334} The results above are all very natural ones;  I have
spelt them out partly for completeness and partly for the sake of an
application in \S346 below.
%346G uses 334E
But note the second alternative in
334Ya;  it is possible, even in an infinite product, for a kernel of
relatively small Maharam type to be preserved.

}%end of notes

\frnewpage

\references{
\frfilename{mt3r.tex}
\versiondate{25.8.17}
\copyrightdate{1999}

% 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 3}

\medskip

%A

Anderson I.\ [87] {\it Combinatorics of Finite Sets.}  Oxford U.P., 1987.
\cmmnt{[332Xk, 3A1K.]}

% Anzai H.\ [51] `Ergodic skew product transformations of the torus',
% % Osaka Math.\ J.\ 3 (1951) 83-99.

Anzai H.\ [51] `On an example of a measure-preserving transformation
which is not conjugate to its inverse',
Proc.\ Japanese Acad.\ Sci.\ 27 (1951) 517-522.
\cmmnt{[\S382 {\it notes\/}.]}

\medskip
%B

Balcar B., G\l\'owczy\'nski W. \& Jech T.\ [98] `The sequential
topology on complete Boolean algebras', Fundamenta Math.\ 155 (1998)
59-78.
\cmmnt{[393L, 393Q.]}

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{[393L, 393Q.]}
% arXiv.math.LO/0312473

Becker H.\ \& Kechris A.S.\ [96] {\it The descriptive set theory of
Polish group actions.}  Cambridge U.P., 1996
(London Math.\ Soc.\ Lecture Note Series 232).
\cmmnt{[\S395 {\it notes\/}.]}

Bekkali M.\ \& Bonnet R.\ [89] `Rigid Boolean algebras', pp.\ 637-678 in
{\smc Monk 89}.
\cmmnt{[384L.]}

Bellow A.\ \& K\"olzow D.\ [76] {\it Measure Theory, Oberwolfach 1975.}
Springer, 1976 (Lecture Notes in Mathematics 541).

Bhaskara Rao, K.P.S.\ \& Bhaskara Rao, M.\ {\it Theory of Charges.}
Academic, 1983.
%326H

Billingsley P.\ [65] {\it Ergodic Theory and Information.}  Wiley, 1965.
\cmmnt{[\S372 {\it notes\/}.]}

Bollob\'as B.\ [79] {\it Graph Theory.}   Springer, 1979.
\cmmnt{[332Xk, 3A1K.]}

Bose R.C.\ \& Manvel B.\ [84] {\it Introduction to Combinatorial Theory.}   Wiley, 1984.
\cmmnt{[3A1K.]}

Bourbaki N.\ [66] {\it General Topology.}  Hermann/Addison-Wesley, 1968.
\cmmnt{[\S3A3, \S3A4.]}

Bourbaki N. [68] {\it Theory of Sets.}   Hermann/Addison-Wesley, 1968.
\cmmnt{[\S315 {\it notes\/}.]}

Bourbaki N. [87] {\it Topological Vector Spaces.}  Springer, 1987.
\cmmnt{[\S3A5.]}

Bukhvalov A.V.\ [95] `Optimization without compactness, and its
applications', pp.\ 95-112 in
{\smc Huijsmans Kaashoek Luxemburg \& Pagter 95}.
\cmmnt{[367U.]}

Burke M.R.\ [93] `Liftings for Lebesgue measure', pp.\ 119-150 in
{\smc Judah 93}.   \cmmnt{[341L, 345F.]}

Burke M.R.\ [n95] `Consistent liftings', privately circulated, 1995.
\cmmnt{[346Ya.]}

Burnside W.\ [1911] {\it Theory of Groups of Finite Order.}   Cambridge
U.P., 1911 (reprinted by Dover, 1955).
\cmmnt{[\S384 {\it notes\/}.]}

\medskip
%C

Chacon R.V. [69]
`Weakly mixing transformations which are not strongly mixing',
Proc.\ Amer.\ Math.\ Soc.\ 22 (1969) 559-562.
\cmmnt{[372R.]}

Chacon R.V.\ \& Krengel U.\ [64] `Linear modulus of a linear
operator', Proc.\ Amer.\ Math.\ Soc.\ 15 (1964) 553-559.
\cmmnt{[\S371 {\it notes\/}.]}

Choksi J.R.\ \& Prasad V.S.\ [82] `Ergodic theory of homogeneous measure
algebras', pp.\ 367-408 of {\smc K\"olzow \& Maharam-Stone 82}.
\cmmnt{[383I.]}

%Christenson C.O.\ \& Voxmann L.\ [77] {\it Aspects of Topology.}
%Dekker, 1977.

Cohn H.\ [06] `A short proof of the simple continued fraction expansion of
$e$', Amer.\ Math.\ Monthly 113 (2006) 57-62;
{\tt arXiv:math.NT/0601660}.
\cmmnt{[372L.]}

Coleman A.J.\ \& Ribenboim P.\ [67] (eds.) {\it Proceedings of the Symposium in Analysis, Queen's University, June 1967.}  Queen's University, Kingston, Ontario, 1967.

Comfort W.W.\ \& Negrepontis S.\ [82] {\it Chain Conditions in
Topology.}   Cambridge U.P., 1982.   \cmmnt{[\S391 {\it notes\/}.]}

\leaveitout{
Cornfeld I.P., Fomin S.V.\ \& Sina\v\i Ya.G.\ [82] {\it Ergodic Theory.}
Springer, 1982 (Grundlehren der mathematischen Wissenschaften 245)
}

Czisz\'ar I.\ [67] `Information-type measures of difference of
probability distributions and indirect observations',
Studia Scientiarum Math.\ Hungarica 2 (1967) 299-318.   \cmmnt{[386G.]}

\medskip
%D

Davey B.A. \& Priestley H.A. [90] `Introduction to Lattices and Order',
Cambridge U.P., 1990.  \cmmnt{[3A6C.]}

%Dow A.\ \& Stepr\=ans J.\ [93] `The $\sigma$-linkedness of the measure algebra', Canad.\ Math.\ Bulletin 37 (1993) 42-45.

Dugundji J.\ [66] {\it Topology.}  Allyn \& Bacon, 1966.
\cmmnt{[\S3A3, 3A4Bb.]}

Dunford N.\ [1936] `Integration and linear operators',
Trans.\ Amer.\ Math.\ Soc.\ 40 (1936) 474-494.
\cmmnt{[\S376 {\it notes\/}.]}

Dunford N.\ \& Schwartz J.T.\ [57] {\it Linear Operators I.}  Wiley,
1957 (reprinted 1988).
\cmmnt{[\S356 {\it notes\/}, \S371 {\it notes\/}, \S376 {\it notes\/},
\S3A5.]}

Dye H.A.\ [59] `On groups of measure preserving transformations I',
Amer.\ J.\ Math.\ 81 (1959) 119-159.   \cmmnt{[\S388 {\it notes\/}.]}

\medskip
%E

Eigen S.J.\ [82] `The group of measure-preserving transformations of
$[0,1]$ has no outer automorphisms', Math.\ Ann.\ 259 (1982) 259-270.
\cmmnt{[384D.]}

Engelking R.\ [89] {\it General Topology.} Heldermann, 1989 (Sigma
Series in Pure Mathematics 6).  \cmmnt{[\S3A3, \S3A4.]}

Enderton H.B.\ [77] {\it Elements of Set Theory.}   Academic, 1977.
\cmmnt{[\S3A1.]}
%QA248.E6  7850 2996 6

Erd\H{o}s P.\ \& Oxtoby J.C.\ [55] `Partitions of the plane into sets
having positive measure in every non-null product set', Trans.\ Amer.\
Math.\ Soc.\ 79 (1955) 91-102.   \cmmnt{[\S325 {\it notes\/}.]}

\medskip
%F

%Farah I.\ [04] `Examples of $\epsilon$-exhaustive pathological
%submeasures', Fundamenta Math.\ 181 (2004) 257-272.

Fathi A.\ [78] `Le groupe des transformations de $[0,1]$ qui
pr\'eservent la mesure de Lebesgue est un groupe simple', Israel J.\
Math.\ 29 (1978) 302-308.  \cmmnt{[\S382 {\it notes\/}, 383I.]}

Fremlin D.H.\ [74a] {\it Topological Riesz Spaces and
Measure Theory.}   Cambridge U.P., 1974.
\cmmnt{[Chap.\ 35 {\it intro\/}., 354Yb, \S355 {\it notes\/},
\S356 {\it notes\/}, \S363 {\it notes\/}, \S365 {\it notes\/},
\S371 {\it notes\/}, \S376 {\it notes\/}.]}

Fremlin D.H.\ [74b] `A characterization of $L$-spaces', Indag.\ Math.\
36 (1974) 270-275.  \cmmnt{[\S371 {\it notes\/}.]}

%Fremlin D.H.\ [84] {\it Consequences of Martin's Axiom.}
%Cambridge U.P., 1984.

\leaveitout{Fremlin D.H.\ [89] `Measure algebras', pp.\ 876-980 in
{\smc Monk 89}.}

Fremlin D.H., de Pagter B.\ \&  Ricker W.J.\ [05]
`Sequential closedness of Boolean algebras of projections in Banach
spaces', Studia Math.\ 167 (2005) 45-62.
\cmmnt{[\S323 {\it notes\/}.]}

Frol\'\i k Z.\ [68] `Fixed points of maps of extremally disconnected
spaces and complete Boolean algebras', Bull.\ Acad.\ Polon.\ Sci.\ 16
(1968) 269-275.  \cmmnt{[382E.]}

\medskip
%G

Gaal S.A.\ [64] {\it Point Set Topology.}  Academic, 1964.
\cmmnt{[\S3A3, \S3A4.]}

Gaifman H.\ [64] `Concerning measures on Boolean algebras', Pacific J.\
Math.\ 14 (1964) 61-73.  \cmmnt{[\S391 {\it notes\/}.]}

Gale D. [60] {\it The theory of linear economic models.}  McGraw-Hill,
1960.  \cmmnt{[3A5D.]}

Gnedenko B.V.\ \& Kolmogorov A.N.\ [54] {\it Limit Distributions for
Sums of Independent Random Variables.}  Addison-Wesley, 1954.
\cmmnt{[\S342 {\it notes\/}.]}

Graf S.\ \& Weizs\"acker H.von [76] `On the existence of lower densities in non-complete measure spaces',
pp.\ 155-158 in {\smc Bellow \& K\"olzow 76}.  \cmmnt{[341L.]}

\medskip%H

Hajian A.\ \& Ito Y.\ [69]
`Weakly wandering sets and invariant measures for a group of
transformations', J.\ of Math.\ and Mech.\ 18 (1969) 1203-1216.
\cmmnt{[396B.]}

Hajian A., Ito Y.\ \& Kakutani S.\ [75] `Full groups and a theorem of
Dye', Advances in Math.\ 17 (1975) 48-59.
\cmmnt{[\S388 {\it notes\/}.]}

Halmos P.R.\ [1948] `The range of a vector measure', Bull.\ Amer.\ Math.\
Soc.\ 54 (1948) 416-421.
\cmmnt{[326Yk.]}

Halmos P.R.\ [60] {\it Naive Set Theory.} Van Nostrand, 1960.
\cmmnt{[3A1D.]}

\leaveitout{
Hopf E.\ [1932] `Theory of measures and invariant integrals', Trans.\
Amer.\ Math.\ Soc.\ 34 (1932) 372-393.
}

Huijsmans C.B., Kaashoek M.A., Luxemburg W.A.J.\ \& de Pagter B.\ [95]
(eds.) {\it Operator Theory in Function Spaces and Banach Lattices.}
Birkh\"auser, 1995.

\medskip %I

Ionescu Tulcea C.\ \& Ionescu Tulcea A.\ [69] {\it Topics in the Theory
of Lifting.}  Springer, 1969.  \cmmnt{[\S341 {\it notes\/}.]}

\medskip %J

James I.M.\ [87] {\it Topological and Uniform Spaces.}  Springer, 1987.
\cmmnt{[\S3A3, \S3A4.]}

Jech T.\ [03] {\it Set Theory, Millennium Edition.}  Springer, 2002.
\cmmnt{[\S3A1.]}

Jech T.\ [08] `Algebraic characterizations of measure algebras',
Proc.\ Amer.\ Math.\ Soc.\ 136 (2008) 1285-1294.
%{\tt http://www.math.psu.edu/jech/preprints/papers.html}
\cmmnt{[393Xj.]}

Johnson R.A.\ [80] `Strong liftings which are not Borel liftings',
Proc.\ Amer.\ Math.\ Soc.\ 80 (1980) 234-236.   \cmmnt{[345F.]}
%Russell A. Johnson

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.

\medskip %K

Kakutani S.\ [1941] `Concrete representation of abstract $L$-spaces
and the mean ergodic theorem', Annals of Math. 42 (1941) 523-537.
\cmmnt{[369E.]}

Kalton N.J., Peck N.T.\ \& Roberts J.W.\ [84] `An F-space sampler',
Cambridge U.P., 1984.  \cmmnt{[\S375 {\it notes\/}.]}

Kalton N.J.\ \& Roberts J.W.\ [83] `Uniformly exhaustive submeasures and
nearly additive set functions', Trans.\ Amer.\ Math.\ Soc.\ 278 (1983)
803-816.  \cmmnt{[392D, \S392 {\it notes\/}.]}

Kantorovich L.V., Vulikh B.Z. \& Pinsker A.G.\ [50]
{\it Functional Analysis in
Partially Ordered Spaces}, Gostekhizdat, 1950.
\cmmnt{[391D.]}

Kawada Y.\ [1944] `\"Uber die Existenz der invarianten Integrale', Jap.\
J.\ Math.\ 19 (1944) 81-95.  \cmmnt{[\S395 {\it notes\/}.]}

Kelley J.L.\ [59] `Measures on Boolean algebras', Pacific J.\ Math.\ 9
(1959) 1165-1177.  \cmmnt{[\S391 {\it notes\/}.]}

\leaveitout{
Kelley J.L. \& Namioka I.\ [76] {\it Linear Topological Spaces.}
Springer, 1976.
}

\leaveitout{
King J.L.F.\ [00] `The generic transformation has roots of all orders',
Colloquium Mathematicum 84/85  (2000) 521-547.
  % if  \frak A  is the Lebesgue measure algebra
  % a comeager set in the Polish group
  % \Aut_{\bar\mu}\frak A   has roots of all orders
}

Kolmogorov A.N. [58] `New metric invariants of transitive dynamical
systems and automorphisms of Lebesgue spaces', Dokl.\ Akad.\ Nauk SSSR
119 (1958) 861-864.  \cmmnt{[385P.]}

K\"olzow D.\ \& Maharam-Stone D.\ [82] (eds.) {\it Measure Theory
Oberwolfach 1981.}   Springer, 1982 (Lecture Notes in Math.\ 945).

Koppelberg S.\ [89] {\it General Theory of Boolean
Algebras\/}, vol.\ 1 of {\smc Monk 89}.
\cmmnt{[Chap.\ 31 {\it intro\/}., \S332 {\it notes\/}.]}

K\"othe G.\ [69] {\it Topological Vector Spaces I.}   Springer, 1969.
\cmmnt{[\S356 {\it notes\/}, 3A5N.]}

Kranz P.\ \& Labuda I.\ [93] (eds.) {\it Proceedings of the Orlicz
Memorial Conference, 1991\/}, unpublished manuscript, available from
first editor ({\tt mmkranz{\@}olemiss.edu}).

Krengel, U.\ [63] `"Uber den Absolutbetrag stetiger linearer
Operatoren und seine Anwendung auf ergodische Zerlegungen', Math.\
Scand.\ 13 (1963) 151-187.   \cmmnt{[371Xb.]}

Krieger W.\ [76] `On ergodic flows and the isomorphism of factors',
Math.\ Ann.\ 223 (1976) 19-70.   [\S388 {\it notes}.]
   %Dye's theorem in general form, von-Neumann-algebra version

Krivine J.-L.\ [71] {\it Introduction to Axiomatic Set Theory.}
D.Reidel, 1971.  \cmmnt{[\S3A1.]}

Kullback S.\ [67]
`A lower bound for discrimination information in terms of variation',
IEEE Trans.\ on Information Theory 13 (1967) 126-127.
\cmmnt{[386G.]}
%Wendt S IN5 EL24 T IT

Kunen K.\ [80] {\it Set Theory.}  North-Holland, 1980.
\cmmnt{[\S3A1.]}

Kwapien S.\ [73] `On the form of a linear operator on the space of
all measurable functions', Bull.\ Acad.\ Polon.\ Sci.\ 21 (1973)
951-954.  \cmmnt{[\S375 {\it notes\/}.]}

\medskip %L

Lang S.\ [93] {\it Real and Functional Analysis.}  Springer, 1993.
\cmmnt{[\S3A5.]}

Liapounoff A.A.\ [1940] `Sur les fonctions-vecteurs compl\`etement
additives', Bull.\ Acad.\ Sci.\ URSS (Izvestia Akad.\ Nauk SSSR) 4 (1940)
465-478.
\cmmnt{[326H.]}

Lindenstrauss J.\ \& Tzafriri L.\ [79] {\it Classical Banach Spaces II.}
Springer, 1979, reprinted in {\smc Lindenstrauss \& Tzafriri 96}.
\cmmnt{[\S354 {\it notes\/}, 374Xj.]}

Lindenstrauss J.\ \& Tzafriri L.\ [96] {\it Classical Banach Spaces I \& II.}  Springer, 1996.

Lipschutz S.\ [64] {\it Set Theory and Related Topics.}  McGraw-Hill,
1964 (Schaum's Outline Series).  \cmmnt{[3A1D.]}

Luxemburg W.A.J.\ [67a] `Is every integral normal?',
Bull.\ Amer.\ Math.\ Soc.\ 73 (1967) 685-688.   \cmmnt{[363S.]}

Luxemburg W.A.J.\ [67b] `Rearrangement-invariant Banach function spaces',
pp.\ 83-144 in {\smc Coleman \& Ribenboim 67}.
\cmmnt{[\S374 {\it notes}.]}

Luxemburg W.A.J.\ \& Zaanen A.C.\ [71] {\it Riesz Spaces
I.} North-Holland, 1971.  \cmmnt{[Chap.\ 35 {\it intro\/}.]}

\medskip %M

Macheras N.D., Musia\l\ K.\ \& Strauss W.\ [99] `On products of
admissible liftings and densities',
J.\ for Analysis and its Applications 18 (1999) 651-668. \cmmnt{[346G.]}

Macheras N.D.\ \& Strauss W.\ [95] `Products of lower densities', J.\
for Analysis and its Applications 14 (1995) 25-32.  \cmmnt{[346Xf.]}

Macheras N.D.\  \& Strauss W.\ [96a] `On products of almost strong
liftings', J.\ Australian Math.\ Soc.\ (A) 60 (1996) 1-23.
\cmmnt{[346Yc.]}

Macheras N.D.\ \& Strauss W.\ [96b] `The product lifting for arbitrary products of complete
probability spaces', Atti Sem.\ Math.\ Fis.\ Univ.\ Modena 44 (1996) 485-496.
\cmmnt{[346H, 346Yd.]}

Maharam D.\ [1942] `On homogeneous measure algebras', Proc.\ Nat.\ Acad.\
Sci.\ U.S.A.\ 28 (1942) 108-111.  \cmmnt{[331F, 332B.]}

Maharam D.\ [1947] `An algebraic characterization of measure algebras',
Ann.\ Math.\ 48 (1947) 154-167.
\cmmnt{[393J.]}

Maharam D.\ [58] `On a theorem of von Neumann', Proc.\ Amer.\ Math.\
Soc.\ 9 (1958) 987-994.
\cmmnt{[\S341 {\it notes\/}, \S346 {\it notes\/}.]}

Marczewski E.\ [53] `On compact measures',
Fund.\ Math.\ 40 (1953) 113-124.  \cmmnt{[342A.]}

McCune W.\ [97] `Solution of the Robbins problem',
J.\ Automated Reasoning 19 (1997) 263-276.
\cmmnt{[311Yc.]}

Miller B.D.\ [04] PhD Thesis, University of California, Berkeley, 2004.
%\query
\cmmnt{[382Xc, \S382 {\it notes\/}.]}

Monk J.D.\ [89] (ed.) {\it Handbook of Boolean Algebra.}
North-Holland, 1989.

\medskip %N

Nadkarni M.G.\ [90] `On the existence of a finite invariant measure',
Proc.\ Indian Acad.\ Sci., Math.\ Sci.\ 100 (1990) 203-220.
\cmmnt{[\S395 {\it notes\/}.]}

\medskip %O

Ornstein D.S.\ [74] {\it Ergodic Theory, Randomness and Dynamical Systems.}  Yale U.P., 1974.
\cmmnt{[\S387 {\it notes\/}.]}

Ornstein D.S.\ \& Shields P.C.\ [73] `An uncountable family of
$K$-automorphisms', Advances in Math.\ 10 (1973) 63-88.
\cmmnt{[\S382 {\it notes\/}.]}

\medskip%P

%Pedersen G.K.\ [89] {\it Analysis Now.}  Springer, 1989.

Perovi\'c \v{Z}.\ \& Veli\v{c}kovi\'c B.\ [18] `Ranks of Maharam algebras',
Advances in Math.\ 330 (2018) 253-279.
\cmmnt{[394A.]}

Petersen K.\ [83] {\it Ergodic Theory.}   Cambridge U.P., 1983.
\cmmnt{[328Xa, 385C, \S385 {\it notes\/}, 386E.]}

\medskip %Q%R

Roberts J.W.\ [93] `Maharam's problem', in {\smc Kranz \& Labuda 93}.
\cmmnt{[\S394 {\it notes\/}.]}

Rotman J.J.\ [84] `An Introduction to the Theory of Groups', Allyn \&
Bacon, 1984.
\cmmnt{[\S384 {\it notes\/}, 3A6B.]}

Rudin W. [91] {\it Functional Analysis.}  McGraw-Hill, 1991.
\cmmnt{[\S3A5.]}

Ryzhikov V.V.\ [93] `Factorization of an automorphism of a complete Boolean algebra into a product of three involutions', Mat.\ Zametki (=Math.\ Notes of Russian Acad.\ Sci.) 54 (1993) 79-84.
\cmmnt{[\S382 {\it notes\/}.]}

\medskip %S

Sazonov V.V.\ [66] `On perfect measures', A.M.S.\ Translations (2) 48
(1966) 229-254.   \cmmnt{[\S342 {\it notes\/}.]}

Schaefer H.H.\ [66] {\it Topological Vector Spaces.}  MacMillan, 1966;  reprinted with corrections Springer, 1971.
\cmmnt{[3A4A, 3A5J, 3A5N.]}
%Store A	QA 611.S3    6640 0159 9

Schaefer H.H.\ [74] {\it Banach Lattices and Positive Operators.}
Springer, 1974.   \cmmnt{[Chap.\ 35 {\it intro\/}., \S354 {\it notes\/}.]}

Schubert H.\ [68] {\it Topology.}  Allyn \& Bacon, 1968.
\cmmnt{[\S3A3, \S3A4.]}

Shelah S.\ [98] `The lifting problem with the full ideal',
J.\ Applied Analysis 4 (1998) 1-17.   \cmmnt{[341L.]}

Sikorski R.\ [64] {\it Boolean Algebras.}  Springer, 1964.
\cmmnt{[Chap.\ 31 {\it intro\/}.]}

Sina\v\i\ Ya.G.\ [59] `The notion of entropy of a dynamical system',
Dokl.\ Akad.\ Nauk SSSR 125 (1959) 768-771.  \cmmnt{[385P.]}

Sina\v\i\ Ya.G.\ [62] `Weak isomorphism of transformations with an invariant measure', Soviet Math.\ 3 (1962) 1725-1729.
\cmmnt{[387E.]}
% ({\cyr Dokl.\ Akad.\ Nauk SSSR 147 (1962) 797-800}).

Smorodinsky M.\ [71] {\it Ergodic Theory, Entropy.}
Springer, 1971 (Lecture Notes in Math., 214).
\cmmnt{[\S387 {\it notes\/}.]}

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\medskip %T

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\medskip %U%V

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\medskip %W

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\medskip %X%Y%Z

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