% 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{mt42.tex}
\versiondate{30.9.08}
\copyrightdate{1998}

\def\chaptername{Descriptive set theory}
\def\sectionname{Introduction}

\newchapter{42}

At this point, I interpolate an auxiliary chapter, in the same spirit as
Chapters 31 and 35 in the last volume.   As with Boolean algebras and
Riesz spaces, it is not just that descriptive set theory provides
essential tools for modern measure theory;  it also offers deep
intuitions, and for this reason demands study well beyond an occasional
glance at an appendix.   Several excellent accounts have been published;
the closest to what we need here is probably {\smc Rogers 80}; at a
deeper level we have {\smc Moschovakis 80}, and an admirable recent
treatment is {\smc Kechris 95}.   Once again, however, I indulge myself
by extracting those parts of the theory which I shall use directly,
giving proofs and exercises adapted to the ideas I am trying to
emphasize in this volume and the next.

The first section describes Souslin's operation and its basic
set-theoretic properties up to first steps in the theory of `constituents'
(421N-421Q), %421N 421O 421P 421Q
mostly steering away from topological ideas,
but with some remarks on $\sigma$-algebras and Souslin-F sets.   \S422
deals with usco-compact relations and K-analytic spaces, working through
the topological properties which will be useful later, and giving a
version of the First Separation Theorem (422I-422J).   \S423 looks at
`analytic' or `Souslin' spaces, treating them primarily as a special
kind of K-analytic space, with the von Neumann-Jankow selection theorem
(423P).   \S424 is devoted to `standard Borel spaces';  it is largely a
series of easy applications of results in \S423, but there is a
substantial theorem on Borel measurable actions of Polish groups (424H).
Finally, I add a note on A.T\"ornquist's theorem on representation of
groups of automorphisms of quotient algebras (425D).

\discrpage


\frfilename{mt421.tex}
\versiondate{14.12.07}
\copyrightdate{2001}

\def\NN{\BbbN^{\Bbb N}}

\def\chaptername{Descriptive set theory}
\def\sectionname{Souslin's operation}

\newsection{421}

I introduce Souslin's operation $\Cal S$ (421B) and show that it is
idempotent (421D).   I describe alternative characterizations of members
of $\Cal S(\Cal E)$, where $\Cal E\subseteq\Cal PX$, as projections of
sets in $\NN\times X$ (421G-421J).   I briefly mention Souslin-F sets
(421J-421L) and a special property of `inner Souslin kernels' (421M).
At the end of the section I set up an abstract theory of `constituents'
for kernels of Souslin schemes and their complements (421N-421Q).

\leader{421A}{Notation}\cmmnt{ Throughout this chapter, and frequently
in the next, I shall regard
a member of $\Bbb N$ as the set of its predecessors, so that a
finite power $X^k$
can be identified with the set of functions from $k$ to $X$, and if
$\phi\in X^{\Bbb N}$ and $k\in\Bbb N$, we can speak of the restriction
$\phi\restr k\in X^k$.   In the same spirit, identifying functions
with their graphs, I can write `$\sigma\subseteq\phi$' when
$\sigma\in X^k$,
$\phi\in X^{\Bbb N}$ and $\phi$ extends $\sigma$.   On occasion I may
write $\#(\sigma)$ for the `length' of a finite function $\sigma$ --
again identifying $\sigma$ with its graph -- so that $\#(\sigma)=k$ if
$\sigma\in X^k$.   And if $k=0$, identified with $\emptyset$, then the
only function from $k$ to $X$ is the empty function, so $X^0$ becomes
$\{\emptyset\}$.

I shall sometimes refer to the `usual topology of $\NN$';
this is the product topology if each copy of $\Bbb N$ is given its
discrete topology.}   $S$ will always be the set
$\bigcup_{k\in\Bbb N}\BbbN^k$, and $S^*=S\setminus\{\emptyset\}$ the set
$\bigcup_{k\ge 1}\BbbN^k$;  for $\sigma\in S$,
$I_{\sigma}$ will be
$\{\phi:\phi\in\BbbN^{\Bbb N},\,\phi\supseteq\sigma\}$.\cmmnt{  Then
$I_{\emptyset}=\NN$ and
$\{I_{\sigma}:\sigma\in S^*\}$ is a base for the topology of
$\BbbN^{\BbbN}$ consisting of open-and-closed sets.}
If $\sigma\in\BbbN^k$ and $i\in\Bbb N$ I write
$\sigma^{\smallfrown}\fraction{i}$ for
the member $\tau$ of $\BbbN^{k+1}$ such that $\tau(k)=i$ and
$\tau(j)=\sigma(j)$ for $j<k$.

\leader{421B}{Definition}
If $\Cal E$ is a family of sets, I write $\Cal S(\Cal E)$ for the family
of sets expressible in the form

$$\bigcup_{\phi\in\NN}\bigcap_{k\ge 1}E_{\phi\restr k}$$

\noindent for some family $\family{\sigma}{S^*}{E_{\sigma}}$ in $\Cal E$.

A family $\family{\sigma}{S^*}{E_{\sigma}}$ is called a {\bf Souslin
scheme};  the corresponding set
$\bigcup_{\phi\in\NN}\bigcap_{k\ge 1}E_{\phi\restr k}$ is its
{\bf kernel};  the operation

$$\family{\sigma}{S^*}{E_{\sigma}}
\mapsto\bigcup_{\phi\in\BbbN^{\Bbb N}}\bigcap_{k\ge 1}E_{\phi\restr k}$$

\noindent is {\bf Souslin's operation} or {\bf operation $\Cal A$}.
Thus $\Cal S(\Cal E)$ is the
family of sets obtainable from sets in $\Cal E$ by Souslin's operation.
If $\Cal E=\Cal S(\Cal E)$, we say that $\Cal E$ is {\bf closed under
Souslin's operation}.

\cmmnt{\medskip

\noindent{\bf Remark} I should perhaps warn you that some authors use
$\bigcup_{k\in\Bbb N}\BbbN^k$ here in place of $S^*$;  so that their
Souslin kernels are of the form
$\bigcup_{\phi\in\NN}\bigcap_{k\ge 0}E_{\phi\restr k}
\subseteq E_{\emptyset}$.   Consequently, for such
authors, any member of $\Cal S(\Cal E)$ is included in some member of
$\Cal E$.   If $\Cal E$ has a greatest member (or, fractionally more
generally, if any sequence in $\Cal E$ is bounded above in $\Cal E$)
this makes no difference;  but if, for instance, $\Cal E$ is the family
of compact subsets of a topological space, the two definitions of
$\Cal S$ may not quite coincide.
I believe that on this point, for once, I am following the majority.}

\leader{421C}{Elementary facts (a)} It is worth noting straight away
that if $\Cal E$ is any family of sets, then $\bigcup_{n\in\BbbN}E_n$
and $\bigcap_{n\in\BbbN}E_n$ belong to $\Cal S(\Cal E)$ for any sequence
$\sequencen{E_n}$ in $\Cal E$.   \prooflet{\Prf\ Set

\Centerline{$F_{\sigma}=E_{\sigma(0)}$ for every $\sigma\in S^*$,}

\Centerline{$G_{\sigma}=E_k$ whenever $k\in\Bbb N$,
$\sigma\in\Bbb N^{k+1}$;}

\noindent then

\Centerline{$\bigcup_{n\in\BbbN}E_n=
\bigcup_{\phi\in\NN}\bigcap_{k\ge 1}F_{\phi\restr k}
\in\Cal S(\Cal E)$,}

\Centerline{$\bigcap_{n\in\BbbN}E_n=
\bigcup_{\phi\in\NN}\bigcap_{k\ge 1}G_{\phi\restr k}\in\Cal
S(\Cal E)$.  \Qed}
}%end of prooflet

In particular, $\Cal E\subseteq\Cal S(\Cal E)$.   \cmmnt{But note that
there is no reason why $E\setminus F$ should belong to $\Cal S(\Cal E)$
for $E$, $F\in\Cal E$.}

\spheader 421Cb Let $X$ and $Y$ be sets, and $f:X\to Y$ a function.
Let $\family{\sigma}{S^*}{F_{\sigma}}$ be a Souslin scheme in $\Cal PY$,
with kernel $B$.   Then $f^{-1}[B]$ is the kernel of the Souslin scheme
$\family{\sigma}{S^*}{f^{-1}[F_{\sigma}]}$.   \prooflet{\Prf\

\Centerline{$f^{-1}[B]
=f^{-1}[\bigcup_{\phi\in\NN}\bigcap_{n\ge 1}F_{\phi\restr n}]
=\bigcup_{\phi\in\NN}\bigcap_{n\ge 1}f^{-1}[F_{\phi\restr n}]$.\ \Qed}
}%end of prooflet

\spheader 421Cc Let $X$ and $Y$ be sets, and $f:X\to Y$ a function.
Let $\Cal F$ be a family of subsets of $Y$.   Then

\Centerline{$\{f^{-1}[B]:B\in\Cal S(\Cal F)\}
=\Cal S(\{f^{-1}[F]:F\in\Cal F\})$.}

\prooflet{\noindent\Prf\ For a set $A\subseteq X$, $A\in
\Cal S(\{f^{-1}[F]:F\in\Cal F\})$ iff there is some Souslin scheme
$\family{\sigma}{S^*}{E_{\sigma}}$ in $\{f^{-1}[F]:F\in\Cal F\}$ such that
$A$ is the kernel of $\family{\sigma}{S^*}{E_{\sigma}}$, that is, iff
there is some Souslin scheme $\family{\sigma}{S^*}{F_{\sigma}}$ in $\Cal
F$ such that $A$ is the kernel of
$\family{\sigma}{S^*}{f^{-1}[F_{\sigma}]}$, that
is, iff $A=f^{-1}[B]$ where $B$ is the kernel of some Souslin scheme in
$\Cal F$.\ \Qed
}%end of prooflet

\spheader 421Cd Let $X$ and $Y$ be sets, and $f:X\to Y$ a surjective
function.   Let $\Cal F$ be a family of subsets of $Y$.   Then

\Centerline{$\Cal S(\Cal F)
=\{B:B\subseteq Y,\,f^{-1}[B]\in\Cal S(\{f^{-1}[F]:F\in\Cal F\})\}$.}

\prooflet{\noindent\Prf\ If $B\in\Cal S(\Cal F)$, then
$f^{-1}[B]\in\Cal S(\{f^{-1}[F]:F\in\Cal F\})$, by (c) above.   If $B\subseteq Y$ and
$f^{-1}[B]\in\Cal S(\{f^{-1}[F]:F\in\Cal F\})$, then there is a Souslin
scheme  $\family{\sigma}{S^*}{F_{\sigma}}$ in $\Cal F$ such that
$f^{-1}[B]$ is the kernel of
$\family{\sigma}{S^*}{f^{-1}[F_{\sigma}]}$, that is, $f^{-1}[B]=f^{-1}[C]$
where $C$ is the kernel of $\family{\sigma}{S^*}{F_{\sigma}}$.   Because
$f$ is surjective, $B=C\in\Cal S(\Cal F)$.\ \Qed
}%end of prooflet

\spheader 421Ce\cmmnt{ Souslin's operation can be thought of as a
projection operator, as follows.}   Let $\family{\sigma}{S^*}{E_{\sigma}}$
be a Souslin scheme with kernel $A$.   Set

\Centerline{$R=\bigcap_{n\ge
1}\bigcup_{\sigma\in\BbbN^n}I_{\sigma}\times E_{\sigma}$.}

\noindent Then $R[\NN]=A$.   \prooflet{\Prf\ For any $x$, and any
$\phi\in\NN$,

$$\eqalign{(\phi,x)\in R
&\iff\text{ for every }n\ge 1\text{ there is a }\sigma\in \BbbN^n
  \text{ such that }x\in E_{\sigma},\,\phi\in I_{\sigma}\cr
&\iff x\in E_{\phi\restr n}\text{ for every }n\ge 1.\cr}$$

\noindent But this means that

$$\eqalign{x\in R[\NN]
&\iff\text{ there is a }\phi\in\NN
  \text{ such that }(\phi,x)\in R\cr
&\iff\text{ there is a }\phi\in\NN
  \text{ such that }x\in\bigcap_{n\ge 1}E_{\phi\restr n}
\iff x\in A.\text{ \Qed}}$$
}%end of prooflet

\spheader 421Cf\dvAformerly{4{}21Xm}{\bf (i)}
I will say that a Souslin scheme
$\family{\sigma}{S^*}{E_{\sigma}}$ is {\bf fully regular} if
$E_{\sigma}\subseteq E_{\tau}$ whenever $\sigma$, $\tau\in S^*$,
$\#(\tau)\le\#(\sigma)$ and
$\sigma(i)\le\tau(i)$ for every $i<\#(\sigma)$.

\medskip

\quad{\bf (ii)} Let $\Cal E$ be a
family of sets such that $E\cup F$ and $E\cap F$ belong to
$\Cal E$ for all $E$,
$F\in\Cal E$.   Then every member of $\Cal S(\Cal E)$ can be
expressed as the kernel of a regular Souslin scheme in $\Cal E$.
\prooflet{\Prf\
Let $\family{\sigma}{S^*}{E_{\sigma}}$ be a Souslin scheme in
$\Cal E$ with kernel $A$.   Set
$F_{\sigma}=\bigcap_{1\le k\le n}E_{\sigma\restr k}$,
$G_{\sigma}=\bigcup_{\tau\in\BbbN^n,\tau\le\sigma}F_{\tau}$ for
$n\ge 1$ and $\sigma\in\BbbN^n$;  then $F_{\sigma}$,
$G_{\sigma}\in\Cal E$ for every $\sigma\in S^*$.
Write $B$, $C$ for the kernels of
$\family{\sigma}{S^*}{F_{\sigma}}$, $\family{\sigma}{S^*}{G_{\sigma}}$
respectively.   If $\phi\in\NN$, then
$\bigcap_{n\ge 1}F_{\phi\restr n}=\bigcap_{n\ge 1}E_{\phi\restr n}$, so
$A'=A$.   Because $F_{\sigma}\subseteq G_{\sigma}$ for
every $\sigma$, $A'\subseteq A''$.   If $x\in A''$, let
$\phi\in\NN$ be such that $x\in\bigcap_{n\ge 1}G_{\phi\restr n}$.
Then for each $k\ge 1$ there is a $\tau_k\in\BbbN^k$ such that
$\tau_k\le\phi\restr k$ and $x\in F_{\tau_k}$.   For $k\ge 1$, set

$$\eqalign{\psi_k(i)
&=\tau_k(i)\text{ for }i<k,\cr
&=0\text{ otherwise};\cr}$$

\noindent then $\psi_k$ belongs to the compact set
$\prod_{i\in\Bbb N}\phi(i)+1$ for every $n$, so
$\langle\psi_k\rangle_{k\ge 1}$ has a cluster point $\psi$ in $\NN$.
For any $n\ge 1$, there are infinitely many $k$ such that
$\psi\restr n=\psi_k\restr n$, so there is such a $k$ with $k\ge n$,
in which case $\psi\restr n=\tau_k\restr n$ and

\Centerline{$x\in F_{\tau_k}\subseteq E_{\tau_k\restr n}
=E_{\psi\restr n}$.}

\noindent Thus $x\in\bigcap_{n\ge 1}E_{\psi\restr n}\subseteq A$.
As $x$ is arbitrary, $A''\subseteq A$ and $A''=A$.

On the other hand, if $\sigma$, $\tau\in S^*$, $m=\#(\tau)\le\#(\sigma)=n$
and $\sigma\restr m\le\tau$, then take any $\sigma'\in\BbbN^n$ such that
$\sigma'\le\sigma$;  in this case $\sigma'\restr m\le\tau$ so

\Centerline{$F_{\sigma'}\subseteq F_{\sigma'\restr m}\subseteq G_{\tau}$.}

\noindent As $\sigma'$ is arbitrary, $G_{\sigma}\subseteq G_{\tau}$;
as $\sigma$ and $\tau$ are arbitrary, $\family{\sigma}{S^*}{G_{\sigma}}$
is fully regular.

Thus $A=A''$ is the kernel of a fully regular Souslin scheme in $\Cal E$.\
\Qed}

\leader{421D}{}\cmmnt{ The first fundamental theorem is that the
operation $\Cal S$ is idempotent.

\medskip

\noindent}{\bf Theorem}\cmmnt{ ({\smc Souslin 1917})} For any family
$\Cal E$ of sets, $\Cal S(\Cal E)$ is closed under Souslin's operation.

\proof{{\bf (a)} Let $\family{\sigma}{S^*}{A_{\sigma}}$ be a family in
$\Cal S(\Cal E)$, and set
$A=\bigcup_{\phi\in\NN}\bigcap_{k\ge 1}A_{\phi\restr k}$;  I have to
show that $A\in\Cal S(\Cal E)$.   For each
$\sigma\in S$, let $\family{\tau}{S^*}{E_{\sigma\tau}}$ be a family in
$\Cal E$ such that $A_{\sigma}
=\bigcup_{\psi\in\NN}\bigcap_{m\ge 1}E_{\sigma,\psi\restr m}$.   Then

$$A
=\bigcup_{\phi\in\NN}\bigcap_{k\ge 1}\bigcup_{\psi\in\NN}
   \bigcap_{m\ge 1}E_{\phi\restr k,\psi\restr m}
=\bigcup_{\Atop{\phi\in\NN}{\pmb{\psi}\in(\NN)^{\BbbN\setminus\{0\}}}}
   \bigcap_{k,m\ge 1}E_{\phi\restr k,\psi_k\restr m},$$

\noindent writing $\pmb{\psi}=\langle\psi_k\rangle_{k\ge 1}$ for
$\pmb{\psi}\in(\NN)^{\BbbN\setminus\{0\}}$.
The idea of the proof is simply that
$\NN\times(\BbbN^{\Bbb N})^{\BbbN\setminus\{0\}}$ is essentially
identical to $\NN$,
so that all we have to do is to organize new names for the
$E_{\sigma\tau}$.   But as it is by no means a trivial matter to devise
a coding scheme which really works, I give the details at length.

\medskip

{\bf (b)} The first step is to note that $S^*$ and $(S^*)^2$ are countable, so
there is a sequence $\sequencen{H_n}$ running over
$\{E_{\sigma\tau}:\sigma$, $\tau\in S^*\}$.   Next, choose any injective
function $q:\BbbN\times\BbbN\to\BbbN\setminus\{0\}$ such that $q(0,0)=1$
and $q(0,1)=2$.   For $k$, $m\ge 1$ set
$J_{km}=\{(i,0):i<k\}\cup\{(i,k):i<m\}$, so that
$J_{11}=\{(0,0),(0,1)\}$,
and choose a family $\langle(k_n,m_n)\rangle_{n\ge 3}$ running over
$(\Bbb N\setminus\{0\})^2$ such that $q[J_{k_n,m_n}]\subseteq n$ for
every $n\ge 3$.   (The pairs $(k_n,m_n)$ need not all be distinct, so
this is easy to achieve.)

Now, for $\upsilon\in\BbbN^n$, where $n\ge 3$, set
$F_{\upsilon}=E_{\sigma\tau}$ where

\Centerline{$\sigma\in\BbbN^{k_n}$, $\sigma(i)=\upsilon(q(i,0))$ for
$i<k_n$,}

\Centerline{$\tau\in\BbbN^{m_n}$, $\tau(i)=\upsilon(q(i,k_n))$ for
$i<m_n$;}

\noindent these are well-defined because $q[J_{k_n,m_n}]\subseteq n$.
For
$\upsilon\in\BbbN^1\cup\BbbN^2$, set $F_{\upsilon}=H_{\upsilon(0)}$.

\medskip

{\bf (c)} This defines a Souslin scheme
$\family{\upsilon}{S^*}{F_{\upsilon}}$ in $\Cal E$.   Let $A'$ be its
kernel,
so that $A'\in\Cal S(\Cal E)$.   The point is that $A'=A$.

\medskip

\Prf\ {\bf (i)} If $x\in A$, there must be $\phi\in\NN$,
$\pmb{\psi}\in(\NN)^{\BbbN\setminus\{0\}}$ such that
$x\in\bigcap_{k,m\ge 1}E_{\phi\restr k,\psi_k\restr m}$.
Choose $\theta\in\NN$ such that

$$\eqalign{H_{\theta(0)}&=E_{\phi\restr 1,\psi_1\restr 1},\cr
\theta(q(i,0))&=\phi(i)\text{ for every }i\in\BbbN,\cr
\theta(q(i,k))&=\psi_k(i)\text{ for every }k\ge 1,\,i\in\BbbN.\cr}$$

\noindent (This is possible because $q:\BbbN^2\to\BbbN\setminus\{0\}$ is
injective.)   Now

\Centerline{$F_{\theta\restr 1}
=F_{\theta\restr 2}
=H_{\theta(0)}
=E_{\phi\restr 1,\psi_1\restr 1}$}

\noindent certainly contains $x$.   And for $n\ge 3$,
$F_{\theta\restr n}=E_{\sigma\tau}$ where $\sigma(i)=\theta(q(i,0))$ for
$i<k_n$, $\tau(i)=\theta(q(i,k_n))$ for $i<m_n$, that is,
$\sigma=\phi\restr k_n$ and $\tau=\psi_{k_n}\restr m_n$, so again
$x\in F_{\theta\restr n}$.   Thus

\Centerline{$x\in\bigcap_{n\ge 1}F_{\theta\restr n}\subseteq A'$.}

\noindent As $x$ is arbitrary, $A\subseteq A'$.

\medskip

\quad{\bf (ii)} Now take any $x\in A'$.   Let $\theta\in\NN$
be
such that $x\in\bigcap_{n\ge 1}F_{\theta\restr n}$.   Define
$\phi\in\Bbb
N^{\BbbN}$, $\pmb{\psi}\in(\NN)^{\BbbN\setminus\{0\}}$ by setting

$$\eqalign{\phi(i)&=\theta(q(i,0))\text{ for }i\in\BbbN,\cr
\psi_k(i)&=\theta(q(i,k))\text{ for }k\ge 1,\,i\in\BbbN.\cr}$$

\noindent If $k$, $m\ge 1$, let $n\ge 3$ be such that $k=k_n$, $m=m_n$.
Then $x\in F_{\theta\restr n}=E_{\sigma\tau}$, where

\Centerline{$\sigma(i)=\theta(q(i,0))$ for $i<k_n$,
\quad$\tau(i)=\theta(q(i,k_n))$ for $i<m_n$,}

\noindent that is, $\sigma=\phi\restr k_n=\phi\restr k$ and
$\tau=\psi_{k_n}\restr m_n=\psi_k\restr m$.   As $m$ and $n$ are
arbitrary,

\Centerline{$x\in\bigcap_{m,n\ge 1}E_{\phi\restr k,\psi_k\restr m}
\subseteq A$.}

\noindent As $x$ is arbitrary, $A'\subseteq A$.\ \Qed

Accordingly we must have $A\in\Cal S(\Cal E)$, and the proof is
complete.
}%end of proof of 421D


\leader{421E}{Corollary} For any family $\Cal E$ of sets,
$\Cal S(\Cal E)$ is closed under countable unions and intersections.

\proof{ For 421Ca tells us that the union and intersection of any
sequence in $\Cal S(\Cal E)$ will belong to
$\Cal S\Cal S(\Cal E)=\Cal S(\Cal E)$.
}%end of proof of 421E

\leader{421F}{Corollary} Let $X$ be a set and $\Cal E$ a family of
subsets of $X$.   Suppose that $X$ and $\emptyset$ belong to
$\Cal S(\Cal E)$ and that $X\setminus E\in\Cal S(\Cal E)$ for every
$E\in\Cal E$.   Then $\Cal S(\Cal E)$ includes the $\sigma$-algebra of subsets of $X$ generated by $\Cal E$.

\proof{ The set

\Centerline{$\Sigma
=\{F:F\in\Cal S(\Cal E),\,X\setminus F\in\Cal S(\Cal E)\}$}

\noindent is closed under complements (necessarily), contains
$\emptyset$
(because $\emptyset$ and $X$ belong to $\Cal S(\Cal E)$), and is also
closed under countable unions, by 421E.   So it is a $\sigma$-algebra;
but the hypotheses also ensure that $\Cal E\subseteq\Sigma$, so that the
$\sigma$-algebra generated by $\Cal E$ is included in $\Sigma$ and in
$\Cal S(\Cal E)$.
}%end of proof of 421F

\leader{421G}{Proposition} Let $\Cal E$ be a family of sets such that
$\emptyset\in\Cal E$.   Then

$$\eqalign{\Cal S(\Cal E)
&=\{R[\NN]:R\in\Cal S(\{I_{\sigma}\times E:\sigma\in S^*,
  \,E\in\Cal E\})\}\cr
&=\{R[\NN]:R\in\Cal S(\{I_{\sigma}\times E:\sigma\in S^*,
  \,E\in\Cal E\}),\,R^{-1}[\{x\}]\text{ is closed for every }x\}.\cr}$$

\proof{ Set $\Cal F=\{I_{\sigma}\times E:\sigma\in S^*,\,E\in\Cal E\}$.

\medskip

{\bf (a)} Suppose first that $A\in\Cal S(\Cal E)$.   Let
$\family{\sigma}{S^*}{E_{\sigma}}$ be a Souslin scheme in $\Cal E$ with
kernel $A$.   Set

\Centerline{$R=\bigcap_{k\ge 1}\bigcup_{\sigma\in\BbbN^k}
  I_{\sigma}\times E_{\sigma}$.}

\noindent Then $R\in\Cal S(\Cal F)$, by 421E, and $R[\NN]=A$,
by 421Ce.   Also

\Centerline{$R^{-1}[\{x\}]
=\bigcap_{k\ge 1}\bigcup\{I_{\sigma}:
  \sigma\in\BbbN^k$, $x\in E_{\sigma}\}$}

\noindent is closed, for every $x$.

\medskip

{\bf (b)} Now suppose that $A=R[\NN]$ for some
$R\in\Cal S(\Cal F)$.   Let
$\family{\sigma}{S^*}{I_{\tau(\sigma)}\times E_{\sigma}}$ be a
Souslin scheme in $\Cal F$ with kernel $R$.   For $k\ge 1$,
$\sigma\in\Bbb N^k$ set

$$\eqalign{F_{\sigma}
&=E_{\sigma}
  \text{ if }\bigcap_{1\le n\le k}I_{\tau(\sigma\restr n)}
    \ne\emptyset,\cr
&=\emptyset\text{ otherwise}.\cr}$$

\noindent Then $\family{\sigma}{S^*}{F_{\sigma}}$ is a Souslin scheme in
$\Cal E$, so its kernel $A'$ belongs to $\Cal S(\Cal E)$.

The point is that $A'=A$.   \Prf\ (i) If $x\in A$, there are a
$\phi\in\Bbb N^{\BbbN}$ such that $(\phi,x)\in R$ and a $\psi\in\NN$
such that $(\phi,x)\in
\bigcap_{n\ge 1}I_{\tau(\psi\restr n)}\times E_{\psi\restr n}$.   Now,
for any $k\ge 1$, we have

\Centerline{$\phi
\in\bigcap_{1\le n\le k}I_{\tau(\psi\restr n)}
=\bigcap_{1\le n\le k}I_{\tau((\psi\restr k)\restr n)}$,}

\noindent so that
$F_{\psi\restr k}=E_{\psi\restr k}$ contains $x$;  thus
$x\in\bigcap_{k\ge 1}F_{\psi\restr k}\subseteq A'$.   As $x$ is
arbitrary, $A\subseteq A'$.
(ii) If $x\in A'$, take $\psi\in\NN$ such that
$x\in\bigcap_{n\ge 1}F_{\psi\restr n}$.   In this case we must have
$F_{\psi\restr k}\ne\emptyset$, so
$\bigcap_{1\le n\le k}I_{\tau(\psi\restr n)}\ne\emptyset$, for every
$k\ge 1$.   But what this means is that, setting
$\tau_n=\tau(\psi\restr n)$ for each $n\ge 1$, $\tau_n(i)=\tau_m(i)$
whenever $i\in\Bbb N$ is such that both are defined.   So
$\{\tau_n:n\ge 1\}$ must have a common extension $\phi\in\NN$, and
$\phi\in\bigcap_{n\ge 1}I_{\tau(\psi\restr n)}$.   Now

\Centerline{$(\phi,x)\in\bigcap_{n\ge 1}I_{\tau(\psi\restr n)}\times
E_{\psi\restr n}\subseteq R$,}

\noindent so $x\in A$.   Thus $A'\subseteq A$ and the two are equal.\
\Qed

This shows that

\Centerline{$\{R[\NN]:R\in\Cal S(\Cal F)\}\subseteq\Cal S(\Cal E)$,}

\noindent and the proof is complete.
}%end of proof of 421G

\leader{421H}{}\cmmnt{ When the class $\Cal E$ is a $\sigma$-algebra,
the last proposition can be extended.

\medskip

\noindent}{\bf Proposition} Let $X$ be a set, and $\Sigma$ a
$\sigma$-algebra of subsets of $X$.   Let $\Cal B$ be the algebra of
Borel subsets of $\NN$.   Then

$$\eqalign{\Cal S(\Sigma)
&=\{R[\NN]:R\in\Cal B\tensorhat\Sigma\}\cr
&=\{R[\NN]:R\in\Cal S(\{I_{\sigma}\times E:
  \sigma\in S^*,\,E\in\Sigma\})\}\cr
&=\{R[\NN]:R\in\Cal S(\Cal B\tensorhat\Sigma)\}.\cr}$$

\cmmnt{\medskip

\noindent{\bf Notation} Recall that $\Cal B\tensorhat\Sigma$ is the
$\sigma$-algebra of subsets of $\NN\times X$ generated by
$\{H\times E:H\in\Cal B,\,E\in\Sigma\}$.
}%end of comment

\proof{{\bf (a)} Suppose first that $A\in\Cal S(\Sigma)$.   As in 421G,
let
$\family{\sigma}{S^*}{E_{\sigma}}$ be a Souslin scheme in $\Sigma$ with
kernel $A$, and set

\Centerline{$R=\bigcap_{k\ge
1}\bigcup_{\sigma\in\BbbN^k}I_{\sigma}\times
E_{\sigma}$,}

\noindent so that $A=R[\NN]$ (421Ce again).
Because every $I_{\sigma}$ is
an open-and-closed set in $\NN$, $R\in\Cal B\tensorhat\Sigma$.   Thus

\Centerline{$\Cal S(\Sigma)
\subseteq\{R[\NN]:R\in\Cal B\tensorhat\Sigma\}$.}

\medskip

{\bf (b)} Set $\Cal F=\{I_{\sigma}\times E:\sigma\in S^*,\,E\in\Sigma\}$.
Then $\Cal S(\Cal B\tensorhat\Sigma)=\Cal S(\Cal F)$.   \Prf\ If
$E\in\Sigma$ and $\sigma\in\BbbN^k$ then

\Centerline{$(\NN\times X)\setminus(I_{\sigma}\times E)
=(I_{\sigma}\times(X\setminus E))\cup\bigcup_{\tau\in\Bbb
N^k,\tau\ne\sigma}I_{\tau}\times X\in\Cal S(\Cal F)$.}

\noindent Also

\Centerline{$\NN\times X=\bigcup_{\sigma\in\Bbb
N^1}I_{\sigma}\times X$,
\quad $\emptyset=I_{\tau}\times\emptyset$}

\noindent (where $\tau$ is any member of $S^*$) belong to $\Cal S(\Cal F)$.
By 421F, $\Cal S(\Cal F)$ includes the $\sigma$-algebra $\Lambda$ of
sets
generated by $\Cal F$.   Now if $E\in\Sigma$ and $H\subseteq\BbbN^{\Bbb
N}$ is open, $H=\bigcup_{\sigma\in T}I_{\sigma}$ for some
$T\subseteq S^*$;  as $T$ is necessarily countable,

\Centerline{$H\times E
=\bigcup_{\sigma\in T}I_{\sigma}\times E\in\Lambda$.}

\noindent Since $\{F:F\subseteq\NN,\,F\times E\in\Lambda\}$ is
a $\sigma$-algebra of subsets of $\NN$, and we have just seen
that it contains all the open sets, it must include $\Cal B$;  thus
$F\times E\in\Lambda$ for every $F\in\Cal B$, $E\in\Sigma$.   So
$\Cal B\tensorhat\Sigma\subseteq\Lambda\subseteq\Cal S(\Cal F)$, and

\Centerline{$\Cal S(\Cal F)
\subseteq\Cal S(\Cal B\tensorhat\Sigma)
\subseteq\Cal S\Cal S(\Cal F)=\Cal S(\Cal F)$}

\noindent (421D).\ \Qed

\medskip

{\bf (c)} Now we have

$$\eqalignno{\Cal S(\Sigma)
&\subseteq\{R[\NN]:R\in\Cal B\tensorhat\Sigma\}\cr
\noalign{\noindent (by (a))}
&\subseteq\{R[\NN]:R\in\Cal S(\Cal B\tensorhat\Sigma)\}
=\{R[\NN]:R\in\Cal S(\Cal F)\}\cr
\noalign{\noindent (by (b))}
&=\Cal S(\Sigma)\cr}$$

\noindent by 421G.
}%end of proof of 421H

\cmmnt{\medskip

\noindent{\bf Remark} A more general form of this result is in 423O below.
}

\leader{421I}{}\cmmnt{ There is a particularly simple description of
sets obtainable by Souslin's operation from closed sets in a topological
space.

\medskip

\noindent}{\bf Lemma} Let $X$ be a topological space and
$R\subseteq\NN\times X$ a closed set.   Then

\Centerline{$R[A]
=\bigcup_{\phi\in A}\bigcap_{n\ge 1}\overline{R[I_{\phi\restr n}]}$.}

\noindent for any $A\subseteq\NN$.
In particular, $R[\NN]$ is the kernel of the Souslin scheme
$\family{\sigma}{S^*}{\overline{R[I_{\sigma}]}}$.

\wheader{421I}{0}{0}{0}{36pt}

\proof{ Set

\Centerline{$B=\bigcup_{\phi\in A}
  \bigcap_{n\ge 1}\overline{R[I_{\phi\restr n}]}$.}

\noindent(i) If $x\in R[A]$, there is a $\phi\in A$ such that
$(\phi,x)\in R$.   In this case, $\phi\in I_{\phi\restr n}$ so

\Centerline{$x\in R[I_{\phi\restr n}]
\subseteq\overline{R[I_{\phi\restr n}]}$}

\noindent for every $n$, and $x\in B$.   Thus $R[A]\subseteq B$.  (ii)
If $x\in B$, let $\phi\in A$ be such that $x\in\overline{R[I_{\phi\restr
n}]}$ for every $n\in\Bbb N$.   \Quer\ If $(\phi,x)\notin R$, then
(because $R$ is closed) there are a $\sigma\in S^*$ and an open
$G\subseteq X$ such that $\phi\in I_{\sigma}$, $x\in G$ and
$(I_{\sigma}\times G)\cap R=\emptyset$.   But this means that $G\cap
R[I_{\sigma}]=\emptyset$ so $G\cap \overline{R[I_{\sigma}]}=\emptyset$
and $x\notin\overline{R[I_{\sigma}]}$;  which is absurd, because
$\sigma=\phi\restr n$ for some $n\ge 1$.\ \BanG\  Thus $(\phi,x)\in R$
and $x\in R[A]$.   As $x$ is arbitrary, $B\subseteq R[A]$ and $B=R[A]$,
as required.
}%end of proof of 421I

\leader{421J}{Proposition} Let $X$ be a topological space,
and $\Cal F$ the family of closed subsets of $X$.   Then a set
$A\subseteq X$ belongs to $\Cal S(\Cal F)$ iff there is a closed set
$R\subseteq\NN\times X$ such that $A$ is the projection of $R$
on $X$.

\proof{{\bf (a)} Suppose that $A\in\Cal S(\Cal F)$.   Let
$\family{\sigma}{S^*}{F_{\sigma}}$ be a Souslin scheme in $\Cal F$ with
kernel $A$.   Set

\Centerline{$R=\bigcap_{n\ge 1}\bigcup_{\sigma\in\BbbN^n}
I_{\sigma}\times F_{\sigma}$.}

\noindent For each $n\ge 1$,

\Centerline{$\bigcup_{\sigma\in\BbbN^n}I_{\sigma}\times F_{\sigma}
=(\NN\times X)\setminus\bigcup_{\sigma\in\Bbb
N^n}I_{\sigma}\times(X\setminus F_{\sigma})$}

\noindent is closed in $\NN\times X$, so $R$ is closed;  and
the projection $R[\NN]$ is $A$, by 421Ce.

\medskip

{\bf (b)} Suppose that $R\subseteq\NN$ is a closed set with
projection $A$.   Then $A$ is the kernel of the Souslin scheme
$\family{\sigma}{S^*}{\overline{R[I_{\sigma}]}}$, by 421I, so belongs to
$\Cal S(\Cal F)$.
}%end of proof of 421J

\leader{421K}{Definition} Let $X$ be a topological space.   A subset of
$X$ is a {\bf Souslin-F} set in $X$ if it is obtainable from closed
subsets of $X$ by Souslin's operation\cmmnt{;  that is, is the
projection of a closed subset of $\NN\times X$}.

\cmmnt{For a subset of $\BbbR^r$, or, more generally, of any Polish
space, it is common to say \lq Souslin set' for \lq Souslin-F
set';  see 421Xl.
}%end of comment

\leader{421L}{Proposition} Let $X$ be any topological space.   Then
every Baire subset of $X$ is Souslin-F.

\proof{ Let $\Cal Z$ be the family of zero sets in $X$.   If
$F\in\Cal Z$ then $X\setminus F$ is a countable union of zero sets
(4A2C(b-vi)), so belongs to $\Cal S(\Cal Z)$.   By 421F, the
$\sigma$-algebra generated by
$\Cal Z$ is included in $\Cal S(\Cal Z)\subseteq\Cal S(\Cal F)$, where
$\Cal F$ is the family of closed subsets of $X$;  that is, every Baire
set is Souslin-F.
}%end of proof of 421L

\leader{421M}{Proposition} Let $\Cal E$ be any family of sets such that
$\emptyset\in\Cal E$ and $E\cup E'$, $\bigcap_{n\in\BbbN}E_n$
belong to $\Cal E$ for every $E$, $E'\in\Cal E$ and all sequences
$\sequencen{E_n}$ in $\Cal E$.   \cmmnt{(For instance, $\Cal E$ could
be the family of closed subsets of a topological space, or a
$\sigma$-algebra of sets.)}   Let $\family{\sigma}{S^*}{E_{\sigma}}$ be
a Souslin scheme in $\Cal E$, and $K\subseteq\NN$ a set which
is compact for the usual topology on $\NN$.   Then
$\bigcup_{\phi\in K}\bigcap_{n\ge 1}E_{\phi\restr n}\in\Cal E$.

\proof{ Set $A=\bigcup_{\phi\in K}\bigcap_{n\ge 1}E_{\phi\restr n}$.
For $k\in\Bbb N$, set $K_k=\{\phi\restr k:\phi\in K\}$;  note that
$K_k\subseteq\BbbN^k$ is compact, because $\phi\mapsto\phi\restr k$ is
continuous, therefore finite, because the topology of $\BbbN^k$ is
discrete.   Set

\Centerline{$H=\bigcap_{k\ge 1}\bigcup_{\phi\in K_k}
  \bigcap_{1\le n\le k}E_{\phi\restr n}$.}

\noindent Because $\Cal E$ is closed under finite unions and countable
intersections, $H\in\Cal E$.   Now $A=H$.   \Prf\ (i) If $x\in A$, take
$\phi\in K$ such that $x\in E_{\phi\restr n}$ for every $n\ge 1$;  then
$\phi\restr k\in K_k$ and $x\in\bigcap_{1\le n\le k}E_{(\phi\restr
k)\restr n}$ for every $k\ge 1$, so $x\in H$.   Thus $A\subseteq H$.
(ii) If $x\in
H$, then for each $k\in\Bbb N$ we have a $\sigma_k\in K_k$ such that
$x\in\bigcap_{1\le n\le k}E_{\sigma_k\restr n}$.   Choose $\phi_k\in K$
such that $\phi_k\restr k=\sigma_k$ for each $k$.   Now $K$ is supposed
to be compact, so the sequence $\sequence{k}{\phi_k}$ has a cluster
point $\phi$ in $K$.

If $n\ge 1$, then $I_{\phi\restr n}$ is a neighbourhood of $\phi$ in
$\NN$, so must
contain $\phi_k$ for infinitely many $k$;  let $k\ge n$ be such that
$\phi_k\restr n=\phi\restr n$.   In this case

\Centerline{$x\in E_{\sigma_k\restr n}=E_{\phi_k\restr n}
=E_{\phi\restr n}$.}

\noindent As $n$ is arbitrary,

\Centerline{$x\in\bigcap_{n\ge 1}E_{\phi\restr n}\subseteq A$.}

\noindent As $x$ is arbitrary, $H\subseteq A$ and $H=A$, as claimed.\
\Qed

So $A\in\Cal E$.
}%end of proof of 421M

\leader{*421N}{}\cmmnt{ I now embark on preparations for the theory of
`constituents' of analytic and coanalytic sets.   It turns out that much
of the work can be done in the abstract context of this section.

\medskip

\noindent}{\bf Trees and derived trees (a)} Let $\Cal T$ be the family
of subsets $T$ of $S^*\cmmnt{\mskip5mu=\bigcup_{n\ge 1}\BbbN^n}$ such
that $\sigma\restr k\in T$ whenever
$\sigma\in T$ and $1\le k\le\#(\sigma)$.   Note that the intersection
and union of any non-empty family of members of $\Cal T$ again belong
to $\Cal T$.   \cmmnt{Members of $\Cal T$ are often called {\bf
trees}.}

\spheader 421Nb For $T\in\Cal T$, set

\Centerline{$\partial T
=\{\sigma:\sigma\in S^*,\,\Exists i\in\Bbb N$,
$\sigma^{\smallfrown}\fraction{i}\in T\}$,}

\noindent so that $\partial T\in\Cal T$ and $\partial T\subseteq T$.
\cmmnt{Of course} $\partial T_0\subseteq\partial T_1$ whenever $T_0$,
$T_1\in\Cal T$ and $T_0\subseteq T_1$.

\spheader 421Nc For $T\in\Cal T$, define
$\langle\partial^{\xi}T\rangle_{\xi<\omega_1}$ inductively by setting
$\partial^0T=T$ and, for $\xi>0$,
$\partial^{\xi}T=\bigcap_{\eta<\xi}\partial(\partial^{\eta}T)$.
\cmmnt{An easy induction shows that} $\partial^{\xi}T\in\Cal T$,
$\partial^{\xi}T\subseteq\partial^{\eta}T$ and
$\partial^{\xi+1}T=\partial(\partial^{\xi}T)$ whenever
$\eta\le\xi<\omega_1$.

\spheader 421Nd For any $T\in\Cal T$, there is a $\xi<\omega_1$ such
that $\partial^{\xi}T=\partial^{\eta}T$ whenever $\xi\le\eta<\omega_1$.
\prooflet{\Prf\ Set $T_1=\bigcap_{\xi<\omega_1}\partial^{\xi}T$.   For
each $\sigma\in S^*\setminus T_1$, there is a $\xi_{\sigma}<\omega_1$
such that $\sigma\notin\partial^{\xi_{\sigma}}T$.   Set
$\xi=\sup\{\xi_{\sigma}:\sigma\in S^*\setminus T_1\}$;  because $S^*$ is
countable, $\xi<\omega_1$, and we must now have $\partial^{\xi}T=T_1$,
so that $\partial^{\xi}T=\partial^{\eta}T$ whenever
$\xi\le\eta<\omega_1$.\ \Qed}

\spheader 421Ne For $T\in\Cal T$, its {\bf rank} is the first ordinal
$r(T)<\omega_1$ such that $\partial^{r(T)}T=\partial^{r(T)+1}T$;
\cmmnt{of course} $\partial^{r(T)}T=\partial^{\eta}T$ whenever
$r(T)\le\eta<\omega_1$, and
$\partial(\partial^{r(T)}T)=\partial^{r(T)}T$.

\spheader 421Nf For $T\in\Cal T$, the following are equiveridical:
($\alpha$) $\partial^{r(T)}T\ne\emptyset$;  ($\beta$) there is a
$\phi\in\NN$ such that $\phi\restr n\in T$ for every $n\ge 1$.
\prooflet{\Prf\ (i) If $\sigma\in\partial^{r(T)}T$ then
$\sigma\in\partial(\partial^{r(T)}T)$ so there is an $i\in\Bbb N$ such
that $\sigma^{\smallfrown}\fraction{i}\in\partial^{r(T)}T$.
We can therefore choose
$\sequencen{\sigma_n}$ inductively so that
$\sigma_n\in\partial^{r(T)}T$ and $\sigma_{n+1}$ properly extends
$\sigma_n$ for every $n$.   At the end of the induction,
$\phi=\bigcup_{n\in\Bbb N}\sigma_n$ belongs to $\NN$ and

\Centerline{$\phi\restr n=\sigma_n\restr n\in\partial^{r(T)}T\subseteq T$}

\noindent for every $n\ge 1$.   (ii) If $\phi\in\NN$ is such that
$\phi\restr n\in T$ for every $n\ge 1$, then an easy induction shows
that
$\phi\restr n\in\partial^{\xi}T$ for every $\xi<\omega_1$ and every
$n\ge 1$, so that $\partial^{r(T)}T$ is non-empty.\ \Qed}

\spheader 421Ng Now suppose that $\family{\sigma}{S^*}{A_{\sigma}}$ is a
Souslin scheme.   For any $x$ we have a tree $T_x\in\Cal T$ defined by
saying that

\Centerline{$T_x=\{\sigma:\sigma\in S^*,
  \,x\in\bigcap_{1\le i\le\#(\sigma)}A_{\sigma\restr i}\}$.}

\noindent Now the kernel of $\family{\sigma}{S^*}{A_{\sigma}}$ is just

$$\eqalign{A&=\{x:\,\Exists\phi\in\NN,\,
  x\in\bigcap_{n\ge 1}A_{\phi\restr n}\}\cr
&=\{x:\,\Exists\phi\in\NN,\,\phi\restr n\in T_x\Forall n\ge 1\}
=\{x:\partial^{r(T)}T\ne\emptyset\}\dvro{.}{}\cr}$$

\cmmnt{\noindent by (f).}

The sets

\Centerline{$\{x:x\in X\setminus A,\,r(T_x)=\xi\}
=\{x:x\in X,\,r(T_x)=\xi,\,\partial^{\xi}T_x=\emptyset\}$,}

\noindent for $\xi<\omega_1$, are called {\bf constituents} of
$X\setminus A$.   \cmmnt{(Of course they should properly be called
`the constituents of the Souslin scheme
$\family{\sigma}{S^*}{A_{\sigma}}$'.)}

\vleader{72pt}{*421O}{Theorem} Let $X$ be a set and $\Sigma$ a
$\sigma$-algebra
of subsets of $X$.   Let $\family{\sigma}{S^*}{A_{\sigma}}$ be a Souslin
scheme in $\Sigma$ with kernel $A$, and for $x\in X$ set

\Centerline{$T_x=\{\sigma:\sigma\in S^*,\,
  x\in\bigcap_{1\le i\le\#(\sigma)}
  A_{\sigma\restr i}\}\dvro{.}{\mskip5mu \in\Cal T}$}

\cmmnt{\noindent as in 421Ng.}

(a) For every $\xi<\omega_1$ and $\sigma\in S^*$,
$\{x:x\in X,\,\sigma\in\partial^{\xi}T_x\}\in\Sigma$.

(b) For every $\xi<\omega_1$, $\{x:x\in A,\,r(T_x)\le\xi\}$ and
$\{x:x\in X\setminus A,\,r(T_x)\le\xi\}$ belong to $\Sigma$.
In particular, all the constituents of $X\setminus A$ belong to
$\Sigma$.

\proof{{\bf (a)} Induce on $\xi$.   For $\xi=0$, we have

\Centerline{$\{x:x\in X,\,\sigma\in\partial^0T_x\}
=\{x:x\in X,\,\sigma\in T_x\}
=\bigcap_{1\le i\le\#(\sigma)}A_{\sigma\restr i}\in\Sigma$.}

\noindent For the inductive step to $\xi>0$, we have

$$\eqalign{\{x:\sigma\in\partial^{\xi}T_x\}
&=\{x:\sigma\in\bigcap_{\eta<\xi}\partial(\partial^{\eta}T_x)\}\cr
&=\bigcap_{\eta<\xi}\bigcup_{i\in\Bbb N}
  \{x:\sigma^{\smallfrown}\fraction{i}\in\partial^{\eta}T_x\}
\in\Sigma\cr}$$

\noindent because $\xi$ is countable and all the sets
$\{x:\sigma^{\smallfrown}\fraction{i}\in\partial^{\eta}T_x\}$ belong to
$\Sigma$ by the inductive hypothesis.

\medskip

{\bf (b)} Now, given $\xi<\omega_1$, we see that $r(T_x)\le\xi$ iff
$\partial^{\xi+1}T_x\supseteq\partial^{\xi}T_x$, so that if we set
$E_{\xi}=\{x:x\in X,\,r(T_x)\le\xi\}$ then

\Centerline{$E_{\xi}
=\bigcap_{\sigma\in S^*}\{x:x\in X,\,\sigma\in\partial^{\xi+1}T_x
  \text{ or }\sigma\notin\partial^{\xi}T_x\}$}

\noindent belongs to $\Sigma$.   If $x\in E_{\xi}$, so that
$\partial^{r(T_x)}T_x=\partial^{\xi}T_x$, 421Ng tells us that $x\in A$
iff $\partial^{\xi}T_x\ne\emptyset$;  so that

\Centerline{$E_{\xi}\cap A
=E_{\xi}\cap\bigcup_{\sigma\in S^*}\{x:\sigma\in\partial^{\xi}T_x\}$}

\noindent and $E_{\xi}\setminus A$ both belong to $\Sigma$.

Now the constituents of $X\setminus A$ are the sets
$(E_{\xi}\setminus A)\setminus\bigcup_{\eta<\xi}E_{\eta}$ for
$\xi<\omega_1$, which all belong to $\Sigma$.
}%end of proof of 421O

\leader{*421P}{Corollary} Let $X$ be a set and $\Sigma$ a
$\sigma$-algebra of subsets of $X$.   If $A\in\Cal S(\Sigma)$ then both
$A$ and $X\setminus A$ can be expressed as the union of at most
$\omega_1$ members of $\Sigma$.

\proof{ In the language of 421O, we have

\Centerline{$A=\bigcup_{\xi<\omega_1}E_{\xi}\cap A$,
\quad$X\setminus A=\bigcup_{\xi<\omega_1}E_{\xi}\setminus A$.}
}%end of proof of 421P

\leader{*421Q}{Lemma} Let $X$ be a set and
$\family{\sigma}{S^*}{A_{\sigma}}$ and $\family{\sigma}{S^*}{B_{\sigma}}$
two Souslin schemes of subsets of $X$.   Suppose that whenever $\phi$,
$\psi\in\NN$ there is an $n\ge 1$ such that
$\bigcap_{1\le i\le n}A_{\phi\restr i}\cap B_{\psi\restr i}=\emptyset$.
For $x\in X$ set

\Centerline{$T_x=\bigcup_{n\ge 1}\{\sigma:
  \sigma\in\BbbN^n,\,x\in\bigcap_{1\le i\le n}A_{\sigma\restr i}\}$}

\noindent\cmmnt{as in 421Ng, }and let $B$ be the kernel of
$\family{\sigma}{S^*}{B_{\sigma}}$.   Then $\sup_{x\in B}r(T_x)<\omega_1$.

\proof{ For $\sigma\in S^*$ set
$A'_{\sigma}=\bigcap_{1\le i\le\#(\sigma)}A_{\sigma\restr i}$,
$B'_{\sigma}=\bigcap_{1\le i\le\#(\sigma)}B_{\sigma\restr i}$.   Then
$T_x=\{\sigma:\sigma\in S^*,\,x\in A'_{\sigma}\}$ for each $x\in X$, $B$
is the kernel of $\family{\sigma}{S^*}{B'_{\sigma}}$, and for every
$\phi$, $\psi\in\NN$ there is an $n\in\Bbb N$ such that
$A'_{\phi\restr n}\cap B'_{\psi\restr n}=\emptyset$.

Define $\ofamily{\xi}{\omega_1}{Q_{\xi}}$ inductively by setting

\Centerline{$Q_0=\{(\sigma,\tau):\sigma,\,\tau\in S^*,
  \,A'_{\sigma}\cap B'_{\tau}\ne\emptyset\}$,}

\noindent and, for $0<\xi<\omega_1$,

\Centerline{$Q_{\xi}
=\bigcap_{\eta<\xi}\{(\sigma,\tau):
  \sigma,\,\tau\in S^*,\,\Exists i,\,j\in\Bbb N,
  \,(\sigma^{\smallfrown}\fraction{i},\tau^{\smallfrown}\fraction{j})
  \in Q_{\eta}\}$.}

\noindent Then the same arguments as in 421Na-421Nd show that there is a
$\zeta<\omega_1$ such that $Q_{\zeta+1}=Q_{\zeta}$.   \Quer\ If
$Q_{\zeta}\ne\emptyset$, then, just as in 421Nf, there must be $\phi$,
$\psi\in\NN$ such that
$(\phi\restr m,\psi\restr n)\in Q_{\zeta}\subseteq Q_0$ for every $m$,
$n\ge 1$;  but this means that
$A'_{\phi\restr n}\cap B'_{\psi\restr n}\ne\emptyset$ for every
$n\ge 1$, which is supposed to be impossible.\ \Bang

Now suppose that $x\in B$.   Then there is a $\psi\in\NN$ such that
$x\in B'_{\psi\restr n}$ for every $n\ge 1$.   But this means that
$(\sigma,\psi\restr n)\in Q_0$ for every $\sigma\in T_x$ and every
$n\ge 1$.  An easy induction shows that
$(\sigma,\psi\restr n)\in Q_{\xi}$ whenever $\xi<\omega_1$,
$\sigma\in\partial^{\xi}T_x$ and $n\ge 1$.   But as
$Q_{\zeta}=\emptyset$ we must have $\partial^{\zeta}T_x=\emptyset$ and
$r(T_x)\le\zeta$.   Thus $\sup_{x\in B}r(T_x)\le\zeta<\omega_1$, and the
proof is complete.
}%end of proof of 421Q

\exercises{\leader{421X}{Basic exercises (a)}
%\spheader 421Xa
Let $X$ be a set and $\Cal E$ a family of subsets of $X$.
(i) Show that $\emptyset\in\Cal S(\Cal E)$ iff there is a sequence in
$\Cal E$ with empty intersection.   (ii) Show that $X\in\Cal S(\Cal E)$
iff there is a sequence in $\Cal E$ with union $X$.
%421C

\spheader 421Xb Let $\Cal E$ be a family of sets and $F$ any set.   Show
that

\Centerline{$\Cal S(\{E\cap F:E\in\Cal E\})
=\{A\cap F:A\in\Cal S(\Cal E)\}$,}

\Centerline{$\Cal S(\{E\cup F:E\in\Cal E\})
=\{A\cup F:A\in\Cal S(\Cal E)\}$.}
%421C


\spheader 421Xc Suppose that $\Cal E$ is a family of sets with $\#(\Cal
E)\le\frak c$.   Show that $\#(\Cal S(\Cal E))\le\frak c$.
\Hint{$\#(\Cal E^{S^*})\le\#((\Cal P\BbbN)^{S^*})
=\#(\Cal P(\BbbN\times S^*))$.}
%421C

\spheader 421Xd Let $\Cal E$ be the family of half-open intervals
$\coint{2^{-n}k,2^{-n}(k+1)}$, where $n\in\Bbb N$, $k\in\Bbb Z$;  let
$\Cal G$ be the set of open subsets of $\Bbb R$;  let $\Cal F$ be the
set of closed subsets of $\Bbb R$;  let $\Cal K$ be the set of compact
subsets of $\Bbb R$;  let $\Cal B$ be the Borel $\sigma$-algebra of
$\Bbb R$.
Show that $\Cal S(\Cal E)=\Cal S(\Cal F)=\Cal S(\Cal G)=\Cal S(\Cal
K)=\Cal S(\Cal B)$.
\Hint{421F.}
%421F

\spheader 421Xe Let $\Cal I$ be the family
$\{I_{\sigma}:\sigma\in\bigcup_{k\in\BbbN}\BbbN^k\}$ (421A);   let $\Cal
G$ be the set of open subsets of $\NN$;  let $\Cal F$ be the
set of closed subsets of $\NN$;  let $\Cal K$ be the set of
compact subsets of $\NN$;  let $\Cal B$ be the Borel $\sigma$-algebra
of $\NN$.   Show that $\Cal S(\Cal I)=\Cal S(\Cal F)=\Cal
S(\Cal G)=\Cal S(\Cal B)$, but that $\Cal S(\Cal K)$ is strictly smaller
than these.   \Hint{if $\sequencen{K_n}$ is any sequence in $\Cal K$,
set $\phi(i)=1+\sup_{\psi\in K_i}\psi(i)$ for each $i\in\Bbb N$, so that
$\phi\notin\bigcup_{n\in\BbbN}K_n$;  hence show that $\BbbN^{\Bbb
N}\notin\Cal S(\Cal K)$.}
%421F, 421Xd

\spheader 421Xf Let $X$ be a separable metrizable space with at least
two points;  let $\Cal U$ be any base for its topology, and $\Cal B$ its
Borel $\sigma$-algebra.   Show that $\Cal S(\Cal U)=\Cal S(\Cal B)$.
What can happen if $\#(X)\le 1$?   What about hereditarily Lindel\"of
spaces?
%421F, 421Xd

\spheader 421Xg Let $X$ be a topological space;  let $\Cal Z$ be the set
of zero sets in $X$, $\Cal G$ the set of cozero sets, and $\CalBa$ the
Baire $\sigma$-algebra.   Show that
$\Cal S(\Cal Z)=\Cal S(\Cal G)=\Cal S(\CalBa)$.
%421F, 421Xd

\spheader 421Xh Let $X$ be a set, $\Cal E$ a family of subsets of $X$,
and $\Sigma$ the $\sigma$-algebra of subsets of $X$ generated by
$\Cal E$.   Show that if $\#(\Cal E)\le\frak c$ then
$\#(\Sigma)\le\frak c$.
\Hint{$\#(\frak c^{S^*})=\#(\Cal P(\Bbb N\times\Bbb N))=\frak c$ and
$\Sigma\subseteq\Cal S(\Cal E\cup\{X\setminus E:E\in\Cal E\})$.}
%421F, 421Xc

\spheader 421Xi Let $X$ be a topological space such that every open set
is Souslin-F.   Show that every Borel set is Souslin-F.
%421F

\spheader 421Xj Let $X$ be a topological space and $\Cal B(X)$ its Borel
$\sigma$-algebra.   Show that $\Cal S(\Cal B(X))$ is just the set of
projections on $X$ of Borel subsets of $\NN\times X$.   \Hint{4A3G.}
%421H

\spheader 421Xk Let $X$ and $Y$ be topological spaces, $f:X\to Y$ a
continuous function and $F\subseteq Y$ a Souslin-F set.   Show that
$f^{-1}[F]$ is a Souslin-F set in $X$.
%421K

\spheader 421Xl Let $X$ be any perfectly normal topological space (e.g.,
any metrizable space);  let $\Cal G$ be the
set of open subsets of $X$, $\Cal F$ the set of closed subsets, and
$\Cal B$ the Borel $\sigma$-algebra.   Show that
$\Cal S(\Cal G)=\Cal S(\Cal F)=\Cal S(\Cal B)$.
%421L, 421Xd

%\query no 421Xm

\sqheader 421Xn Let $X$ be a Hausdorff topological space and
$\family{\sigma}{S^*}{K_{\sigma}}$ a Souslin scheme in which every
$K_{\sigma}$ is a compact subset in $X$.   Show that
$\bigcup_{\phi\in K}\bigcap_{n\ge 1}K_{\phi\restr n}$ is compact for any
compact $K\subseteq\NN$.
%421M

\leader{421Y}{Further exercises (a)}
%\spheader 421Ya
Let $X$ be a topological space, $Y$ a Hausdorff space and
$f:X\to Y$ a continuous function.   Let $\Cal K$ be the family of closed
countably compact subsets of $X$.   Show that for any
$\Cal E\subseteq\Cal K$ such that $E\cap F\in\Cal E$ for all $E$,
$F\in\Cal E$,

\Centerline{$\{f[A]:A\in\Cal S(\Cal E)\}=\Cal S(\{f[E]:E\in\Cal E\})$.}
%421C

\spheader 421Yb Let $\Cal E$ be a family of sets and $F$ any set.   Show
that

$$\eqalign{\Cal S(\Cal E\cup\{F\})
&=\{F\}\cup\{A\cap F:A\in\Cal S(\Cal E)\}
  \cup\{B\cup F:B\in\Cal S(\Cal E)\}\cr
&\qquad\qquad\qquad\cup\{(A\cap F)\cup B:A,\,B\in\Cal S(\Cal E)\}.\cr}$$
%421C  mt42bits

\spheader 421Yc Let $X$ be a topological space, and $\CalBa$ its Baire
$\sigma$-algebra.   Show that $\Cal S(\CalBa)$ is just the
family of sets expressible as $f^{-1}[B]$ where $f$ is a continuous
function from $X$ to some metrizable space $Y$ and
$B\subseteq Y$ is Souslin-F.
%421F, 421Xg

\spheader 421Yd Let $X$ be a set, $\Cal E$ a family of subsets of $X$,
and $\Sigma$ the smallest $\sigma$-algebra of subsets of $X$ including
$\Cal E$ and closed under Souslin's operation.   Show that if
$\#(\Cal E)\le\frak c$ then $\#(\Sigma)\le\frak c$.   \Hint{define
$\ofamily{\xi}{\omega_1}{\Cal E_{\xi}}$ by setting
$\Cal E_{\xi}
=\Cal S(\{X\setminus E:E\in\Cal E\cup\bigcup_{\eta<\xi}\Cal E_{\xi}\})$
for each $\xi$.   Show that $\#(\Cal E_{\xi})\le\frak c$ for every $\xi$
and that $\Sigma=\bigcup_{\xi<\omega_1}\Cal E_{\xi}$.}

\spheader 421Ye Let $X$ be a compact space and $A$ a Souslin-F set in
$X$.   Show that there is a family $\ofamily{\xi}{\omega_1}{F_{\xi}}$ of
Borel sets such that $X\setminus A=\bigcup_{\xi<\omega}F_{\xi}$ and
whenever $B\subseteq X\setminus A$ is a Souslin-F set there is a
$\xi<\omega_1$ such that $B\subseteq F_{\xi}$.   \Hint{take
$F_{\xi}=\{x:r(T_x)\le\xi\}\setminus A$ as in 421Ob, and apply 421Q.}
}%end of exercises

\endnotes{
\Notesheader{421} In 111G, I defined the Borel sets of $\Bbb R$ to be
the members of the
smallest $\sigma$-algebra containing every open set.   In 114E, I
defined a set to be Lebesgue measurable if it behaves in the right way
with respect to Lebesgue outer measure.   The latter formulation, at
least, provides some sort of testing principle to determine whether a
set is Lebesgue measurable.   But the definition of `Borel set' does
not.   The only tool so far available for proving that a set
$E\subseteq\Bbb R$ is {\it not}
Borel is to find a $\sigma$-algebra containing all open sets and not
containing $E$;  conversely, the only method we have for proving
properties of Borel sets is to show that a property is possessed by
every member of some $\sigma$-algebra containing every open set.   The
revolutionary insight of {\smc Souslin 1917} was a construction which
could build every Borel set from rational intervals.   (See 421Xd.)
For fundamental reasons, no construction of this kind can provide all
Borel sets without also producing other sets, and to actually
characterize the Borel $\sigma$-algebra a
further idea is needed (423Fa);  but the class of analytic sets, being
those constructible by Souslin's operation from rational intervals (or
open sets, or closed sets, or Borel sets -- the operation is robust
under such variations), turns out to have remarkable properties which
make it as important in modern real analysis as the Borel algebra
itself.

The guiding principle of `descriptive set theory' is that
the properties of a
set may be analysed in the light of a construction for that set.   Thus
we can think of a closed set $F\subseteq\Bbb R$ as

\Centerline{$\Bbb R\setminus\bigcup_{(q,q')\in I}\ooint{q,q'}$}

\noindent where $I\subseteq\Bbb Q\times\Bbb Q$.   The principle can be
effective because we often have such descriptions in terms of objects
fundamentally simpler than the set being described.   In the formula
above, for instance, $\Bbb Q\times\Bbb Q$ is simpler than the set $F$,
being a countable set with a straightforward description from $\Bbb N$.
The set $\Cal P(\Bbb Q\times\Bbb Q)$ is relatively complex;  but a
single subset $I$ of $\Bbb Q\times\Bbb Q$ can easily be coded as a
single subset of $\Bbb N$ (taking some more or less natural enumeration
of $\Bbb Q^2$ as a
sequence $\sequencen{(q_n,q'_n)}$, and matching $I$ with
$\{n:(q_n,q'_n)\in I\}$).   So, subject to an appropriate coding, we
have a description of
closed subsets of $\Bbb R$ in terms of subsets of $\Bbb N$.   At the
most elementary level, this shows that there are at most $\frak c$
closed subsets of $\Bbb R$.   But we can also set out to analyse such
operations as intersection, union, closure in terms of these
descriptions.   The
details are complex, and I shall go no farther along this path until
Chapter 56 in Volume 5;
but investigations of this kind are at the heart of some of the most
exciting developments of twentieth-century real analysis.

The particular descriptive method which concerns us in the present
section is Souslin's operation.   Starting from a relatively simple
class $\Cal E$, we proceed to the larger class $\Cal S(\Cal E)$.   The
most fundamental property of $\Cal S$ is 421D:
$\Cal S\Cal S(\Cal E)=\Cal S(\Cal E)$.
This means, for instance, that if $\Cal E\subseteq\Cal S(\Cal F)$ and
$\Cal F\subseteq\Cal S(\Cal E)$, then $\Cal S(\Cal E)$ will be equal to
$\Cal S(\Cal F)$;  consequently, different classes of sets will often
have the same Souslin closures, as in 421Xd-421Xg.   After a little
practice you will find that it is often easy to see when two classes
$\Cal E$ and $\Cal F$ are at the same level in this sense;  but watch
out for traps like the
class of compact subsets of $\NN$ (421Xe) and odd technical
questions (421Xf).

Souslin's operation, and variations on it, will be the basis of much of
the next chapter;  it has dramatic applications in general topology and
functional analysis as well as in real analysis and measure theory.   An
important way of looking at the kernel of a Souslin scheme
$\family{\sigma}{S^*}{E_{\sigma}}$ is to regard it as the projection on
the second coordinate of the corresponding set
$R=\bigcap_{k\ge 1}\bigcup_{\sigma\in\BbbN^k}I_{\sigma}\times E_{\sigma}$
(421Ce).   We find that many other sets $R\subseteq\NN\times X$ will
also have projections in $\Cal S(\Cal E)$ (421G, 421H).   Let me remark
that it is essential here that the first coordinate should be of the
right type.   In one sense, indeed, $\NN$ is the only thing that
will do;  but its virtue transfers to analytic spaces, as we shall see in
423N-423Q below.  %423N 423P 423Q
We shall often
want to deal with members of $\Cal S(\Cal E)$ which are most naturally
defined in terms of some such auxiliary space.

I have moved into slightly higher gear for 421N-421Q
%421N 421O 421P 421Q
because these are not essential for most of the work of the next
chapter.   From the point of view of this section 421P is very striking
but the significance of 421Q is unlikely to be apparent.   It becomes
important in contexts in which the condition

\Centerline{$\Forall\phi$, $\psi\in\NN\Exists n\ge 1,\,
\bigcap_{1\le i\le n}A_{\phi\restr i}\cap B_{\psi\restr i}=\emptyset$}

\noindent is satisfied for natural reasons.   I will expand on these in
the next two sections.   In the meantime, I offer 421Ye as an example
of what 421O and 421Q together can tell us.
}%end of notes

\discrpage

\frfilename{mt422.tex}
\versiondate{12.4.16}
\copyrightdate{2001}

\def\NN{\BbbN^{\Bbb N}}

\def\chaptername{Descriptive set theory}
\def\sectionname{K-analytic spaces}

\newsection{422}

I introduce K-analytic spaces, defined in terms of usco-compact
relations.   The first step is to define the latter (422A) and give
their fundamental properties (422B-422E).   I reach K-analytic spaces
themselves in 422F, with an outline of the most important facts about
them in 422G-422K.

\leader{422A}{Definition} Let $X$ and $Y$ be Hausdorff spaces.   A
relation $R\subseteq X\times Y$ is {\bf usco-compact} if

\inset{($\alpha$) $R[\{x\}]$ is a compact subset of $Y$ for every
$x\in X$,}

\inset{($\beta$) $R^{-1}[F]$ is a closed subset of $X$ for every closed
set $F\subseteq Y$.}

\cmmnt{\noindent (Relations satisfying condition ($\beta$) are
sometimes called `upper semi-continuous'.)}

\leader{422B}{}\cmmnt{ The following elementary remark will be useful.

\medskip

\noindent}{\bf Lemma} Let $X$ and $Y$ be Hausdorff spaces and
$R\subseteq X\times Y$ an usco-compact relation.   If $x\in X$ and
$H$ is an open subset of $Y$ including $R[\{x\}]$, there is an open set
$G\subseteq X$, containing $x$, such that $R[G]\subseteq H$.

\proof{ Set $G=X\setminus R^{-1}[Y\setminus H]$.   Because $Y\setminus
H$ is closed, so is $R^{-1}[Y\setminus H]$, and $G$ is open.   Of course
$R[G]\subseteq H$, and $x\in G$ because $R[\{x\}]\subseteq H$.
}%end of proof of 422B

\leader{422C}{Proposition} Let $X$ and $Y$ be Hausdorff spaces.   Then a
subset $R$ of $X\times Y$ is an usco-compact relation iff whenever
$\Cal F$ is an ultrafilter on $X\times Y$, containing $R$, such that the
first-coordinate image
$\pi_1[[\Cal F]]$ of $\Cal F$ has a limit in $X$, then $\Cal F$ has a
limit in $R$.

\proof{ Recall that, writing $\pi_1(x,y)=x$ and $\pi_2(x,y)=y$ for
$(x,y)\in X\times Y$,

\Centerline{$\pi_1[[\Cal F]]
=\{A:A\subseteq X,\,\pi_1^{-1}[A]\in\Cal F\}
=\{A:A\subseteq X,\,A\times Y\in\Cal F\}$}

\noindent (2A1Ib), and that $\Cal F\to(x,y)$ iff $\pi_1[[\Cal F]]\to x$
and $\pi_2[[\Cal F]]\to y$ (3A3Ic).

\medskip

{\bf (a)} Suppose that $R$ is usco-compact and that $\Cal F$ is an
ultrafilter on $X\times Y$, containing $R$, such that $\pi_1[[\Cal F]]$
has a limit $x\in X$.   \Quer\ If $\Cal F$ has no limit in $R$, then, in
particular, it does not converge to $(x,y)$ for any $y\in R[\{x\}]$;
that is, $\pi_2[[\Cal F]]$ does not converge to any point of $R[\{x\}]$,
that is, every point of $R[\{x\}]$ belongs to an open set not belonging
to $\pi_2[[\Cal F]]$.   Because $R[\{x\}]$ is compact, it is covered by
a finite union of open sets not belonging to $\pi_2[[\Cal F]]$;  but as
$\pi_2[[\Cal F]]$ is an ultrafilter (2A1N), there is an open set
$H\supseteq R[\{x\}]$ such that $Y\setminus H\in\pi_2[[\Cal F]]$.

Now 422B tells us that there is an open set $G$ containing $x$ such that
$R[G]\subseteq H$.   In this case, $G\in\pi_1[[\Cal F]]$ so
$G\times Y\in\Cal F$;  at the same time, $X\times(Y\setminus H)\in\Cal F$.
So

\Centerline{$R\cap(G\times Y)\cap(X\times(Y\setminus H))\in\Cal F$.}

\noindent But this is an empty set, by the choice of $G$;  which is
intolerable.\ \Bang

Thus $\Cal F$ has a limit in $R$, as required.

\wheader{422C}{4}{2}{2}{48pt}

{\bf (b)} Now suppose that $R$ has the property described.

\medskip

\quad{\bf (i)} Let $x\in X$, and suppose that $\Cal G$ is an ultrafilter
on $Y$ containing $R[\{x\}]$.   Set $h(y)=(x,y)$ for $y\in Y$;  then
$\Cal F=h[[\Cal G]]$ is an ultrafilter on $X\times Y$ containing $R$.
The image $\pi_1[[\Cal F]]$ is just the principal filter generated by
$\{x\}$, so certainly converges to $x$;  accordingly $\Cal F$ must
converge to some point $(x,y)\in R$, and

\Centerline{$\pi_2[[\Cal F]]=\pi_2h[[\Cal G]]=\Cal G$}

\noindent converges to $y\in R[\{x\}]$.
As $\Cal G$ is arbitrary, $R[\{x\}]$ is compact (2A3R).

\medskip

\quad{\bf (ii)} Let $F\subseteq Y$ be closed, and take
$x\in\overline{R^{-1}[F]}\subseteq X$.   Consider

\Centerline{$\Cal E=\{R,X\times F\}\cup\{G\times Y:G\subseteq X$
is open, $x\in G\}$.}

\noindent Then $\Cal E$ has the finite intersection property.   \Prf\ If
$G_0,\ldots,G_n$ are open sets containing $x$, then $R^{-1}[F]$ meets
$G_0\cap\ldots\cap G_n$ in $z$ say, and now
$(z,y)\in R\cap(X\times F)\cap\bigcap_{i\le n}(G_i\times Y)$ for some
$y\in F$.\ \QeD\   Let
$\Cal F$ be an ultrafilter on $X\times Y$ including $\Cal E$
(4A1Ia).   Because $G\times Y\in\Cal E\subseteq\Cal F$ for every
open set $G$ containing $x$, $\pi_1[[\Cal F]]\to x$, so $\Cal F$
converges to some point
$(x,y)$ of $R$.   Because $X\times F$ is a
closed set belonging to $\Cal E\subseteq\Cal F$, $y\in F$ and
$x\in R^{-1}[F]$.   As $x$ is arbitrary, $R^{-1}[F]$ is closed;  as $F$
is arbitrary, $R$ satisfies condition ($\beta$) of 422A, and is
usco-compact.
}%end of proof of 422C

\leader{422D}{Lemma} (a) Let $X$ and $Y$ be Hausdorff spaces.
If $R\subseteq X\times Y$ is an usco-compact relation, then $R$ is
closed in $X\times Y$.

(b) Let $X$ and $Y$ be Hausdorff spaces.
If $R\subseteq X\times Y$ is an usco-compact relation and
$R'\subseteq R$ is a closed set, then $R'$ is usco-compact.

(c) Let $X$ and $Y$ be Hausdorff spaces.   If $f:X\to Y$ is a continuous
function, then its graph is an usco-compact relation.

(d) Let $\familyiI{X_i}$ and $\familyiI{Y_i}$ be families of Hausdorff
spaces, and $R_i\subseteq X_i\times Y_i$ an usco-compact relation for
each $i$.   Set $X=\prod_{i\in I}X_i$, $Y=\prod_{i\in I}Y_i$ and

\Centerline{$R=\{(x,y):x\in X,\,y\in Y,\,(x(i),y(i))\in R_i$ for every
$i\in I\}$.}

\noindent Then $R$ is usco-compact in $X\times Y$.

(e) Let $X$ and $Y$ be Hausdorff spaces, and $R\subseteq X\times Y$ an
usco-compact relation.   Then (i) $R[K]$ is a compact subset of $Y$ for
any compact subset $K$ of $X$ (ii) $R[L]$ is a Lindel\"of subset of $Y$
for any Lindel\"of subset $L$ of $X$.

(f) Let $X$, $Y$ and $Z$ be Hausdorff spaces, and
$R\subseteq X\times Y$, $S\subseteq Y\times Z$ usco-compact relations.
Then the composition

\Centerline{$S\frsmallcirc R=\{(x,z):$ there is some $y\in Y$ such that
$(x,y)\in R$ and $(y,z)\in S\}$}

\noindent is usco-compact in $X\times Z$.

(g) Let $X$ and $Y$ be Hausdorff spaces and $Y_0$ any subset of $Y$.
Then a relation $R\subseteq X\times Y_0$ is usco-compact when regarded
as a relation between $X$ and $Y_0$ iff it is usco-compact when regarded
as a relation between $X$ and $Y$.

(h) Let $Y$ be a Hausdorff space and $R\subseteq\NN\times Y$ an
usco-compact relation.   Set

\Centerline{$R'=\{(\alpha,y):\alpha\in\NN$, $y\in Y$ and there is a
$\beta\le\alpha$ such that $(\beta,y)\in R\}$.}

\noindent Then $R'$ is usco-compact.

\proof{{\bf (a)} If $(x,y)\in\overline{R}$, there is an ultrafilter
$\Cal F$ containing $R$ and converging to $(x,y)$ (4A2Bc).   By 422C,
$\Cal F$ must have a limit in $R$;  but as $X\times Y$ is Hausdorff,
this limit
must be $(x,y)$, and $(x,y)\in R$.   As $(x,y)$ is arbitrary, $R$ is
closed.

\medskip

{\bf (b)} It is obvious that $R'$ will satisfy the condition of 422C if
$R$ does.

\medskip

{\bf (c)} $f[\{x\}]=\{f(x)\}$ is surely compact for every $x\in X$, and
$f^{-1}[F]$ is closed for every closed set $F\subseteq Y$ because $f$ is
continuous.

\medskip

{\bf (d)} For $i\in I$, $x\in X$, $y\in Y$ set
$\phi_i(x,y)=(x(i),y(i))$.
If $\Cal F$ is an ultrafilter on $X\times Y$ containing $R$ such that
$\pi_1[[\Cal F]]$ has a limit in $X$, then

\Centerline{$\pi_1\phi_i[[\Cal F]]=\psi_i\pi_1[[\Cal F]]$}

\noindent has a limit in $X_i$ for every $i\in I$, writing
$\psi_i(x)=x(i)$
for $i\in I$ and $x\in X$.   But $\phi_i[[\Cal F]]$ is an ultrafilter
containing $R_i$, so has
a limit $(x_0(i),y_0(i))$ in $R_i$, for each $i$.
Accordingly $(x_0,y_0)$ is a limit of $\Cal F$ in $X\times Y$ (3A3Ic),
while $(x_0,y_0)\in R$.   As $\Cal F$ is arbitrary, $R$ is usco-compact.

\medskip

{\bf (e)} For the moment, let $L$ be any subset of $X$.    Let $\Cal H$
be a family of open sets in $Y$ covering $R[L]$.   Let $\Cal G$ be the
family of those open sets $G\subseteq X$ such that $R[G]$ can be covered
by finitely many members of $\Cal H$.   Then $\Cal G$ covers $L$.
\Prf\ If $x\in L$, then $R[\{x\}]$ is a compact subset of
$R[L]\subseteq\bigcup\Cal H$, so there is a finite set
$\Cal H'\subseteq\Cal H$ covering
$R[\{x\}]$.   Now there is an open set $G$ containing $x$ such that
$R[G]\subseteq\bigcup\Cal H'$, by 422B.\ \Qed

\medskip

\quad{\bf (i)} If $L$ is compact, then there must be a finite subfamily
$\Cal G'$ of $\Cal G$ covering $L$;  now $R[L]\subseteq R[\bigcup\Cal
G']$
is covered by finitely many members of $\Cal H$.   As $\Cal H$ is
arbitrary, $R[L]$ is compact.

\medskip

\quad{\bf (ii)} If $L$ is Lindel\"of, then there must be a countable
subfamily $\Cal G'$ of $\Cal G$ covering $L$;  now
$R[L]\subseteq R[\bigcup\Cal G']$ is covered by countably many members
of $\Cal H$.   As $\Cal H$ is arbitrary, $R[L]$ is Lindel\"of.

\medskip

{\bf (f)} If $x\in X$ then $R[\{x\}]\subseteq Y$ is compact, so
$(SR)[\{x\}]=S[R[\{x\}]]$ is compact, by (e-i).   If $F\subseteq Z$ is
closed then $S^{-1}[F]\subseteq Y$ is closed so
$(SR)^{-1}[F]=R^{-1}[S^{-1}[F]]$ is closed.

\medskip

{\bf (g)(i)} Suppose that $R$ is usco-compact when regarded as a subset
of $X\times Y_0$.   Set $S=\{(y,y):y\in Y_0\}$;  by (c), $S$ is
usco-compact when regarded as a subset of $Y_0\times Y$, so by (f)
$R=SR$ is usco-compact when regarded as a subset of $X\times Y$.

\medskip

\quad{\bf (ii)} If $R$ is usco-compact when regarded as a subset of
$X\times Y$, and $x\in X$, then $R[\{x\}]$ is a subset of $Y_0$ which is
compact for the topology of $Y$, therefore for the subspace topology of
$Y_0$.   If $F\subseteq Y_0$ is closed for the subspace topology, it is
of the form $F'\cap Y_0$ for some closed $F'\subseteq Y$,
so $R^{-1}[F]=R^{-1}[F']$ is closed in $X$.   As $x$ and $F$ are
arbitrary, $R$ is usco-compact in $X\times Y_0$.

\medskip

{\bf (h)} Set
$S=\{(\alpha,\beta):\beta\le\alpha\in\NN\}$.   Then $S$ is usco-compact in
$\NN\times\NN$.   \Prf\
$S[\{\alpha\}]=\{\beta:\beta\le\alpha\}$ is a product of finite sets, so is
compact, for every $\alpha\in\NN$.
If $F\subseteq\NN$ is closed and $\sequencen{\alpha_n}$ is a convergent
sequence in $S^{-1}[F]$ with limit $\alpha\in\NN$,
then for every $n\in\Bbb N$ there is a
$\beta_n\in F$ such that $\beta_n\le\alpha_n$.   For any $i\in\Bbb N$,
$\sup_{n\in\Bbb N}\beta_n(i)\le\sup_{n\in\Bbb N}\alpha_n(i)$ is finite, so
$\{\beta_n:n\in\Bbb N\}$ is relatively compact and $\sequencen{\beta_n}$
has a cluster point $\beta$ say;  now

\Centerline{$\beta(i)\le\limsup_{n\to\infty}\beta_n(i)
\le\limsup_{n\to\infty}\alpha_n(i)=\alpha(i)$}

\noindent for every $i\in\Bbb N$, so $\beta\le\alpha$.   Also, of course,
$\beta\in F$, so $\alpha\in S^{-1}[F]$.   As $\sequencen{\alpha_n}$ is
arbitrary, $S^{-1}[F]$ is closed;  as $F$ is arbitrary, $S$ is
usco-compact.\ \Qed

Now $R'=R\frsmallcirc S$ is usco-compact, by (f) above.
}%end of proof of 422D

\leader{422E}{}\cmmnt{ The following lemma is actually very important
in the structure theory of K-analytic spaces (see 422Yc).   It will be
useful to us in 423C below.

\medskip

\noindent}{\bf Lemma} Let $X$ and $Y$ be Hausdorff spaces, and
$R\subseteq X\times Y$ an usco-compact relation.   If $X$ is regular, so
is $R$\cmmnt{ (in its subspace topology)}.

\proof{ \Quer\ Suppose, if possible, otherwise;  that there are a closed
set $F\subseteq R$ and an $(x,y)\in R\setminus F$ which cannot be
separated from $F$ by open sets (in $R$).   If $G$, $H$ are open sets
containing $x$, $y$ respectively, then $R\cap(G\times H)$,
$R\setminus(\overline{G}\times\overline{H})$ are disjoint relatively
open sets in $R$, so the latter cannot include $F$;  that is,
$F\cap(\overline{G}\times\overline{H})\ne\emptyset$ whenever $G$, $H$
are open, $x\in G$ and $y\in H$.   Accordingly there is an ultrafilter
$\Cal F$ on $X\times Y$ such that
$F\cap(\overline{G}\times\overline{H})\in\Cal F$
whenever $G\subseteq X$ and $H\subseteq Y$ are open sets containing $x$,
$y$ respectively.   In this case $R\in\Cal F$, and
$\overline{G}\in\pi_1[[\Cal F]]$ for every open set $G$ containing $x$.
Because the topology of
$X$ is regular, every open set containing $x$ includes $\overline{G}$
for some smaller open set $G$ containing $x$, and belongs to
$\pi_1[[\Cal F]]$;  thus $\pi_1[[\Cal F]]\to x$ in $X$.   Because $R$ is
usco-compact, $\Cal F$
has a limit in $R$, which must be of the form $(x,y')$ (422C).   Because
$F\in\Cal F$ is closed (in $R$), $(x,y')\in F$.   But also
$y'\in\overline{H}$ for
every open set $H$ containing $y$, since $X\times\overline{H}$ is a
closed set belonging to $\Cal F$;  because the topology of $Y$ is
Hausdorff, $y'$ must be equal to $y$, and $(x,y)\in F$, which is
absurd.\ \Bang
}%end of proof of 422E


\leader{422F}{Definition}\cmmnt{ ({\smc Frol\'\i k 61})} Let $X$ be a
Hausdorff space.   Then $X$ is
{\bf K-analytic} if there is an usco-compact relation
$R\subseteq\NN\times X$ such that $R[\NN]=X$.

If $X$ is a Hausdorff space, we call a subset of $X$ K-analytic if
it is a K-analytic space in its subspace topology.


\leader{422G}{Theorem} (a) Let $X$ be a Hausdorff space.   Then a subset
$A$ of $X$ is K-analytic iff there is an usco-compact relation
$R\subseteq\NN\times X$ such that $R[\NN]=A$.

(b) $\NN$ is K-analytic.

(c) Compact Hausdorff spaces are K-analytic.

(d) If $X$ and $Y$ are Hausdorff spaces and $R\subseteq X\times Y$ is an
usco-compact relation, then $R[A]$ is K-analytic whenever $A\subseteq X$
is K-analytic.  In particular, a Hausdorff continuous image of a
K-analytic Hausdorff space is K-analytic.

(e) A product of countably many K-analytic Hausdorff spaces is
K-analytic.

(f) A closed subset of a K-analytic Hausdorff space is K-analytic.

(g) A K-analytic Hausdorff space is Lindel\"of, so a regular K-analytic
Hausdorff space is completely regular.

\proof{{\bf (a)} $A$ is K-analytic iff there is an usco-compact relation
$R\subseteq\NN\times A$ with projection $A$.   But a subset of
$\NN\times X$ with projection $A$ is usco-compact in $\NN\times A$ iff
it is usco-compact in $\NN\times X$, by 422Dg.

\medskip

{\bf (b)} The identity function from $\NN$ to itself is an usco-compact
relation, by 422Dc.

\medskip

{\bf (c)} If $X$ is compact, then $R=\NN\times X$ is an
usco-compact relation (because $R[\{\phi\}]=X$ is compact for every
$\phi\in\NN$, while $R^{-1}[F]$ is either $\NN$ or $\emptyset$ for every
closed $F\subseteq X$), so $X=R[\NN]$ is K-analytic.

\medskip

{\bf (d)} By (a), there is an usco-compact relation
$S\subseteq\NN\times X$ such that $S[\NN]=A$.   Now
$R\frsmallcirc S\subseteq\NN\times Y$ is usco-compact, by 422Df, and
$RS[\NN]=R[A]$.

In particular, if $X$ itself is K-analytic and $f:X\to Y$ is a
continuous surjection, $f$ is an
usco-compact relation (422Dc), so $Y=f[X]$ is K-analytic.

\medskip

{\bf (e)} Let $\familyiI{X_i}$ be a countable family of K-analytic
Hausdorff spaces with product $X$.   If $I=\emptyset$ then
$X=\{\emptyset\}$ is compact,
therefore K-analytic.   Otherwise, choose for each $i\in I$ an
usco-compact relation $R_i\subseteq\NN\times X_i$ such that
$R_i[\NN]=X_i$.   Set

\Centerline{$R=\{(\phi,x):\phi\in(\NN)^I,\,x\in X,\,
(\phi(i),x(i))\in R_i$ for every $i\in I\}$.}

\noindent By 422Dd, $R$ is an usco-compact relation in
$(\NN)^I\times X$, and it is easy to see that $R[(\NN)^I]=X$.   But
$(\NN)^I\cong\BbbN^{\Bbb N\times I}$ is homeomorphic to $\NN$, because
$I$ is countable,
so we can identify $R$ with a relation in $\NN\times X$ which is still
usco-compact, and $X$ is K-analytic.

\medskip

{\bf (f)} Let $X$ be a K-analytic Hausdorff space and $F$ a closed
subset.   Let $R\subseteq\NN\times X$ be an usco-compact relation such
that $R[\NN]=X$.   Set $R'=R\cap(\NN\times F)$.   Then $R'$ is a closed
subset of $R$, so is
usco-compact (422Db).   By (a) here, $F=R'[\NN]$ is K-analytic.

\medskip

{\bf (g)} Let $X$ be a K-analytic Hausdorff space.   $\NN$ is
Lindel\"of (4A2Ub), and there is an usco-compact relation $R$ such
that $R[\NN]=X$, so that $X$ is Lindel\"of, by 422D(e-ii).   4A2H(b-i)
now tells us that if $X$ is regular it is completely regular.
}%end of proof of 422G

\leader{422H}{Theorem} (a) If $X$ is a Hausdorff space, then any
K-analytic subset of $X$ is Souslin-F in $X$.

(b) If $X$ is a K-analytic Hausdorff space, then a subset of $X$ is
K-analytic iff it is Souslin-F in $X$.

(c) For any Hausdorff space $X$, the family of K-analytic subsets of $X$
is closed under Souslin's operation.

\proof{{\bf (a)} If $A\subseteq X$ is K-analytic, there is an
usco-compact relation $R\subseteq\NN\times X$ such that $R[\NN]=A$, by
422Ga.   By 422Da, $R$ is a closed set;  so $A$ is Souslin-F by 421J.

\medskip

{\bf (b)} Now suppose that $X$ itself is K-analytic, and that
$A\subseteq X$ is Souslin-F in $X$.   Then there is a closed set
$R\subseteq\NN\times X$ such that $R[\NN]=A$ (421J, in the other
direction).   $\NN\times X$ is
K-analytic (422Gb, 422Ge), and $R$ is closed, therefore itself
K-analytic (422Gf);  so
its continuous image $A$ is
K-analytic, by 422Gd.

\medskip

{\bf (c)(i)} The first step is to show that the union of a sequence of
K-analytic subsets of $X$ is K-analytic.   \Prf\ Let $\sequencen{A_n}$
be a sequence of K-analytic sets, with union $A$.   For each
$n\in\Bbb N$, let $R_n\subseteq\NN\times X$ be an usco-compact relation
such that
$R_n[\NN]=A_n$.   In $(\Bbb N\times\NN)\times X$ let $R$ be the set

\Centerline{$\{((n,\phi),x):n\in\Bbb N,\,(\phi,x)\in R_n\}$.}

\noindent If $(n,\phi)\in\Bbb N\times\NN$, then
$R[\{(n,\phi)\}]=R_n[\{\phi\}]$ is compact;  if $F\subseteq X$ is
closed, then

\Centerline{$R^{-1}[F]=\{(n,\phi):n\in\Bbb N,\,\phi\in R_n^{-1}[F]\}$}

\noindent is closed in $\Bbb N\times\NN$.   So $R$ is
usco-compact, and of course

\Centerline{$R[\Bbb N\times\NN]=\bigcup_{n\in\Bbb N}R_n[\NN]=A.$}

\noindent As $\Bbb N\times\NN$ is K-analytic (in fact, homeomorphic to
$\NN$), $A$ is K-analytic.\ \Qed

\medskip

\quad{\bf (ii)} Now suppose that $\family{\sigma}{S^*}{A_{\sigma}}$ is a
Souslin scheme consisting of K-analytic sets with kernel $A$.   Then
$X'=\bigcup_{\sigma\in S^*}A_{\sigma}$ is K-analytic, by (i).   By (a),
every $A_{\sigma}$ is Souslin-F when regarded as a subset of $X'$.   But
since the family of Souslin-F subsets of $X'$ is closed under Souslin's
operation, by 421D, $A$ also is Souslin-F in $X'$.   By (b) of this
theorem, $A$ is K-analytic.   As $\family{\sigma}{S^*}{A_{\sigma}}$ is
arbitrary, we have the result.
}%end of proof of 422H


\leader{422I}{}\cmmnt{ It seems that for the measure-theoretic
results of \S432, at least, the following result (the `First
Separation Theorem') is not essential.   However I do not think it
possible to get a firm grasp on K-analytic and analytic spaces without
knowing some version of it, so I present it here.   It is most often
used through the forms in 422J and 422Xf below.

\wheader{422I}{4}{2}{2}{48pt}

\noindent}{\bf Lemma} Let $X$ be a Hausdorff space.   Let $\Cal E$ be a
family of subsets of $X$ such that (i) $\bigcup_{n\in\Bbb N}E_n$ and
$\bigcap_{n\in\Bbb N}E_n$ belong to
$\Cal E$ whenever $\sequencen{E_n}$ is a sequence in $\Cal E$
(ii) whenever $x$, $y$ are distinct points of $X$, there are disjoint
$E$, $F\in\Cal E$ such that $x\in\interior E$ and $y\in\interior F$.
Then whenever $A$, $B$ are disjoint non-empty K-analytic subsets of $X$,
there are disjoint $E$, $F\in\Cal E$ such that $A\subseteq E$ and
$B\subseteq F$.

\proof{{\bf (a)} We need to know that if $K$, $L$ are disjoint non-empty
compact subsets of $X$, there are disjoint $E$, $F\in\Cal E$ such that
$K\subseteq\interior E$ and $L\subseteq\interior F$.   \Prf\ For any
point $(x,y)\in K\times L$, we can find disjoint $E_{xy}$,
$F_{xy}\in\Cal E$ such that $x\in\interior E_{xy}$ and $y\in\interior F_{xy}$.
Because $L$ is compact and non-empty, there is for each $x\in K$ a
non-empty finite set
$I_x\subseteq L$ such that $L\subseteq\bigcup_{y\in I_x}\interior
F_{xy}$.   Set $E_x=\bigcap_{y\in I_x}E_{xy}$, $F_x=\bigcup_{y\in
I_x}F_{xy}$;  then $E_x$, $F_x$ are disjoint members of $\Cal E$,
$x\in\interior E_x$ and $L\subseteq\interior F_x$.   Because $K$ is
compact and not empty, there is a non-empty finite set $J\subseteq K$
such that
$K\subseteq\bigcup_{x\in J}\interior E_x$.   Set $E=\bigcup_{x\in
J}E_x$, $F=\bigcap_{x\in J}F_x$;  then $E$, $F\in\Cal E$, $E\cap
F=\emptyset$, $K\subseteq\interior E$ and $L\subseteq\interior F$, as
required.\ \Qed

\medskip

{\bf (b)} Let $Q$, $R\subseteq\NN\times X$ be usco-compact relations
such that $Q[\NN]=A$ and $R[\NN]=B$.    For each
$\sigma\in S=\bigcup_{n\in\Bbb N}\BbbN^n$, set

\Centerline{$I_{\sigma}=\{\phi:\sigma\subseteq\phi\in\NN\}$,
\quad$A_{\sigma}=Q[I_{\sigma}]$,
\quad$B_{\sigma}=R[I_{\sigma}]$,}

\noindent so that $A=A_{\emptyset}$ and $A_{\sigma}=\bigcup_{i\in\Bbb
N}A_{\sigma^{\smallfrown}\fraction{i}}$ for every $\sigma$.

\medskip

{\bf (c)} Write $T$ for the set of pairs

\Centerline{$\{(\sigma,\tau):\sigma,\,\tau\in S$ and there are
disjoint $E$, $F\in\Cal E$ such that $A_{\sigma}\subseteq E$ and
$B_{\tau}\subseteq F\}$.}

\noindent If $\sigma$, $\tau\in S$ are such that
$(\sigma^{\smallfrown}\fraction{i},\tau^{\smallfrown}\fraction{j})\in T$ for every $i$,
$j\in\Bbb N$, then $(\sigma,\tau)\in T$.   \Prf\ For each $i$, $j\in\Bbb
N$ take disjoint $E_{ij}$, $F_{ij}\in\Cal E$ such that

\Centerline{$A_{\sigma^{\smallfrown}\fraction{i}}\subseteq E_{ij}$,
\quad$B_{\tau^{\smallfrown}\fraction{j}}\subseteq F_{ij}$.}

\noindent Then $E=\bigcup_{i\in\Bbb N}\bigcap_{j\in\Bbb N}E_{ij}$,
$F=\bigcap_{i\in\Bbb N}\bigcup_{j\in\Bbb N}F_{ij}$ are disjoint and
belong to $\Cal E$, and $A_{\sigma}\subseteq E$, $B_{\tau}\subseteq
F$.   So $(\sigma,\tau)\in T$.\ \Qed

\medskip

{\bf (d)} \Quer\ Now suppose, if possible, that there are no disjoint
$E$, $F\in\Cal E$ such that $A\subseteq E$ and $B\subseteq F$;  that is,
that $(\emptyset,\emptyset)\notin T$.   By (c),
used repeatedly, we can find sequences
$\sequence{i}{\phi(i)}$, $\sequence{i}{\psi(i)}$ such that
$(\phi\restr n,\psi\restr n)\notin T$ for every $n\in\Bbb N$.   Set
$K=Q[\{\phi\}]$, $L=R[\{\psi\}]$.   These are compact (because $R$ is
usco-compact) and disjoint (because $K\subseteq A$ and $L\subseteq B$).
By (a), there are disjoint $E$, $F\in\Cal E$ such that
$K\subseteq\interior E$ and $L\subseteq\interior F$.

By 422B, there are open sets $U$, $V\subseteq\NN$ such that

\Centerline{$\phi\in U$,
\quad$Q[U]\subseteq\interior E$,
\quad$\psi\in V$,
\quad$R[V]\subseteq\interior F$.}

\noindent But now there is some $n\in\Bbb N$ such that
$I_{\phi\restr n}\subseteq U$ and $I_{\psi\restr n}\subseteq V$, in
which case

\Centerline{$A_{\phi\restr n}\subseteq E$,
\quad$B_{\psi\restr n}\subseteq F$,}

\noindent and $(\phi\restr n,\psi\restr n)\in T$, contrary to the choice
of $\phi$ and $\psi$.\ \Bang

This contradiction shows that the lemma is true.
}%end of proof of 422I

\leader{422J}{Corollary} Let $X$ be a Hausdorff space and $A$, $B$
disjoint K-analytic subsets of $X$.   Then there is a Borel set which
includes $A$ and is disjoint from $B$.

\proof{ Apply 422I with $\Cal E$ the Borel $\sigma$-algebra of $X$.
}%end of proof of 422J

\leader{*422K}{}\cmmnt{ I give the next step in the theory of
`constituents' begun in 421N-421Q.
%421N 421O 421P 421Q

\medskip

\noindent}{\bf Theorem} Let $X$ be a Hausdorff space.

(i) Suppose that $X$ is regular.   Let $A\subseteq X$ be a K-analytic
set.   Then there is a
non-decreasing family $\ofamily{\xi}{\omega_1}{E_{\xi}}$ of Borel sets
in $X$, with union $X\setminus A$, such that every Souslin-F subset of
$X$ disjoint from $A$ is included in some $E_{\xi}$.

(ii) Suppose that $X$ is regular.   Let $A\subseteq X$ be a Souslin-F
set.   Then there is a
non-decreasing family $\ofamily{\xi}{\omega_1}{E_{\xi}}$ of Borel sets
in $X$, with union $X\setminus A$, such that every K-analytic subset of
$X\setminus A$ is included in some $E_{\xi}$.

(iii) Let $A\subseteq X$ be a K-analytic set.   Then there is a
non-decreasing family $\ofamily{\xi}{\omega_1}{E_{\xi}}$ of Borel sets
in $X$, with union $X\setminus A$, such that every K-analytic subset of
$X\setminus A$ is included in some $E_{\xi}$.

%what if X is not regular?  Can we still separate disjoint K-analytic
%sets in this way?

\proof{{\bf (a)} The first two parts depend on the following fact:  if
$X$ is regular, $R\subseteq\NN\times X$ is usco-compact,
$\family{\sigma}{S^*}{F_{\sigma}}$
is a Souslin scheme consisting of closed sets with kernel $B$, and
$R[\NN]\cap B=\emptyset$, then for any $\phi$, $\psi\in\NN$ there is
an $n\ge 1$ such that $\overline{R[I_{\phi\restr n}]}
\cap\bigcap_{1\le i\le n}F_{\psi\restr i}$ is empty, where I write
$I_{\sigma}=\{\theta:\sigma\subseteq\theta\in\NN\}$ for
$\sigma\in S^*=\bigcup_{n\ge 1}\Bbb N^n$.   \Prf\ We know that
$K=R[\{\phi\}]$ is a compact set disjoint from the closed set
$\bigcap_{n\ge 1}F_{\psi\restr n}$.   So there is some $m\ge 1$ such
that $K\cap F=\emptyset$ where
$F=\bigcap_{1\le i\le m}F_{\psi\restr i}$.   Because $X$ is regular,
there are disjoint open sets $G$, $H\subseteq X$ such that
$K\subseteq G$ and $F\subseteq H$ (4A2F(h-ii)).   By 422B, there
is some $n$ such that $R[I_{\phi\restr n}]\subseteq G$.   Of course we
can take $n\ge m$, and in this case

\Centerline{$\overline{R[I_{\phi\restr n}]}
  \cap\bigcap_{1\le i\le n}F_{\psi\restr i}
\subseteq\overline{G}\cap F=\emptyset$,}

\noindent as required.\ \Qed

\medskip

{\bf (b)(i)} Suppose that $A\subseteq X$ is K-analytic.   Then there is
an usco-compact set $R\subseteq\NN\times X$ such that $R[\NN]=A$, and
$R$ is closed (422Da), so that $A$ is the kernel of the Souslin scheme
$\family{\sigma}{S^*}{\overline{R[I_{\sigma}]}}$ (421I).   For $x\in X$
set
$T_x=\{\sigma:\sigma\in S^*,\,x\in\overline{R[I_{\sigma}]}\}$,
as in 421Ng, and let
$r(T_x)<\omega_1$ be the rank of the tree $T_x$ (421Ne).   Then
$E_{\xi}=\{x:x\in X\setminus A,\,r(T_x)\le\xi\}$ is a Borel set for
every $\xi<\omega_1$, by 421Ob.   Now suppose that
$B\subseteq X\setminus A$ is a Souslin-F set.   Then it is the kernel of
a Souslin scheme $\family{\sigma}{S^*}{F_{\sigma}}$ consisting of closed
sets.   If $\phi$, $\psi\in\NN$ then by (a) above there is an $n\ge 1$
such that
$\overline{R[I_{\phi\restr n}]}\cap\bigcap_{1\le i\le n}F_{\psi\restr i}$
is empty.   By 421Q, there
must be some $\xi<\omega_1$ such that $r(T_x)\le\xi$ for every $x\in B$, that
is, $B\subseteq E_{\xi}$.   So
$\ofamily{\xi}{\omega_1}{E_{\xi}}$ is a suitable family.

\medskip

\quad{\bf (ii)} The other part is almost the same.
Suppose that $A\subseteq X$ is Souslin-F.   Then
it is the kernel of a Souslin scheme $\family{\sigma}{S^*}{F_{\sigma}}$
consisting of closed sets.   For $x\in X$ set

\Centerline{$T_x=\bigcup_{n\ge 1}\{\sigma:\sigma\in\Bbb N^n,\,
  x\in\bigcap_{1\le i\le n}F_{\sigma\restr i}\}$,}

\noindent and let $r(T_x)<\omega_1$ be the rank of the tree $T_x$.
Then $E_{\xi}=\{x:x\in X\setminus A,\,r(T_x)\le\xi\}$ is a Borel set for
every $\xi<\omega_1$.   Now let $B\subseteq X\setminus A$ be a
K-analytic set.   There is an usco-compact set $R\subseteq\NN\times X$
such that $R[\NN]=B$, and $B$ is the kernel of the Souslin scheme
$\family{\sigma}{S^*}{\overline{R[I_{\sigma}]}}$.
%as in (i)
If $\phi$, $\psi\in\NN$ then by (a) above there is an $n\ge 1$ such that
$\bigcap_{1\le i\le n}F_{\psi\restr i}\cap\overline{R[I_{\phi\restr n}]}$
is empty.   So there must be some
$\xi<\omega_1$ such that $B\subseteq E_{\xi}$.   Thus here again
$\ofamily{\xi}{\omega_1}{E_{\xi}}$ is a suitable family.

\medskip

{\bf (c)} If $X$ is not regular, we still have a version of the result
in (a), as follows:  if $R$, $S\subseteq\NN\times X$ are usco-compact and
$R[\NN]\cap S[\NN]=\emptyset$, then for any $\phi$, $\psi\in\NN$ there
is an $n\ge 1$ such that
$\overline{R[I_{\phi\restr n}]}\cap S[I_{\psi\restr n}]$ is
empty.   \Prf\ This time, $R[\{\phi\}]$ and $S[\{\psi\}]$ are disjoint
compact sets, so there are disjoint open sets $G$, $H$ with
$R[\{\phi\}]\subseteq G$ and $S[\{\psi\}]\subseteq H$ (4A2F(h-i)).  Now
$S[I_{\psi\restr n}]\subseteq H$, $R[I_{\phi\restr n}]\subseteq G$
and $\overline{R[I_{\phi\restr n}]}\cap H=\emptyset$
for all $n$ large enough.\ \Qed

Now the argument of (b-i), with $F_{\sigma}=S[I_{\sigma}]$,
gives part (iii).
}%end of proof of 422K

\exercises{\leader{422X}{Basic exercises (a)}
%\spheader 422Xa
Let $X$ and $Y$ be Hausdorff spaces and $X_0$ a closed subset of $X$.
Show that a relation $R\subseteq X_0\times Y$ is
usco-compact when regarded as a relation between $X_0$ and $Y$ iff it is
usco-compact when regarded as a relation between $X$ and $Y$.
%422D

\spheader 422Xb Show that a locally compact Hausdorff space is
K-analytic iff it is Lindel\"of iff it is $\sigma$-compact.
%422H

\sqheader 422Xc Prove 422Hc from first principles, without using 421D.
%idempotence of Souslin opn
({\it Hint\/}:  if $\family{\sigma}{S^*}{R_{\sigma}}$ is a Souslin scheme
of usco-compact relations in $\NN\times X$,

\Centerline{$\{((\phi,\family{\sigma}{S^*}{\psi_{\sigma}}),x):
(\psi_{\phi\restr n},x)\in R_{\phi\restr n}$ for every $n\ge 1\}$}

\noindent is usco-compact in $(\NN\times(\NN)^{S^*})\times X$.)
%422H

\spheader 422Xd Let $X$ be a K-analytic Hausdorff space.   (i) Show that
every Baire subset of $X$ is K-analytic.   \Hint{apply
136Xi to the family of K-analytic subsets of $X$.}
(ii) Show that if $X$ is regular, it is
perfectly normal iff it is hereditarily Lindel\"of iff every open subset
of $X$ is K-analytic.
%422H

\spheader 422Xe Let $X$ be a Hausdorff space in which every open set is
K-analytic.   Show that every Borel set is K-analytic.
%422Xd 422H

\spheader 422Xf Let $X$ be a completely regular Hausdorff space and $A$,
$B$ disjoint K-analytic subsets of $X$.   Show that there is
a Baire set including $A$ and disjoint from $B$.
%422J

\leader{422Y}{Further exercises (a)}
%\spheader 422Ya
Let $X$ be a completely regular Hausdorff space, and $\beta X$ its
Stone-\v Cech compactification.   Show that $X$ is K-analytic iff it is
a Souslin-F set in $\beta X$.
%422H

\spheader 422Yb Let $X$ be a set and $\frak S$, $\frak T$ two Hausdorff
topologies on $X$ such that $\frak S\subseteq\frak T$ and $(X,\frak T)$
is K-analytic.   Show that $\frak S$ and $\frak T$ yield the same
K-analytic subspaces of $X$.
%422H

\spheader 422Yc Show that a Hausdorff space is K-analytic iff it is a
continuous image of a K$_{\sigma\delta}$ set in a compact Hausdorff
space, that is, of a set expressible as
$\bigcap_{m\in\Bbb N}\bigcup_{n\in\Bbb N}K_{mn}$ where every $K_{mn}$ is
compact.   \Hint{Write $\Cal K^*$ for
the class of Hausdorff continuous images of K$_{\sigma\delta}$
subsets of compact Hausdorff spaces.   (i) Show that $\NN$ is a
K$_{\sigma\delta}$ set in $Y^{\Bbb N}$, where $Y$ is the one-point
compactification of $\Bbb N$.   (ii) Show that if $X$ is a compact
Hausdorff space and $R\subseteq\NN\times X$ is closed, then $R\in\Cal
K^*$.  (iii) Show that if $X$ is a compact Hausdorff space, then every
Souslin-F subset of $X$ belongs to $\Cal K^*$.   (iv) Show that if $X$
is a regular K-analytic Hausdorff space, then $X\in\Cal K^*$.   (v) Show
that if $X$ is any Hausdorff space and $R\subseteq\NN\times X$ is an
usco-compact relation, then $R\in\Cal K^*$.   See {\smc Jayne 76}.}
%422H

\spheader 422Yd Let $X$ be a normal space and $\Cal C$ the family of
countably compact closed subsets of $X$.   Let $A$, $B$ be disjoint sets
obtainable from $\Cal C$ by Souslin's operation.   (For instance, if $X$
itself is countably compact, $A$ and $B$ could be disjoint Souslin-F
sets.)   Show that there is a Borel set including $A$ and disjoint from
$B$.   %\Hint{adapt the proof of 422I.}
%422I

\spheader 422Ye Let $X$ be a Hausdorff space and $\sequencen{A_n}$ a
sequence of K-analytic subsets of $X$ such that
$\bigcap_{n\in\Bbb N}A_n=\emptyset$.   Show that there is a sequence
$\sequencen{E_n}$ of
Borel sets such that $A_n\subseteq E_n$ for every $n\in\Bbb N$ and
$\bigcap_{n\in\Bbb N}E_n=\emptyset$.   \Hint{for each $n\in\Bbb N$
choose an usco-compact $R_n\subseteq\NN\times X$ with projection $A_n$.
Consider the set $T=\{\sequencen{\sigma_n}:\exists$ Borel
$E_n,\,R_n[I_{\sigma_n}]\subseteq E_n\Forall n$,
$\bigcap_{n\in\Bbb N}E_n=\emptyset\}$.}
%422I

\spheader 422Yf Explain how to prove 422J from 421Q, without using 422I.
%422J

\spheader 422Yg Let $X$ be a Hausdorff space,
$\Cal K$ the family of K-analytic subsets of $X$, $Y$ a set and $\Cal H$ a
family of subsets of $Y$ containing $\emptyset$.
Show that $R[X]\in\Cal S(\Cal H)$
for every $R\in\Cal S(\{K\times H:K\in\Cal K$, $H\in\Cal H\})$.
%421G
}%end of exercises

\endnotes{
\Notesheader{422}
In a sense, this section starts at the deep end of its topic.
`Descriptive set theory' originally developed in the context of the real
line and associated spaces, and this remains the centre of the subject.
But it turns out that some of the same arguments can be used in much
more general contexts, and in particular greatly illuminate the theory
of Radon measures on Hausdorff spaces.   I find that a helpful way to
look at K-analytic spaces is to regard them as a common generalization
of compact Hausdorff spaces and Souslin-F subsets of $\Bbb R$;  if you
like, any theorem which is true of both these classes has a fair chance
of being true of all K-analytic spaces.   In the next section we shall
come to the special properties of the more restricted class of
`analytic' spaces, which are much closer to the separable metric spaces
of the original theory.

The phrase `usco-compact' is neither elegant nor transparent, but is
adequately established and (in view of the frequency with which it is
needed) seems preferable to less concise alternatives.   If we think of
a relation $R\subseteq X\times Y$ as a function $x\mapsto R[\{x\}]$ from
$x$ to $\Cal PY$, then an usco-compact relation is one which takes
compact values and is `upper semi-continuous' in the sense that
$\{x:R[\{x\}]\subseteq H\}$ is open for every open set $H\subseteq Y$;
just as a real-valued function is upper semi-continuous if
$\{x:f(x)<\alpha\}$ is open for every $\alpha$.

This is not supposed to be a book on general topology, and in my account
of the topological properties of K-analytic spaces I have concentrated
on facts which are useful when proving that spaces are K-analytic, on
the assumption that these will be valuable when we seek to apply
the results of \S432 below.   Other properties are mentioned only when
they are relevant to the measure-theoretic results which are my real
concern, and readers already acquainted with this area may be startled
by some of my omissions.   For a proper treatment of the subject, I
refer you to {\smc Rogers 80}.   As usual, however, I take technical
details seriously in
the material I do cover.   I hope you will not find that such results as
422Dg and 422Ga try your patience too far.   I think a moment's thought
will persuade you that it
is of the highest importance that K-analyticity (like compactness) is an
intrinsic property.   In contrast, the property of being `Souslin-F',
like the property of being closed, depends on the surrounding space.   A
completely regular Hausdorff space is compact iff it must be a closed
set in any surrounding Hausdorff space iff it is closed in its
Stone-\v Cech compactification;  and it is K-analytic iff it must be a
Souslin-F set in any surrounding Hausdorff space iff it is a Souslin-F
set in its Stone-\v Cech compactification (422Ya).

For regular spaces, 422K gives us another version of the First
Separation Theorem.   But this one is simultaneously more restricted in
its scope (it does not seem to have applications to Baire
$\sigma$-algebras, for instance) and much more powerful in its
application.   When all Borel sets are Souslin-F, as in the next
section, it tells us something very important about the cofinal
structure of the Souslin-F subsets of the complement of a K-analytic
set.
}%end of notes

\discrpage


\frfilename{mt423.tex}
\versiondate{28.11.16}
\copyrightdate{2002}

\def\NN{\BbbN^{\Bbb N}}

\def\chaptername{Descriptive set theory}
\def\sectionname{Analytic spaces}

\newsection{423}

We come now to the original class of K-analytic spaces, the `analytic'
spaces.   I define these as continuous images of $\NN$ (423A), but move
as quickly as possible to their characterization as K-analytic spaces
with countable networks (423C), so that many other fundamental facts
(423E-423G) can be regarded as simple corollaries of results in \S422.
I give two versions of Lusin's theorem on injective images of Borel sets
(423I), and a form of the von Neumann-Jankow measurable selection
theorem (423P).   I end with notes on constituents of coanalytic sets
(423R-423S).

\leader{423A}{Definition} A Hausdorff space is {\bf
analytic} or {\bf Souslin} if it is either empty or a continuous image
of $\NN$.

\leader{423B}{Proposition} (a) A Polish space is analytic.

(b) A Hausdorff continuous image of an analytic Hausdorff space is
analytic.

(c) A product of countably many analytic Hausdorff spaces is analytic.

(d) A closed subset of an analytic Hausdorff space is analytic.

(e) An analytic Hausdorff space has a countable network consisting of
analytic sets.

\proof{{\bf (a)} Let $X$ be a Polish space.   If $X=\emptyset$, we can
stop.   Otherwise, let $\rho$ be a metric on $X$, inducing its topology,
under which $X$ is complete.   For
$\sigma\in S=\bigcup_{n\in\Bbb N}\BbbN^n$ choose
$X_{\sigma}\subseteq X$ as follows.
$X_{\emptyset}=X$. Given that $X_{\sigma}$ is a closed non-empty subset
of $X$, where $\sigma\in S$, then $X_{\sigma}$ is
separable, because $X$ is separable and metrizable
(4A2P(a-iv)), and we can choose a sequence $\sequence{i}{x_{\sigma i}}$
in $X_{\sigma}$ such that $\{x_{\sigma i}:i\in\Bbb N\}$ is dense in
$X_{\sigma}$.   Set
$X_{\sigma^{\smallfrown}\fraction{i}}=X_{\sigma}\cap B(x_{\sigma i},2^{-n})$ for
each $i\in\Bbb N$, where
$B(x,\delta)=\{y:\rho(y,x)\le\delta\}$, and continue.   Note that
because $\{x_{\sigma i}:i\in\Bbb N\}$ is dense in $X_{\sigma}$,
$X_{\sigma}=\bigcup_{i\in\Bbb N}X_{\sigma^{\smallfrown}\fraction{i}}$, for every
$\sigma\in S$.

For each $\phi\in\NN$, $\sequence{n}{X_{\phi\restr n}}$ is a
non-increasing sequence of non-empty closed sets, and
$\diam(X_{\phi\restr n+1})\penalty-100
\le 2^{-n+1}$ for every $n$.   Because $X$ is
complete under
$\rho$, $\bigcap_{n\in\Bbb N}X_{\phi\restr n}$ is a singleton
$\{f(\phi)\}$ say. ($f(\phi)$ is the limit of the Cauchy sequence
$\sequencen{x_{\phi\restr n,\phi(n)}}$.)   Thus we have a function
$f:\NN\to X$.   $f$ is
continuous because $\rho(f(\psi),f(\phi))\le 2^{-n+1}$ whenever
$\phi\restr n+1=\psi\restr n+1$ (since in this case both $f(\psi)$ and
$f(\phi)$ belong to $X_{\phi\restr n+1}$).   $f$ is surjective because,
given $x\in X$, we can choose $\sequence{i}{\phi(i)}$ inductively so
that $x\in X_{\phi\restr n}$ for every $n$;  at the inductive step, we
have $x\in X_{\phi\restr n}
=\bigcup_{i\in\Bbb N}X_{(\phi\restr n)^{\smallfrown}\fraction{i}}$, so we can
take $\phi(n)$ such that
$x\in X_{(\phi\restr n)^{\smallfrown}\fraction{\phi(n)}}
=X_{\phi\restr n+1}$.

Thus $X$ is a continuous image of $\NN$, as claimed.

\medskip

{\bf (b)} If $X$ is an analytic Hausdorff space and $Y$ is a Hausdorff
continuous image of
$X$, then either $X$ is a continuous image of $\NN$ and $Y$ is a
continuous image of $\NN$, or $X=\emptyset$ and $Y=\emptyset$.

\medskip

{\bf (c)} Let $\familyiI{X_i}$ be a countable family of analytic
Hausdorff spaces,
with product $X$.   Then $X$ is Hausdorff (3A3Id).   If $I=\emptyset$
then $X=\{\emptyset\}$ is a
continuous image of $\NN$, therefore analytic.   If there is some $i\in
I$ such that $X_i=\emptyset$, then $X=\emptyset$ is analytic.
Otherwise, we have for
each $i\in I$ a continuous surjection $f_i:\NN\to X_i$.   Setting
$f(\phi)=\familyiI{f_i(\phi(i))}$ for $\phi\in(\NN)^I$, $f:(\NN)^I\to X$
is a continuous surjection.   But $(\NN)^I\cong\BbbN^{\Bbb N\times I}$
is homeomorphic to $\NN$, so
$X$ is analytic.

\medskip

{\bf (d)} Let $X$ be an analytic Hausdorff space and $F$ a closed subset
of $X$.   Then $F$ is Hausdorff in its subspace topology (4A2F(a-i)).
If $X=\emptyset$ then $F=\emptyset$ is analytic.   Otherwise, there is a
continuous surjection $f:\NN\to X$.   Now $H=f^{-1}[F]$ is a closed
subset of the Polish space $\NN$, therefore Polish in its induced
topology (4A2Qd).   By (a), $H$ is analytic, so its continuous
image $F=f[H]$ also is analytic, by (b).

\medskip

{\bf (e)} Let $X$ be an analytic Hausdorff space.   If it is empty then
of course it has a countable network consisting of analytic sets.
Otherwise, there is a continuous surjection $f:\NN\to X$.   For
$\sigma\in S$ set $I_{\sigma}=\{\phi:\sigma\subseteq\phi\in\NN\}$;
then $\{I_{\sigma}:\sigma\in S\}$ is a base for the topology of $\NN$,
so $\{f[I_{\sigma}]:\sigma\in S\}$ is a network for the topology of
$X$ (see the proof of 4A2Nd).   But $I_{\sigma}$ is homeomorphic to
$\NN$, so $f[I_{\sigma}]$ is analytic, for every $\sigma\in S$, and
$\{f[I_{\sigma}]:\sigma\in S\}$ is a countable network consisting of
analytic sets.
}%end of proof of 423B

\leader{423C}{Theorem} A Hausdorff space is analytic iff it is
K-analytic and has a countable network.

\proof{{\bf (a)} Let $X$ be an analytic Hausdorff space.   By 423Be, it
has a countable network.   If $X=\emptyset$ then
surely it is K-analytic.   Otherwise, $X$ is
a continuous image of $\NN$.   But $\NN$ is K-analytic (422Gb),  so
$X$ also is K-analytic, by 422Gd.

\medskip

{\bf (b)} Now suppose that $X$ is a K-analytic Hausdorff space and has a
countable network.

\medskip

\quad{\bf (i)} If $X\subseteq\NN$ then $X$ is analytic.   \Prf\ Let
$R\subseteq\NN\times X$ be an usco-compact relation such that
$R[\NN]=X$.   Then $R$ is still usco-compact when regarded as a subset
of $\NN\times\NN$ (422Dg), so is closed in $\NN\times\NN$ (422Da).   But
$\NN\times\NN\cong\NN$ is analytic, so $R$ is in itself an analytic
space (423Bd), and its continuous image $X$ is analytic, by 423Bb.\ \Qed

\medskip

\quad{\bf (ii)} Now suppose that $X$ is regular.   By 4A2Ng, $X$ has a
countable network $\Cal E$ consisting of closed sets.   Adding
$\emptyset$ to $\Cal E$ if need be, we may suppose that $\Cal
E\ne\emptyset$.   Let
$\sequencen{E_n}$ be a sequence running over $\Cal E$.   For each
$n\in\Bbb N$, let $\sequence{i}{F_{ni}}$ be a sequence running over
$\{E_n\}\cup\{E:E\in\Cal E,\,E\cap E_n=\emptyset\}$.   Now consider the
relation

\Centerline{$R=\{(\phi,x):\phi\in\NN,\,x\in\bigcap_{n\in\Bbb
N}F_{n,\phi(n)}\}\subseteq\NN\times X$.}

\medskip

\qquad\grheada\ $R$ is closed in $\NN\times X$.   \Prf\ Because every
$F_{ni}$ is closed,

\Centerline{$(\NN\times X)\setminus R
=\bigcup_{i,n\in\Bbb N}\{(\phi,x)$:  $\phi(n)=i$ and $x\notin F_{ni}\}$}

\noindent is open.\ \Qed

\medskip

\qquad\grheadb\ $R[\NN]=X$.   \Prf\  For every $n\in\Bbb N$,

\Centerline{$X\setminus E_n=\bigcup\{E:E\in\Cal E,\,E\subseteq
X\setminus E_n\}$}

\noindent because $\Cal E$ is a network and $E_n$ is closed, so
$\bigcup_{i\in\Bbb N}F_{ni}=X$.   So, given $x\in X$, we can find for
each $n$ a $\phi(n)$ such that $x\in F_{n,\phi(n)}$, and $(\phi,x)\in
R$.\ \Qed

\medskip

\qquad\grheadc\ $R$ is the graph of a function.   \Prf\Quer\ Suppose
that we have $(\phi,x)$ and $(\phi,y)$ in $R$ where $x\ne y$.   Because
the topology of $X$ is Hausdorff, there is an $n\in\Bbb N$ such that
$x\in E_n$ and $y\notin E_n$.   But in this case $x\in E_n\cap
F_{n,\phi(n)}$, so $F_{n,\phi(n)}=E_n$, while $y\in
F_{n,\phi(n)}\setminus E_n$, so
$F_{n,\phi(n)}\ne E_n$;  which is absurd.\ \Bang\Qed

\medskip

\qquad\grheadd\ Set $A=R^{-1}[X]$, so that $R$ is the graph of a
function from $A$ to $X$;  in recognition of its new status, give it a
new name $f$.   Then $f$ is continuous.   \Prf\ Suppose that $\phi\in A$
and that $x=f(\phi)\in G$, where $G\subseteq X$ is open.   Then there is
an $n\in\Bbb N$ such that $x\in E_n\subseteq G$.   In this case $x\in
F_{n,\phi(n)}$, because $(\phi,x)\in R$, so $F_{n,\phi(n)}=E_n$.   Now
if $\psi\in A$ and $\psi(n)=\phi(n)$, we must have

\Centerline{$f(\psi)\in F_{n,\psi(n)}=E_n\subseteq G$.}

\noindent Thus $f^{-1}[G]$ includes a neighbourhood of $\phi$ in $A$.
As $\phi$ and $G$ are arbitrary, $f$ is continuous.\ \Qed

\medskip

\qquad\grheade\ At this point recall that $X$ is K-analytic.   It
follows that $\NN\times X$ is K-analytic (422Ge), so that its closed
subset $R$ is K-analytic (422Gf) and $A$, which is a continuous image of
$R$, is K-analytic (422Gd).   But now $A$ is a K-analytic subset of
$\NN$, so is analytic, by (i) just above.   And, finally, $X$ is a
continuous image of $A$, so is analytic.

\medskip

\quad{\bf (iii)} Thus any regular K-analytic space with a countable
network is analytic.  Now suppose that $X$ is an arbitrary K-analytic
Hausdorff space with a countable network.   Let $R\subseteq\NN\times X$
be an usco-compact relation such that $R[\NN]=X$.   Then $R$ is a closed
subset of $\NN\times X$, so is itself a K-analytic space with a
countable network.   But it is also regular, by 422E.   So $R$ is
analytic and its continuous image $X$ is analytic.

This completes the proof.
}%end of proof of 423C

\leader{423D}{Corollary} (a) An analytic Hausdorff space is
hereditarily Lindel\"of.

(b) In a regular analytic Hausdorff space, closed sets are zero sets
and the Baire and Borel $\sigma$-algebras coincide.
%4@35

(c) A compact subset of an analytic Hausdorff space is metrizable.

(d) A metrizable space is analytic iff it is K-analytic.

\proof{{\bf (a)-(c)} These are true just because there is a countable
network (4A2Nb, 4A3Kb, 4A2Na, 4A2Qh).

\medskip

{\bf (d)} Let $X$ be a metrizable space.   If $X$ is analytic, of course
it is K-analytic.   If $X$ is K-analytic, it is Lindel\"of (422Gg)
therefore separable (4A2Pd) and has a countable network (4A2P(a-iii)),
so is analytic.
}%end of proof of 423D

\leader{423E}{Theorem} (a) For any Hausdorff space $X$, the family of
subsets of $X$ which are analytic in their subspace topologies is closed
under Souslin's operation.

(b) Let $(X,\frak T)$ be an analytic Hausdorff space.   For a subset $A$
of $X$, the following are equiveridical:

\inset{(i) $A$ is analytic;

(ii) $A$ is K-analytic;

(iii) $A$ is Souslin-F;

(iv) $A$ can be obtained by Souslin's operation from the family
of Borel subsets of $X$.}

\noindent In particular, all Borel sets in $X$ are analytic.

\proof{{\bf (a)} Let $X$ be a Hausdorff space and $\Cal A$ the family of
analytic subsets of $X$.   Let
$\family{\sigma}{S^*}{A_{\sigma}}$ be a Souslin scheme in $\Cal A$ with
kernel $A$.   Then every $A_{\sigma}$ is
K-analytic, so $A$ is K-analytic, by 422Hc.   Also every $A_{\sigma}$
has a countable network, so $A'=\bigcup_{\sigma\in S^*}A_{\sigma}$ has a
countable network (4A2Nc);  as $A\subseteq A'$, $A$ also has a countable
network (4A2Na) and is analytic.

\medskip

{\bf (b)} Because $X$ has a countable network, so does $A$.   So 423C
tells us at once that (i)$\Leftrightarrow$(ii).   In particular, $X$ is
K-analytic, so 422Hb tells us that (ii)$\Leftrightarrow$(iii).   Of course
(iii)$\Rightarrow$(iv).

Now suppose that $G\subseteq X$ is open.   Then $G\in\Cal A$.   \Prf\
If $X=\emptyset$ then $G=\emptyset$ is analytic.   Otherwise, there is a
continuous surjection $f:\NN\to X$.   Set $H=f^{-1}[G]$, so that
$H\subseteq\NN$ is open and $G=f[H]$.   Being an open set in a metric
space, $H$ is F$_{\sigma}$ (4A2Lc), so, in particular, is
Souslin-F;  but $\NN$ is analytic, so $H$ is analytic and its continuous
image $G$ is analytic.\ \Qed

We have already seen that closed subsets of $X$ belong to $\Cal A$
(423Bd).   Because $\Cal A$ is closed under Souslin's operation, it
contains every Borel set, by 421F.   It therefore contains every set
obtainable by Souslin's operation from Borel sets, and
(iv)$\Rightarrow$(i).
}%end of proof of 423E

\cmmnt{\medskip

\noindent{\bf Remark} See also 423Yb below.}

\leader{423F}{Proposition} Let $(X,\frak T)$ be an analytic Hausdorff
space.

(a) A set $E\subseteq X$ is Borel iff both $E$ and $X\setminus E$ are
analytic.

(b) If $\frak S$ is a coarser\cmmnt{ (= smaller)} Hausdorff topology on
$X$, then $\frak S$ and $\frak T$ have the same Borel sets.

\proof{{\bf (a)} If $E$ is Borel, then $E$ and $X\setminus E$ are
analytic, by 423Eb.   If $E$ and $X\setminus E$ are analytic, they are
K-analytic (423Eb) and disjoint, so there is a Borel set $F\supseteq E$
which is disjoint from $X\setminus E$ (422J);   but now of course
$F=E$, so $E$ must be Borel.

\medskip

{\bf (b)} Because the identity map from $(X,\frak T)$ to $(X,\frak S)$
is continuous, $\frak S$ is an analytic topology (423Bb) and every
$\frak S$-Borel set is $\frak T$-Borel.   If $E\subseteq X$ is
$\frak T$-Borel, then it and its complement are $\frak T$-analytic,
therefore $\frak S$-analytic (423Bb), and $E$ is $\frak S$-Borel by (a).
}%end of proof of 423F

\vleader{60pt}{423G}{Lemma} Let $X$ and $Y$ be analytic Hausdorff spaces and
$f:X\to Y$ a Borel measurable function.

(a)\cmmnt{ (The graph of)} $f$ is an analytic set.

(b) $f[A]$ is an analytic set in $Y$ for any analytic set\cmmnt{ (in
particular, any Borel set)} $A\subseteq X$.

(c) $f^{-1}[B]$ is an analytic set in $X$ for any analytic
set\cmmnt{ (in particular, any Borel set)} $B\subseteq Y$.

\proof{{\bf (a)} Let $\Cal E$ be a countable network for the topology of
$Y$.   Set

\Centerline{$R=\bigcap_{E\in\Cal E}(X\times\overline
E)\cup((X\setminus f^{-1}[\overline{E}])\times Y)$.}

\noindent Then $R$ is a Borel set in $X\times Y$.   But also $R$ is the
graph of $f$.   \Prf\ If $f(x)=y$, then surely $y\in\overline{E}$
whenever $x\in f^{-1}[\overline{E}]$, so $(x,y)\in R$.   On the other
hand, if $x\in X$, $y\in Y$ and $f(x)\ne y$, there are disjoint open
sets $G$, $H\subseteq Y$ such that $f(x)\in G$ and $y\in H$;  now there
is an $E\in\Cal E$ such that $f(x)\in E\subseteq G$, so that
$f(x)\in\overline{E}$ but
$y\notin\overline{E}$, and $(x,y)\notin R$.\ \Qed

Because $X\times Y$ is analytic (423Bc), $R$ is analytic (423Eb).

\medskip

{\bf (b)} If $A\subseteq X$ is analytic, then $A\times Y$ and
$R\cap(A\times Y)$ are analytic (423Ea), so $f[A]=R[A]$, which is a
continuous image of $R\cap(A\times Y)$, is analytic.

\medskip

{\bf (c)} Similarly, if $B\subseteq Y$ is analytic, then $f^{-1}[B]$ is
a continuous image of $R\cap(X\times B)$, so is analytic.
}%end of proof of 423G

\leader{423H}{Lemma} Let $(X,\frak T)$ be an analytic Hausdorff space,
and $\sequencen{E_n}$ any sequence of Borel sets in $X$.   Then the
topology $\frak T'$ generated by $\frak T\cup\{E_n:n\in\Bbb N\}$ is
analytic.

\proof{ If $X=\emptyset$ this is trivial.   Otherwise, there is a
continuous surjection $f:\NN\to X$.   Set $F_n=f^{-1}[E_n]$ for each
$n$;  then $F_n$ is a Borel subset of $\NN$, so there is a Polish
topology $\frak S'$ on $\NN$, finer than the usual topology, for which
every $F_n$ is open, by 4A3I.   But now $f$
is continuous for $\frak S'$ and $\frak T'$, so $\frak T'$ is analytic, by 423Ba and 423Bb.   (Of course $\frak T'$
is Hausdorff, because it is finer than $\frak T$.)
}%end of proof of 423H

\leader{423I}{Theorem} Let $X$ be a Polish space, $E\subseteq X$ a Borel
set, $Y$ a Hausdorff space and $f:E\to Y$ an injective function.

(a) If $f$ is continuous, then $f[E]$ is Borel.

(b) If $Y$ has a countable network\cmmnt{ (e.g., is an analytic space
or a separable metrizable space),} and $f$ is Borel measurable, then
$f[E]$ is Borel.

\proof{{\bf (a)(i)} Since there is a finer Polish topology on $X$ for
which $E$ is closed (4A3I again), therefore Polish in the subspace topology
(4A2Qd), and $f$ will still be continuous for this topology, we may
suppose that $E=X$.

\medskip

\quad{\bf (ii)} Let $\sequencen{U_n}$ run over a base for the topology
of $X$ (4A2P(a-i)).   For each pair $m$, $n\in\Bbb N$ such that
$U_m\cap U_n$
is empty, $f[U_m]$ and $f[U_n]$ are analytic sets in $Y$ (423Eb, 423Bb)
and are disjoint (because $f$ is injective), so there is a Borel set
$H_{mn}$ including $f[U_m]$ and disjoint from $f[U_n]$ (422J).
Set

\Centerline{$E_n=\overline{f[U_n]}\cap\bigcap\{H_{nm}\setminus
H_{mn}:m\in\Bbb N,\,U_m\cap U_n=\emptyset\}$}

\noindent for each $n\in\Bbb N$;  then $E_n$ is a Borel set in $Y$
including $f[U_n]$.   Note that if $U_m\cap U_n$ is empty, then
$E_m\cap E_n\subseteq(H_{mn}\setminus H_{nm})
\cap(H_{nm}\setminus H_{mn})$ is also empty.

Fix a metric $\rho$ on $X$, inducing its topology, for which $X$ is
complete, and for $k\in\Bbb N$ set

\Centerline{$F_k=\bigcup\{E_n:n\in\Bbb N,\,\diam U_n\le 2^{-k}\}$,}

\noindent so that $F_k$ is Borel.   Let $F=\bigcap_{k\in\Bbb N}F_k$;
then $F$ also is a Borel subset of $Y$.

The point is that $F=f[X]$.   \Prf\ (i) If $x\in X$, then for every
$k\in\Bbb N$ there is an $n\in\Bbb N$ such that

\Centerline{$x\in U_n\subseteq\{y:\rho(y,x)\le 2^{-k-1}\}$;}

\noindent now $\diam U_n\le 2^{-k}$, so

\Centerline{$f(x)\in f[U_n]\subseteq E_n\subseteq F_k$.}

\noindent As $k$ is arbitrary, $f(x)\in F$;  as $x$ is arbitrary,
$f[X]\subseteq F$.   (ii) If $y\in F$, then for each $k\in\Bbb N$ we can
find an $n(k)$ such that $y\in E_{n(k)}$ and $\diam U_{n(k)}\le 2^{-k}$.
Since $\overline{f[U_{n(k)}]}\supseteq E_{n(k)}$ is not empty, nor is
$U_{n(k)}$, and we can choose $x_k\in U_{n(k)}$.   Indeed, for any $j$,
$k\in\Bbb N$, $E_{n(j)}\cap E_{n(k)}$ contains $y$, so is not empty, and
$U_{n(j)}\cap U_{n(k)}$ cannot be empty;  but this means that there is
some $x$ in the intersection, and

\Centerline{$\rho(x_j,x_k)\le\rho(x_j,x)+\rho(x,x_k)\le\diam
U_{n(j)}+\diam U_{n(k)}\le 2^{-j}+2^{-k}$.}

\noindent This means that $\sequence{k}{x_k}$ is a Cauchy sequence.
But $X$ is supposed to be complete, so $\sequence{k}{x_k}$ has a limit
$x$ say.

\Quer\ If $f(x)\ne y$, then (because $Y$ is Hausdorff) there is an open
set $H$ containing $f(x)$ such that $y\notin\overline{H}$.   Now $f$ is
continuous, so there is a $\delta>0$ such that $f(x')\in H$ whenever
$\rho(x',x)\le\delta$.   There is a $k\in\Bbb N$ such that
$2^{-k}+\rho(x_k,x)\le\delta$.   If $x'\in U_{n(k)}$, then

\Centerline{$\rho(x',x)\le\rho(x',x_k)+\rho(x_k,x)\le\delta$;}

\noindent thus $f[U_{n(k)}]\subseteq H$, and

\Centerline{$E_{n(k)}\subseteq\overline{f[U_{n(k)}]}
\subseteq\overline{H}$.}

\noindent But $y\in E_{n(k)}\setminus\overline{H}$.\ \Bang

Thus $y=f(x)$ belongs to $f[X]$;  as $y$ is arbitrary, $F\subseteq
f[X]$.\ \QeD\  Accordingly $f[X]=F$ is a Borel subset of $Y$, as
claimed.

\medskip

{\bf (b)} By 4A2Nf, there is a countable family $\Cal V$ of open sets in
$Y$ such that whenever $y$, $y'$ are distinct points of $Y$ there are
disjoint $V$, $V'\in\Cal V$ such that $y\in V$ and $y'\in V'$.   Let
$\frak S'$ be the topology generated by $\Cal V$;  then $\frak S'$ is
Hausdorff.   For each $V\in\Cal V$, $f^{-1}[V]$ is a Borel set in $X$,
so there is a Polish topology $\frak T'$ on $X$, finer than the original
topology, for which every $f^{-1}[V]$ is open (4A3I once more).
Now $f$ is continuous for $\frak T'$ and
$\frak S'$ (4A2B(a-ii)), and $E$ is $\frak T'$-Borel, so $f[E]$ is a
$\frak S'$-Borel set in $Y$, by (a).   Since $\frak S'$ is coarser than
the original topology $\frak S$ on $Y$, $f[E]$ is also $\frak S$-Borel.
}%end of proof of 423I

\leader{423J}{Proposition}
Let $(X,\frak T)$ be an analytic Hausdorff space, and
$\Sigma$ a countably generated $\sigma$-subalgebra of the Borel
$\sigma$-algebra $\Cal B(X,\frak T)$
which separates the points of $X$.   Then $\Sigma=\Cal B(X,\frak T)$.

\proof{ Let $\sequencen{E_n}$ be a sequence in $\Sigma$ which
$\sigma$-generates $\Sigma$.   If $x$, $y\in X$ are distinct, the set
$\{E:E\subseteq X$, $x\in E\iff y\in E\}$ is a $\sigma$-algebra of subsets
of $X$ not including $\Sigma$, so there is an $n\in\Bbb N$ such that just
one of $x$, $y$ belongs to $E_n$, and the topology
$\frak S$ generated by
$\{E_n:n\in\Bbb N\}\cup\{X\setminus E_n:n\in\Bbb N\}$ is Hausdorff.
Write
$\frak T\vee\frak S$ for the topology generated by $\frak T\cup\frak S$,
that is, the topology generated by
$\frak T\cup\{E_n:n\in\Bbb N\}\cup\{X\setminus E_n:n\in\Bbb N\}$.
By 423H, $\frak T\vee\frak S$ is analytic;  because both $\frak T$ and
$\frak S$ are Hausdorff topologies coarser than $\frak T\vee\frak S$,
the Borel $\sigma$-algebras $\Cal B(X,\frak T)$,
$\Cal B(X,\frak T\vee\frak S)$ and $\Cal B(X,\frak S)$ are all the same
(423Fb).
Next, $\frak S$ is second-countable, therefore hereditarily Lindel\"of
(4A2Oc), with a subbase included in $\Sigma$, so
$\Cal B(X,\frak S)\subseteq\Sigma$ (4A3Da) and $\Cal B(X,\frak T)$ must be
equal to $\Sigma$.
}%end of proof of 423J
%out of order query;  better after 423I?

\leader{423K}{Lemma} If $X$ is an uncountable analytic Hausdorff space,
it has subsets homeomorphic to $\{0,1\}^{\Bbb N}$ and $\NN$.

\proof{{\bf (a)} Let $f:\NN\to X$ be a continuous surjection.   Write
$S=\bigcup_{n\in\Bbb N}\BbbN^n$,
$I_{\sigma}=\{\phi:\sigma\subseteq\phi\in\NN\}$ for $\sigma\in S$,

\Centerline{$T=\{\sigma:\sigma\in S,\,f[I_{\sigma}]$ is
uncountable$\}$.}

\noindent Then if $\sigma\in T$ there are $\tau$, $\tau'\in T$, both
extending $\sigma$, such that $f[I_{\tau}]\cap f[I_{\tau'}]=\emptyset$.
\Prf\ Set

\Centerline{$A=\bigcup\{f[I_{\tau}]:\tau\in S\setminus T\}$.}

\noindent Then $A$ is a countable union of countable sets, so is
countable.   There must therefore be distinct points $x$, $y$ of
$f[I_{\sigma}]\setminus A$;  express $x$ as $f(\phi)$ and $y$ as
$f(\psi)$ where $\phi$ and $\psi$ belong to $I_{\sigma}$.   Because $X$
is Hausdorff, there are disjoint open sets $G$, $H$ such that $x\in G$
and $y\in H$.   Because $f$ is continuous, there are $m$, $n\in\Bbb N$
such that $I_{\phi\restr m}\subseteq f^{-1}[G]$ and $I_{\psi\restr
n}\subseteq f^{-1}[H]$.   Of course both $\tau=\phi\restr m$ and
$\tau'=\psi\restr n$ must extend $\sigma$, and they belong to $T$
because $x\in f[I_{\tau}]\setminus A$ and $y\in f[I_{\tau'}]\setminus A$.\
\Qed

\medskip

{\bf (b)} We can therefore choose inductively a family
$\family{\upsilon}{S_2}{\tau(\upsilon)}$ in $T$, where
$S_2=\bigcup_{n\in\Bbb N}\{0,1\}^n$, such that

\Centerline{$\tau(\emptyset)=\emptyset$,}

\Centerline{$\tau(\upsilon^{\smallfrown}\fraction{i})\supseteq\tau(\upsilon)$
whenever $\upsilon\in S_2$, $i\in\{0,1\}$,}

\Centerline{$f[I_{\tau(\upsilon^{\smallfrown}\fraction{0}})]
  \cap f[I_{\tau(\upsilon^{\smallfrown}\fraction{1}}]=\emptyset$ for every
$\upsilon\in S_2$.}

\noindent Note that $\#(\tau(\upsilon))\ge\#(\upsilon)$ for every
$\upsilon\in S_2$.   For each $z\in\{0,1\}^{\Bbb N}$,
$\sequencen{\tau(z\restr n)}$ is a sequence in $S$ in which each term
strictly extends its predecessor, so there is a unique $g(z)\in\NN$ such
that $\tau(z\restr n)\subseteq g(z)$ for every $n$.    Now
$g(z')\restr n=g(z)\restr n$ whenever $z\restr n=z'\restr n$, so
$g:\{0,1\}^{\Bbb N}\to\NN$ and
$fg:\{0,1\}^{\Bbb N}\to X$ are continuous.   If $w$, $z$ are distinct
points of $\{0,1\}^{\Bbb N}$, there is a first $n$ such that
$w(n)\ne z(n)$, in which case
$fg(w)\in f[I_{\tau(w\restr n)^{\smallfrown}\fraction{w(n)}}]$
and $fg(z)\in f[I_{\tau(w\restr n)^{\smallfrown}\fraction{z(n)}}]$ are distinct.
So $fg:\{0,1\}^{\Bbb N}\to X$ is a continuous injection, therefore a
homeomorphism between $\{0,1\}^{\Bbb N}$ and its image, because
$\{0,1\}^{\Bbb N}$ is compact (3A3Dd).

\medskip

{\bf (c)} Thus $X$ has a subspace homeomorphic to $\{0,1\}^{\Bbb N}$.
Now $\{0,1\}^{\Bbb N}$ has a subspace homeomorphic to $\Bbb N$.   \Prf\
For instance, setting $d_n(n)=1$, $d_n(i)=0$ for $i\ne n$,
$D=\{d_n:n\in\Bbb N\}$ is homeomorphic to $\Bbb N$.\ \QeD\  Now $D^{\Bbb
N}$ is homeomorphic to $\NN$ and is a subspace of $(\{0,1\}^{\Bbb
N})^{\Bbb N}\cong\{0,1\}^{\Bbb N\times\Bbb N}\cong\{0,1\}^{\Bbb N}$, so
$\{0,1\}^{\Bbb N}$ has a subspace homeomorphic to $\NN$.   Accordingly
$X$ also has a subspace homeomorphic to $\NN$.
}%end of proof of 423K

\leader{423L}{Corollary} Any uncountable Borel set in any analytic
Hausdorff space has cardinal $\frak c$.

\proof{ If $X$ is an analytic space and $E\subseteq X$ is an uncountable
Borel set, then $E$ is analytic (423Eb), so includes a copy of
$\{0,1\}^{\Bbb N}$ and must have cardinal at least
$\#(\{0,1\}^{\Bbb N})=\frak c$.   On the other hand, $E$ is also a
continuous image of $\NN$, so has cardinal at most $\#(\NN)=\frak c$.
}%end of proof of 423L

\leader{423M}{Proposition} Let $X$ be an uncountable analytic Hausdorff
space.   Then it has a non-Borel analytic subset.

\proof{{\bf (a)} I show first that there is an analytic set
$A\subseteq\NN\times\NN$ such that every
analytic subset of $\NN$ is a vertical section of $A$.
\Prf\ Let $\Cal U$ be a countable base for the topology of $\BbbN^{\Bbb
N}\times\NN$, containing $\emptyset$, and $\sequencen{U_n}$
an enumeration of $\Cal U$.   Write

\Centerline{$M=(\NN\times\NN\times\NN)
\setminus\bigcup_{m,n\in\Bbb N}(\{x:x(m)=n\}\times U_n)$.}

\noindent Then $M$ is a closed subset of $(\NN\times\Bbb
N^{\Bbb N})\times\NN$, therefore analytic (423Ba, 423Bd), so
its continuous image

\Centerline{$A=\{(x,z):$ there is some $y$ such that $(x,y,z)\in M\}$}

\noindent is analytic (423Bb).

Now let $E$ be any analytic subset of $\NN$.   By 423Eb, $E$
is Souslin-F;  by 421J, there is a closed
set $F\subseteq\NN\times\NN$ such that
$E=\{z:\exists\,y,\,(y,z)\in F\}$.   Let $\sequence{m}{x(m)}$ be a
sequence running over $\{n:n\in\Bbb N,\,U_n\cap F=\emptyset\}$, so that

\Centerline{$F=(\NN\times\BbbN^{\Bbb
N})\setminus\bigcup_{m\in\Bbb N}U_{x(m)}=\{(y,z):(x,y,z)\in M\}$.}

\noindent Now

$$\eqalign{\{z:(x,z)\in A\}
&=\{z:\text{ there is some }y\text{ such that }(x,y,z)\in M\}\cr
&=\{z:\text{ there is some }y\text{ such that }(y,z)\in F\}
=E,\cr}$$

\noindent and $E$ is a vertical section of $A$, as required.\ \Qed

\medskip

{\bf (b)} It follows that there is a non-Borel analytic set
$B\subseteq\NN$.   \Prf\ Take $A$ from (a) above, and try

\Centerline{$B=\{x:(x,x)\in A\}$.}

\noindent Because $B$ is the inverse image of $A$ under the continuous
map $x\mapsto(x,x)$, it is analytic (423Gc).   \Quer\ If
$B$ were a Borel set, then $B'=\NN\setminus B$ would also be
Borel, therefore analytic (423Eb), and there would be an
$x\in\Bbb N^{\Bbb N}$ such that $B'=\{y:(x,y)\in A\}$.   But in this case

\Centerline{$x\in B\iff (x,x)\in A\iff x\in B'$,}

\noindent which is a difficulty you may have met before.\ \Bang\Qed

\medskip

{\bf (c)} Now return to our arbitrary uncountable analytic Hausdorff
space $X$.   By 423K, $X$ has a subset $Z$ homeomorphic to $\NN$.   By
(b), $Z$ has an analytic subset $A$ which is not Borel in $Z$, therefore
cannot be a Borel subset of $X$.
}%end of proof of 423M

\leader{423N}{}\dvAformerly{4{}23M}\cmmnt{ I devote a few paragraphs to an
important method of constructing selectors.

\medskip

\noindent}{\bf Theorem} Let $X$ be an analytic Hausdorff space,  $Y$ a
set, and $\Cal C\subseteq\Cal PY$.   Write $\Tau$ for the
$\sigma$-algebra of subsets of $Y$ generated by $\Cal S(\Cal C)$, where
$\Cal S$ is Souslin's operation, and $\Cal V$ for
$\Cal S(\{F\times C:F\subseteq X$ is closed, $C\in\Cal C\})$.   If
$W\in\Cal V$, then $W[X]\in\Cal S(\Cal C)$ and there is a $\Tau$-measurable
function $f:W[X]\to X$ such that $(f(y),y)\in W$ for every $y\in W[X]$.

\proof{ Write $\Cal F$ for
$\{F\times C:F\subseteq X$ is closed, $C\in\Cal C\}$.

\medskip

{\bf (a)} Consider first the case in which $X=\NN$ and all the
horizontal sections $W^{-1}[\{y\}]$ of $W$ are closed.   Let
$\Cal E$ be the family of closed subsets of $Y$.   For
$\sigma\in S=\bigcup_{n\in\Bbb N}\BbbN^n$ set
$I_{\sigma}=\{\phi:\sigma\subseteq\phi\in\NN\}$.   Then
$W\cap(I_{\sigma}\times Y)\in\Cal V$.   \Prf\
Because Souslin's operation is idempotent (421D),
$\Cal S(\Cal V)=\Cal V$.   The set
$\{V:V\cap(I_{\sigma}\times Y)\in\Cal V\}$ is therefore closed under
Souslin's operation (apply 421Cc to the identity map from
$I_{\sigma}\times Y$ to $X\times Y$, or otherwise);  since it includes
$\Cal F$, it is the whole of $\Cal V$, and contains $W$.\ \Qed

By 421G, $W[I_{\sigma}]=(W\cap(I_{\sigma}\times Y))[\NN]$ belongs to
$\Cal S(\Cal C)\subseteq\Tau$ for every $\sigma$.   In particular,
$W[\NN]=W[I_{\emptyset}]\in\Cal S(\Cal C)$.   Define
$\family{\sigma}{S}{Y_{\sigma}}$ in $\Tau$
inductively, as follows.   $Y_{\emptyset}=W[\NN]$.   Given that
$Y_{\sigma}\in\Tau$ and that $Y_{\sigma}\subseteq W[I_{\sigma}]$, set

\Centerline{$Y_{\sigma^{\smallfrown}\fraction{j}}
=Y_{\sigma}\cap W[I_{\sigma^{\smallfrown}\fraction{j}}]
\setminus\bigcup_{i<j}W[I_{\sigma^{\smallfrown}\fraction{i}}]\in\Tau$}

\noindent for every $j\in\Bbb N$.   Continue.

At the end of the induction, we have

\Centerline{$\bigcup_{j\in\Bbb N}Y_{\sigma^{\smallfrown}\fraction{j}}
=Y_{\sigma}\cap\bigcup_{j\in\Bbb N}W[I_{\sigma^{\smallfrown}\fraction{j}}]
=Y_{\sigma}\cap W[I_{\sigma}]=Y_{\sigma}$}

\noindent for every $\sigma\in S$, while
$\sequence{j}{Y_{\sigma^{\smallfrown}\fraction{j}}}$ is always disjoint.
So for
each $y\in Y_{\emptyset}=W[\NN]$ we have a unique $f(y)\in\NN$ such that
$y\in Y_{f(y)\restr n}$ for every $n$.   Since
$f^{-1}[I_{\sigma}]=Y_{\sigma}\in\Tau$ for every $\sigma\in S$, $f$ is
$\Tau$-measurable (4A3Db).   Also $(f(y),y)\in W$ for every
$y\in W[\NN]$.   \Prf\ For each $n\in\Bbb N$,
$y\in Y_{f(y)\restr n}=W[I_{f(y)\restr n}]$, so there is an $x_n\in\NN$
such that $x_n\restr n=f(y)\restr n$ and $(x_n,y)\in W$.   But this
means that $f(y)=\lim_{n\to\infty}x_n$;  since we are supposing that the
horizontal sections of $W$ are closed, $(f(y),y)\in W$.\ \Qed

Thus the theorem is true if $X=\NN$ and $W$ has closed horizontal
sections.

\medskip

{\bf (b)} Now suppose that $X=\NN$ and that $W\subseteq\NN\times Y$ is
any set in $\Cal V$.   Then there is a Souslin scheme
$\family{\sigma}{S^*}{F_{\sigma}\times C_{\sigma}}$ in $\Cal F$ with
kernel $W$;  of course I mean you to suppose that
$F_{\sigma}\subseteq\NN$ is closed and $C_{\sigma}\in\Cal C$ for every
$\sigma$.   Set

\Centerline{$\tilde W
=\bigcap_{k\ge 1}\bigcup_{\sigma\in\BbbN^k}
  I_{\sigma}\times F_{\sigma}\times C_{\sigma}
\subseteq\NN\times\NN\times Y$.}

\noindent Then $W$ is the projection of $\tilde W$ onto the last two
coordinates, by 421Ce.   If $y\in Y$, then

\Centerline{$\{(\phi,\psi):(\phi,\psi,y)\in\tilde W\}
=\bigcap_{k\ge 1}\bigcup\{I_{\sigma}\times F_{\sigma}:
  \sigma\in\BbbN^k,\,y\in C_{\sigma}\}$}

\noindent is closed in $\NN\times\NN$.   (If $J$ is any subset of
$\BbbN^k$, then

\Centerline{$(\NN\times\NN)
  \setminus\bigcup_{\sigma\in J}I_{\sigma}\times F_{\sigma}
=\bigcup_{\sigma\in J}I_{\sigma}\times(\NN\setminus F_{\sigma})
   \cup\bigcup_{\sigma\in\BbbN^k\setminus J}I_{\sigma}\times\NN$}

\noindent is open.)   Also $I_{\sigma}\times F_{\sigma}$ is a closed
subset of $\NN\times\NN$ for every $\sigma$, and $\NN\times\NN$ is
homeomorphic to $\NN$.   We can therefore apply (a) to $\tilde W$,
regarded as a subset of $(\NN\times\NN)\times Y$, to see that
$W[\NN]=\tilde W[\NN\times\NN]\in\Cal S(\Cal C)$ and that there is a
$\Tau$-measurable function $h=(g,f):W[\NN]\to\NN\times\NN$ such that
$(g(y),f(y),y)\in\tilde W$ for every $y\in W[\NN]$.   Now, of course,
$f:W[\NN]\to\NN$ is $\Tau$-measurable and $(f(y),y)\in W$ for every
$y\in W[\NN]$.

\medskip

{\bf (c)} Finally, suppose only that $X$ is an analytic Hausdorff space
and that $W\in\Cal V$.   If $X$ is empty, so is
$Y$, and the result is trivial.   Otherwise, there is a continuous
surjection $h:\NN\to X$.   Set $\tilde h(\phi,y)=(h(\phi),y)$ for
$\phi\in\NN$ and
$y\in Y$;  then $\tilde h:\NN\times Y\to X\times Y$ is a continuous
surjection, and $\tilde W=\tilde h^{-1}[W]$ is the kernel of a Souslin
scheme in

\Centerline{$\{\tilde h^{-1}[F\times C]:
  F\subseteq\NN\text{ is closed}, C\in\Cal C\}
=\{h^{-1}[F]\times C:F\subseteq\NN\text{ is closed}, C\in\Cal C\}$}

\noindent by 421Cb.   So we can apply (b) to see that
$W[X]=\tilde W[\NN]\in\Cal S(\Cal C)$ and there is a
$\Tau$-measurable $g:W[X]\to\NN$ such that $(g(y),y)\in\tilde W$ for
every $y\in Y$.   Finally $f=hg:W[X]\to X$ is $\Tau$-measurable and
$(f(y),y)\in W$ for every $y\in Y$.   This completes the proof.
}%end of proof of 423N

\leader{423O}{}\cmmnt{ The expression

\Centerline{$\Cal V
=\Cal S(\{F\times C:F\subseteq X$ is closed, $C\in\Cal C\})$}

\noindent in 423N is a new formulation, and I had better describe a couple
of the basic cases in which we can use the result.   The first really is
elementary.

\medskip

\noindent}{\bf Corollary}\dvAnew{2016} 
Let $X$ be an analytic Hausdorff space, $Y$ a set and
$\Tau$ a $\sigma$-algebra of subsets of $Y$ which is closed under Souslin's
operation.   Suppose that $W\in\Cal S(\Cal B(X)\tensorhat\Tau)$ 
where $\Cal B(X)$ is the
Borel $\sigma$-algebra of $X$.   Then $W[X]\in\Tau$ and there is a
$\Tau$-measurable function $f:W[X]\to X$ such that $(f(y),y)\in W$ for
every $y\in W[X]$.

\proof{ Set $\Cal V=\Cal S(\{F\times C:F\subseteq X$ is closed, $C\in\Tau\}$
and
$\Cal E=\{F\times Y:F\subseteq X$ is closed$\}\cup\{X\times C:C\in\Tau\}$.
Then $E\times Y\in\Cal S(\Cal E)$ for every Souslin-F subset $E$ of $X$,
by 421Cc, and in particular for every Borel subset
$E$ of $X$ (423Eb).   Accordingly $\Cal E'
=\{E\times Y:E\in\Cal B(X)\}\cup\{X\times C:C\in\Tau\}$ is included in
$\Cal S(\Cal E)$.   Since $\Cal E'$ contains $\emptyset$ and
the complement of any member of $\Cal E'$ belongs to
$\Cal E'$, $\Cal S(\Cal E')$ includes the $\sigma$-algebra of subsets of
$X\times Y$ generated by $\Cal E'$ (421F), and we have

\Centerline{$W\in\Cal S(\Cal B(X)\tensorhat\Tau)\subseteq\Cal S(\Cal E')
\subseteq\Cal S(\Cal S(\Cal E))=\Cal S(\Cal E)\subseteq\Cal V$.}

\noindent So $W\in\Cal V$ and we can read the conclusion off from 423N.
}%end of proof of 423O

\leader{423P}{Corollary}\dvAformerly{4{}23N} 
Let $X$ be an analytic Hausdorff space and $Y$
any topological space.   Let $\Tau$ be the $\sigma$-algebra of subsets
of $Y$ generated by $\Cal S(\Cal B(Y))$, where $\Cal B(Y)$ is the Borel
$\sigma$-algebra of $Y$.   If $W\in\Cal S(\Cal B(X\times Y))$, then
$W[X]\in\Tau$ and there is a
$\Tau$-measurable function $f:W[X]\to X$ such that $(f(y),y)\in W$ for
every $y\in W[X]$.

\proof{{\bf (a)} Suppose to begin with that $X=\NN$.   In 423N, set
$\Cal C=\Cal B(Y)$.   Then every open subset and every closed subset of
$X\times Y$ belongs to $\Cal V$ as defined in 423N.   \Prf\ For
$\sigma\in S=\bigcup_{k\in\Bbb N}\BbbN^k$, set
$I_{\sigma}=\{\phi:\sigma\subseteq\phi\in\NN\}$.   If
$V\subseteq X\times Y$ is open, set

\Centerline{$H_{\sigma}=\bigcup\{H:H\subseteq Y$ is open,
$I_{\sigma}\times H_{\sigma}\subseteq V\}$}

\noindent for each $\sigma\in S$.   Because
$\{I_{\sigma}:\sigma\in S\}$ is a base for the topology of $\NN$,
$V=\bigcup_{\sigma\in S}I_{\sigma}\times H_{\sigma}\in\Cal V$.

As for the complement of $V$, we have

$$\eqalign{(\NN\times Y)\setminus V
&=\bigcap_{\Atop{k\in\Bbb N}{\sigma\in\BbbN^k}}
  (\NN\times Y)\setminus(I_{\sigma}\times H_{\sigma})\cr
&=\bigcap_{\Atop{k\in\Bbb N}{\sigma\in\BbbN^k}}
  ((\NN\setminus I_{\sigma})\times Y)
  \cup(\NN\times(Y\setminus H_{\sigma}))\cr
&=\bigcap_{\Atop{k\in\Bbb N}{\sigma\in\BbbN^k}}
  \bigcup_{\Atop{\tau\in\BbbN^k}{\tau\ne\sigma}}
  (I_{\tau}\times Y)
  \cup(\NN\times(Y\setminus H_{\sigma}))\cr}$$

\noindent which again belongs to $\Cal V$, because $\Cal V$ is closed under
countable unions and intersections and contains $I_{\tau}\times Y$ and
$\NN\times(Y\setminus H_{\sigma})$ for all $\sigma$, $\tau\in S$.\
\Qed

By 421F, $\Cal V$ contains every Borel subset of $\NN\times Y$, so
includes $\Cal S(\Cal B(\NN\times Y))$.   So in this case we can apply
423N directly to get the result.

\medskip

{\bf (b)} Now suppose that $X$ is any analytic space.   If $X$ is empty,
the result is trivial.   Otherwise, let
$h:\NN\to X$ be a continuous surjection.   Set
$\tilde h(\phi,y)=(h(\phi),y)$ for $\phi\in\NN$ and $y\in Y$, so that
$\tilde h:\NN\times Y\to X\times Y$ is continuous.   Set
$\tilde W=\tilde h^{-1}[W]$.   If
$V\in\Cal B(X\times Y)$ then $\tilde h^{-1}[V]\in\Cal B(\NN\times Y)$
(4A3Cd), so  $\tilde W\in\Cal S(\Cal B(\NN\times Y))$ (421Cc).
By (a), $\tilde W[\NN]\in\Tau$ and there is a $\Tau$-measurable function
$g:\tilde W[\NN]\to\NN$ such that $(g(y),y)\in\tilde W$ for every
$y\in\tilde W[\NN]$.   It is now easy to check that
$W[X]=\tilde W[\NN]\in\Tau$ (this is where we need to know that $h$ is
surjective), that $f=hg:W[X]\to X$ is $\Tau$-measurable, and that
$(f(y),y)\in W$ for every $y\in W[X]$, as required.
}%end of proof of 423P

\cmmnt{\medskip

\noindent{\bf Remark} This is a version of the {\bf von Neumann-Jankow
selection theorem}.
}%end of comment

\leader{423Q}{Corollary}\dvAformerly{4{}23O} Let $X$ and $Y$ be analytic
Hausdorff spaces, $A$ an analytic subset of $X$ and $f:A\to Y$ a Borel
measurable function.   Let
$\Tau$ be the $\sigma$-algebra of subsets of $Y$ generated by the
Souslin-F subsets of $Y$.
Then $f[A]\in\Tau$ and there is a $\Tau$-measurable function
$g:f[A]\to A$ such that $fg$ is the identity on $f[A]$.

\proof{ In this context, every Borel subset of $Y$ is Souslin-F (423Eb),
so every member of $\Cal S(\Cal B(Y))$ is Souslin-F (421D) and
$\Tau\supseteq\Cal S(\Cal B(Y))$.
If we think of $f$ as a subset of $X\times Y$, it is analytic
(423Ga), therefore Souslin-F in
$X\times Y$;  now we can use 423P to find
a $\Tau$-measurable function $g:f[A]\to A$ such that
$(g(y),y)\in f$, that is, $f(g(y))=y$, for every $y\in f[A]$.
}%end of proof of 423Q

\leader{*423R}{Constituents of coanalytic sets:  Theorem}\dvAformerly{4{}23P} 
Let $X$ be a
Hausdorff space, and $A\subseteq X$ an analytic subset of $X$.   Then
there is a non-decreasing family $\ofamily{\xi}{\omega_1}{E_{\xi}}$ of
Borel subsets of $X$, with union $X\setminus A$, such that every
analytic subset of $X\setminus A$ is included in some $E_{\xi}$.

\proof{ Put 422K(iii) and 423C together.
}%end of proof of 423R

\leader{*423S}{Remarks (a)}\dvAformerly{4{}23Q} 
Let $A$ be an analytic set in an analytic
space $X$ and $\ofamily{\xi}{\omega_1}{E_{\xi}}$ a family of Borel sets
as in 423R.   There is nothing unique about the $E_{\xi}$.   But if
$\ofamily{\xi}{\omega_1}{E'_{\xi}}$ is another such family, then
\dvro{there is a cofinal closed set $C$ in $\omega_1$ such that}{every
$E'_{\xi}$
is an analytic subset of $X\setminus A$, by 423Eb, so is included in
some $E_{\eta}$;  and, similarly, every $E_{\xi}$ is included in some
$E'_{\eta}$.   We therefore have a function $f:\omega_1\to\omega_1$ such
that $E'_{\xi}\subseteq E_{f(\xi)}$ and $E_{\xi}\subseteq E'_{f(\xi)}$
for every $\xi<\omega$.   If we set
$C=\{\xi:\xi<\omega_1,\,f(\eta)<\xi$ for every $\eta<\xi\}$, then $C$ is
a closed cofinal set in $\omega_1$ (4A1Bc), and}
$\bigcup_{\eta<\xi}E_{\eta}=\bigcup_{\eta<\xi}E'_{\eta}$ for every
$\xi\in C$.   \cmmnt{If $X\setminus A$ is itself analytic, that is, if
$A$ is a Borel set, then we shall have to have
$X\setminus A=E_{\xi}=E'_{\xi}$ for some $\xi<\omega_1$.}

Another way of expressing the result in 423R is to say that if we write
$\Cal I=\{B:B\subseteq X\setminus A$ is analytic$\}$, then
$\{E:E\in\Cal I,\,E$ is Borel$\}$ is cofinal with $\Cal I$\cmmnt{
(this is the First Separation Theorem)} and $\cf\Cal I\le\omega_1$.

\spstheader 423Sb It is a remarkable fact that, in some models of set theory, we can have
non-Borel coanalytic sets in Polish spaces such that all their
constituents are
countable\cmmnt{ ({\smc Jech 78}, p.\ 529, Cor.\ 2)}.
\cmmnt{(Note that, by (a), this is the same thing as
saying that $X\setminus A$ is uncountable but all its Borel subsets are
countable.)}   But in `ordinary' cases we shall have, \cmmnt{for every
Borel subset $E$ of $X\setminus A$, an uncountable Borel subset of
$(X\setminus A)\setminus E$;  so that} for any family
$\ofamily{\xi}{\omega_1}{G_{\xi}}$ of Borel constituents of
$X\setminus A$, \cmmnt{there must be uncountably many} uncountable
$G_{\xi}$.
\prooflet{To see that this happens at least sometimes, take any
non-Borel analytic subset $A_0$ of $\NN$ (423M), and consider
$A=A_0\times\NN\subseteq(\NN)^2$.
Then $A$ is analytic (423B).   If $E\subseteq(\NN)^2\setminus A$ is
Borel, then $\pi_1[E]=\{x:(x,y)\in E\}$ is an analytic subset of
$\NN\setminus A_0$, so is not the whole of $\NN\setminus A_0$ (by 423Fa).
Taking any $x\in(\NN\setminus A_0)\setminus\pi_1[E]$, $\{x\}\times\NN$
is an uncountable Borel subset of $((\NN)^2\setminus A)\setminus E$.
For an alternative construction, see 423Ye.}

\leader{*423T}{Coanalytic and PCA sets}\dvAformerly{4{}23R}
\cmmnt{ In the case of Polish spaces, we can go a great deal
farther.   I mention only some fragments which will be used in
Volume 5.}   Let $X$ be a Polish space.

\spheader 423Ta A subset $A$ of $X$ is
{\bf coanalytic}\cmmnt{ or $\pmb{\Pi}^1_1$} if $X\setminus A$ is
analytic, and
{\bf PCA}\cmmnt{ or $\pmb{\Sigma}^1_2$} if there is a coanalytic set
$R\subseteq\BbbN^{\Bbb N}\times X$ such that $R[\BbbN^{\Bbb N}]=A$.
\cmmnt{Generally, for $n\ge 1$, $A\subseteq X$ is
$\pmb{\Pi}^1_n$ if $X\setminus A$ is $\pmb{\Sigma}^1_n$ ($A$ is
$\pmb{\Sigma}^1_1$ if it is analytic), and
$\pmb{\Sigma}^1_{n+1}$ if there is a $\pmb{\Pi}^1_n$ set
$R\subseteq\BbbN^{\Bbb N}\times X$ such that $R[\BbbN^{\Bbb N}]=A$.
If $A$ is both $\pmb{\Sigma}^1_n$ and $\pmb{\Pi}^1_n$, we say that $A$ is
$\pmb{\Delta}^1_n$.}

\spheader 423Tb\dvro{ Every}{ Analytic subsets of $X$ are Souslin-F
(423Eb).   Applying 421P to the Borel $\sigma$-algebra
of $X$, we see that a subset of
$X$ which is either analytic or coanalytic can be expressed as the union of
at most $\omega_1$ Borel sets.   It follows that every} PCA set
$A\subseteq X$ can be expressed as the union of at most $\omega_1$ Borel
sets.   \prooflet{\Prf\ Let
$R\subseteq\BbbN^{\Bbb N}\times X$ be a coanalytic set such that
$A=R[\NN]$.   Express $R$ as $\bigcup_{\xi<\omega_1}R_{\xi}$ where every
$R_{\xi}$ is a Borel subset of the Polish space $\NN\times X$.
For each $\xi<\omega_1$,
$R_{\xi}$ is analytic (423Eb) so $A_{\xi}=R_{\xi}[\NN]$ is analytic (423Bb)
and can be expressed as $\bigcup_{\eta<\omega_1}E_{\xi\eta}$ where every
$E_{\xi\eta}$ is a Borel subset of $X$.   Now
$A=\bigcup_{\xi,\eta<\omega_1}E_{\xi\eta}$ is the union of at most
$\omega_1$ Borel sets.\ \Qed}

\spheader 423Tc A subset of $X$ is Borel iff it
is\cmmnt{ $\pmb{\Delta}^1_1$, that is, is} both analytic and
coanalytic\prooflet{ (423Eb, 423Fa)}.
\dvro{ The}{ Since the intersection and union of a
sequence of analytic subsets of $X$ are analytic (423Ea), the} union
and intersection of
a sequence of coanalytic subsets of $X$ are coanalytic.   If $Y$ is
another Polish space and $h:X\to Y$ is Borel measurable,
then\cmmnt{ $h^{-1}[A]$ is analytic for every analytic $A\subseteq Y$
(423Gc), so} $h^{-1}[B]$ is coanalytic in $X$ for every coanalytic
$B\subseteq Y$.   If $Y$ is a G$_{\delta}$ subset of $X$ and
$B\subseteq Y$ is coanalytic in $Y$\cmmnt{ (remember that $Y$, with its
subspace topology, is Polish, by 4A2Qd),} then $B$ is coanalytic in
$X$\prooflet{, because
$X\setminus B=(X\setminus Y)\cup(Y\setminus B)$ is the union of two
analytic sets}.

\spheader 423Td If $X$ and $Y$ are Polish spaces, $A\subseteq Y$ is PCA and
$f:X\to Y$ is Borel measurable, then $f^{-1}[A]$ is a PCA set in $X$.
\prooflet{\Prf\ Let $R\subseteq\NN\times X$ be a coanalytic set such that
$R[\NN]=A$.   Set $g(\phi,x)=(\phi,f(x))$ for $\phi\in\NN$ and $x\in X$;
writing $\Cal B(X)$ for the Borel $\sigma$-algebra of $X$, etc.,
$g$ is $(\Cal B(\NN)\tensorhat\Cal B(X),
\Cal B(\NN)\tensorhat\Cal B(Y))$-measurable (use 4A3Bc).   But
$\Cal B(\NN)\tensorhat\Cal B(X)=\Cal B(\NN\times X)$ and
$\Cal B(\NN)\tensorhat\Cal B(Y)=\Cal B(\NN\times Y)$ (4A3Ga), so
$g$ is Borel measurable.   By (c), $R'=g^{-1}[R]$ is a coanalytic set in
$\NN\times X$.   Now

\Centerline{$R'[\NN]
=\{x:\,\Exists\phi,\,(\phi,x)\in g^{-1}[R]\}
=\{x:\,\Exists\phi,\,(\phi,f(x))\in R\}=f^{-1}[A]$,}

\noindent so $f^{-1}[A]$ is PCA.\ \Qed}

\leaveitout{\spheader 423T?
If $X$ and $Y$ are Polish spaces, $A\subseteq X$ is PCA and
$f:X\to Y$ is a Borel measurable function, then $f[A]$ is a PCA subset of
$Y$.   \prooflet{\Prf\ If $X$ is empty, so is $f[A]$, and we can stop.
Otherwise, let $R\subseteq\NN\times X$ be a coanalytic set such that
$R[\NN]=A$, and $g:\NN\to X$ a continuous surjection.
Consider

\Centerline{$S
=\{((\phi,\psi),y):\phi$, $\psi\in\NN$, $y\in Y$, $(\phi,g(\psi))\in R$,
$f(g(\psi))=y\}$.}

\noindent The complement $((\NN\times\NN)\times Y)\setminus S$ is equal to
$C\cup D$ where

\Centerline{$C=\{((\phi,\psi),y):(\phi,g(\psi))\notin R\}$,
\quad$D=\{((\phi,\psi),y):f(g(\psi))\ne y\}$.}

\noindent Setting $\hat g((\phi,\psi),y)=(\phi,g(\psi))$,
$\hat g:(\NN\times\NN)\times Y\to\NN\times X$ is continuous, so
$C=\hat g^{-1}[(\NN\times X)\setminus R]$ is analytic (423Gc).
On the other hand, setting $\bar g((\phi,\psi),y)=(g(\psi),y)$,
$\bar g:(\NN\times\NN)\times Y\to X\times Y$ is continuous, while
$D_0=\{(x,y):f(x)\ne y\}$ is a Borel subset of $X\times Y$ (4A3Gb),
so $D=\bar g^{-1}[D_0]$ is Borel (4A3Cd), therefore analytic (423Eb
again).   Thus $C\cup D$ is analytic (423Ea) and $S$ is coanalytic, while
$S[\NN\times\NN]=f[A]$.   Since $\NN\times\NN$ is homeomorphic to $\NN$,
this is enough to show that $f[A]$ is PCA.\ \Qed}
}

\cmmnt{\spheader 423Te For a fuller account of this material, see
{\smc Kechris 95} or {\smc Kuratowski 66}.}

\exercises{\leader{423X}{Basic exercises $\pmb{>}$(a)}
%\spheader 423Xa
For a Hausdorff space $X$, show that the following are
equiveridical:  (i) $X$ is analytic;  (ii) $X$ is a continuous image of
a Polish space;  (iii) $X$ is a continuous image of a closed subset of
$\NN$.
%423B

\spheader 423Xb Write out a direct proof of 423Ea, not quoting 423C or
421D.
%423E

\sqheader 423Xc Let $X$ be a set, $\frak S$ a
Hausdorff topology on $X$ and $\frak T$ an analytic topology on $X$ such
that $\frak S\subseteq\frak T$.   Show that $\frak S$ and $\frak T$ have
the same analytic sets.   \Hint{423F.}
%423F

\spheader 423Xd Let $X$ be an analytic Hausdorff space.   (i) Show that
its Borel $\sigma$-algebra $\Cal B(X)$ is countably generated as
$\sigma$-algebra.  \Hint{use 4A2Nf
%countable network Hausdorff > coarser second countable Hausdorff topy
and 423Fb.}   (ii) Show that there is an analytic subset $Y$ of
$\Bbb R$ such that $(X,\Cal A_X)$ is isomorphic to $(Y,\Cal A_Y)$, where
$\Cal A_X$, $\Cal A_Y$ are the families of Souslin-F subsets of $X$, $Y$
respectively.   \Hint{show
that there is an injective Borel measurable function from $X$ to
$\Bbb R$ (cf.\ 343E), and use 423G.}   (iii) Show that $(X,\Cal B(X))$
is isomorphic to $(Y,\Cal B(Y))$, where $\Cal B(Y)$ is the Borel
$\sigma$-algebra of
$Y$.   \Hint{423Fa.}   (iv) Let $\Tau_X$, $\Tau_Y$ be the
$\sigma$-algebras generated by $\Cal A_X$, $\Cal A_Y$ respectively.
Show that $(X,\Tau_X)$ and $(Y,\Tau_Y)$ are isomorphic.
%423F 423G

\spheader 423Xe Let $\frak S$ be the right-facing Sorgenfrey topology on
$\Bbb R$ (415Xc).   Show that $\frak S$ has the same Borel sets as the
usual topology $\frak T$ on $\Bbb R$.   \Hint{$\frak S$ is hereditarily
Lindel\"of (419Xf) and has a base consisting of $\frak T$-Borel sets.}
Show that $\frak S$ is not analytic.   \Hint{in the product topology on
$\BbbR^2$ the counter-diagonal $\{(x,-x)\}:x\in\Bbb R\}$ is closed and
discrete.}
%423F

\spheader 423Xf Let $X$ be an analytic Hausdorff space and $Y$
any topological space.   Let $\Tau$ be the $\sigma$-algebra of subsets
of $Y$ generated by the Souslin-F sets.   Show that if
$W\subseteq X\times Y$ is Souslin-F, then $W[X]\in\Tau$ and there is a
$\Tau$-measurable function $f:W[X]\to X$ such that $(f(y),y)\in W$ for
every $y\in W[X]$.   \Hint{start with $X=\NN$, as in 423P.}
%423P 422Ha

\spheader 423Xg Let $X$ and $Y$ be Hausdorff spaces, and
$R\subseteq X\times Y$ an analytic set such that $R^{-1}[Y]=X$.   Show
that there is
a function $g:X\to Y$, measurable with respect to the $\sigma$-algebra
generated by the Souslin-F subsets of $X$, such that $(x,g(x))\in R$ for
every $x\in X$.
%423Q

\sqheader 423Xh(i) Show that the family of analytic subsets of $[0,1]$
has cardinal $\frak c$.   \Hint{421Xc.}   (ii) Show that the
$\sigma$-algebra $\Tau$ of subsets of $[0,1]$ generated by the analytic
sets has cardinal $\frak c$.  \Hint{421Xh.}   (iii) Show that
there is a set $A\subseteq[0,1]$ which does not belong to $\Tau$.
%423Q

\spheader 423Xi Let $X=Y=[0,1]$.   Give $Y$ the usual topology, and give
$X$ the topology corresponding to the one-point compactification of the
discrete topology on $\coint{0,1}$, that is, a set $G\subseteq X$ is
open if either $1\notin G$ or $G$ is cofinite.   Show that the identity
map $f:X\to Y$ is a Borel measurable bijection, but that $f^{-1}$ is not
measurable for the $\sigma$-algebra of subsets of $Y$ generated by the
Souslin-F sets.
%423Q %423Xh

\leader{423Y}{Further exercises (a)}
%\spheader 423Ya
Show that a space with a countable network is hereditarily
separable, therefore countably tight.

\spheader 423Yb
Show that if $X$ is a Hausdorff space with a countable network, then
every analytic subset of $X$ is
obtainable by Souslin's operation from the open subsets of $X$.
%423E %mt42bits

\spheader 423Yc Let $(X,\frak T)$ and $(Y,\frak S)$ be analytic
Hausdorff
spaces and $f:X\to Y$ a Borel measurable function.   (i) Show that there
is a zero-dimensional separable metrizable topology $\frak S'$ on $Y$
with the same Borel sets and the same analytic sets as $\frak S$.
\Hint{423Xd.}  (ii) Show that there is a zero-dimensional
separable metrizable topology $\frak T'$ on $X$, with the same Borel
sets and the same analytic sets as $\frak T$, such that $f$ is
continuous for the topologies $\frak T'$ and
$\frak S'$.   (iii) Explain how to elaborate these ideas
to deal with any countable family of analytic spaces and Borel
measurable functions between them.
%423G, 423Xd

\spheader 423Yd Let $X$ be an analytic Hausdorff space, $Y$ a Hausdorff
space with a countable network, and $f:X\to Y$ a Borel measurable
surjection.   Let
$\Tau$ be the $\sigma$-algebra of subsets of $Y$ generated by the
Souslin-F sets
in $Y$.    Show that there is a $\Tau$-measurable function $g:Y\to X$
such that $fg$ is the identity on $Y$.
%423Q

\spheader 423Ye Set $S^*=\bigcup_{n\ge 1}\BbbN^n$ and consider
$\Cal P(S^*)$
with its usual topology.   Let $\Cal T\subseteq\Cal PS^*$ be the set of
trees (421N);  show that $\Cal T$ is closed, therefore a compact
metrizable space.   Set $F_{\sigma}=\{T:\sigma\in T\in\Cal T\}$ for
$\sigma\in S^*$, and let $A$ be the kernel of the Souslin scheme
$\family{\sigma}{S^*}{F_{\sigma}}$.   Show that the constituents of
$\Cal T\setminus A$ for this scheme are just the sets
$G_{\xi}=\{T:T\in\Cal T\setminus A$, $r(T)=\xi\}$,
where $r$ is the rank function of 421Ne.
Show by induction on $\xi$ that all the $G_{\xi}$ are non-empty, so that
$A$ is not a Borel set.   Show that $\#(G_{\xi})=\frak c$ for
$1\le\xi<\omega_1$.   Show that if $X$ is any topological space and
$B\subseteq X$ is a Souslin-F set, there is a Borel measurable function
$f:X\to\Cal T$ such that $B=f^{-1}[A]$.
%423S

}%end of exercises

\endnotes{
\Notesheader{423}
We have been dealing, in this section and the last, with three classes
of topological space:  the class of analytic spaces, the class of
K-analytic spaces and the class of spaces with countable networks.   The
first is more important than the other two put together, and I am sure
many people would find it more comfortable, if more time-consuming, to
learn the theory of
analytic spaces thoroughly first, before proceeding to the others.
This was indeed my own route into the subject.   But I think that the
theory of K-analytic spaces has now matured to the point that it can
stand on its own, without constant reference to its origin as an
extension of descriptive set theory on the real line;  and that our
understanding of analytic spaces is usefully advanced by seeing how
easily their properties
can be deduced from the fact that they are K-analytic spaces with
countable networks.

As in \S422, I have made no attempt to cover the general theory of
analytic spaces, nor even to give a balanced introduction.   I have
tried instead to give a condensed account of the principal methods for
showing that spaces are analytic, with some of the ideas which can
be applied to make them more accessible to the imagination (423K,
423Xc-423Xd, 423Yb-423Yc).   Lusin's theorem 423I does not
mention `analytic' sets in its statement, but it depends essentially on
the separation theorem 422J, so cannot really be put with the other
results on Polish spaces in 4A2Q.   You must of course know that not all
analytic sets are Borel (423M) and that not all sets are analytic
(423Xh).   For further information about this fascinating subject, see
{\smc Rogers 80}, {\smc Kechris 95} and {\smc Moschovakis 80}.

`Selection theorems' appear everywhere in mathematics.   The axiom of
choice is a selection theorem;  it says that whenever
$R\subseteq X\times Y$ is a relation and $R[X]=Y$, there is a function
$f:Y\to X$ such that $(f(y),y)\in R$ for every $y\in Y$.   The Lifting
Theorem (\S341) asks for a selector which is a Boolean homomorphism.   In
general topology we look for continuous selectors.   In measure theory,
naturally, we are interested in measurable selectors, as in
423N-423Q.  %423N 423P 423Q
Any selection theorem will be expressible either as a theorem on right
inverses of functions, as in 423Q, or as a theorem on selectors for
relations, as in 423N-423P.  In the language here, however, we get
better theorems by examining relations, because the essence of the
method is that we can find measurable functions into analytic spaces,
and the relations of 423N can be very far from being analytic, even when
there is a natural topology on the space $Y$.   The value of these
results will become clearer in \S433, when we shall see that the
$\sigma$-algebras $\Tau$ of 423N-423Q %423N 423P 423Q
are often included in familiar $\sigma$-algebras.   Typical applications
are in 433F-433G below.
}%end of notes

\leaveitout{The irrational slope topology ({\smc Steen \& Seebach 78},
\S75) is a countable, second-countable connected Hausdorff space;  of
course it is analytic.
}%end of leaveitout

\discrpage

\frfilename{mt424.tex}
\versiondate{21.3.08}
\copyrightdate{1998}

\def\NN{\BbbN^{\Bbb N}}

\def\chaptername{Descriptive set theory}
\def\sectionname{Standard Borel spaces}

\newsection{424}

This volume is concerned with topological measure spaces, and it will
come as no surprise that the topological properties of
Polish spaces are central to the theory.   But even from the
point of view of unadorned measure theory, not looking for
topological structures on the underlying spaces, it turns
out that the Borel algebras of Polish spaces have a very
special position.   It will be useful later on to be able to
refer to some fundamental facts concerning them.

\leader{424A}{Definition} Let $X$ be a set and $\Sigma$ a
$\sigma$-algebra of subsets of $X$.   We say that $(X,\Sigma)$ is a {\bf
standard Borel space} if there is a Polish topology on $X$ for which
$\Sigma$ is the algebra of Borel sets.

\cmmnt{{\bf Warning!}  Many authors reserve the phrase `standard Borel
space' for the case in which $X$ is uncountable.   I have seen the
phrase `Borel space' used for what I call a `standard Borel space'.}

\leader{424B}{Proposition} (a) If $(X,\Sigma)$ is a standard Borel
space, then $\Sigma$ is countably generated as $\sigma$-algebra of sets.

(b) If $\familyiI{(X_i,\Sigma_i)}$ is a countable family of standard
Borel spaces, then
$(\prod_{i\in I}X_i,\Tensorhat_{i\in I}\Sigma_i)$\cmmnt{ (definition:
254E)} is a standard Borel space.

(c) Let $(X,\Sigma)$ and $(Y,\Tau)$ be standard Borel spaces and
$f:X\to Y$ a $(\Sigma,\Tau)$-measurable surjection.   Then

\quad(i) if $E\in\Sigma$ is such that
$f[E]\cap f[X\setminus E]=\emptyset$, then $f[E]\in\Tau$;

\quad(ii) $\Tau=\{F:F\subseteq Y,\,f^{-1}[F]\in\Sigma\}$;

\quad(iii) if $f$ is a bijection it is an isomorphism.

(d) Let $(X,\Sigma)$ and $(Y,\Tau)$ be standard Borel spaces and
$f:X\to Y$ a $(\Sigma,\Tau)$-measurable injection.   Then
$Z=f[X]\in\Tau$ and
$f$ is an isomorphism between $(X,\Sigma)$ and $(Z,\Tau_Z)$, where
$\Tau_Z$ is the subspace $\sigma$-algebra.

\proof{{\bf (a)} Let $\frak T$ be a Polish topology on $X$ such that
$\Sigma$ is the algebra of Borel sets.   Then $\frak T$ has a countable
base $\Cal U$, which generates $\Sigma$ (4A3Da/4A3E).

\medskip

{\bf (b)} For each $i\in I$ let $\frak T_i$ be a Polish topology on
$X_i$ such that $\Sigma_i$ is the algebra of $\frak T_i$-Borel sets.
Then $X=\prod_{i\in I}\frak T_i$, with the product topology
$\frak T$, is Polish (4A2Qc).   By 4A3Dc/4A3E,
$\Sigma=\Tensorhat_{i\in I}\Sigma_i$ is just
the Borel $\sigma$-algebra of $X$, so $(X,\Sigma)$ is a standard Borel space.

\medskip

{\bf (c)} Let $\frak T$, $\frak S$ be Polish topologies on $X$, $Y$
respectively for which $\Sigma$ and $\Tau$ are the Borel
$\sigma$-algebras.   Then $f$ is Borel measurable.

\medskip

\quad{\bf (i)} By 423Eb and 423Gb, $f[E]$ and $f[X\setminus E]$ are
analytic subsets of $Y$.   But they are complementary, so they are Borel
sets, by 423Fa.

\medskip

\quad{\bf (ii)} $f^{-1}[F]\in\Sigma$ for every $F\in\Tau$, just
because $f$ is measurable.   On the other hand, if $F\subseteq Y$ and
$E=f^{-1}[F]\in\Sigma$, then $F=f[E]\in\Tau$ by (i).

\medskip

\quad{\bf (iii)} follows at once.

\medskip

{\bf (d)} Give $X$ and $Y$ Polish topologies for which $\Sigma$, $\Tau$
are the Borel $\sigma$-algebras.   By 423Ib, $f[E]\in\Tau$ for every
$E\in\Sigma$;
in particular, $Z=f[X]$ belongs to $\Tau$.   Also $f^{-1}[F]\in\Sigma$
for every $F\in\Tau_Z$, so $f$ is an isomorphism between $(X,\Sigma)$
and $(Z,\Tau_Z)$.
}%end of proof of 424B

\leader{424C}{Theorem} Let $(X,\Sigma)$ be a standard Borel space.

(a) If $X$ is countable then $\Sigma=\Cal PX$.

(b) If $X$ is uncountable then $(X,\Sigma)$ is isomorphic to
$(\NN,\Cal B(\NN))$, where $\Cal B(\NN)$ is the algebra of Borel subsets of
$\NN$.

\proof{ Let $\frak T$ be a Polish topology on $X$ such that $\Sigma$ is
its Borel $\sigma$-algebra.

\medskip

{\bf (a)} Every singleton subset of $X$ is closed, so must belong to
$\Sigma$.   If $X$ is countable, every subset of $X$ is a countable
union of singletons, so belongs to $\Sigma$.

\medskip

{\bf (b)} ({\smc Rao \& Srivastava 94}) The strategy of the proof is to
find Borel sets $Z\subseteq
X$, $W\subseteq\NN$ such that $(Z,\Sigma_Z)\cong(\NN,\Cal B(\NN))$ and
$(W,\Cal B(W))\cong(X,\Sigma)$ (writing
$\Sigma_Z$, $\Cal B(W)$ for the subspace $\sigma$-algebras), and use a
form of the Schr\"oder-Bernstein theorem.

\medskip

\quad{\bf (i)} By 423K, $X$ has a subset $Z$ homeomorphic to $\NN$;  let
$h:\NN\to Z$ be a homeomorphism.   By 424Bd, $h$ is an isomorphism
between $(\NN,\Cal B(\NN))$ and $(Z,\Sigma_Z)$.

\medskip

\quad{\bf (ii)} Let $\sequencen{U_n}$ run over a base for the topology
of $X$.   Define $g:X\to\{0,1\}^{\Bbb N}\subseteq\NN$ by setting
$g(x)=\sequencen{\chi U_n(x)}$ for every $x\in\Bbb N$.   Then $g$ is
injective, because $X$ is Hausdorff.   Also $g$ is Borel measurable,
by 4A3D(c-ii).   By 424Bd, $g$ is an isomorphism between $(X,\Sigma)$ and
$(W,\Cal B(W))$, where $W=g[X]$ belongs to $\Cal B(\NN)$.

\medskip

\quad{\bf (iii)} We have $Z\in\Sigma$,
$W\in\Cal B(\NN)$ such that $(Z,\Sigma_Z)\cong(\NN,\Cal B(\NN))$ and
$(W,\Cal B(W))\cong(X,\Sigma)$.   By 344D,
$(X,\Sigma)\cong(\NN,\Cal B(\NN))$, as claimed.
}%end of proof of 424C

\leader{424D}{Corollary} (a) If $(X,\Sigma)$ and $(Y,\Tau)$ are standard
Borel spaces and $\#(X)=\#(Y)$, then $(X,\Sigma)$ and $(Y,\Tau)$ are
isomorphic.

(b) If $(X,\Sigma)$ is an uncountable standard Borel space then
$\#(X)=\#(\Sigma)=\frak c$.

\proof{ These follow immediately from 424C, if we recall that
$\#(\Cal B(\NN))=\frak c$ (4A3Fb).
}%end of proof of 424D

\leader{424E}{Proposition} Let $X$ be a set and $\Sigma$ a
$\sigma$-algebra of subsets of $X$;  suppose that $(X,\Sigma)$ is
countably separated in the sense that there is a countable set
$\Cal E\subseteq\Sigma$ separating the points of $X$.   If
$A\subseteq X$ is
such that $(A,\Sigma_A)$ is a standard Borel space, where $\Sigma_A$ is
the subspace $\sigma$-algebra, then $A\in\Sigma$.

\proof{ Give $A$ a Polish topology $\frak T$ such that $\Sigma_A$ is the
Borel $\sigma$-algebra of $A$, and let $\frak S$ be the topology on $X$
generated
by $\Cal E\cup\{X\setminus E:E\in\Cal E\}$.   Then $\frak S$ is
second-countable (4A2Oa), so has a countable network (4A2Oc),
and is Hausdorff because $\Cal E$ separates the points of $X$.   The
identity map from $A$ to $X$ is Borel measurable for $\frak T$ and
$\frak S$, so
423Ib tells us that $A$ is $\frak S$-Borel;  but of course the
$\frak S$-Borel $\sigma$-algebra is just the $\sigma$-algebra generated
by $\Cal E$ (4A3Da), so is included in $\Sigma$.
}%end of proof of 424E

\leader{424F}{Corollary} Let $X$ be a Polish space and $A\subseteq X$
any set which is not Borel.   Let $\Cal B(A)$ be the Borel
$\sigma$-algebra of
$A$.   Then $(A,\Cal B(A))$ is not a standard Borel space.

\leader{424G}{Proposition} Let $(X,\Sigma)$ be a standard Borel space.
Then $(E,\Sigma_E)$ is a standard Borel space for every
$E\in\Sigma$\cmmnt{, writing $\Sigma_E$ for the subspace
$\sigma$-algebra}.

\proof{ Let $\frak T$ be a Polish topology on $X$ for which $\Sigma$ is
the Borel $\sigma$-algebra.   Then there is a Polish topology
$\frak T'\supseteq\frak T$ for which $E$ is closed (4A3I), therefore
itself Polish in the subspace topology $\frak T'_E$ (4A2Qd).   But
$\frak T'$ and $\frak T$ have the same Borel sets (423Fb), so $\Sigma_E$
is just the Borel $\sigma$-algebra of $E$ for $\frak T'_E$, and
$(E,\Sigma_E)$ is a standard Borel space.
}%end of proof of 424G

\leader{*424H}{}\cmmnt{ For the full strength of a theorem in \S448
%448P
we need a remarkable result concerning group actions on Polish spaces.

\medskip

\noindent}{\bf Theorem}\cmmnt{ ({\smc Becker \& Kechris 96})} Let $G$
be a Polish group, $(X,\frak T)$ a Polish space and $\action$ a Borel
measurable action of $G$ on $X$.   Then there is a Polish topology
$\frak T'$ on $X$, yielding the same Borel sets as $\frak T$, such that
the action is continuous for $\frak T'$ and the given topology of $G$.

\proof{{\bf (a)} Fix on a right-translation-invariant metric $\rho$ on
$G$ defining the topology of $G$ (4A5Q), and let $D$ be a countable
dense subset of $G$;  write $e$ for the identity of $G$.   Let
$Z$ be the set of $1$-Lipschitz functions from $G$ to $[0,1]$, that is,
functions $f:G\to[0,1]$ such that $|f(g)-f(h)|\le\rho(g,h)$ for all $g$,
$h\in G$.   Then $Z$, with the topology of pointwise convergence
inherited from the
product topology of $[0,1]^G$, is a compact metrizable space.   \Prf\ It
is a closed subset of $[0,1]^G$, so is a compact Hausdorff space.
Writing $q(f)=f\restr D$ for $f\in Z$, $q:Z\to[0,1]^D$ is injective,
because $D$ is dense and every member of $Z$ is continuous;   but this
means that $Z$ is homeomorphic to $q[Z]$, which is metrizable, by
4A2Pc.\ \Qed

We see also that the Borel $\sigma$-algebra of $Z$ is the
$\sigma$-algebra
generated by sets of the form $W_{g\alpha}=\{f:f(g)<\alpha\}$ where
$g\in G$ and $\alpha\in[0,1]$.   \Prf\ This $\sigma$-algebra contains
every set of the form $\{f:f\in Z,\,\alpha<f(g)<\beta\}$, where $g\in G$
and $\alpha$, $\beta\in\Bbb R$;  since these sets
generate the topology of $Z$, the $\sigma$-algebra they generate is the
Borel $\sigma$-algebra of $Z$, by 4A3Da.\ \Qed

\medskip

{\bf (b)} There is a continuous action of $G$ on $Z$ defined by setting

\Centerline{$(g\action_rf)(h)=f(hg)$}

\noindent for $f\in Z$ and $g$, $h\in G$.   \Prf\
(i) If $f\in Z$ and $g$, $h_1$, $h_2\in G$, then

\Centerline{$|(g\action_r f)(h_1)-(g\action_r f)(h_2)|
=|f(h_1g)-f(h_2g)|
\le\rho(h_1g,h_2g)=\rho(h_1,h_2)$}

\noindent because $\rho$ is right-translation-invariant.   So
$g\action_rf\in Z$ for every $f\in Z$, $g\in G$.   (ii)
As in 4A5Cc-4A5Cd, $\action_r$ is an action of $G$ on $Z$.
(iii) Suppose that
$g_0\in G$, $f_0\in Z$, $h\in G$ and $\epsilon>0$.   Set

\Centerline{$V=\{g:g\in G,\,\rho(hg,hg_0)<\bover12\epsilon\}$,}

\Centerline{$W=\{f:f\in Z,\,|f(hg_0)-f_0(hg_0)|<\bover12\epsilon\}$.}

\noindent Then $V$ is an open set in $G$ containing $g_0$ (because
$g\mapsto\rho(hg,hg_0)$ is continuous) and $W$ is an open set in $Z$
containing $f_0$.   If $g\in V$ and $f\in W$,

$$\eqalign{|(g\action_r f)(h)-(g_0\action_r f_0)(h)|
&=|f(hg)-f_0(hg_0)|\cr
&\le|f(hg)-f(hg_0)|+|f(hg_0)-f_0(hg_0)|\cr
&\le\rho(hg,hg_0)+\Bover12\epsilon
\le\epsilon.\cr}$$

\noindent As $f_0$, $g_0$, $\epsilon$ are arbitrary, the map
$(g,f)\mapsto (g\action_r f)(h)$ is continuous;  as $h$ is arbitrary, the
map $(g,f)\mapsto g\action_r f$ is continuous.\ \Qed

\medskip

{\bf (c)} Let $\Cal B(X)$ be the Borel $\sigma$-algebra of $X$.   For
$x\in X$ and $B\in\Cal B(X)$, set

\Centerline{$P_B(x)=\{g:g\in G,\,g\action x\in B\}$,}

\Centerline{$Q_B(x)=\bigcup\{V:V\subseteq G$ is open,
$V\setminus P_B(x)$ is meager$\}$,}

\Centerline{$f_B(x)(g)
=\inf(\{1\}\cup\{\rho(g,h):h\in G\setminus Q_B(x)\})$}

\noindent for $g\in G$.   It is easy to check that

\Centerline{$f_B(x)(g')\le\rho(g,g')+f_B(x)(g')$}

\noindent for all $g$, $g'\in G$, so that every $f_B(x)$ belongs to $Z$.

Every $P_B(x)$ is a Borel set, because $\action$ is Borel measurable, so
has the Baire property in $X$ (4A3S(b-i)).
Because $X$ is a Baire space (4A2Ma), $Q_B(x)\subseteq\overline{P_B(x)}$
(4A3Sa).

Let $\Cal V$ be a countable base for the topology of $G$ containing $G$.

\medskip

{\bf (d)} For each $B\in\Cal B(X)$, the map $f_B:X\to Z$ is Borel
measurable.   \Prf\ Because the Borel $\sigma$-algebra of $Z$ is
generated by the
sets $W_{g\alpha}$ of (a) above, it is enough to show that

\Centerline{$\{x:f_B(x)\in W_{g\alpha}\}=\{x:f_B(x)(g)<\alpha\}$}

\noindent always belongs to $\Cal B(X)$, because $\{W:f_B^{-1}[W]\in\Cal
B\}$ is surely a $\sigma$-algebra of subsets of $Z$.   But if $\alpha>1$
this set is $X$, while if $\alpha\le 1$ it is

$$\eqalignno{\{x:\text{ there is some }&h\in X\setminus Q_B(x)
  \text{ such that }\rho(g,h)<\alpha\}\cr
&=\{x:\text{ there is some }V\in\Cal V
  \text{ such that }V\setminus Q_B(x)\ne\emptyset\cr
&\qquad\qquad\qquad\text{ and }\rho(g,h)<\alpha\text{ for every }h\in
V\}\cr
&=\bigcup_{V\in\Cal V'}\{x:V\not\subseteq Q_B(x)\}\cr
\noalign{\noindent (where $\Cal V'=\{V:V\in\Cal V,\,\rho(g,h)<\alpha$
for every $h\in V\}$)}
&=\bigcup_{V\in\Cal V'}\{x:V\setminus P_B(x)\text{ is not
meager}\}.\cr}$$

\noindent But for any fixed $V\in\Cal V$,

\Centerline{$\{x:V\setminus P_B(x)$ is not meager$\}
=X\setminus\{x:W[\{x\}]$ is meager$\}$}

\noindent where

\Centerline{$W=\{(y,g):y\in X,\,g\in V,\,g\action y\in X\setminus B\}$}

\noindent is a Borel subset of $X\times G$, because $\action$ is
supposed to be Borel measurable;  and therefore
$W\in\Cal B(X)\tensorhat\Cal B(G)$, writing $\Cal B(G)$ for the Borel
$\sigma$-algebra of $G$ (4A3Ga).
Now $\Cal B(G)\subseteq\widehat{\Cal B}(G)$, the Baire-property algebra
of $G$ (4A3S(b-i) again), so $W\in\Cal B(X)\tensorhat\widehat{\Cal B}(G)$.
By 4A3Sc, the quotient
algebra $\widehat{\Cal B}/\Cal M$ has a countable order-dense set,
where $\Cal M$ is the $\sigma$-ideal of meager sets, so
4A3Ta tells us that $\{x:W[\{x\}]$ is meager$\}$ is a Borel subset of
$X$.   Accordingly
$\{x:V\setminus P_B(x)$ is not meager$\}$ is Borel for every $V$, and
$\{x:f_B(x)(g)<\alpha\}$ is Borel.\ \Qed

\medskip

{\bf (e)} If $g\in G$, $B\in\Cal B(X)$ and $x\in X$, then
$g\action_rf_B(x)=f_B(g\action x)$.   \Prf\

\Centerline{$P_B(g\action x)=\{h:h\action(g\action x)\in B\}
=\{h:hg\in P_B(x)\}
=P_B(x)g^{-1}$.}

\noindent Because the map $h\mapsto hg^{-1}:G\to G$ is a homeomorphism,
$Q_B(g\action x)=Q_B(x)g^{-1}$;  because it is an isometry,

$$\eqalign{f_B(g\action x)(h)
&=\min(1,\rho(h,X\setminus Q_B(g\action x)))
=\min(1,\rho(h,X\setminus Q_B(x)g^{-1}))\cr
&=\min(1,\rho(hg,X\setminus Q_B(x)))
=f_B(x)(hg)
=(g\action_rf_B(x))(h)\cr}$$

\noindent for every $h\in G$.\ \Qed

\medskip

{\bf (f)} Let $\sequence{m}{B_{0m}}$ run over a base for
$\frak T$ containing $X$.   We can now find a countable set
$\Cal E\subseteq\Cal B(X)$ such
that (i) the topology $\frak T^*$ generated by $\Cal E$ is a Polish
topology finer than $\frak T$ (ii) $f_E$ is $\frak T^*$-continuous for
every $E\in\Cal E$ (iii) $X\in\Cal E$.   \Prf\ Let $\Cal W$ be a
countable base for the topology of $Z$.   Enumerate
$\Bbb N\times\Bbb N\times\Cal W$ as $\sequencen{(k_n,m_n,W_n)}$ in such
a way that $k_n\le n$ for every $n$.   Having chosen Borel sets
$B_{ij}\subseteq X$
for $i\le n$, $j\in\Bbb N$ in such a way that the topology $\frak S_n$
generated by $\{B_{ij}:i\le n,\,j\in\Bbb N\}$ is a Polish topology finer
than $\frak T$, consider the set

\Centerline{$C_n=\{x:f_{B_{k_n,m_n}}(x)\in W_n\}$.}

\noindent This is $\frak T$-Borel, therefore $\frak S_n$-Borel, so by
4A3H there is a Polish topology $\frak S_{n+1}\supseteq\frak S_n$ such
that $C_n\in\frak S_{n+1}$.   Let $\sequence{m}{B_{n+1,m}}$ run over a
base for $\frak S_{n+1}$;  by 423Fb, every $B_{n+1,m}$ belongs to
$\Cal B(X)$.   Continue.

Let $\frak T^*$ be the topology generated by
$\Cal E=\{B_{ij}:i,\,j\in\Bbb N\}$.
By 4A2Qf, $\frak T^*$ is Polish, because it is the topology
generated by $\bigcup_{n\in\Bbb N}\frak S_n$.   If $W\in\Cal W$ and
$E\in\Cal E$ there are $i$, $j\in\Bbb N$ such that $B=B_{ij}$ and an
$n\in\Bbb N$ such that $(i,j,W)=(k_n,m_n,W_n)$;  now

\Centerline{$f_E^{-1}[W]=C_n\in\frak S_{n+1}\subseteq\frak T^*$.}

\noindent As $W$ is arbitrary, $f_E$ is $\frak T^*$-continuous.   Also

\Centerline{$X\in\{B_{0m}:m\in\Bbb N\}\subseteq\Cal E$,}

\noindent as required.\ \Qed

\medskip

{\bf (g)} Define $\theta:X\to Z^{\Cal E}$ by setting
$\theta(x)(E)=f_E(x)$ for $x\in X$ and $E\in\Cal E$.   Then $\theta$ is
injective.   \Prf\ Suppose that $x$, $y\in X$ and that $x\ne y$.   For
every $g\in G$,

\Centerline{$g^{-1}\action(g\action x)=x\ne y=g^{-1}\action(g\action
y)$,}

\noindent so $g\action x\ne g\action y$ and there is some $m\in\Bbb N$
such that $g\action x\in B_{0m}$ while $g\action y\notin B_{0m}$, that
is, $g\in P_{B_{0m}}(x)\setminus P_{B_{0m}}(y)$.   Thus
$G=\bigcup_{m\in\Bbb N}P_{B_{0m}}(x)\setminus P_{B_{0m}}(y)$;  by
Baire's theorem, there is some $m\in\Bbb N$ such that
$P_{B_{0m}}(x)\setminus P_{B_{0m}}(y)$ is non-meager.    Because
$Q_{B_{0m}}(x)\symmdiff P_{B_{0m}}(x)$ and
$Q_{B_{0m}}(y)\symmdiff P_{B_{0m}}(y)$ are both meager,

\Centerline{$\{g:f_{B_{0m}}(x)(g)>0\}=Q_{B_{0m}}(x)\ne Q_{B_{0m}}(y)
=\{g:f_{B_{0m}}(y)(g)>0\}$,}

\noindent and

\Centerline{$\theta(x)(B_{0m})=f_{B_{0m}}(x)\ne f_{B_{0m}}(y)
=\theta(y)(B_{0m})$.}

\noindent So $\theta(x)\ne\theta(y)$.\ \QeD\  Because $f_E$ is $\frak
T^*$-continuous for every $E\in\Cal E$, $\theta$ is
$\frak T^*$-continuous.

\medskip

{\bf (h)} Let $\frak T'$ be the topology on $X$ induced by $\theta$;
that is, the topology which renders $\theta$ a homeomorphism between $X$
and $\theta[X]$.   Because $\theta[X]\subseteq Z^{\Cal E}$ is separable
and metrizable, $\frak T'$ is separable and metrizable.   Because
$\theta$ is
$\frak T^*$-continuous, $\frak T'\subseteq\frak T^*$.

\medskip

{\bf (i)} The action of $G$ on $X$ is continuous for the given topology
$\frak S$ on $G$ and $\frak T'$ on $X$.   \Prf\ For any $E\in\Cal E$,

\Centerline{$(g,x)\mapsto\theta(g\action x)(E)
=f_E(g\action x)=g\action_rf_E(x)$}

\noindent ((e) above) is $\frak S\times\frak T'$-continuous because the
action of $G$ on $Z$ is continuous ((b) above) and $f_E:X\to Z$ is
$\frak T'$-continuous (by the definition of $\frak T'$).   But this
means that $(g,x)\mapsto\theta(g\action x)$ is
$\frak S\times\frak T'$-continuous, so that $(g,x)\mapsto g\action x$ is
$(\frak S\times\frak T',\frak T')$-continuous.\ \Qed

\medskip

{\bf (j)} Let $\sigma$ be a complete metric on $G$ defining the topology
$\frak S$, and $\tau$ a complete metric on $X$ defining the topology
$\frak T^*$.   (We do not need to relate $\sigma$ to $\rho$ in any way
beyond the fact that they both give rise to the same topology $\frak
S$.)   For $E\in\Cal E$, $V\in\Cal V$ and $n\in\Bbb N$ let $S_{EVn}$ be
the set of those $\phi\in Z^{\Cal E}$ such that {\it either}
$\phi(E)(g)=0$ for every $g\in V$ {\it or} there is an $F\in\Cal E$ such
that $F\subseteq E$, $\diam_{\tau}(F)\le 2^{-n}$ and $\phi(F)(g)>0$ for
some $g\in V$.   Then $S_{EVn}$ is the union of a closed set and an open
set, so is a G$_{\delta}$ set in $Z^{\Cal E}$ (4A2C(a-i)).
Consequently

\Centerline{$Y=\{\phi:\phi\in Z^{\Cal E},\,\phi(X)=\chi G\}
\cap\overline{\theta[X]}\cap\bigcap_{E\in\Cal E,V\in\Cal V,n\in\Bbb
N}S_{EVn}$}

\noindent is a G$_{\delta}$ subset of $Z^{\Cal E}$, being the
intersection of countably many G$_{\delta}$ sets.

\medskip

{\bf (k)} $\theta[X]\subseteq Y$.   \Prf\ Let $x\in X$.   (i) $P_X(x)=G$
so $Q_X(x)=G$ and $\theta(x)(X)=f_X(x)=\chi G$.   (ii) Of course
$\theta(x)\in\overline{\theta[X]}$.   (iii) Suppose that $E\in\Cal E$,
$V\in\Cal V$ and $n\in\Bbb N$.   If $Q_E(x)\cap V=\emptyset$ then
$\theta(x)(E)(g)=f_E(x)(g)=0$ for every $g\in V$, and
$\theta(x)(E)\in S_{EVn}$.   Otherwise, $V\cap P_E(x)$ is non-meager.
But

\Centerline{$E
=\bigcup\{F:F\in\Cal E,\,F\subseteq E,\,\diam_{\tau}(F)\le 2^{-n}\}$,}

\noindent so

\Centerline{$P_E(x)=\bigcup\{P_F(x):F\in\Cal E,\,F\subseteq E,\,
\diam_{\tau}(F)\le 2^{-n}\}$;}

\noindent because $\Cal E$ is countable, this is a countable union and
there is an $F\in\Cal E$ such that $F\subseteq E$,
$\diam_{\tau}(F)\le 2^{-n}$ and $P_F(x)\cap V$ is non-meager.   In this
case $Q_F(x)\cap V$
is non-empty and $\theta(x)(F)=f_F(x)$ is non-zero at some point of
$V$;  thus again $\theta(x)\in S_{EVn}$.   As $E$, $V$ and $n$ are
arbitrary, we have the result.\ \Qed

\medskip

{\bf (l)} (The magic bit.)   $Y\subseteq\theta[X]$.   \Prf\ Take any
$\phi\in Y$.   Choose $\sequencen{E_n}$ in $\Cal E$, $\sequencen{V_n}$
in $\Cal V$ and $\sequencen{\tilde g_n}$ in $G$ as follows.   $E_0=X$
and $V_0=G$.   Given that $\phi(E_n)$ is non-zero at some point of
$V_n$, then, because $\phi\in S_{E_nV_nn}$, there is an
$E_{n+1}\in\Cal E$ such that $E_{n+1}\subseteq E_n$,
$\diam_{\tau}(E_{n+1})\le 2^{-n}$
and $\phi(E_{n+1})$ is non-zero at some point of $V_n$;  say
$\tilde g_n\in V_n$ is such that $\phi(E_{n+1})(\tilde g_n)>0$.
Now we can
find a $V_{n+1}\in\Cal V$ such that $\tilde g_n\in V_{n+1}\subseteq V_n$
and $\diam_{\sigma}(V_{n+1})\le 2^{-n}$.   Continue.

We are supposing also that $\phi\in\overline{\theta[X]}$, so we have
a sequence $\sequence{i}{x_i}$ in $X$ such that
$\sequence{i}{\theta(x_i)}\to\phi$, that is,
$\sequence{i}{f_E(x_i)}\to\phi(E)$ for every $E\in\Cal E$.   In
particular,

\Centerline{$\lim_{i\to\infty}f_{E_{n+1}}(x_i)(\tilde
g_n)=\phi(E_{n+1})(\tilde g_n)>0$}

\noindent for every $n$.    Let
$\sequence{n}{i_n}$ be a strictly increasing sequence in $\Bbb N$ such
that $f_{E_{n+1}}(x_{i_n})(\tilde g_n)>0$ for every $n$.   Then

\Centerline{$\tilde g_n\in V_{n+1}\cap Q_{E_{n+1}}(x_{i_n})\subseteq
V_{n+1}\cap\overline{P_{E_{n+1}}(x_{i_n})}$;}

\noindent there is therefore some $g_n\in V_{n+1}\cap
P_{E_{n+1}}(x_{i_n})$, so that $g_n\action x_{i_n}\in E_{n+1}$.

$\sequencen{V_n}$ is a non-increasing sequence of sets with
$\sigma$-diameters
converging to $0$, so $\sequencen{g_n}$ is a Cauchy sequence for the
complete metric $\sigma$.   Similarly, $\sequencen{g_n\action x_{i_n}}$
is a Cauchy sequence for the complete metric $\tau$, because
$\diam_{\tau}(E_{n+1})\le 2^{-n}$.   We therefore have $g\in G$,
$y\in X$ such that $\sequencen{g_n}\to g$ for $\frak S$ and
$\sequencen{g_n\action x_{i_n}}\to y$ for $\frak T^*$.   In this case,
$\sequencen{g_n\action x_{i_n}}\to y$ for the coarser topology
$\frak T'$, while
$\sequencen{g_n^{-1}}\to g^{-1}$ for $\frak S$.   Because the action is
$(\frak S\times\frak T',\frak T')$-continuous,

\Centerline{$\sequencen{x_{i_n}}
=\sequencen{g_n^{-1}\action(g_n\action x_{i_n})}\to g^{-1}\action y$}

\noindent for $\frak T'$.   But $\theta$ is continuous for $\frak T'$,
by the definition of $\frak T'$, so

\Centerline{$\theta(g^{-1}\action y)
=\lim_{n\to\infty}\theta(x_{i_n})=\phi$,}

\noindent and $\phi\in\theta[X]$.   As $\phi$ is arbitrary,
$Y\subseteq\theta[X]$.\ \Qed

\medskip

{\bf (m)} Thus $\theta[X]=Y$ is a G$_{\delta}$ set in the compact metric
space $Z^{\Cal E}$, and is a Polish space in its induced topology
(4A2Qd).   But this means that $(X,\frak T')$, which is homeomorphic to
$\theta[X]$, is also Polish.

\medskip

{\bf (n)} I have still to check that $\frak T'$ has the same Borel sets
as $\frak T$.   But $\frak T$, $\frak T^*$ and $\frak T'$ are all Polish
topologies and $\frak T^*$ is finer than both the other two.   By 423Fb,
$\frak T^*$ has the same Borel sets as either of the others.

This completes the proof.
}%end of proof of 424H

\exercises{\leader{424X}{Basic exercises $\pmb{>}$(a)}
%\spheader 424Xa
Let $\family{i}{I}{(X_i,\Sigma_i)}$ be a countable family of standard
Borel spaces, and $(X,\Sigma)$ their direct sum, that is,
$X=\{(x,i):i\in I,\,x\in X_i\}$, $\Sigma=\{E:E\subseteq X,\,\{x:(x,i)\in
E\}\in\Sigma_i$ for every $i\}$.   Show that $(X,\Sigma)$ is a standard
Borel space.
%424B

\sqheader 424Xb Let $(X,\Sigma)$ be a standard Borel space and $\Tau$ a
countably generated $\sigma$-subalgebra of $\Sigma$.   Show that there
is an analytic Hausdorff space $Z$ such that $\Tau$ is isomorphic to the
Borel $\sigma$-algebra of $Z$.   \Hint{by 4A3I, we can suppose that $X$
is a Polish space and $\Tau$ is generated by a sequence of
open-and-closed
sets, corresponding to a continuous function from $X$ to
$\{0,1\}^{\Bbb N}$.}

\sqheader 424Xc Let $(X,\Sigma)$ be a standard Borel space and $\Tau_1$,
$\Tau_2$ two countably generated $\sigma$-subalgebras of $\Sigma$ which
separate the same points, in the sense that if $x$, $y\in X$ then there
is an $E\in\Tau_1$ such that $x\in E$ and $y\in X\setminus E$ iff there is
an $E'\in\Tau_2$ such that $x\in E'$ and $y\in X\setminus E'$.   Show that
$\Tau_1=\Tau_2$.   \Hint{424Xb, 423Fb.}   In particular, if $\Tau_1$
separates the points of $X$ then $\Tau_1=\Sigma$.
%424Xb

\spheader 424Xd Let $\frak A$ be a Dedekind $\sigma$-complete Boolean
algebra.   Show that $\frak A$ is isomorphic to the Borel
$\sigma$-algebra of an
analytic Hausdorff space iff it is isomorphic to a countably generated
$\sigma$-subalgebra of the Borel $\sigma$-algebra of $[0,1]$.
%424Xb

\sqheader 424Xe Let $U$ be a separable Banach space.   Show that its
Borel $\sigma$-algebra is generated, as $\sigma$-algebra, by the sets of
the form
$\{u:h(u)\le\alpha\}$ as $h$ runs over the dual $U^*$ and $\alpha$ runs
over $\Bbb R$.

\sqheader 424Xf Let $X$ be a compact metrizable space.   Show that the
Borel $\sigma$-algebra of $C(X)$ (the Banach space of continuous
real-valued
functions on $X$) is generated, as $\sigma$-algebra,  by the sets
$\{u:u\in C(X),\,u(x)\ge\alpha\}$ as $x$ runs over $X$ and $\alpha$ runs
over $\Bbb R$.
%424Xe

\spheader 424Xg Let $(X,\Sigma)$ be a standard Borel space, $Y$ any set,
and $\Tau$ a $\sigma$-algebra of subsets of $Y$.
Write $\Tau^*$ for the $\sigma$-algebra of subsets of $Y$ generated by
$\Cal S(\Tau)$, where $\Cal S$ is Souslin's operation.
Let $W\in\Cal S(\Sigma\tensorhat\Tau)$.
Show that $W[X]\in\Cal S(\Tau)$ and that there is a
$(\Tau^*,\Sigma)$-measurable function $f:W[X]\to X$ such that
$(f(y),y)\in W$ for every $y\in W[X]$.
\Hint{423O.}

\spheader 424Xh Let $(X,\Sigma)$ be a standard Borel space, $Y$ any set,
and $\Tau$ a countably generated $\sigma$-algebra of subsets of $Y$.
Let $f:X\to Y$ be a $(\Sigma,\Tau)$-measurable function, and write
$\Tau^*$ for the $\sigma$-algebra of subsets of $Y$ generated by
$\Cal S(\Tau)$, where $\Cal S$ is Souslin's
operation.    Show that there is a
$(\Tau^*,\Sigma)$-measurable function $g:f[X]\to X$ such that $gf$ is
the identity on $X$.   \Hint{start with the case in which $\Tau$
separates the points of $Y$, so that the graph of $f$ belongs to
$\Sigma\tensorhat\Tau$.}
%424Xg

\sqheader 424Xi Show that 424Xc and 424Xh are both false if we omit the
phrase `countably generated' from the hypotheses.   \Hint{consider (i) the
countable-cocountable algebra of $\Bbb R$ (ii) the split interval.}
%424Xc %424Xh

\spheader 424Xj\dvAnew{2007} Let $(X,\Sigma,\mu)$ be a
$\sigma$-finite measure space in which $\Sigma$
is countably generated.   Let $\Cal A$ be the set of atoms $A$ of
the Boolean algebra $\Sigma$
such that $\mu A>0$, and set $H=X\setminus\bigcup\Cal A$.   Show that the
subspace measure on $H$ is atomless.
%424?

\leader{424Y}{Further exercises (a)}
%\spheader 424Ya
Let $(X,\frak T)$ be a Polish space and $\Cal C$ the family of closed
subsets of $X$.   Let $\Sigma$ be the $\sigma$-algebra of subsets of
$\Cal C$ generated by the sets
$\Cal E_H=\{F:F\in\Cal C,\,F\cap H\ne\emptyset\}$ as $H$ runs over the
open subsets of $X$.   (i) Show
that $(\Cal C,\Sigma)$ is a standard Borel space.   \Hint{take a
complete metric $\rho$ defining the topology of $X$.   Set
$S=\bigcup_{n\in\Bbb N}\BbbN^n$ and choose a family
$\family{\sigma}{S}{U_{\sigma}}$ of
open sets in $X$ such that $U_{\emptyset}=X$,
$\diam U_{\sigma}\le 2^{-n}$ whenever $\#(\sigma)=n+1$,
$U_{\sigma}=\bigcup_{i\in\Bbb N}U_{\sigma^{\smallfrown}\fraction{i}}$
for every $\sigma$,
$\overline{U}_{\sigma^{\smallfrown}\fraction{i}}\subseteq U_{\sigma}$ for
every $\sigma$, $i$.   Define $f:\Cal C\to\{0,1\}^S$ by setting
$f(F)(\sigma)=1$ if $F\cap U_{\sigma}\ne\emptyset$, $0$ otherwise.
Show that $Z=f[\Cal C]$ is a Borel set and that $f$ is an isomorphism
between $\Sigma$ and the Borel $\sigma$-algebra of $Z$.}   (This is the
{\bf Effros Borel structure} on $\Cal C$.)   (ii) Show that
$[X]^n\in\Sigma$ for every $n\in\Bbb N$.   (iii)\dvAnew{2014} Show that
$\Sigma$ is the Borel $\sigma$-algebra of $\Cal C$ when $\Cal C$ is given
its Fell topology.
%mt42bits

\spheader 424Yb Let $(X,\Sigma)$ be a standard Borel space.   Let
$\Bbb T$ be the family of Polish topologies on $X$ for which $\Sigma$ is
the Borel $\sigma$-algebra.   Show that any sequence in $\Bbb T$ has an
upper bound in $\Bbb T$, and that any sequence with a lower bound has a
least upper bound.

\spheader 424Yc Let $(X,\Sigma)$ be a standard Borel space.   Say that
$C\subseteq X$ is {\bf coanalytic} if its complement belongs to
$\Cal S(\Sigma)$.   Show that for any such $C$ the partially ordered set
$\Sigma\cap\Cal PC$ has cofinality $1$ if $C\in\Sigma$ and cofinality
$\omega_1$ otherwise.  \Hint{423R.}
%423R

\spheader 424Yd Let $I^{\|}$ be the split interval.   Show that there is
a $\sigma$-algebra $\Sigma$ of subsets of $I^{\|}$ such that
$(I^{\|},\Sigma)$ is a standard Borel space and
$\{(x,y):x$, $y\in I^{\|}$, $x\le y\}\in\Sigma\tensorhat\Sigma$.
%+

\spheader 424Ye Let $(X,\Sigma)$ be a standard Borel space.    Show that
if $X$ is uncountable, $\Sigma$ has a countably generated
$\sigma$-subalgebra not isomorphic either to $\Sigma$ or to $\Cal PI$
for any set $I$.
%mt42bits

\spheader 424Yf Let $(X,\Sigma,\mu)$ be a $\sigma$-finite countably
separated perfect measure space (definition:  342K/451Ad).
Show that there is a standard Borel space $(Y,\Tau)$ such that
$Y\in\Sigma$, $\Tau\subseteq\Sigma$ and $\mu$ is inner regular with respect
to $\Tau$.
}%end of exercises

\endnotes{
\Notesheader{424} In this treatise I have generally indulged my
prejudice in favour of `complete' measures.   Consequently Borel
$\sigma$-algebras, as such, have taken subordinate roles.
But important parts of the theory of Lebesgue measure, and Radon
measures on Polish spaces in general, are associated with the fact that
these are completions of measures defined on standard Borel spaces.
Moreover, such spaces provide a suitable framework for a large part of
probability theory.   Of course they become deficient in contexts where
we need to look at uncountable independent families of random variables,
and there are also difficulties with
$\sigma$-subalgebras, even countably generated ones, since these can
correspond to the Borel algebras of general analytic spaces, which will
not always be standard Borel structures (424F, 423M).   424Xf and 424Ya
suggest the ubiquity of standard Borel structures;  the former shows
that they are not always presented as
countably generated algebras, while the latter is an example in which we
have to make a special construction in order to associate a topology
with the algebra.   The theory is of course dominated by the results of
\S423, especially 423Fb and 423I.

I include 424H in this section because there is no other convenient
place for it, but I have an excuse:  the idea of `Borel measurable
action' can, in this context, be described entirely in terms of
$\sigma$-algebras, since the Borel algebra of $G\times X$ is just the
$\sigma$-algebra product of the Borel algebras of the factors (as in
424Bb).   Of course for the theorem as expressed here we do need to know
that $G$ has a Polish group structure;  but $X$ could be presented just
as a standard Borel space.   The result is a dramatic expression of the
fact that, given a standard Borel space $(X,\Sigma)$, we have a great
deal of freedom in defining a corresponding Polish topology on $X$.
}%end of notes

\frnewpage

\frfilename{mt425.tex}
\versiondate{9.8.13}
\copyrightdate{2009}

\def\chaptername{Descriptive set theory}
\def\sectionname{Realization of automorphisms}

\newsection{*425\dvAnew{2009}}

In \S344 I presented some results on the representation of a countable
semigroup of Boolean homomorphisms in a measure algebra by a semigroup of
functions on the measure space underlying the algebra.   \S424 provides us
with the
tools needed for a remarkable extension, in the case of the Lebesgue
measure algebra, to groups with cardinal $\omega_1$
(Theorem 425D).   The expression of the ideas is made smoother by using the
language of group actions\cmmnt{ (4A5B-4A5C)}.

\leader{425A}{}\dvAnew{2011}
I begin with what amounts to a special case of the main
theorem, with some refinements which will be useful elsewhere.

\medskip

\noindent{\bf Proposition} (a) Let $(X,\Sigma)$ and $(Y,\Tau)$ be
non-empty standard Borel
spaces, and $\Cal I$, $\Cal J\,\,\sigma$-subalgebras of $\Sigma$, $\Tau$
respectively;  write $\frak A=\Sigma/\Cal I$ and $\frak B=\Tau/\Cal J$ for
the quotient algebras.   For $E\in\Sigma$, $F\in\Tau$ write $\Sigma_E$,
$\Tau_F$ for the subspace $\sigma$-algebras on $E$, $F$ respectively.

(a) If $\pi:\frak A\to\frak B$ is a sequentially order-continuous
Boolean
homomorphism, there is a $(\Tau,\Sigma)$-measurable $f:Y\to X$ which
represents $\pi$ in the sense that
$\pi E^{\ssbullet}=f^{-1}[E]^{\ssbullet}$ for every $E\in\Sigma$.

(b) If $\pi:\frak A\to\frak B$ is a Boolean
isomorphism, there are $G\in\Cal I$, $H\in\Cal J$ and a bijection
$h:Y\setminus H\to X\setminus G$ which is an isomorphism between
$(Y\setminus H,\Tau_{Y\setminus H})$ and
$(X\setminus G,\Sigma_{X\setminus G})$, and represents $\pi$ in the sense that
$\pi E^{\ssbullet}=h^{-1}[E\setminus G]^{\ssbullet}$ for every
$E\in\Sigma$.

(c) If $\pi:\frak A\to\frak A$ is a Boolean automorphism, there is a
bijection $h:X\to X$ which is an
automorphism of $(X,\Sigma)$ and represents $\pi$ in the sense of (a).

(d) If $\#(X)=\#(Y)=\frak c$, $\frak A$ and $\frak B$ are ccc, and
$\pi:\frak A\to\frak B$ is a Boolean isomorphism, there is a bijection
$h:Y\to X$ which is an isomorphism between $(Y,\Tau)$ and $(X,\Sigma)$, and
represents $\pi$ in the sense of (a).

\proof{{\bf (a)(i)} If either $\frak A$ or $\frak B$ is $\{0\}$, so
is the other, and we can take $f$ to be a constant function.   So
henceforth let us suppose that $X\notin\Cal I$ and $Y\notin\Cal J$.

\medskip

\quad{\bf (ii)} If $X\subseteq\Bbb N$ and $\Sigma=\Cal PX$, set
$Z=\{n:n\in X$, $\{n\}\notin\Cal I\}$.   For $n\in Z$, set
$a_n=\{n\}^{\ssbullet}$ and $b_n=\pi a_n$, and choose $F_n\in\Tau$ such
that $F_n^{\ssbullet}=b_n$.   Since $\sup_{n\in Z}a_n=1$ in
$\frak A$, $\sup_{n\in Z}b_n=1$ in $\frak B$ and
$Y\setminus\bigcup_{n\in Z}F_n\in\Cal J$.   Define $f:Y\to X$ by saying
that

$$\eqalign{f(y)
&=\min\{n:n\in Z,\,y\in F_n\}\text{ if }y\in\bigcup_{n\in Z}F_n,\cr
&=\min \text{ otherwise}.\cr}$$

\noindent Then $f$ represents $\pi$ in the required sense.

\medskip

\quad{\bf (iii)} If $X=\{0,1\}^{\Bbb N}$ and $\Sigma$ is
its Borel $\sigma$-algebra, then for each $n\in\Bbb N$ set
$E_n=\{x:x(n)=1\}$ and $e_n=E_n^{\ssbullet}$
and choose $F_n\in\Tau$ such that
$F_n^{\ssbullet}=\pi e_n$.
Set $f(y)=\sequencen{\chi F_n(y)}$ for $y\in Y$.
Then $f^{-1}[E_n]^{\ssbullet}=\pi E_n^{\ssbullet}$ for every $n$;  as
$\{E:f^{-1}[E]^{\ssbullet}=\pi E^{\ssbullet}\}$ is a $\sigma$-subalgebra of
$\Sigma$ containing every $E_n$, it is the whole of $\Sigma$, and again $f$
represents $\pi$.

\medskip

\quad{\bf (iv)} By 424C, any standard Borel space is isomorphic to either
that in (iii) or one of those in (ii), so these cases together
are sufficient to prove the general result.

\medskip

{\bf (b)} By (a), we have $f:Y\to X$ and $g:X\to Y$ representing
$\pi$, $\pi^{-1}$ respectively.   Now we see that $gf:Y\to Y$ represents
$\pi\pi^{-1}:\frak B\to\frak B$, that is,
$F\symmdiff(gf)^{-1}[F]\in\Cal J$ for every $F\in\Tau$.   Let
$\sequencen{F_n}$ be a sequence in $\Tau$ which separates the points of
$Y$, and set $H_0=\bigcup_{n\in\Bbb N}F_n\symmdiff(gf)^{-1}[F_n]$;
then $H_0\in\Cal J$ and $g(f(y))=y$ for every $y\in Y\setminus H_0$.

Similarly, there is a $G_0\in\Cal I$ such that $f(g(x))=x$ for every
$x\in X\setminus G_0$.   Set $G=G_0\cup g^{-1}[H_0]$ and
$H=H_0\cup f^{-1}[G_0]$.   Then $g[X\setminus G]\subseteq Y\setminus H$.
\Prf\ If $x\in X\setminus G$ then $g(x)\notin H_0$;  moreover,
$f(g(x))=x\notin G_0$ so $g(x)\notin f^{-1}[G_0]$ and $g(x)\notin H$.\
\QeD\  Similarly, $f(y)\in X\setminus G$ for every $y\in Y\setminus H$.
As $f(g(x))=x$ for $x\in X\setminus G$ and $g(f(y))=y$ for
$y\in Y\setminus H$, $h=f\restr Y\setminus H$ is a bijection with inverse
$g\restr X\setminus G$.   Because $f$ is $(\Tau,\Sigma)$-measurable,
$h$ is
$(\Tau_{Y\setminus H},\Sigma_{X\setminus G})$-measurable;
because $g$ is $(\Sigma,\Tau)$-measurable, $h^{-1}$ is
$(\Sigma_{X\setminus G},\Tau_{Y\setminus H})$-measurable, and $h$ is an
isomorphism between $(Y\setminus H,\Tau_{Y\setminus H})$
and $(X\setminus G,\Sigma_{X\setminus G})$.
Finally, if $E\in\Sigma$,

\Centerline{$\pi(E^{\ssbullet})=\pi((E\setminus G)^{\ssbullet})
=(f^{-1}[E\setminus G])^{\ssbullet}
=(h^{-1}[E\setminus G])^{\ssbullet}$,}

\noindent so $h$ represents $\pi$ in the sense declared.

\medskip

{\bf (c)} We can repeat the proof of (b) with an additional idea.   The
point is that there is an element $E_0$ of $\Cal I$ with maximal
cardinality.   \Prf\ Because $(G,\Sigma_G)$ is a standard Borel space
(424G), $\#(G)$ is either $\frak c$ or countable (424Db), 
for every $G\in\Cal I$.
If $\Cal I$ contains arbitrarily large finite sets, it must contain an
infinite set, because it is a $\sigma$-algebra.   So the supremum
$\sup\{\#(G):G\in\Cal I\}$ is attained.\ \Qed

Now, in (b), take $(Y,\Tau)=(X,\Sigma)$ and $\frak B=\frak A$, and choose
$f$, $H_0$, $g$ and $G_0$ as before;  but this time set
$G=(G_0\cup E_0)\cup g^{-1}[H_0\cup E_0]$ and
$H=(H_0\cup E_0)\cup f^{-1}[G_0\cup E_0]$.   The same arguments as before
tell us that $h_0=f\restr Y\setminus H$ is an isomorphism between
$(Y\setminus H,\Tau_{Y\setminus H})$ and
$(X\setminus G,\Sigma_{X\setminus G})$ representing $\pi$.   We now,
however, have $G$, $H\in\Cal I$ and both include $E_0$;  so
we must have $\#(G)=\#(E_0)=\#(H)$.   By 424Da, there is an isomorphism
$h_1$ between $(H,\Sigma_H)$ and $(G,\Sigma_G)$.   So if we set

$$\eqalign{h(y)
&=h_0(y)\text{ if }y\in X\setminus H,\cr
&=h_1(y)\text{ if }y\in H,\cr}$$

\noindent then $h:X\to X$ is a bijection which is an automorphism
of $(X,\Sigma)$, and

\Centerline{$h^{-1}[E]^{\ssbullet}
=(H\cap h^{-1}[E])^{\ssbullet}
=(h^{-1}[E\cap G])^{\ssbullet}
=(h_0^{-1}[E\cap G])^{\ssbullet}
=\pi E^{\ssbullet}$}

\noindent for every $E\in\Sigma$, as required.

\medskip

{\bf (d)} An adaptation of the ideas of (c) works in this case too.
First note that as $\#(X)=\frak c$,
$(X,\Sigma)\cong(\BbbR^2,\Cal B(\BbbR^2))$ and there is a partition
of $X$ into $\frak c$ members of $\Sigma$ all with cardinal $\frak c$.
As $\frak A$ is ccc, all but countably many
of these must belong to $\Cal I$, and we have an $E_0\in\Cal I$ with
$\#(E_0)=\frak c$.   Similarly, there is an $F_0\in\Cal J$ with
$\#(F_0)=\frak c$.

Now choose
$f$, $H_0$, $g$ and $G_0$ as in (b);  but this time set
$G=(G_0\cup E_0)\cup g^{-1}[H_0\cup F_0]$ and
$H=(H_0\cup F_0)\cup f^{-1}[G_0\cup E_0]$.   Again,
$h_0=f\restr Y\setminus H$ is an isomorphism between
$(Y\setminus H,\Tau_{Y\setminus H})$ and
$(X\setminus G,\Sigma_{X\setminus G})$ representing $\pi$.   As
$\#(G)=\frak c=\#(H)$, there is an isomorphism
$h_1$ between $(H,\Tau_H)$ and $(G,\Sigma_G)$.   So if we set

$$\eqalign{h(y)
&=h_0(y)\text{ if }y\in Y\setminus H,\cr
&=h_1(y)\text{ if }y\in H,\cr}$$

\noindent we get a bijection $h:Y\to X$ which is an isomorphism between
$(Y,\Tau)$ and $(X,\Sigma)$, and represents $\pi$, just as in (c).
}%end of proof of 425A
%should be at the beginning of the section;  useful for (c) in the proof
%of 425D

\leader{425B}{Lemma} Let $G$ be a group, $G_0$ a
subgroup of $G$, $H$ another group, and $X$, $Z$ sets;  let
$\action_r$ be the right shift action of $H$ on $Z^H$\cmmnt{ (4A5C(c-ii))}.
Suppose we are given a group homomorphism $\theta:G\to H$,
an injective function $f:\Bbb N\times Z^H\to X$
and an action $\action_0$ of $G_0$ on $X$ such that
$\pi\action_0f(n,z)=f(n,\theta(\pi)\action_rz)$ whenever
$n\in\Bbb N$ and $z\in Z^H$.

(a) If $\#(X\setminus f[\Bbb N\times Z^H])\le\#(Z)$, there is an
action $\action$ of $G$ on $X$ extending $\action_0$.

(b) Suppose moreover 
that $H$ is countable, $X$ and $Z$ are Polish spaces, and
$f$ is Borel measurable when $\Bbb N\times Z^H$ is given the product
topology.   If $x\mapsto\pi\action_0x$ is
Borel measurable for every $\pi\in G_0$, then $\action$ can be chosen in
such a way that $x\mapsto\psi\action x$ is Borel measurable
for every $\psi\in G$.

\proof{{\bf (a)(i)}
Let $D\subseteq H$ be a selector for the left cosets of the subgroup
$\theta[G_0]$,
so that every member of $H$ is uniquely expressible as $\psi\theta(\pi)$
where $\psi\in D$ and $\pi\in G_0$.   Observe that if $\pi\in G_0$ and
$x\in f[\Bbb N\times Z^H]$, then
$\pi^{-1}\action_0x\in f[\Bbb N\times Z^H]$;  consequently, setting
$Y=X\setminus f[\Bbb N\times Z^H]$, $\pi\action_0y\in Y$ for
every $y\in Y$,
because $\pi^{-1}\action_0(\pi\action_0y)=y$.
Let $g_0:Y\to Z$ be any injection, and define
$g:Y\to Z^H$ by setting
$g(y)(\psi\theta(\pi))=g_0(\pi\action_0y)$ whenever $\psi\in D$,
$\pi\in G_0$ and $y\in Y$.   In this case,
$\theta(\pi)\action_rg(y)=g(\pi\action_0y)$ whenever $y\in Y$ and
$\pi\in G_0$.   \Prf\ Take any $\psi\in D$ and $\phi\in G_0$.
Then

$$\eqalign{(\theta(\pi)\action_rg(y))(\psi\theta(\phi))
&=g(y)(\psi\theta(\phi)\theta(\pi))
=g(y)(\psi\theta(\phi\pi))\cr
&=g_0(\phi\pi\action_0y)
=g_0(\phi\action_0(\pi\action_0y))
=g(\pi\action_0y)(\psi\theta(\phi)).\cr}$$

\noindent As $D\theta[G_0]=H$, this shows that
$\theta(\pi)\action_rg(y)=g(\pi\action_0y)$.  \Qed

Note that as $g(y)(\psi)=g_0(y)$ whenever $\psi\in D$ and $y\in Y$,
$g$ also is injective.

\medskip

\quad{\bf (ii)} Now define $h:\Bbb N\times Z^H\to X$ by setting

$$\eqalignno{h(n,z)
&=g^{-1}(z)\text{ if }n=0\text{ and }z\in g[Y]\cr
&\mskip150mu
  \text{matching }\{0\}\times g[Y]\text{ with }Y,\cr
&=f(n-1,z)\text{ if }n\ge 1\text{ and }z\in g[Y]\cr
&\mskip150mu
  \text{matching }(\Bbb N\setminus\{0\})\times g[Y]\text{ with }
    f[\Bbb N\times g[Y]],\cr
&=f(n,z)\text{ if }n\in\Bbb N\text{ and }z\notin g[Y]\cr
&\mskip150mu
  \text{matching }\Bbb N\times(Z^H\setminus g[Y])
    \text{ with }f[\Bbb N\times(Z^H\setminus g[Y])].\cr}$$

\noindent Clearly $h$ is a bijection.   If $n\in\Bbb N$, $z\in Z^H$ and
$\pi\in G_0$, then $h(n,\theta(\pi)\action_rz)=\pi\action_0h(n,z)$.   \Prf\
If $z\in g[Y]$, then

\Centerline{$\theta(\pi)\action_rz
=\theta(\pi)\action_rg(g^{-1}(z))
=g(\pi\action_0g^{-1}(z))\in g[Y]$,}

\noindent so

\Centerline{$h(0,\theta(\pi)\action_rz)
=\pi\action_0g^{-1}(z)=\pi\action_0h(0,z)$,}

\noindent while if $n\ge 1$, then

\Centerline{$h(n,\theta(\pi)\action_rz)
=f(n-1,\theta(\pi)\action_rz)
=\pi\action_0f(n-1,z)
=\pi\action_0h(n,z)$.}

\noindent On the other hand,
if $n\in\Bbb N$ and $z\in Z^H\setminus g[Y]$, then
$\theta(\pi)\action_rz\notin g[Y]$, because
$\theta(\pi^{-1})\action_r(\theta(\pi)\action_rz)\notin g[Y]$;   so

\Centerline{$h(n,\theta(\pi)\action_rz)
=f(n,\theta(\pi)\action_rz)
=\pi\action_0f(n,z)
=\pi\action_0h(n,z)$.  \Qed}

\medskip

\quad{\bf (iii)} Try

\Centerline{$\psi\action x=h(n,\theta(\psi)\action_rz)$}

\noindent whenever $x\in X$, $h^{-1}(x)=(n,z)$ and $\psi\in G$.
If $x\in X$, $\psi$, $\psi'\in G$ and $\pi\in G_0$,
express $h^{-1}(x)$ as $(n,z)$;  then, writing $\iota$ for the identity of
$G$,

$$\eqalign{\iota\action x
&=h(n,\theta(\iota)\action_rz)
=h(n,z)
=x,\cr
\psi'\psi\action x
&=h(n,\theta(\psi'\psi)\action_rz)
=h(n,\theta(\psi')\action_r(\theta(\psi)\action_rz))
=\psi'\action h(n,\theta(\psi)\action_rz)
=\psi'\action(\psi\action x),\cr
\pi\action x
&=h(n,\theta(\pi)\action_rz)
=\pi\action_0h(n,z)
=\pi\action_0x,\cr}$$

\noindent so we have an action
of $G$ on $X$ extending $\action_0$, as required.

\medskip

{\bf (b)} Under the topological hypotheses, we follow the same line of
argument, but taking time for checks at each stage.
Because $Z$, and therefore $\Bbb N\times Z^H$, are Polish spaces,
and $f$ is a measurable injection,
$f[\Bbb N\times Z^H]$ and $Y$ are Borel sets (423Ib).
In (i), we must of
course take $g_0$ to be Borel measurable;  since all the
functions $x\mapsto\pi\action_0x$ are Borel measurable,
all the functions
$y\mapsto g(y)(\psi)$ will be Borel measurable, and
$g:Y\to Z^H$ will be Borel measurable.
Consequently all the sets $\{n\}\times g[Y]$ will be Borel,
and $h$ will be Borel measurable, therefore a Borel
isomorphism between $\Bbb N\times Z^H$ and $X$.
Finally, because $z\mapsto\theta(\psi)\action_rz:Z^H\to Z^H$
is Borel measurable for every $\psi\in G$,
$(n,z)\mapsto(n,\theta(\psi)\action_rz)$ is Borel measurable for every
$n$, and $x\mapsto\psi\action x$ is Borel measurable,
for every $\psi\in G$.   So $\action$ is measurable in the
required sense.
}%end of proof of 425B

\leader{425C}{Master actions (a)} For each $R\subseteq\BbbN^2$,
consider the family $F_R$ of injective functions $f$ from countable
ordinals to $\Bbb N$ such that

\Centerline{for every $\beta\in\dom f$, $f(\beta)$ is the unique member of
$\Bbb N$ such that
$R[\{f(\beta)\}]=f[\beta]$.}

\noindent\cmmnt{If $f$, $g\in F_R$ have domains $\alpha$, $\alpha'$
respectively
where $\alpha\le\alpha'$, then $f=g\restr\alpha$.
\prooflet{\Prf\ If $\beta<\alpha$
is such that $f\restr\beta=g\restr\beta$, then both $f(\beta)$ and
$g(\beta)$ are the unique $m\in\Bbb N$ such that
$R[\{m\}]=f[\beta]$.\ \QeD}   Consequently }%
$f_R=\bigcup F_R$ is the unique maximal element of $F_R$.
\cmmnt{(Compare 2A1B.)}

For a countable ordinal $\alpha$, let
$\Cal R_{\alpha}$ be the set of those $R\subseteq\BbbN^2$ such that
$\alpha\le\dom f_R$.   Note that if $f:\alpha\to\Bbb N$ is injective, then
there is an $R\in\Cal R_{\alpha}$ such that
$f=f_R\restr\alpha$\prooflet{ (set
$R=\{(f(\beta),f(\gamma)):\gamma<\beta<\alpha\}
  \cup((\Bbb N\setminus f[\alpha])\times\Bbb N)$)}.

\medskip

{\bf (b)} Let $\maltese$ be the family of group operations $\star$ on 
$\Bbb N$.   We are going to need the natural Borel structure on
$\maltese$ corresponding to the identification of each $\star\in\maltese$,
which is a function from $\Bbb N\times\Bbb N$ to $\Bbb N$, with the set

\Centerline{$\{(i,j,k):i\star j=k\}\subseteq\BbbN^3$.}

\noindent\cmmnt{ So we can think of $\maltese$ as the set of subsets 
$\star$ of $\BbbN^3$ such that

\inset{for all $i$, $j\in\Bbb N$ there is just one $k\in\Bbb N$ such that
$(i,j,k)\in\star$,

if $i$, $j$, $k$, $l$, $m$, $n\in\Bbb N$, and $(i,j,l)$,
$(l,k,n)$, $(j,k,m)$ belong to $\star$, then $(i,m,n)\in\star$,

$(0,i,i)$, $(i,0,i)\in\star$ for every $i\in\Bbb N$,

for every $i\in\Bbb N$ there is a $j\in\Bbb N$ such that $(i,j,0)$ and
$(j,i,0)$ belong to $\star$.}

\noindent} For $\star\in\maltese$, let $\action^{\star}_r$ be the 
corresponding
right shift action of $\Bbb N$ on $\BbbR^{\Bbb N}$, so that
$(m\action^{\star}_rz)(i)=z(i\star m)$
whenever $z\in\BbbR^{\Bbb N}$ and $i$, $m\in\Bbb N$.

\medskip

{\bf (c)} Let $G$ be a group, with cardinal $\omega_1$, with
identity $\iota$;  let $\ofamily{\alpha}{\omega_1}{\pi_{\alpha}}$
enumerate $G$, with $\pi_0=\iota$.   Let $F\subseteq\omega_1$ be the
set of those $\alpha$ such that $G_{\alpha}=\{\pi_{\beta}:\beta<\alpha\}$
is a subgroup of $G$.
%note $1=\min F$
For $\alpha\in F$, set

$$\eqalign{\Cal S_{\alpha}
&=\{(R,\star):R\in \Cal R_{\alpha},\,\star\in\maltese\text{ and }
f_R(\beta)\star f_R(\gamma)=f_R(\delta)\cr
&\mskip160mu
\text{whenever }\beta,\,\gamma,\,\delta<\alpha\text{ and }
\pi_{\beta}\pi_{\gamma}=\pi_{\delta}\}\dvro{.}{;}\cr}$$

\cmmnt{\noindent so that if $(R,\star)\in\Cal S_{\alpha}$,
$f_R\restr\alpha$ codes a group homomorphism from
$G_{\alpha}$ to $(\Bbb N,\star)$.}

\medskip

{\bf (d)} For $\alpha\in F$, set

\Centerline{$\Cal M_{\alpha}
=\{(R,\star,z):(R,\star)\in\Cal S_{\alpha}$, $z\in\BbbR^{\Bbb N}\}$.}

\noindent Then
we have an action $\action'_{\alpha}$ of $G_{\alpha}$ on $\Cal M_{\alpha}$
defined by saying that

\Centerline{$\pi_{\beta}\action'_{\alpha}(R,\star,z)
=(R,\star,f_R(\beta)\action^{\star}_rz)$}

\noindent whenever $(R,\star)\in\Cal S_{\alpha}$, $z\in\BbbR^{\Bbb N}$
and $\beta<\alpha$.   \prooflet{\Prf\
$\action'_{\alpha}$ is well-defined as a
function on $G_{\alpha}\times\Cal M_{\alpha}$ because
$f_R(\beta)=f_R(\gamma)$ whenever $(R,\star)\in\Cal S_{\alpha}$ and
$\pi_{\beta}=\pi_{\gamma}$.
If $\beta$, $\gamma$, $\delta<\alpha$ and
$\pi_{\delta}=\pi_{\beta}\pi_{\gamma}$, then

$$\eqalignno{
\pi_{\beta}\action'_{\alpha}(\pi_{\gamma}\action'_{\alpha}(R,\star,z))
&=\pi_{\beta}\action'_{\alpha}(R,\star,f_R(\gamma)\action^{\star}_rz)\cr
&=(R,\star,f_R(\beta)\action^{\star}_r(f_R(\gamma)\action^{\star}_rz))
=(R,\star,f_R(\delta)\action^{\star}_rz)\cr
\displaycause{because $(f_R(\beta),f_R(\gamma),f_R(\delta))\in\star$}
&=\pi_{\delta}\action'_{\alpha}(R,\star,z),\cr
\iota\action'_{\alpha}(R,\star,z)
&=\pi_0\action'_{\alpha}(R,\star,z)
=(R,\star,f_R(0)\action^{\star}_rz)
=(R,\star,z).  \text{ \Qed}\cr}$$
}

\medskip

{\bf (e)} If $\alpha$, $\beta\in F$ and $\alpha\le\beta$, then\cmmnt{ it
is elementary to check that}
$\Cal R_{\beta}\subseteq \Cal R_{\alpha}$,
$\Cal S_{\beta}\subseteq\Cal S_{\alpha}$,
$\Cal M_{\beta}\subseteq\Cal M_{\alpha}$ and
$\pi\action'_{\beta}(R,\star,z)=\pi\action'_{\alpha}(R,\star,z)$ whenever
$(R,\star,z)\in\Cal M_{\beta}$ and $\pi\in G_{\alpha}$.
If $\beta\in F$ and $\beta=\sup(\beta\cap F)$, then
$\Cal R_{\beta}=\bigcap_{\alpha\in\beta\cap F}\Cal R_{\alpha}$ and
$\Cal M_{\beta}=\bigcap_{\alpha\in\beta\cap F}\Cal M_{\alpha}$.

\medskip

{\bf (f)(i)} For $\alpha<\omega_1$, $\Cal R_{\alpha}$ belongs to
the Borel $\sigma$-algebra $\Cal B(\Cal P\BbbN^2)$\cmmnt{ when
$\Cal P(\BbbN^2)$ is
given its usual compact Hausdorff topology\cmmnt{ (4A2Ud)}},
and\cmmnt{ moreover}
$\{(R,m):R\subseteq\BbbN^2$, $(\beta,m)\in f_R\}
\in\Cal B(\Cal P\BbbN^2\times\Bbb N)$ whenever $m\in\Bbb N$ and
$\beta<\alpha$.   \prooflet{\Prf\
Induce on $\alpha$.   We start with $\Cal R_0=\Cal P(\BbbN^2)$.
For the inductive step to a successor ordinal $\alpha+1$, given
$R\subseteq\BbbN^2$ and $m\in\Bbb N$, then

$$\eqalign{(\alpha,m)\in f_R
&\iff R\in\Cal R_{\alpha}\text{ and }m\text{ is the unique member of }
\BbbN\cr
&\mskip100mu\text{ such that }R[\{m\}]=f_R[\alpha]\cr
&\iff R\in\Cal R_{\alpha},\,R[\{m\}]\ne R[\{n\}]\text{ for every }
  n\ne m,\cr
&\mskip100mu\Forall i\in\Bbb N,\,(m,i)\in R\iff\Exists\beta<\alpha,
\,(\beta,i)\in f_R.\cr}$$

\noindent So $\{(R,m):(\alpha,m)\in f_R\}$ is

$$\eqalign{(\Cal R_{\alpha}\times\Bbb N)
&\cap\bigcap_{n\in\Bbb N}\bigcup_{i\in\Bbb N}
  \{(R,m):n=m\text{ or }(m,i)\in R\,\&\,(n,i)\notin R
  \text{ or }(m,i)\notin R\,\&\,(n,i)\in R\}\cr
&\cap\bigcap_{i\in\Bbb N}\bigcup_{\beta<\alpha}
  \{(R,m):(m,i)\notin R\text{ or }(\beta,i)\in f_R\}\cr
&\cap\bigcap_{i\in\Bbb N}\bigcap_{\beta<\alpha}
  \{(R,m):(m,i)\in R\text{ or }(\beta,i)\notin f_R\},\cr}$$

\noindent and is a Borel set.   Now

\Centerline{$\Cal R_{\alpha+1}
=\bigcup_{m\in\Bbb N}\{R:(\alpha,m)\in f_R\}
\in\Cal B(\Cal P\BbbN^2)$.}

\noindent For the inductive step to a countable limit
ordinal $\alpha>0$,
$\Cal R_{\alpha}=\bigcap_{\gamma<\alpha}\Cal R_{\gamma}$.\ \Qed}

\medskip

\quad{\bf (ii)} $\maltese$ is a Borel subset of $\Cal P(\BbbN^3)$,\cmmnt{ so}
$\Cal S_{\alpha}$ is a Borel subset of
$\Cal P(\BbbN^2)\times\Cal P(\BbbN^3)$,
and $\Cal M_{\alpha}$ is a Borel subset of
$\Cal P(\BbbN^2)\times\Cal P(\BbbN^3)\times\BbbR^{\Bbb N}$, for every
$\alpha\in F$.   \cmmnt{Next,}
$(R,\star,z)\mapsto\pi\action'_{\alpha}(R,\star,z):
  \Cal M_{\alpha}\to\Cal M_{\alpha}$ is Borel
measurable whenever $\alpha\in F$ and $\pi\in G_{\alpha}$.
\prooflet{\Prf\ Let
$\beta<\alpha$ be such that $\pi=\pi_{\beta}$.   If
$E_0\subseteq\Cal P(\BbbN^2)\times\Cal P(\BbbN^3)$ is a Borel set, and
$E=\{(R,\star,z):(R,\star,z)\in\Cal M_{\alpha}$, $(R,\star)\in E_0\}$,
then

\Centerline{$\{(R,\star,z):\pi\action'_{\alpha}(R,\star,z)\in E\}=E$}

\noindent is Borel;  if $n\in\Bbb N$ and $E_1\subseteq\Bbb R$ is a Borel
set, and $E=\{(R,\star,z):(R,\star,z)\in\Cal M_{\alpha}$, $z(n)\in E_1\}$, then

$$\eqalign{\{(R,\star,z):\pi\action'_{\alpha}(R,\star,z)\in E\}
&=\{(R,\star,z):(f_R(\beta)\action^{\star}_rz)(n)\in E_1\}\cr
&=\bigcup_{i,j\in\Bbb N}\{(R,\star,z):f_R(\beta)=i,\,(n,i,j)\in\star,\,
   z(j)\in E_1\}\cr}$$

\noindent is Borel.   Since
$\Cal B(\Cal P\BbbN^2\times\Cal P\BbbN^3\times\BbbR^{\Bbb N})$ is the
product $\sigma$-algebra

\Centerline{$\Cal B(\Cal P\BbbN^2)\tensorhat\Cal B(\Cal P\BbbN^3)
  \tensorhat\Tensorhat_{\Bbb N}\Cal B(\Bbb R)$,}

\noindent $(R,\star,z)\mapsto\pi\action'_{\alpha}(R,\star,z)$ is 
Borel measurable.\ \Qed}

\leader{425D}{T\"ornquist's theorem}\cmmnt{ ({\smc T\"ornquist 11})}
Let $(X,\Sigma)$ be a standard Borel space and
$\Cal I$ a $\sigma$-ideal of $\Sigma$ containing an uncountable
set.   Let $\frak A$ be the quotient algebra $\Sigma/\Cal I$, and
$G\subseteq\Aut\frak A$ a subgroup with cardinal at most $\omega_1$.
Then there is an action $\action$ of $G$ on $X$ which represents $G$ in the
sense that $\pi\action E\cmmnt{\mskip5mu=\{\pi\action x:x\in E\}}$
belongs to $\Sigma$, and
$(\pi\action E)^{\ssbullet}=\pi(E^{\ssbullet})$,
for every $E\in\Sigma$ and $\pi\in G$.

\proof{{\bf (a)} It may help if I try to describe the line of argument I
mean to follow.   The important case is when $G$ has an enumeration
$\ofamily{\alpha}{\omega_1}{\pi_{\alpha}}$;  let
$F$ be the set of those $\alpha<\omega_1$ such that
$G_{\alpha}=\{\pi_{\beta}:\beta<\alpha\}$ is a subgroup of $G$.
For each $\pi\in G$, choose $g_{\pi}:X\to X$ representing $\pi$.   
For $\alpha\in F$, set

\Centerline{$Y_{\alpha}=\{x:x\in X$,
$g_{\phi}(g_{\pi}(x))=g_{\pi\phi}(x)\in X\setminus M$ for all $\pi$,
$\phi\in G_{\alpha}\}$,}

\noindent where $M$ is an uncountable member of $\Cal I$.
Choose $\family{\alpha}{F}{\action_{\alpha}}$ inductively such
that $\action_{\alpha}$ is an action of $G_{\alpha}$ on $X$,
$\pi\action_{\alpha}x=g_{\pi^{-1}}(x)$ whenever $\pi\in G_{\alpha}$ and
$x\in Y_{\alpha}$, and $\pi\action_{\alpha}x=\pi\action_{\beta}x$ whenever
$\beta<\alpha$, $\pi\in G_{\beta}$ and $x\in X$.   At the end of the
construction, set $\action=\bigcup_{\alpha\in F}\action_{\alpha}$.

It is straightforward to show that $X\setminus Y_{\alpha}$
always belongs to $\Cal I$ (part (c) of the proof);
consequently $\action$
will represent $G$.   The non-trivial part of the proof is in the
extension of a given action of $G_{\alpha}$ to an action of $G_{\beta}$
where $\beta$ is the next element of $F$ above $\alpha$, and this is where
we shall need 425B-425C.

Now for the details.

\medskip

{\bf (b)} Give $X$ a Polish topology for which $\Sigma$ is the Borel
$\sigma$-algebra $\Cal B(X)$.   
For nearly the whole of the proof (down to the end of
(g) below), suppose that $\#(G)=\omega_1$, that
$X=\{0,1\}^{\Bbb N}$ and that
$\Sigma=\Cal B(X)$ is the Borel $\sigma$-algebra of $X$.
Enumerate $G$ as
$\ofamily{\alpha}{\omega_1}{\pi_{\alpha}}$ starting with $\pi_0$ equal to
the identity $\iota$.   We need to know that, setting 
$G_{\alpha}=\{\pi_{\beta}:\beta<\alpha\}$, the set
$F=\{\alpha:\alpha<\omega_1$, $G_{\alpha}$ is a subgroup of $G\}$ is
a closed cofinal subset of $\omega_1$.   \Prf\ This is elementary.
If $\alpha\in\overline{F}$, then
$\{G_{\beta}:\beta\in F$, $\beta\le\alpha\}$ is a
non-empty upwards-directed family
of subgroups of $G$, so
$G_{\alpha}=\bigcup_{\beta\in F,\beta\le\alpha}G_{\beta}$ is a subgroup of
$G$, and $\alpha\in F$.   If $\alpha<\omega_1$, let $\sequencen{\alpha_n}$
be a non-decreasing sequence in $\omega_1$ such that
$\alpha_0=\max(1,\alpha)$ and, for each $n\in\Bbb N$,

\inset{----- for all $\beta$, $\gamma<\alpha_n$ there is a
$\delta<\alpha_{n+1}$ such that $\pi_{\delta}=\pi_{\beta}\pi_{\gamma}$,

----- for every $\beta<\alpha_n$ there is a $\delta<\alpha_{n+1}$ such that
$\pi_{\delta}=\pi_{\beta}^{-1}$.}

\noindent Setting $\alpha^*=\sup_{n\in\Bbb N}\alpha_n$, we see that
$\alpha\le\alpha^*\in F$.   So $F$ is cofinal with $\omega_1$.\ \Qed

Of course $1=\min F$ and $G_1=\{\iota\}$, because we started with
$\pi_0=\iota$.

\medskip

{\bf (c)} Next, for every $\pi\in\Aut\frak A$, we can choose a
Borel measurable $g_{\pi}:X\to X$ representing $\pi$ in the
sense that $\pi E^{\ssbullet}=g_{\pi}^{-1}[E]^{\ssbullet}$ for every
$E\in\Cal B(X)$ (425Ac).   Of course when $\pi=\iota$ we take
$g_{\iota}(x)=x$ for every $x\in X$.   Fix an uncountable $M\in\Cal I$, and
for $\alpha\in F$ set

\Centerline{$Y_{\alpha}=\{x:x\in X$,
$g_{\phi}(g_{\pi}(x))=g_{\pi\phi}(x)\in X\setminus M$ for all $\pi$,
$\phi\in G_{\alpha}\}$,}

\noindent as declared in (a).   If $\alpha\in F$ and $\pi\in G_{\alpha}$,
$X\setminus Y_{\alpha}\in\Cal I$ and
$g_{\pi}\restr Y_{\alpha}$ is a permutation of $Y_{\alpha}$.
\Prf\ (Compare the proof of 344B.)
There is a sequence $\sequence{k}{E_k}$ in $\Cal B(X)$ separating the
points of $X$.   So, for any $\psi$, $\phi\in G_{\alpha}$, the set

\Centerline{$\{x:g_{\psi}g_{\phi}(x)\ne g_{\phi\psi}(x)\}
=\bigcup_{k\in\Bbb N}g_{\phi}^{-1}[g_{\psi}^{-1}[E_k]]
   \symmdiff g_{\phi\psi}^{-1}[E_k]$}

\noindent is Borel, and moreover, transferring the formulae
to the quotient algebra,

$$\eqalign{\{x:g_{\psi}g_{\phi}(x)\ne g_{\phi\psi}(x)\}^{\ssbullet}
&=\sup_{k\in\Bbb N}g_{\phi}^{-1}[g_{\psi}^{-1}[E_k]]^{\ssbullet}
   \Bsymmdiff g_{\phi\psi}^{-1}[E_k]^{\ssbullet}\cr
&=\sup_{k\in\Bbb N}\phi(\psi E_k^{\ssbullet})
   \Bsymmdiff(\phi\psi)E_k^{\ssbullet}
=0,\cr}$$

\noindent so $\{x:g_{\psi}g_{\phi}(x)\ne g_{\phi\psi}(x)\}\in\Cal I$.
Accordingly

\Centerline{$X\setminus Y_{\alpha}
=\bigcup_{\psi,\phi\in G_{\alpha}}
     \{x:g_{\psi}g_{\phi}(x)\ne g_{\phi\psi}(x)\}
   \cup\bigcup_{\psi\in G_{\alpha}}g_{\psi}^{-1}[M]$}

\noindent belongs to $\Cal I$.

If $\pi\in G_{\alpha}$ and $x\in Y_{\alpha}$, then

\Centerline{$g_{\psi}g_{\phi}g_{\pi}(x)=g_{\psi}g_{\pi\phi}(x)
  =g_{\pi\phi\psi}(x)=g_{\phi\psi}g_{\pi}(x)$}

\noindent and $g_{\pi\phi\psi}(x)\notin M$, for all
$\phi$, $\psi\in G_{\alpha}$;  so $g_{\pi}(x)\in Y_{\alpha}$.
Similarly, $g_{\pi^{-1}}[Y_{\alpha}]\subseteq Y_{\alpha}$.
Moreover,

\Centerline{$g_{\pi}g_{\pi^{-1}}(x)=g_{\iota}(x)=x
=g_{\pi^{-1}}g_{\pi}(x)$}

\noindent for every $x\in Y_{\alpha}$, so that $g_{\pi}\restr Y_{\alpha}$
must be a permutation of $Y_{\alpha}$.\ \Qed

\medskip

{\bf (d)} From the group $G$ and the enumeration
$\ofamily{\alpha}{\omega_1}{\pi_{\alpha}}$, construct $\maltese$ and
families $\family{R}{\Cal P\BbbN^2}{f_R}$,
$\ofamily{\alpha}{\omega_1}{\Cal R_{\alpha}}$, 
$\family{\star}{\maltese}{\action^{\star}_r}$ and
$\family{\alpha}{F}{(\Cal S_{\alpha},\Cal M_{\alpha},\action'_{\alpha})}$
as in 425C.   Let $h$ be a Borel isomorphism from
$\Bbb R\times\Bbb N\times\Cal M_1$ to $M$ (424Da again).
Fix a family $\family{\delta}{F}{t_{\delta}}$ of distinct members of
$\Bbb R$, and set $J_{\alpha}=\Bbb R\setminus\{t_{\delta}:\delta<\alpha\}$
for $\alpha\in F$.

\medskip

{\bf (e)} (The key.  Some readers may wish at this point to provide
themselves with
coffee and a large scratch-pad.)  Let $\alpha<\beta<\gamma$ be members of 
$F$, and suppose that
$\action_0$ is an action of $G_{\alpha}$ on $X$ such that

\inset{$x\mapsto\pi\action_0x$ is Borel measurable
for every $\pi\in G_{\alpha}$,

$\pi\action_0x=g_{\pi^{-1}}(x)$ whenever $x\in Y_{\alpha}$ and
$\pi\in G_{\alpha}$,

$\pi\action_0h(t,n,q)=h(t,n,\pi\action'_{\beta}q)$ whenever
$t\in J_{\alpha}$,
$n\in\Bbb N$, $q\in\Cal M_{\beta}$ and $\pi\in G_{\alpha}$.}

\noindent Then there is an action $\action_1$ of $G_{\beta}$ on $X$ such
that

\inset{$x\mapsto\pi\action_1x$ is Borel measurable
for every $\pi\in G_{\beta}$,

$\pi\action_1x=g_{\pi^{-1}}(x)$ whenever $x\in Y_{\beta}$ and
$\pi\in G_{\beta}$,

$\pi\action_1h(t,n,q)=h(t,n,\pi\action'_{\gamma}q)$ whenever
$t\in J_{\beta}$, $n\in\Bbb N$, $q\in\Cal M_{\gamma}$ and
$\pi\in G_{\beta}$,

$\pi\action_1x=\pi\action_0x$ whenever $\pi\in G_{\alpha}$ and $x\in X$.}

\noindent\Prf\ Choose $\star\in\maltese$ such that
there is an injective group homomorphism $\theta$ from
$G_{\beta}$ to $(\Bbb N,\star)$,
and set $f'(\delta)=\theta(\pi_{\delta})$ for
$\delta<\beta$;  let $R\in \Cal R_{\beta}$ be such that
$f_R\restr\beta=f'$.   (In the normal case,
when $\beta\ge\omega$, we can choose $f'$
first, as an arbitrary bijection from $\beta$ to $\Bbb N$, and use this to
define $\theta$, $\star$ and $R$.   If $\beta<\omega$, the first step is to
take an injective group homomorphism from $G_{\beta}$ to a countably
infinite group, e.g., $G_{\beta}\times\Bbb Z$.)   
Then $(R,\star)\in\Cal S_{\beta}$.
Define $h_0:\Bbb N\times\BbbR^{\Bbb N}\to M$ by setting
$h_0(n,z)=h(t_{\alpha},n,(R,\star,z))$ for $n\in\Bbb N$ and
$z\in\BbbR^{\Bbb N}$.   Then $h_0$ is injective and Borel
measurable, and if $\pi\in G_{\alpha}$ then

$$\eqalignno{h_0(n,\theta(\pi)\action^{\star}_rz)
&=h(t_{\alpha},n,(R,\star,\theta(\pi)\action^{\star}_rz))
=h(t_{\alpha},n,(R,\star,f_{R}(\delta)\action^{\star}_rz))\cr
\displaycause{where $\delta<\alpha$ is such that $\pi=\pi_{\delta}$}
&=h(t_{\alpha},n,\pi\action'_{\alpha}(R,\star,z))
=h(t_{\alpha},n,\pi\action'_{\beta}(R,\star,z))\cr
&=\pi\action_0h(t_{\alpha},n,(R,\star,z))
=\pi\action_0h_0(n,z)
\cr}$$

\noindent for every $n\in\Bbb N$ and $z\in\BbbR^{\Bbb N}$.
Set

\Centerline{$V=h[J_{\beta}\times\Bbb N\times\Cal M_{\gamma}]
\subseteq M\subseteq X\setminus Y_{\beta}$,
\quad$X'=X\setminus(Y_{\beta}\cup V)$.}

\noindent  Note that
if $\pi\in G_{\alpha}$, $t\in J_{\beta}$, $n\in\Bbb N$ and
$q\in\Cal M_{\gamma}$, then

\Centerline{$\pi\action_0h(t,n,q)
=h(t,n,\pi\action'_{\beta}q)=h(t,n,\pi\action'_{\gamma}q)\in V$;}

\noindent thus $V$ is invariant under the action $\action_0$.
The same is true of $Y_{\beta}$, because 
$g_{\pi^{-1}}\restr Y_{\beta}$ is
a permutation of $Y_{\beta}$ for every $\pi\in G_{\alpha}$,
therefore also of $X'$.   Note that as $t_{\alpha}\notin J_{\beta}$,
$h_0[\Bbb N\times\BbbR^{\Bbb N}]\subseteq X'$, while
there is certainly a Borel measurable injection from
$X'\setminus h_0[\Bbb N\times\BbbR^{\Bbb N}]$ into $\Bbb R$.   So
425Bb tells us that there is an action $\hat{\action}_1$ of $G_{\beta}$
on $X'$, extending $\action_0\restr G_{\alpha}\times X'$, such that
$x\mapsto\pi\hat{\action}_1x:X'\to X'$ is
Borel measurable for every $\pi\in G_{\beta}$.

We can therefore define $\action_1$ by setting

$$\eqalign{\pi\action_1x
&=g_{\pi^{-1}}(x)\text{ if }x\in Y_{\beta},\cr
&=h(t,n,\pi\action'_{\gamma}q)\text{ whenever }t\in J_{\beta},\,
   n\in\Bbb N,\,q\in\Cal M_{\gamma}\text{ and }x=h(t,n,q),\cr
&=\pi\hat{\action}_1x\text{ if }x\in X'\cr}$$

\noindent for every $\pi\in G_{\beta}$.   It is easy to check that
$\action_1$ is a function from $G_{\beta}\times X$ to $X$ extending
$\action_0$, and that $x\mapsto\pi\action_1x$ is
Borel measurable for every $\pi\in G_{\beta}$.   If $\pi$,
$\phi\in G_{\beta}$ and $x\in X$, then

$$\eqalign{\pi\action_1(\phi\action_1x)
&=\pi\action_1g_{\phi^{-1}}(x)
=g_{\pi^{-1}}g_{\phi^{-1}}(x)
=g_{(\pi\phi)^{-1}}(x)
=(\pi\phi)\action_1x\text{ if }x\in Y_{\beta},\cr
&=\pi\action_1h(t,n,\phi\action'_{\gamma}q)
=h(t,n,\pi\action'_{\gamma}(\phi\action'_{\gamma}q))
=h(t,n,(\pi\phi)\action'_{\gamma}q)\cr
&\mskip100mu=(\pi\phi)\action_1h(t,n,q)
=(\pi\phi)\action_1x\cr
&\mskip50mu\text{ whenever }t\in J_{\beta},\,
   n\in\Bbb N,\,q\in\Cal M_{\gamma}\text{ and }x=h(t,n,q),\cr
&=\pi\hat{\action}_1(\phi\hat{\action}_1x)
=(\pi\phi)\hat{\action}_1x
=(\pi\phi)\action_1x
\text{ if }x\in X'.\cr}$$

\noindent So $\action_1$ is an action of $G_{\beta}$ on $X$, as required.\
\Qed

\medskip

{\bf (f)} Accordingly we can build $\family{\alpha}{F}{\action_{\alpha}}$
inductively, as follows.   The
inductive hypothesis will be that, for each $\alpha\in F$,

\inset{$\action_{\alpha}$ is an action of $G_{\alpha}$ on $X$,

$x\mapsto\pi\action_{\alpha}x$ is Borel measurable for every
$\pi\in G_{\alpha}$,

$\pi\action_{\alpha}x=g_{\pi^{-1}}(x)$ whenever $x\in Y_{\alpha}$ and
$\pi\in G_{\alpha}$,

$\pi\action_{\alpha}h(t,n,q)=h(t,n,\pi\action'_{\beta}q)$ whenever
$t\in J_{\alpha}$, $n\in\Bbb N$, $q\in\Cal M_{\beta}$,
$\beta\in F$, $\beta>\alpha$ and $\pi\in G_{\alpha}$,

$\action_{\delta}=\action_{\alpha}\restr G_{\delta}\times X$ whenever
$\delta\in F\cap\alpha$.
}

\noindent The induction starts with $G_1=\{\iota\}$ and $\iota\action_1x=x$
for every $x\in X$.

Given $\alpha\in F$ and $\action_{\alpha}$, let $\beta$ be the next
element of $F$ above $\alpha$ and $\gamma$ the next element of $F$ above
$\beta$.   By (d), we have an action
$\action_{\beta}$ of $G_{\beta}$ on $X$ such that

\inset{$x\mapsto\pi\action_{\beta}x$ is Borel measurable for every
$\pi\in G_{\beta}$,

$\pi\action_{\beta}x=g_{\pi^{-1}}(x)$ whenever $x\in Y_{\beta}$ and
$\pi\in G_{\beta}$,

$\pi\action_{\beta}h(t,n,q)=h(t,n,\pi\action'_{\gamma}q)$ whenever
$t\in J_{\beta}$, $n\in\Bbb N$, $q\in\Cal M_{\gamma}$ and
$\pi\in G_{\alpha}$,

$\action_{\alpha}=\action_{\beta}\restr G_{\alpha}\times X$.}

\noindent It follows at once that if $\delta\in F$ and $\delta\le\alpha$,

\Centerline{$\pi\action_{\delta}x=\pi\action_{\alpha}x
=\pi\action_{\beta}x$}

\noindent whenever $\pi\in G_{\delta}$ and $x\in X$;  on the other side,
if $\delta\in F$ and $\delta\ge\gamma$,

\Centerline{$\pi\action_{\beta}h(t,n,q)=h(t,n,\pi\action'_{\gamma}q)
=h(t,n,\pi\action'_{\delta}q)$}

\noindent whenever $t\in J_{\beta}$, $n\in\Bbb N$, $q\in\Cal M_{\delta}$
and $\pi\in G_{\beta}$.   So the induction continues to the next step.

If $\alpha\in F$ and $\alpha=\sup(F\cap\alpha)$, then
$G_{\alpha}=\bigcup_{\beta\in F\cap\alpha}G_{\beta}$, so we have an action
$\action_{\alpha}=\bigcup_{\beta\in F\cap\alpha}\action_{\beta}$ of
$G_{\alpha}$ on $X$;  and it is elementary to check that the inductive
hypothesis is satisfied at the new level.

\medskip

{\bf (g)} At the end of the induction,
$\action=\bigcup_{\alpha\in F}\action_{\alpha}$ will be an action of $G$ on
$X$ such that
$x\mapsto\pi\action x$ is Borel measurable for every
$\pi\in G$.   Moreover,
$\pi E^{\ssbullet}=(\pi\action E)^{\ssbullet}$ for every Borel set
$E\subseteq X$ and
$\pi\in G$.   \Prf\ Let $\alpha\in F$ be such that $\pi\in G_{\alpha}$.
Then

$$\eqalignno{\pi(E^{\ssbullet})
&=\pi((E\cap Y_{\alpha})^{\ssbullet})\cr
\displaycause{because $X\setminus Y_{\alpha}\in\Cal I$}
&=(g_{\pi}^{-1}[E\cap Y_{\alpha}])^{\ssbullet}
=(Y_{\alpha}\cap g_{\pi}^{-1}[E\cap Y_{\alpha}])^{\ssbullet}
=(g_{\pi^{-1}}[E\cap Y_{\alpha}])^{\ssbullet}\cr
\displaycause{because $g_{\pi}\restr Y_{\alpha}$ is a permutation with
inverse $g_{\pi^{-1}}\restr Y_{\alpha}$}
&=(\pi\action(E\cap Y_{\alpha}))^{\ssbullet}
\Bsubseteq(\pi\action E)^{\ssbullet}.\cr}$$

\noindent(Because $x\mapsto\pi\action x$ is a Borel measurable
permutation of $X$, $\pi\action E$ is certainly a Borel set.)
Since equally we must have

\Centerline{$\pi((X\setminus E)^{\ssbullet})
\Bsubseteq(\pi\action(X\setminus E))^{\ssbullet}$,}

\noindent while
$\pi(E^{\ssbullet})\Bcup\pi((X\setminus E)^{\ssbullet})=1_{\frak A}$ and
$(\pi\action E)^{\ssbullet}
  \Bcap(\pi\action(X\setminus E))^{\ssbullet}=0_{\frak A}$,
both the inclusions here are equalities, and
$\pi E^{\ssbullet}=(\pi\action E)^{\ssbullet}$.\ \Qed

\medskip

{\bf (h)} As for the elementary case in which $G$ is countable, we can use
arguments already presented, as follows.   For each $\pi\in G$,
choose $g_{\pi}$ representing $\pi$.   This time, go
straight to
$Y=\{x:g_{\pi\phi}(x)=g_{\phi}g_{\pi}(x)$ for all $\pi$, $\phi\in G\}$;
as in (c), $X\setminus Y\in\Cal I$ and
$g_{\pi}\restr Y$ is a permutation of $Y$ for every $\pi\in G$.
So if we set

$$\eqalign{\pi\action x
&=g_{\pi^{-1}}(x)\text{ for }x\in Y,\cr
&=x\text{ for }x\in X\setminus Y,\cr}$$

\noindent $\action$ will be an appropriate action of $G$ on $X$.
}%end of proof of 425D

\leader{425E}{Scholium}
The theorem here applies to groups with cardinal at
most $\omega_1$.   So it is worth noting that in the context of 425D the
whole group $\Aut\frak A$ has cardinal at most $\frak c$.
\prooflet{\Prf\ Since
$\Sigma$ is countably $\sigma$-generated, there is a countable set
$D\subseteq\frak A\,\,\sigma$-generating $\frak A$.   If $\pi$,
$\phi\in\Aut\frak A$ and $\pi\restr D=\phi\restr D$, then $\pi=\phi$;  so
$\#(\Aut\frak A)\le\#(\frak A^D)$.   As
$\#(\frak A)\le\#(\Sigma)\le\frak c$ (424Db), $\#(\Aut\frak A)$ is at most
$\frak c$ (4A1A(c-ii)).\ \Qed}

We therefore have\cmmnt{ a corollary of 425D, as follows:}

\inset{Suppose the continuum hypothesis is true.   Let $(X,\Sigma)$ be a
standard Borel space and
$\Cal I$ a $\sigma$-ideal of subsets of $\Sigma$ containing an uncountable
set.   Then there is an action $\action$ of
$\Aut(\Sigma/\Cal I)$ on $X$ such that
$\pi E^{\ssbullet}=(\pi\action E)^{\ssbullet}$
whenever $E\in\Sigma$ and $\pi\in\Aut(\Sigma/\Cal I)$.}

\exercises{\leader{425X}{Basic exercises (a)}
%\sqheader 425Xa
(Cf.\ 382Xc.)
Let $(X,\Sigma)$ be a standard Borel space, $\Cal I$ a
$\sigma$-ideal of subsets of $X$ and $\frak A=\Sigma/\Cal I$ the quotient
algebra.   (i) Show that every member of
$\Aut\frak A$ has a separator (definition:  382Aa).   (ii) Show that
if $G$ is a countably full subgroup of $\Aut\frak A$, then every
member of $G$ is expressible as the product of at most three involutions
belonging to $G$.
%425- 382M

\spheader 425Xb Let $(X,\Sigma)$ be a standard Borel space and set
$\frak A=\Sigma/[X]^{\le\omega}$.
(i)\dvAformerly{4{}24Xk} Show that the Boolean
algebra $\frak A$ is homogeneous (definition:  316N).
(ii) Show that $\Aut\frak A$ is simple.
\Hint{382Yc.}
%425Xa

\spheader 425Xc Let $(X,\Sigma)$ be a standard Borel
space and $\Cal I$ a $\sigma$-subalgebra of $\Sigma$ with associated
quotient algebra $\frak A=\Sigma/\Cal I$.
Suppose that $G$ is a countable semigroup of sequentially
order-continuous Boolean homomorphisms from $\frak A$ to itself.
Show that there is a family
$\family{\pi}{G}{f_{\pi}}$ of $(\Sigma,\Sigma)$-measurable functions from
$X$ to itself such that ($\alpha$)
$\pi E^{\ssbullet}=f_{\pi}^{-1}[E]^{\ssbullet}$
for every $\pi\in G$ and $E\in\Sigma$ ($\beta$)
$f_{\pi\phi}=f_{\phi}f_{\pi}$ for all $\pi$, $\phi\in G$.
%425A
%should be early in 425X, see 425Xd

\spheader 425Xd Let $X$ be a set,
$\Sigma$ a $\sigma$-algebra of subsets of $X$, $\Cal I$ a $\sigma$-ideal of
$\Sigma$ and $\frak A$ the quotient $\Sigma/\Cal I$.
Suppose that
$(X,\Sigma)$ is countably separated in the sense that there is a
countable subset of $\Sigma$ separating the points of $X$.
Let $G$ be a countable subsemigroup of the semigroup of
Boolean homomorphisms from $\frak A$ to itself such that
for every $\pi\in G$
there is a $(\Sigma,\Sigma)$-measurable $g:X\to X$ such that
$\pi E^{\ssbullet}=g^{-1}[E]^{\ssbullet}$ for every $E\in\Sigma$.
Show that there is a
family $\family{\pi}{G}{f_{\pi}}$ of $(\Sigma,\Sigma)$-measurable functions
from $X$ to itself such that
$f_{\pi}^{-1}[E]^{\ssbullet}=\pi E^{\ssbullet}$ and
$f_{\pi\phi}=f_{\phi}f_{\pi}$ whenever $\pi$, $\phi\in G$ and $E\in\Sigma$.
\Hint{344B.}
%425A

\spheader 425Xe Show that the set $\maltese$ in 425Cb is a G$_{\delta}$ set
in the compact metrizable space $\Cal P(\BbbN^3)$.
%425C

\spheader 425Xf Give an example of a standard Borel space $(X,\Sigma)$, a
$\sigma$-ideal $\Cal I$ of $\Sigma$, a finite subgroup $G$ of
$\Aut(\Sigma/\Cal I)$, a subgroup $H$ of $G$ and an action $\action_0$ of
$H$ on $X$ such that
$\pi E^{\ssbullet}=(\pi\action_0E)^{\ssbullet}$ whenever
$\pi\in H$ and $x\in X$, but there is no action $\action$ of $G$ on $X$,
extending $\action_0$, such that
$\pi E^{\ssbullet}=(\pi\action E)^{\ssbullet}$ whenever
$\pi\in G$ and $x\in X$.  \Hint{$\#(X)=6$, $\#(H)=2$.}
%\Aut\frak A\cong S_4, H\subseteq A_4 mt42bits
%425D

\spheader 425Xg Let $I^{\|}$ be the split interval with its usual topology
and measure, and $\frak A$ its measure algebra.   Let $G$ be a subgroup of
$\Aut\frak A$ with cardinal at most $\omega_1$.   Show that there is
an action $\action$ of $G$ on $I^{\|}$ such that
$\pi\action E$ is a Borel set and
$(\pi\action E)^{\ssbullet}=\pi E^{\ssbullet}$
for every $E\in\Cal B(I^{\|})$ and $\pi\in G$.
%425D take \pi\action(t^{+/-})=(\pi\action't)^{+/-}

\leader{425Y}{Further exercises (a)}
%\spheader 425Ya
({\smc T\"ornquist 11}) Let $X$ be a set,
$\Sigma$ a $\sigma$-algebra of subsets of $X$, $\Cal I$ a $\sigma$-ideal of
$\Sigma$ and $\frak A$ the quotient $\Sigma/\Cal I$.   Suppose that
$(X,\Sigma,\Cal I)$ is countably separated in the sense that there is a
countable subset of $\Sigma$ separating the points of $X$, and complete in
the sense that $A\in\Cal I$ whenever $A\subseteq B\in\Cal I$.
Let $G\subseteq\Aut\frak A$ be a group with cardinal
at most $\omega_1$ such that for every $\pi\in G$ there is a
$(\Sigma,\Sigma)$-measurable function $g:X\to X$ such that
$\pi E^{\ssbullet}=g^{-1}[E]^{\ssbullet}$ for every $E\in\Sigma$.
Show that if $\Cal I$ contains a set with cardinal $\frak c$
there is an action $\action$ of $G$ on $X$ such that
$\pi E^{\ssbullet}=(\pi\action E)^{\ssbullet}$ for every $\pi\in G$ and
$E\in\Sigma$.
%425D mt42bits

\spheader 425Yb
Let $(X,\Sigma,\mu)$ be a countably separated
perfect complete strictly localizable measure space,
$\frak A$ its measure algebra and $G$ a subgroup of $\Aut\frak A$ of
cardinal at most $\omega_1$.   Show that there is an action
$\action$ of $G$ on $X$ such that
$\pi\action E\in\Sigma$ and
$(\pi\action E)^{\ssbullet}=\pi(E^{\ssbullet})$ whenever
$\pi\in G$ and $E\in\Sigma$.
%425Ya 425D

}%end of exercises

\leader{425Z}{Problems (a)} %425Za
Suppose that $(X,\Sigma)$ is an uncountable
standard Borel space and $\Cal I$ the ideal $[X]^{\le\omega}$\cmmnt{ of
countable
subsets of $X$}.   Which subgroups $G$ of $\Aut(\Sigma/\Cal I)$ can be
represented by actions of $G$ on $X$?

\spheader 425Zb Let $(\frak A,\bar\mu)$ be the measure algebra of Lebesgue
measure $\mu$ on $[0,1]$, and $G$ a semigroup of measure-preserving Boolean
homomorphisms from $\frak A$ to itself with $\#(G)=\omega_1$.
Must there be a family $\family{\pi}{G}{f_{\pi}}$ of \imp\ functions from
$[0,1]$ to itself such that $f_{\pi\phi}=f_{\phi}f_{\pi}$ for all $\pi$,
$\phi\in G$ and $f_{\pi}$ represents $\pi$, in the sense
of 425A, for every $\pi\in G$?   \cmmnt{(See 344B.)}

\spheader 425Zc Let $(\frak B_{\omega_1},\bar\nu_{\omega_1})$ be the
measure algebra of the usual measure on $\{0,1\}^{\omega_1}$, and $G$ a
group of measure-preserving automorphisms of $\frak B_{\omega_1}$ with
$\#(G)=\omega_1$.   Must there be a family $\family{\pi}{G}{f_{\pi}}$ of
\imp\ functions from
$\{0,1\}^{\omega_1}$ to itself such that $f_{\pi\phi}=f_{\phi}f_{\pi}$ for
all $\pi$,
$\phi\in G$ and $f_{\pi}$ represents $\pi$ for every $\pi\in G$?
\cmmnt{(See 344E.)}

\endnotes{
\Notesheader{425} I have starred this section because (apart from 425A)
it deals with a very
special topic.   From the point of view of measure theory, 425D 
applies only
to copies of Borel measures on $\Bbb R$ (and not quite all of those),
and the limitation to groups of
cardinal $\omega_1$ means that we need to assume the continuum hypothesis,
or at least $\frak p=\frak c$ (535Yd),
to get a theorem we really want.   However the result
connects naturally with an
important theme from Chapter 34, and the general question of simultaneous
representation of many automorphisms has significant implications for
the ergodic theory treated in Chapter 38 and \S494 of this volume.

What makes 425D difficult is the ambitious target:  we
want to represent the automorphisms in $G$ by
a consistent family of Borel measurable
functions.   We know from Chapter 34 that we can hope to handle countable
groups, so it is natural to start by expressing $G$ as an inductive limit
of a family $\family{\alpha}{F}{G_{\alpha}}$, as in parts (a)-(b) of the
proof of 425D, and to try to define the action of $G$ from actions of the
$G_{\alpha}$.   Since any $\pi\in G$ must eventually determine a
Borel measurable function on $X$, we are going to have to
freeze its action at some point;  we could afford to change it once or
twice, or even countably often, as the induction continued, but sometime
we must stop tinkering, and really we want to have
$\pi\action x=\pi\action_{\alpha}x$ whenever $x\in X$ and
$\pi\in G_{\alpha}$.   In this case, $\action_{\beta}$ will have to be a
direct extension of $\action_{\alpha}$ whenever $\beta>\alpha$.
We are in a context in which arbitrary actions are not always
extensible (425Xf), and something like Lemma 425B is going to be needed.
This demands a plentiful supply of copies of shift actions,
at the very least including representations of all the shift actions on
$X^{G_{\alpha}}$, which will have to be built in from the very beginning.
The trouble is that these have to be assembled in a way which will
give us Borel measurable functions.   Now while $X$ can be taken to be a
fixed Polish space, the groups $G_{\alpha}$ are more or less arbitrary
countable groups.   They can all be represented by group structures on
$\Bbb N$, but if we go by that road, we seem to need to choose
injective functions from countable ordinals into $\Bbb N$.   (Of course
each $G_{\alpha}$ comes with a bijection between it and the ordinal
$\alpha$.)
No direct enumeration of these is going to lead to Borel structures.
Instead, we have to look at the whole set of group structures on
$\Bbb N$, the set $\maltese$ of 425Cb, and nearly all injective
functions from countable ordinals to $\Bbb N$, coded by subsets of
$\BbbN^2$, as in 425Ca.   Fortunately it does not matter that there is a
great deal of redundancy in this coding;  it can all be fitted naturally
into a standard Borel structure, and we just need to include, in the
hypotheses of 425D, a negligible set of size $\frak c$.
When $\Cal I\subseteq[X]^{\le\omega}$, the problem changes (425Za).

In order to ensure that each of the actions $\action_{\alpha}$ correctly
represents the action of $G_{\alpha}$ on $\frak A$, we can use essentially
the same method as that of 344B (part (c) of the proof of 425D).
This means, of course, that 425B has to be applied to a carefully chosen
fragment of $X$, the set $X'$ of part (d) of the proof of 425D.
There is an awkward shift here between the representation of an
automorphism $\pi$ by the function $g_{\pi}$, where I follow the
conventions used in Chapter 34 and 425A here with the contravariant formula
$\pi(E^{\ssbullet})=(g_{\pi}^{-1}[E])^{\ssbullet}$,
and the representation in the statement of this theorem, 
with the covariant formula
$\pi E^{\ssbullet}=(\pi\action E)^{\ssbullet}$.   The latter is
forced by the rule of 4A5Ba that
$(\pi\phi)\action x=\pi\action(\phi\action x)$.   By allowing `reverse
actions', in which $(\pi\phi)\action x=\phi\action(\pi\action x)$, we could
escape this conflict, and in the formulae of parts (d)-(e)
of the proof of
425D we should have $\pi\action_{\alpha}x=g_{\pi}(x)$ for $x\in Y_{\alpha}$
and $\pi\in G_{\alpha}$.   This would be essential if we wanted to use
the formulae here on semigroups of Boolean homomorphisms, as in \S344,
so that the $g_{\pi}$ were no longer injective on conegligible sets.   But
there seem to be more substantial obstacles (425Zb).

The proof of 425D appeals repeatedly to the special properties of standard
Borel spaces.   But conceivably enough of it can be applied to the Baire
$\sigma$-algebras of powers of $\{0,1\}$
to give a similar result for other important probability spaces (425Zc).
If we change the rules, and assume that $\Cal I$ is an ideal of $\Cal PX$
as well as of $\Sigma$,
we can dispense with the ideas of 425C and work directly from 425Ba,
using copies of
$\Bbb N\times X^{G_{\alpha}}$ inside a negligible set $M$ (425Ya);
this gives us an approach to use on complete measure spaces (425Yb,
535Yd).

Another way of looking at 425D is to think of it as a kind of
lifting theorem.   Let $\Phi$ be the group of 
$(\Sigma,\Sigma)$-bimeasurable
$\Cal I$-invariant permutations of $X$.   Then each $f\in\Phi$ induces an
automorphism $f^{\smallcirc}$ of $\frak A$ defined by saying that
$f^{\smallcirc}(E^{\ssbullet})=(f[E])^{\ssbullet}$ for each $E\in\Sigma$.
(I am using the push-forward rather than the pull-back representation
here.)   425Ac is enough to show that
the group homomorphism
$f\mapsto f^{\smallcirc}:\Phi\to\Aut\frak A$ is surjective.   Under the
conditions of 425D, the action $\action$ corresponds to a group
homomorphism $\theta:G\to\Phi$ such that $(\theta\pi)^{\smallcirc}=\pi$ for
every $\pi\in G$;  and subject to the continuum hypothesis, we have a
lifting for the whole of $\Aut\frak A$.   The word `lifting' in this
context should remind you of the Lifting Theorem of measure theory (341K).
That theorem demands a complete measure space, and does not
ordinarily apply to Borel measures.   However, subject to the continuum
hypothesis, there is a lifting theorem applicable to a variety of
non-complete measure spaces, including any $(X,\Sigma,\mu)$
where $(X,\Sigma)$ is a standard Borel space and $\mu$ is $\sigma$-finite
(535E(b-i) of Volume 5).   I am not sure that it is really helpful to think
of 425E and 535E together;  certainly the manoeuvres of 425C have no
analogues in the lifting theorems of measure theory.   But the
correspondence is striking and suggests directions of enquiry which may be
worth exploring.
}

\discrpage

\frfilename{mt42conc.tex}
\versiondate{30.11.16}
\copyrightdate{2016}

\gdef\topparagraph{}
\gdef\bottomparagraph{}
\gdef\newparagraph{}

\def\chaptername{}
\def\sectionname{Concordance}

\newsection{}

I list here the section and paragraph numbers which have (to my
knowledge) appeared in print in references to this chapter, and which
have since been changed.

\medskip

\leaveitout{4{}21H 421H projection of set in
$\Sigma\tensorhat\Cal B$ is kernel of a Souslin scheme in $\Sigma$
{\smc Fremlin 03}
}

\leaveitout{4{}21P 421P Let $X$ be a set and $\Sigma$ a
$\sigma$-algebra of subsets of $X$.   If $A\in\Cal S(\Sigma)$ then both
$A$ and $X\setminus A$ can be expressed as the union of at most
$\omega_1$ members of $\Sigma$.   {\smc Fremlin 03}
}

\leader{4{}23J}{} Extra results have been interpolated into \S4{}23, so
the second half of that section (4{}23J-4{}23R), referred to in the 2008 and
2015 editions of Volume 5, is now 423K-423T.
% 4{}23K 4{}23L 4{}23M 4{}23N 4{}23O 4{}23Q 4{}23S

\leaveitout{4{}24Cb 424Cb
An uncountable standard Borel space $(X,\Sigma)$ is
isomorphic to $(\NN,\Cal B(\NN))$ {\smc Fremlin 03}
}

\frnewpage
%\fi


\references{
\frfilename{mt4r.tex}
\versiondate{9.2.13}
\copyrightdate{2002}

% 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}

\Loadcyrillic

\Centerline{\bf References for Volume 4}


\medskip%A

Aldaz J.M.\ [95] `On compactness and Loeb measures', Proc.\ Amer.\
Math.\ Soc.\ 123 (1995) 173-175.
\cmmnt{[439Xa.]}

Aldaz J.M.\ [97] `Borel extensions of Baire measures', Fund.\ Math.\ 154
(1997) 275-293.
\cmmnt{[434Yn.]}

Aldaz J.M.\ \& Render H.\ [00] `Borel measure extensions of measures
defined on sub-$\sigma$-algebras', Advances in Math.\ 150 (2000)
233-263.
\cmmnt{[432D.]}

Alexandroff P.S.\ \& Urysohn P.S.\ [1929] {\it  M\'emoire sur les espaces
topologiques compacts},
Verhandelingen Akad.\ Wetensch.\ Amsterdam 14 (1929) 1-96.
\cmmnt{[419L.]}

Anderson I.\ [87] {\it Combinatorics of Finite Sets.}  Oxford U.P.,
1987.
\cmmnt{[4A4N.]}

Andretta A.\ \& Camerlo R.\ [13] `The descriptive set theory of the
Lebesgue density theorem', Advances in Math.\ 234 (2013).
%{\tt http://arxiv.org/abs/1105.3355}.
\cmmnt{[475Yg.]}

Arkhangel'skii A.V.\ [92] {\it Topological Function Spaces.}  Kluwer,
1992.
\cmmnt{[\S462 {\it intro.}, \S467 {\it notes\/}.]}

\leaveitout{
Armitage D.H.\ \& Gardiner S.J.\ [01] {\it Classical Potential Theory.}
Springer, 2001.
478Yd}

Aronov B., Basu S., Pach J.\ \& Sharir M.\ [03] (eds.)
{\it Discrete and Computational Geometry;  the Goodman-Pollack
Festschrift,}  Springer, 2003.

Asanov M.O.\ \& Velichko N.V.\ [81] `Compact sets in $C_p(X)$', Comm.\
Math.\ Helv.\ 22 (1981) 255-266.
\cmmnt{[462Ya.]}

Austin T.\ [10a] `On the norm convergence of nonconventional ergodic
averages', Ergodic Theory and Dynamical Systems 30 (2010) 321-338.
%({\tt http://{\penalty-100}arxiv.org/abs/{\penalty-100}0805.0320}).
\cmmnt{[497M.]}

Austin T.\ [10b] `Deducing the multidimensional Szemer\'edi theorem from
the infinitary hypergraph removal lemma',
J.\ d'Analyse Math.\ 111 (2010) 131-150.
%({\tt http://arxiv.org{\bsp}abs/0808.2267}).
\cmmnt{[497M.]}

\leaveitout{
Avil\'es A., Plebanek G.\ \& Rodr\'{\i}guez J.\ [p11] `Measurability in
$C(2^{\kappa})$ and Kunen cardinals', 2011.
%material on  C(\{0,1\}^{\kappa}) ;  Baire and Borel \sigma-algebras
%in weak and pointwise topologies
}

\leaveitout{Axler S. \& Ramey W.\ [95]
`Harmonic polynomials and Dirichlet-type problems',
Proc.\ Amer.\  Math.\ Soc.\ 123 (1995) 3765-3773.
http://www.axler.net/HarmonicPolynomials.html
478Yd}

\medskip%B

Balogh Z.\ [96] `A small Dowker space in ZFC', Proc.\ Amer.\ Math.\
Soc.\ 124 (1996) 2555-2560.
\cmmnt{[439O.]}

Banach S.\ [1923] `Sur le probl\`eme de la mesure',
Fundamenta Math.\ 4 (1923) 7-33.
\cmmnt{[449O.]}

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{[424H, 448P, \S448 {\it notes\/}.]}

Bellow A.\ {\it see} Ionescu Tulcea A.

Bellow A.\ \& K\"olzow D.\ [76] {\it Measure Theory, Oberwolfach 1975.}
Springer, 1976 (Lecture Notes in Mathematics 541).
%4r

Bergelson V.\ [85] `Sets of recurrence of $\Bbb Z^m$-actions and properties of
sets of differences in $\Bbb Z^m$',
J.\ London Math.\ Soc.\ (2) 31 (1985) 295-304.
%http://people.dm.unipi.it/dinasso/ULTRA2015/BUMN/Bergelson%20-%20Sets%20of%20recurrence%20of%20Zm-actions%20and%20properties%20of%20sets%20of%20differences%20in%20Zm%20(1985).pdf
\cmmnt{[491Yt.]}

Bergman G.M.\ [06] `Generating infinite symmetric groups', Bull.\ London
Math.\ Soc.\ 38 (2006) 429-440.
%http://math.berkeley.edu/~gbergman/papers/
%({\tt http:/{\bsp}arxiv.org{\bsp}abs{\bsp}math{\bsp}0401304v2}).
\cmmnt{[\S494 {\it notes}.]}

Billingsley P.\ [99] {\it Convergence of Probability Measures.}  Wiley,
1999.
\cmmnt{[\S4A3 {\it notes}.]}
%second edition ISBN 0-471-19745-9

Blackwell D.\ [56] `On a class of probability spaces',
pp.\ 1-6 in {\smc Neyman 56}.
\cmmnt{[419K, 452P.]}

Bochner S.\ [1933] `Monotone Funktionen, Stieltjessche Integrale, und
harmonische Analyse', Math.\ Ann.\ 108 (1933) 378-410.
\cmmnt{[445N.]}

\indexheader{Bogachev}
Bogachev V.I.\ [07] {\it Measure theory.}  Springer, 2007.
\cmmnt{[437Q, 437Xs, 437Yo, 437Yu, 457K.]}

Bollob\'as B.\ [79] {\it Graph Theory.}   Springer, 1979.
\cmmnt{[4A1G, 4A4N.]}

Bongiorno B., Piazza L.di \ \& Preiss D.\ [00]
`A constructive minimal integral which includes Lebesgue integrable
functions and derivatives', J.\ London Math.\ Soc.\ (2) 62 (2000) 117-126.
\cmmnt{[483Yi.]}

Borodulin-Nadzieja P.\ \& Plebanek G.\ [05] `On compactness of measures on
Polish spaces', Illinois J.\ Math.\ 49 (2005) 531-545.
\cmmnt{[451L.]}

Bourbaki N.\ [65] {\it Int\'egration,} chaps.~1-4.   Hermann, 1965
(Actualit\'es Scientifiques et Industrielles 1175).
\cmmnt{[\S416 {\it notes\/}, \S436 {\it notes\/}.]}

Bourbaki N.\ [66] {\it Espaces Vectoriels Topologiques}, chaps.~1-2,
$2^{\text{e}}$ \'ed.
Hermann, 1966 (Actualit\'es Scientifiques et Industrielles 1189).
\cmmnt{[4A2J.]}
%Store A QA 37.B6 no 7792 0070 5

Bourbaki N.\ [69] {\it Int\'egration,} chap.~9.   Hermann, 1969
(Actualit\'es Scientifiques et Industrielles 1343).
\cmmnt{[\S436 {\it notes.}]}

Bourbaki N.\ [87] {\it Topological Vector Spaces.}  Springer, 1987.
\cmmnt{[\S461 {\it notes\/}, \S4A4.]}

\indexheader{Bourgain}
Bourgain J., Fremlin D.H.\ \& Talagrand M.\ [78] `Pointwise compact
sets of Baire-measurable functions', Amer.\ J.\ Math.\ 100 (1978)
845-886.
\cmmnt{[462Ye.]}

\indexheader{Brodski\v{\i}}
Brodski\v{\i} M.L.\ [1949] `On some properties of sets of positive measure',
Uspehi Matem.\ Nauk (N.S.) 4 (1949) 136-138.
\cmmnt{[498B.]}

Brook R.B.\ [70] `A construction of the greatest ambit',
Math.\ Systems Theory 4 (1970) 243-248.
\cmmnt{[449D.]}

Burke D.K.\ \& Pol R.\ [05] `Note on separate continuity and the Namioka
property', Top.\ Appl.\ 152 (2005) 258-268.
\cmmnt{[\S463 {\it notes.}]}

Burke M.R., Macheras N.D.\ \& Strauss W.\ [p21] `The strong marginal lifting
problem for hyperstonian spaces', preprint (2021).
%bms5s.pdf

\medskip%C

Carrington D.C.\ [72] `The generalised Riemann-complete integral', PhD
thesis, Cambridge, 1972.
\cmmnt{[481L.]}

\v{C}ech E.\ [66] {\it Topological Spaces.}  Wiley, 1966.
\cmmnt{[\S4A2, 4A3S.]}

Chacon R.V. [69]
`Weakly mixing transformations which are not strongly mixing',
Proc.\ Amer.\ Math.\ Soc.\ 22 (1969) 559-562.
\cmmnt{[494F.]}

\indexheader{Choquet}
Choquet G.\ [55] `Theory of capacities', Ann.\ Inst.\ Fourier (Grenoble)
5 (1955) 131-295.
\cmmnt{[432K.]}

\indexheader{Chung}
Chung K.L.\ [95] {\it Green, Brown and Probability.}   World Scientific,
1995.
% 978-9810225339  HA33.4.C5  479Xt

\indexheader{Ciesielski}
Ciesielski K.\ \& Pawlikowski J.\ [03]
`Covering Property Axiom CPA$_{\text{cube}}$ and its consequences',
Fundamenta Math. 176 (2003) 63-75.
\cmmnt{[498C.]}

Cs\'asz\'ar \'A.\ [78] {\it General Topology.}  Adam Hilger, 1978.
\cmmnt{[\S4A2, \S4A5.]}

\medskip%D

Davies R.O.\ [70] `Increasing sequences of sets and Hausdorff measure', Proc.\ London Math.\ Soc.\ (3) 20 (1970) 222-236.
\cmmnt{[471G.]}

Davies R.O.\ [71] `Measures not approximable or not specifiable by means
of balls', Mathematika 18 (1971) 157-160.
\cmmnt{[\S466 {\it notes\/}.]}

Davis W.J., Ghoussoub N.\ \& Lindenstrauss J.\ [81] `A lattice-renorming
theorem and applications to vector-valued processes', Trans.\ Amer.\
Math.\ Soc.\ 263 (1981) 531-540.
\cmmnt{[\S467 {\it notes\/}.]}

\indexheader{de Caux}
de Caux P.\ [76] `A collectionwise normal, weakly $\theta$-refinable
Dowker space which is neither irreducible nor realcompact', Topology
Proc.\ 1 (1976) 67-77.
\cmmnt{[439O.]}

Dellacherie C.\ [80] `Un cours sur les ensembles analytiques', pp.\
183-316 in {\smc Rogers 80}.
\cmmnt{[\S432 {\it notes\/}.]}

de Maria J.L.\ \& Rodriguez-Salinas B.\ [91] `The space
$(\ell_{\infty}/c_0)$,weak) is not a Radon space', Proc.\ Amer.\ Math.\
Soc.\ 112 (1991) 1095-1100.
\cmmnt{[466I.]}

Deville R., Godefroy G.\ \& Zizler V.\ [93] {\it Smoothness and
Renormings in Banach Spaces.}  Pitman, 1993.
\cmmnt{[\S467 {\it intro.}, 467Ye, \S467 {\it notes\/}.]}

Diaconis, P.\ \& Freedman, D.\ [80] `Finite exchangeable sequences.'
Ann. of Probability 8 (1980) 745-764.
\cmmnt{[459Xd.] }

Droste M., Holland C.\ \& Ulbrich G.\ [08]
`On full groups of measure preserving and ergodic transformations with
uncountable cofinalities',
Bull.\ London Math. Soc.\ 40 (2008) 463-472.
%{\tt http://web.mac.com/chollan1{\bsp}iWeb/Site{\bsp}publications.html}.
\cmmnt{[494Q.]}

Dubins L.E.\ \& Freedman D.A.\ [79]
`Exchangeable processes need not be mixtures of independent,
identically distributed random variables',
Z.\ Wahrscheinlichkeitstheorie und verw.\ Gebiete 48 (1979) 115-132.
\cmmnt{[459K.]}

Dunford N.\ \& Schwartz J.T.\ [57] {\it Linear Operators I.}  Wiley,
1957 (reprinted 1988).
\cmmnt{[\S4A4.]}

Du Plessis, N.\ [70] {\it Introduction to Potential Theory.}  Oliver \&
Boyd, 1970.
%used for \S478

\medskip%E

Eggleston H.G.\ [54] `Two measure properties of Cartesian product sets',
Quarterly J.\ Math.\ (2) 5 (1954) 108-115.
\cmmnt{[498B.]}

Emerson W.\ \& Greenleaf F.\ [67] `Covering properties and F{\o}lner
conditions for locally compact groups',
Math.\ Zeitschrift 102 (1967) 370-384.
\cmmnt{[449J.]}

Enderton H.B.\ [77] {\it Elements of Set Theory.}   Academic, 1977.
\cmmnt{[4A1A.]}

Engelking R.\ [89] {\it General Topology.} Heldermann, 1989 (Sigma
Series in Pure Mathematics 6).
\cmmnt{[Vol.\ 4 {\it intro.}, 415Yb, \S434 {\it notes\/},
\S435 {\it notes\/}, 462L, \S4A2, 4A4B.]}

Evans L.C.\ \& Gariepy R.F.\ [92] {\it Measure Theory and Fine
Properties of Functions.}   CRC Press, 1992.
\cmmnt{[Chap.\ 47 {\it intro.}, \S473 {\it intro.}, \S473 {\it notes\/},
\S474 {\it notes\/}, \S475 {\it notes\/}.]}

\medskip%F

Falconer K.J.\ [90] {\it Fractal Geometry.}  Wiley, 1990.
\cmmnt{[\S471 {\it intro.}]}

Federer H.\ [69] {\it Geometric Measure Theory.}
Springer, 1969 (reprinted 1996).
\cmmnt{[\S441 {\it notes\/}, 442Ya,
\S443 {\it notes\/}, Chap.\ 47 {\it intro.}, 471Yk,
\S471 {\it notes\/}, \S472 {\it notes\/}, \S475 {\it notes\/}.]}

Fernique X.\ [97] {\it Fonctions al\'eatoires gaussiennes,
vecteurs al\'eatoires gaussiens}, CRM Publications, 1997.
\cmmnt{[\S456 {\it notes\/}.]}

Folland G.B.\ [95] {\it A Course in Abstract Harmonic Analysis.}   CRC
Press, 1995
\cmmnt{[\S4A5, \S4A6.]}

\indexheader{F{\o}lner}
F{\o}lner E.\ [55] `On groups with full Banach mean value',
Math.\ Scand.\ 3 (1955) 243-254.
\cmmnt{[\S449 {\it notes\/}.]}

\indexheader{Ford}
Ford L.R.\ \& Fulkerson D.R.\ [56] `Maximal flow through a network',
Canadian J.\ Math.\ 8 (1956) 399-404.
\cmmnt{[4A4N.]}

Frankl P.\ \& R\"odl V.\ [02] `Extremal problems on set systems',
Random Structures Algorithms 20 (2002) 131-164.
\cmmnt{[497L.]}

\indexheader{Fremlin}
Fremlin D.H.\ [74] {\it Topological Riesz Spaces and
Measure Theory.}   Cambridge U.P., 1974.
\cmmnt{[\S462 {\it notes\/}.]}

Fremlin D.H.\ [75a] `Pointwise compact sets of measurable functions',
Manuscripta Math.\ 15 (1975) 219-242.
\cmmnt{[463K.]}

Fremlin D.H.\ [75b] `Topological measure spaces:  two counter-examples',
Math.\ Proc.\ Cambridge Phil.\ Soc.\ 78 (1975) 95-106.
\cmmnt{[419C, 419D.]}

Fremlin D.H.\ [76] `Products of Radon measures:  a counterexample',
Canadian Math.\ Bull.\ 19 (1976) 285-289.
\cmmnt{[419E.]}

Fremlin D.H.\ [81] `Measurable functions and almost continuous
functions', Manuscripta Math.\ 33 (1981) 387-405.
\cmmnt{[451T.]}

Fremlin D.H.\ [84] {\it Consequences of Martin's Axiom.}
Cambridge U.P., 1984.
\cmmnt{[\S434 {\it notes\/}, 439Yd, 4A2E.]}

Fremlin D.H.\ [93] `Real-valued-measurable cardinals', pp.\ 151-304
in {\smc Judah 93}.
\cmmnt{[\S438 {\it notes\/}.]}

Fremlin D.H.\ [95] `The generalized McShane integral',
Illinois J.\ Math.\ 39 (1995) 39-67.
\cmmnt{[481N.]}

Fremlin D.H.\ [00] `Weakly $\alpha$-favourable measure spaces',
Fundamenta Math.\ 165 (2000) 67-94.
\cmmnt{[451V.]}

Fremlin D.H.\ [n05] `Strictly localizable Borel measures', note of 7.5.05
({\tt http:/{\bsp}www.\penalty-100essex.ac.uk{\bsp}maths/people{\bsp}fremlin/preprints.htm}).
\cmmnt{[434Yr.]}

Fremlin D.H.\ [n15] `The Sorgenfrey line is not a Prokhorov space',
note of 5.8.15
({\tt http:/{\bsp}www.\penalty-100essex.ac.uk{\bsp}maths/people{\bsp}fremlin/preprints.htm}).
\cmmnt{[439Yk.]}

Fremlin D.H.\ \& Grekas S.\ [95] `Products of completion regular
measures', Fund.\ Math.\ 147 (1995) 27-37.
\cmmnt{[434Q.]}

Fremlin D.H.\ \& Talagrand M.\ [78] `On the representation of increasing
linear functionals on Riesz spaces by measures',
Mathematika 25 (1978) 213-215.
\cmmnt{[439I.]}

Fremlin D.H.\ \& Talagrand M.\ [79] `A decomposition theorem for
additive set functions and applications to Pettis integral and ergodic
means', Math.\ Zeitschrift 168 (1979) 117-142.
\cmmnt{[\S464.]}  %{\it 464C}, 464I, 464M, 464N, \S464 {\it notes\/}

Fristedt B.\ \& Gray L.\ [97] {\it A Modern Approach to Probability
Theory.}  Birkh\"auser, 1997.
\cmmnt{[\S455 {\it notes\/}, \S495 {\it notes\/}.]}
%HA33.4.F7

Frol{\'\i}k Z.\ [61] `On analytic spaces', Bull.\ Acad.\ Polon.\ Sci.\ 
%8 (1961) 747-750.
9 (1961) 721-726.   %Paulo Monteiro, email 8.9.22
\cmmnt{[422F.]}

Frol\'\i k Z.\ [88] (ed) {\it General Topology and its Relations to
Modern Analysis and Algebra VI} (Proc.\ Sixth Prague Topological
Symposium), Heldermann, 1988.

Furstenberg H.\ [77]
`Ergodic behavior of diagonal measures and a theorem of Szemer\'edi
on arithmetic progressions', J. d'Analyse Math.\ 31 (1977), 204-256.
\cmmnt{[\S497 {\it intro.}, \S497 {\it notes\/}.]}

Furstenberg H.\ [81]
{\it Recurrence in Ergodic Theory and Combinatorial Number Theory.}
Princeton U.P., 1981.
\cmmnt{[\S497 {\it intro.}, 497N.]}
% QA614

Furstenberg H.\ \& Katznelson Y.\ [85]
`An ergodic Szemer\'edi theorem for IP-systems and combinatorial theory',
J.\ d'Analyse Math.\ 45 (1985) 117-168.
\cmmnt{[\S497 {\it intro.}]}

\medskip%G

Gaal S.A.\ [64] {\it Point Set Topology.}  Academic, 1964.
\cmmnt{[\S4A2.]}

Gardner R.J.\ [75] `The regularity of Borel measures and Borel
measure-compactness', Proc.\ London Math.\ Soc.\ (3) 30 (1975) 95-113.
\cmmnt{[438M.]}

Gardner R.J.\ \& Pfeffer W.F.\ [84] `Borel measures', pp. 961-1043 in
{\smc Kunen \& Vaughan 84}.
\cmmnt{[434G.]}

Giordano T.\ \& Pestov V.\ [02] `Some extremely amenable groups',
C.R.A.S.\ Paris (S\'er.\ I) 334 (2002) 273-278.
\cmmnt{[494I.]}

Glasner S.\ [98] `On minimal actions of Polish groups', Topology and its Applications 85 (1998) 119-125.
\cmmnt{[\S493 {\it notes\/}.]}

Gordon R.A.\ [94] {\it The Integrals of Lebesgue, Denjoy, Perron and
Henstock.}   Amer.\ Math.\ Soc., 1994 (Graduate Studies in Mathematics
4).
\cmmnt{[481Q, \S483 {\it notes\/}.]}

Greenleaf F.P.\ [69] {\it Invariant Means on Topological Groups and Their
Applications.}   van Nostrand, 1969.
\cmmnt{[\S449 {\it notes.}]}

Grothendieck A. [92] {\it Topological Vector Spaces.}  Gordon \& Breach,
1992.
\cmmnt{[\S462 {\it notes\/}.]}

\indexheader{Grzegorek}
Grzegorek E.\ [81] `On some results of Darst and Sierpi\'nski concerning
universal null and universally negligible sets', Bull.\ Acad.\ Polon.\
Sci.\ (Math.) 29 (1981) 1-5.
\cmmnt{[439F, \S439 {\it notes\/}.]}

\medskip%H

\indexheader{Halmos}
Halmos P.R.\ [1944] `In general a measure-preserving transformation is
mixing', Annals of Math.\ 45 (1944) 786-792.
\cmmnt{[494E.]}

Halmos P.R.\ [50] {\it Measure Theory.}   Springer, 1974.
\cmmnt{[\S441 {\it notes\/}, \S442 {\it notes\/}, 443M, \S443 {\it notes\/}.]}

Halmos P.R.\ [56] {\it Lectures on ergodic theory.}  Chelsea, 1956.
\cmmnt{[494A.]}

Hansell R.W.\ [01] `Descriptive sets and the topology of non-separable
Banach spaces', Serdica Math.\ J.\ 27 (2001) 1-66.
\cmmnt{[466D.]}

Hart J.E.\ \& Kunen K.\ [99] `Orthogonal continuous functions',
Real Analysis Exchange 25 (1999) 653-659.
\cmmnt{[416Yh.]}

Hartman S.\ \& Mycielski J.\ [58] `On the imbedding of topological groups into connected topological groups', Colloq.\ Math.\ 5 (1958) 167-169.
\cmmnt{[493Ya.]}

Haydon R.\ [74] `On compactness in spaces of measures and
measure-compact spaces', Proc.\ London Math.\ Soc.\ (3) 29 (1974) 1-6.
\cmmnt{[438J.]}

Henry J.P.\ [69] `Prolongement des mesures de Radon', Ann.\ Inst.\
Fourier 19 (1969) 237-247.
\cmmnt{[416N.]}

Henstock R.\ [63] {\it Theory of Integration.}  Butterworths, 1963.
\cmmnt{[Chap.\ 48 {\it intro.}, 481J.]}

Henstock R.\ [91] {\it The General Theory of Integration.}   Oxford
U.P., 1991.
\cmmnt{[\S481 {\it notes\/}.]}

Herer W.\ \& Christensen J.P.R.\ [75] `On the existence of pathological submeasures and the construction of exotic topological groups', Math.\ Ann.\ 213 (1975) 203-210.
\cmmnt{[\S493 {\it notes\/}.]}

Herglotz G.\ [1911] `\"Uber Potenzreihen mit positivem reellen Teil im
Einheitskreis', Leipziger Berichte 63 (1911) 501-511.
\cmmnt{[445N.]}

Hewitt E.\ \& Ross K.A.\ [63] {\it Abstract Harmonic Analysis I.}
Springer, 1963 and 1979.
\cmmnt{[\S442 {\it notes\/}, \S444 {\it intro.}, \S444 {\it notes\/},
\S4A5, \S4A6.]}

Hewitt E.\ \& Savage L.J.\ [55]
`Symmetric measures on Cartesian products',
Trans.\ Amer.\ Math.\ Soc.\ 80 (1955) 470-501.
\cmmnt{[459Xe.]}

%Holick\'y P.\ \& Spurn\'y J.\ [03] `Perfect images of absolute Souslin
%and absolute Borel Tychonoff spaces', Topology and its Appl.\ 131 (2003)
%281-294.
%\cmmnt{[434Yr.]}

\indexheader{Howroyd}
Howroyd J.D.\ [95] `On dimension and on the existence of sets of finite
positive Hausdorff measure', Proc.\ London Math.\ Soc.\ (3) 70 (1995)
581-604.
\cmmnt{[471R, 471S.]}

\medskip%I

\indexheader{Ionescu Tulcea}
Ionescu Tulcea A.\ [73] `On pointwise convergence, compactness and
equicontinuity I', Z.\ Wahrscheinlichkeitstheorie und verw.\ Gebiete 26
(1973) 197-205.
\cmmnt{[463C.]}

Ionescu Tulcea A.\ [74] `On pointwise convergence, compactness, and
equicontinuity II', Advances in Math.\ 12 (1974) 171-177.
\cmmnt{[463G.]}

Ionescu Tulcea A.\ \& Ionescu Tulcea C.\ [67] `On the existence of a
lifting commuting with the left translations of an arbitrary locally
compact group', pp.\ 63-97 in {\smc LeCam \& Neyman 67}.
\cmmnt{[\S447 {\it intro.}, 447I, \S447 {\it notes\/}.]}

Ionescu Tulcea A.\ \& Ionescu Tulcea C.\ [69] {\it Topics in the Theory
of Lifting.}  Springer, 1969.
\cmmnt{[\S453 {\it intro.}]}

\indexheader{Ismail}
Ismail M.\ \& Nyikos P.J.\ [80] `On spaces in which countably compact
sets are closed, and hereditary properties', Topology and its Appl.\ 11
(1980) 281-292.
\cmmnt{[\S434 {\it notes\/}.]}

\medskip%J

%Jackson S.\ \& Mauldin R.D.\ [99] `On the $\sigma$-class generated by
%open balls', Math.\ Proc.\ Cambridge Phil.\ Soc.\ 127 (1999) 99-108.

Jameson G.J.O.\ [74] {\it Topology and Normed Spaces.}  Chapman \& Hall,
1974.
\cmmnt{[\S4A4.]}

\indexheader{Jayne}
Jayne J.E.\ [76] `Structure of analytic Hausdorff spaces', Mathematika
23 (1976) 208-211.
\cmmnt{[422Yc.]}

Jayne J.E.\ \& Rogers C.A.\ [95] `Radon measures on Banach spaces with
their weak topologies', Serdica Math.\ J.\ 21 (1995) 283-334.
\cmmnt{[466H, \S466 {\it notes\/}.]}

\indexheader{Jech}
Jech T.\ [78] {\it Set Theory.}  Academic, 1978.
\cmmnt{[423S, \S4A1.]}

Jech T.\ [03] {\it Set Theory, Millennium Edition.}  Springer, 2003.
%ISBN 3540440852
\cmmnt{[\S4A1.]}

Johnson R.A.\ [82] `Products of two Borel measures', Trans.\ Amer.\
Math.\ Soc.\ 269 (1982) 611-625.
\cmmnt{[434R.]}

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.

Juh\'asz I., Kunen K.\ \& Rudin M.E.\ [76] `Two more hereditarily
separable non-Lindel\"of spaces', Canadian J.\ Math.\ 29 (1976)
998-1005.
\cmmnt{[\S437 {\it notes\/}, 439K, \S439 {\it notes\/}.]}

Just W.\ \& Weese M.\ [96] {\it Discovering Modern Set Theory I.}  Amer.\ Math.\ Soc., 1996 (Graduate Studies in Mathematics 8).
\cmmnt{[4A1A.]}

Just W.\ \& Weese M.\ [97] {\it Discovering Modern Set Theory II.}  Amer.\ Math.\ Soc., 1997 (Graduate Studies in Mathematics 18).
\cmmnt{[\S4A1.]}

\medskip%K

Kampen, E.R.van [1935] `Locally bicompact abelian groups and their character groups',
Ann.\ of Math.\ (2) 36 (1935) 448-463.
\cmmnt{[445U.]}

\indexheader{Kaplansky}
Kaplansky I.\ [71] {\it Lie Algebras and Locally Compact Groups.}
Univ.\ of Chicago Press, 1971.
\cmmnt{[\S446 {\it notes.}]}

Kechris A.S.\ [95] {\it Classical Descriptive Set Theory.}  Springer,
1995.
\cmmnt{[Chap.\ 42 {\it intro.}, 423T, \S423 {\it notes\/},
\S448 {\it notes\/}, \S4A2, 4A3S, 4A3T.]}

%Kechris A.S.\ [p09] {\it Global Aspects of Ergodic Group Actions.}
%Amer.\ Math.\ Soc.\

Kellerer H.G.\ [84] `Duality theorems for marginal problems', Zeitschrift
f\"ur Wahrscheinlichkeitstheorie verw.\ Gebiete 67 (1984) 399-432.
\cmmnt{[457M.]}

Kelley J.L.\ [55] {\it General Topology.} Van Nostrand, 1955.
\cmmnt{[438R.]}

Kelley J.L.\ \& Srinivasan T.P.\ [71] `Pre-measures on lattices of sets',
Math.\ Annalen 190 (1971) 233-241.
\cmmnt{[413Xr.]}

Kittrell J.\ \& Tsankov T.\ [09] `Topological properties of full groups',
Ergodic Theory and Dynamical Systems 2009 (doi:10.1017/S0143385709000078).
\cmmnt{[494O, \S494 {\it notes}.]}

Kolmogorov A.N.\ [1933] {\it Grundbegriffe der
Wahrscheinlichkeitsrechnung.}  Springer, 1933;  translated as {\it
Foundations of Probability Theory,}  Chelsea, 1950.
\cmmnt{[454D.]}

K\"othe G.\ [69] {\it Topological Vector Spaces I.}   Springer, 1969.
\cmmnt{[\S462 {\it notes\/}, \S4A4, 4A6C.]}

Koumoullis G.\ \& Prikry K.\ [83] `The Ramsey property and measurable
selectors', J.\ London Math.\ Soc.\ (2) 28 (1983) 203-210.
\cmmnt{[451T.]}

Kuipers L.\ \& Niederreiter H.\ [74] {\it Uniform Distribution of
Sequences.}   Wiley, 1974.
\cmmnt{[491Z.]}

Kunen K.\ [80] {\it Set Theory.}  North-Holland, 1980.
\cmmnt{[\S4A1, 4A2E.]}

Kunen K.\ \& Vaughan J.E.\ [84] (eds.) {\it Handbook of Set-Theoretic
Topology.}   North-Holland, 1984.
%4r

Kuratowski K.\  [66] {\it Topology,} vol.\ I.   Academic, 1966.
\cmmnt{[423T, 462Ye, \S4A2, 4A3S.]}

Kurzweil J.\ [57] `Generalized ordinary differential equations and
continuous dependence on a parameter', Czechoslovak Math.\ J.\ 7 (1957)
418-446.
\cmmnt{[Chap.\ 48 {\it intro.}]}

Kwiatowska A. \& Solecki S.\ [11] `Spatial models of Boolean actions and
groups of isometries', Ergodic Theory Dynam.\ Systems 31 (2011) 405-421.
%{\tt http://www.math.uiuc.edu/\penalty-100$^{\sim}$ssolecki/papers.html}.
\cmmnt{[\S448 {\it notes\/}.]}

\medskip%L

Laczkovich M.\ [02] `Paradoxes in measure theory', pp.\ 83-123 in
{\smc Pap 02}.
\cmmnt{[\S449 {\it notes\/}.]}

LeCam L.M.\ \& Neyman J.\ [67] (eds.) {\it Proc.\ Fifth Berkeley
Symposium in Mathematical Statistics and Probability}, vol.\ II.
Univ.\ of California Press, 1967.

Levy A.\ [79] {\it Basic Set Theory.}  Springer, 1979.
\cmmnt{[\S4A1.]}

Lo\`eve M.\ [77] {\it Probability Theory I.} Springer, 1977.
\cmmnt{[\S495 {\it notes\/}.]}

\indexheader{Losert}
Losert V.\ [79] `A measure space without the strong lifting property',
Math. Ann. 239 (1979) 119-128.
\cmmnt{[453N, \S453 {\it notes\/}.]}

\indexheader{Luke\v{s}}
Luke\v{s} J., Mal\'y J.\ \& Zaj\'\i\v{c}ek L.\ [86]
{\it Fine topology methods
in real analysis and potential theory,} Springer, 1986 (Lecture Notes in
Mathematics 1189).
\cmmnt{[414Ye.]}

\medskip%M

\indexheader{Mackey}
Mackey G.W.\ [62] `Point realizations of transformation
groups', Illinois J.\ Math.\ 6 (1962) 327-335.
\cmmnt{[448S.]}

Marczewski E.\ [53] `On compact measures', Fundamenta Math.\ 40 (1953)
113-124.
\cmmnt{[413T, Chap.\ 45 {\it intro.}, \S451 {\it intro.}, 451J.]}

Marczewski E.\ \& Ryll-Nardzewski C.\ [53] `Remarks on the compactness
and non-direct products of measures', Fundamenta Math.\ 40 (1953)
165-170.
\cmmnt{[454C.]}

Ma\v{r}\'{\i}k J.\ [57] `The Baire and Borel measure', Czechoslovak
Math.\ J.\ 7 (1957) 248-253.
\cmmnt{[435C.]}

Mattila P.\ [95] {\it Geometry of Sets and Measures in Euclidean
Spaces.}   Cambridge U.P., 1995.
\cmmnt{[Chap.\ 47 {\it intro.}, \S471 {\it intro.}, \S472 {\it notes\/},
\S475 {\it notes\/}.]}
%QA311.M2 9520 6456 1

Mauldin R.D. \& Stone A.H.\ [81] `Realization of maps', Michigan Math.\
J.\ 28 (1981) 369-374.
\cmmnt{[418T.]}

Maurey B.\ [79] `Construction des suites sym\'etriques', C.R.A.S.\ (Paris) 288 (1979) 679-681.
\cmmnt{[492H.]}
  % one m in sym\'etrique

McShane E.J. [73] `A unified theory of integration', Amer.\ Math.\
Monthly 80 (1973) 349-359.
\cmmnt{[481M.]}

Meester R.\ \& Roy R.\ [96] {\it Continuum percolation.}  Cambridge
U.P., 1996.
\cmmnt{[\S495 {\it notes\/}.]}

Milman V.D.\ \& Schechtman G.\ [86]
{\it Asymptotic Theory of Finite Dimensional Normed Spaces.}
Springer, 1986 (Lecture Notes in Mathematics 1200).
\cmmnt{[492G, \S492 {\it notes\/}.]}

\leaveitout{Mohammed D.\ \& Saadi K.\ [p09#]
`Probl\`emes de mesurabilit\'es sur
l'espace des fonctions continues sur un compact', 2009.
\cmmnt{[466Y?.]}
%refereeing for Canad Math Bull;  rejected
}

Molt\'o A., Orihuela J., Troyanski S.\ \& Valdivia M.\ [09] {\it A
Non-linear Transfer Technique for Renorming.}   Springer, 2009 (Lecture
Notes in Mathematics 1951).
\cmmnt{[\S467 {\it notes.}]}

Montgomery D.\ \& Zippin L.\ [55] {\it Topological Transformation
Groups.}   Interscience, 1955.
\cmmnt{[\S446 {\it notes.}]}

\indexheader{Moran}
Moran W.\ [68] `The additivity of measures on completely regular
spaces', J.\ London Math.\ Soc.\ 43 (1968) 633-639.
\cmmnt{[439P.]}

Moran W.\ [69] `Measures and mappings on topological spaces', Proc.\
London Math.\ Soc.\ (3) 19 (1969) 493-508.
\cmmnt{[435Xk.]}

Moran W.\ [70] `Measures on metacompact spaces', Proc.\ London Math.\
Soc.\ (3) 20 (1970) 507-524.
\cmmnt{[438J.]}

\indexheader{Moschovakis}
Moschovakis Y.N.\ [80] {\it Descriptive Set Theory.}   Springer, 1980.
\cmmnt{[Chap.\ 42 {\it intro.}, \S423 {\it notes\/}.]}

Muldowney P.\ [87] {\it A General Theory of Integration in Function
Spaces.}   Longmans, 1987 (Pitman Research Notes in Mathematics 153).
\cmmnt{[481P, \S482 {\it notes\/}.]}

Mushtari D.H.\ [96] {\it Probabilities and Topologies on Linear Spaces.}
Kazan Mathematics Foundation, 1996.
%ISBN 5-900975-01-0

Musia\l\ K.\ [76] `Inheritness of compactness and perfectness of
measures by thick subsets',  pp.\ 31-42 in {\smc Bellow \& K\"olzow 76}.
\cmmnt{[451U.]}

\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{[448P, \S448 {\it notes\/}.]}

Nagle B., R\"odl V.\ \& Schacht M.\ [06]
`The counting lemma for regular k-uniform hypergraphs',
Random Structures and Algorithms 28 (2006) 113-179.
\cmmnt{[497J.]}

%Naimark M.A. [70] {\it Normed Rings.}  Wolters-Noordhoff, 1970.


Neyman J.\ [56] (ed.) {\it Proceedings of the Third Berkeley Symposium
on Mathematical Statistics and Probability, 1954-55,} vol.\ II.   Univ.\
of California, 1956.

\medskip%O
Oxtoby J.C.\ [70] `Homeomorphic measures on metric spaces',
Proc.\ Amer.\ Math.\ Soc.\ 24 (1970) 419-423.
\cmmnt{[434Yq.]}

\medskip%P

Pachl J.K.\ [78] `Disintegration and compact measures', Math.\ Scand.\
43 (1978) 157-168.
\cmmnt{[452I, 452S, 452Ye, \S452 {\it notes\/}.]}

Pachl J.K.\ [79] `Two classes of measures', Colloquium Math.\ 42 (1979)
331-340.
\cmmnt{[452R.]}

Pap E.\ [02] (ed.) {\it Handbook of Measure Theory.}  North-Holland, 2002.

Paterson A.L.T.\ [88] {\it Amenability.}   Amer.\ Math.\ Soc., 1988.
\cmmnt{[449K, \S449 {\it notes\/}.]}
% QA 403.P2 store A

Perron O.\ [1914] `\"Uber den Integralbegriff', Sitzungsberichte der
Heidelberger Akad.\ Wiss.\ A14 (1914) 1-16.
\cmmnt{[Chap.\ 48 {\it intro.}, 483J.]}

Pestov V.\ [99] `Topological groups:  where to from here?',
Topology Proc.\ 24 (1999) 421-502.
\cmmnt{[\S493 {\it notes\/}.]}

Pestov V.\ [02] `Ramsey-Milman phenomenon, Urysohn metric spaces,
and extremely amenable groups', Israel J.\ Math.\ 127 (2002) 317-357.
\cmmnt{[493E, 493Ya, \S493 {\it notes\/}.]}

Pfeffer W.F.\ [91a] `The Gauss-Green theorem', Advances in Math.\ 87
(1991) 93-147.
\cmmnt{[484Xb.]}

Pfeffer W.F.\ [91b] `Lectures on geometric integration and the
divergence theorem', Rend.\ Ist.\ Mat.\ Univ.\ Trieste 23 (1991)
263-314.
\cmmnt{[484B.]}

Pfeffer W.F.\ [01] {\it Derivation and Integration.}  Cambridge U.P.,
2001.
\cmmnt{[\S484 {\it notes\/}.]}

Phelps R.R.\ [66] {\it Lectures on Choquet's Theorem.}  van Nostrand,
1966.
\cmmnt{[\S461 {\it notes\/}.]}

\indexheader{Plewik}
Plewik S.\ \& Voigt B.\ [91] `Partitions of reals:  a measurable approach',
J.\ Combinatorial Theory (A) 58 (1991) 136-140.
\cmmnt{[443Yl.]}

\indexheader{Pontryagin}
Pontryagin L.S.\ [1934] `The theory of topological commutative groups',
Ann.\ Math.\ 35 (1934) 361-388.
\cmmnt{[445U.]}

\indexheader{Preiss}
Preiss D.\ [73] `Metric spaces in which Prohorov's theorem is not valid',
Z.\ Wahrscheinlichkeitstheorie und verw.\ Gebiete 27 (1973) 109-116.
\cmmnt{[439S.]}
%ILL 31.7.15

Preiss D.\ \& Ti\v{s}er J.\ [91]
`Measures in Banach spaces are determined by their values on balls',
Mathematika 38 (1991) 391-397.
\cmmnt{[\S466 {\it notes\/}.]}

Pryce J.D.\ [71] `A device of R.J.Whitley's applied to pointwise
compactness in spaces of continuous functions', Proc.\ London Math.\
Soc.\ (3) 23 (1971) 532-546.
\cmmnt{[462B, 462C.]}

\medskip%Q\medskip%R

Radon J.\ [1913] `Theorie und Anwendungen der absolut additivien
Mengenfunktionen,' Sitzungsbereichen der Kais.\ Akad.\ der Wiss.\ in
Wien 122 (1913) 28-33.
\cmmnt{[\S416 {\it notes\/}.]}

Ramachandran D.\ [02] `Perfect measures and related topics', pp.\
765-786 in {\smc Pap 02}.
\cmmnt{[\S451 {\it notes\/}.]}

\indexheader{Rao B.V.}
Rao B.V.\ [69] `On discrete Borel spaces and projective hierarchies',
Bull.\ Amer.\ Math.\ Soc.\ 75 (1969) 614-617.
\cmmnt{[419F.]}

Rao B.V. \& Srivastava S.M.\ [94] `Borel isomorphism theorem',
privately circulated, 1994.
\cmmnt{[424C.]}

\indexheader{Rao M.}
Rao M.\ [77] {\it Brownian motion and classical potential theory.}
Aarhus Universitet Matematisk Institut, 1977 (Lecture Notes Series 47);
{\tt http://www1.essex.ac.uk/maths/people{\bsp}fremlin{\bsp}rao.htm}.
%used for \S477

\indexheader{Ressel}
Ressel P.\ [77] `Some continuity and measurability results on spaces of
measures', Math.\ Scand.\ 40 (1977) 69-78.
\cmmnt{[417C, 437M.]}

Riesz F.\ \& Sz.-Nagy B.\ [55] {\it Functional Analysis.}  Frederick Ungar,
1955.
\cmmnt{[\S494 {\it notes\/}.]}

Rogers C.A.\ [70] {\it Hausdorff Measures.}   Cambridge, 1970.
\cmmnt{[\S471 {\it notes\/}.]}

Rogers C.A.\ [80] (ed.) {\it Analytic Sets.}   Academic, 1980.
\cmmnt{[Chap.\ 42 {\it intro.}, \S422 {\it notes\/}, \S423 {\it notes\/}.]}

Rogers L.C.J.\ \& Williams D.\ [94] {\it Diffusions, Markov Processes
and Martingales,} vol.\ I.   Wiley, 1994.
\cmmnt{[\S455 {\it notes\/}.]}
%HA33.4.W5

Rokhlin V.A.\ [1948] `A ``general'' measure-preserving transformation is not
mixing', Doklady Akademii Nauk SSSR 60 (1948) 349-351.
\cmmnt{[494E.]}

Rosendal C.\ [09]
`The generic isometry and measure preserving homeomorphism are conjugate
to their powers',  Fundamenta Math.\ 205 (2009) 1-27.
%({\tt http://www.math.uic.edu{\bsp}$\sptilde$rosendal/}).
\cmmnt{[494Ye.]}

Rosendal C.\ \& Solecki S.\ [07] `Automatic continuity of homomorphisms and
fixed points on metric compacta', Israel J.\ Math.\ 162 (2007) 349-371.
\cmmnt{[494Z, \S494 {\it notes}.]}

Rudin M.E.\ [71] `A normal space $X$ for which $X\times I$ is not
normal', Fund.\ Math.\ 73 (1971) 179-186.
\cmmnt{[439O.]}

Rudin M.E.\ [84] `Dowker spaces', pp.\ 761-780 in
{\smc Kunen \& Vaughan 84}.
\cmmnt{[439O.]}

Rudin W.\ [67] {\it Fourier Analysis on Groups.} Wiley, 1967 and 1990.
\cmmnt{[\S445 {\it intro\/}.]}

Rudin W.\ [91] {\it Functional Analysis.}  McGraw-Hill, 1991.
\cmmnt{[\S4A4, \S4A6.]}

Ryll-Nardzewski C.\ [53] `On quasi-compact measures', Fundamenta Math.\
40 (1953) 125-130.
\cmmnt{[451C.]}

\medskip%S

Sazonov V.V.\ [66] `On perfect measures', A.M.S.\ Translations (2) 48
(1966) 229-254.
\cmmnt{[451F.]}

Schachermayer W.\ [77] `Eberlein-compacts et espaces de Radon',
C.R.A.S.\ (Paris) (A) 284 (1977) 405-407.
\cmmnt{[467P.]}

Schaefer H.H.\ [71] {\it Topological Vector Spaces.}  Springer, 1971.
\cmmnt{[\S4A4.]}

Schwartz L.\ [73] {\it Radon Measures on Arbitrary Topological Spaces
and Cylindrical Measures.}   Oxford U.P., 1973.
\cmmnt{[\S434 {\it notes\/}.]}

Shelah S.\ \& Fremlin D.H.\ [93] `Pointwise compact and stable sets of
measurable functions', J.\ Symbolic Logic 58 (1993) 435-455.
\cmmnt{[\S465 {\it notes\/}.]}

\indexheader{Sierpi\'nski}
Sierpi\'nski W.\ [1945] `Sur une suite infinie de fonctions de classe 1 dont toute fonction d'accumulation est non mesurable', Fundamenta Math.\ 33 (1945) 104-105.
\cmmnt{[464C.]}

\indexheader{Solecki}
Solecki S.\ [01] `Haar null and non-dominating sets',
Fundamenta Math.\ 170 (2001) 197-217.
\cmmnt{[444Ye.]}

Solymosi J.\ [03]
`Note on a generalization of Roth's theorem', pp.\ 825-827 in
{\smc Aronov Basu Pach \& Sharir 03}.
\cmmnt{[497M.]}

Souslin M.\ [1917] `Sur une d\'efinition des ensembles mesurables B sans
nombres infinis', C.R.Acad.Sci.\ (Paris) 164 (1917) 88-91.
\cmmnt{[421D, \S421 {\it notes\/}.]}

Steen L.A.\ \& Seebach J.A.\ [78] {\it Counterexamples in Topology.}
Springer, 1978.
\cmmnt{[434Ya, \S439 {\it notes\/}.]}

Steinlage R.C.\ [75] `On Haar measure in locally compact T$_2$ spaces',
Amer.\ J.\ Math.\ 97 (1975) 291-307.
\cmmnt{[441C.]}

Strassen V.\ [65] `The existence of probability measures with given
marginals', Ann.\ Math.\ Statist.\ 36 (1965) 423-439.
\cmmnt{[457D.]}

Sullivan J.M.\ [94] `Sphere packings give an explicit bound for the
Besicovitch covering theorem', J.\ Geometric Analysis 4 (1994) 219-231.
\cmmnt{[\S472 {\it notes\/}.]}

\'Swierczkowski S.\ [58] `On a free group of rotations of the Euclidean
space', Indagationes Math.\ 20 (1958) 376-378.
\cmmnt{[449Yi.]}

\indexheader{Szemer\'edi}
Szemer\'edi, E.\ [75]
`On sets of integers containing no k elements in arithmetic progression',
Acta Arithmetica 27 (1975) 199-245.
\cmmnt{[497L.]}
\indexheader{Talagrand} %? before \medskip to act correctly

\medskip%T

Talagrand M.\ [75] `Sur une conjecture de H.H.Corson', Bull.\ Sci.\
Math.\ 99 (1975) 211-212.
\cmmnt{[467M.]}

Talagrand M.\ [78a] `Sur un th\'eor\`eme de L.Schwartz', C.R.Acad.\
Sci.\ Paris 286 (1978) 265-267.
\cmmnt{[466I.]}

Talagrand M.\ [78b] `Comparaison des boreliens pour les topologies
fortes et faibles', Indiana Univ.\ Math.\ J.\ 21 (1978) 1001-1004.
\cmmnt{[466Za.]}

Talagrand M.\ [80] `Compacts de fonctions mesurables et filtres
non-mesurables', Studia Math.\ 67 (1980) 13-43.
\cmmnt{[464C, 464D.]}

Talagrand M.\ [81] `La $\tau$-r\'egularit\'e de mesures gaussiennes',
Z.\ Wahrscheinlichkeitstheorie und verw.\ Gebiete 57 (1981) 213-221.
\cmmnt{[\S456 {\it intro.}, 456O.]}

Talagrand M.\ [82] `Closed convex hull of set of measurable functions,
Riemann-measurable functions and measurability of translations', Ann.\
Insitut Fourier (Grenoble) 32 (1982) 39-69.
\cmmnt{[465M.]}

Talagrand M.\ [84] {\it Pettis integral and measure theory.}
Mem.\ Amer.\ Math.\ Soc.\  307 (1984).
\cmmnt{[463Ya, 463Yd, 463Za, \S465 {\it intro.}, 465B, 465N,
465R-465T, %465R 465S 465T
466I, \S466 {\it notes\/}.]}

Talagrand M.\ [87] `The Glivenko-Cantelli problem', Ann.\ of Probability
15 (1987) 837-870.
\cmmnt{[\S465 {\it intro.}, 465L, 465M, {\it 465Yd},
\S465 {\it notes\/}.]}

Talagrand M.\ [88] `On liftings and the regularization of stochastic
processes', Prob.\ Theory and Related Fields 78 (1988) 127-134.
\cmmnt{[465U.]}

Talagrand M.\ [89] `A ``regular oscillation'' property of stable sets of
measurable functions', Contemporary Math.\ 94 (1989) 309-313.
\cmmnt{[\S465 {\it notes\/}.]}

Talagrand M.\ [95]
`Concentration of measure and isoperimetric inequalities in product spaces',
Publ.\ Math.\ Inst.\ Hautes \'Etudes Scientifiques 81 (1995) 73-205.
\cmmnt{[492D.]}

Talagrand M.\ [96] `The Glivenko-Cantelli problem, ten years later', J.\
Theoretical Probability 9 (1996) 371-384.
\cmmnt{[\S465 {\it notes\/}.]}
%465Xm

Tamanini I.\ \& Giacomelli C.\ [89] `Approximation of Caccioppoli sets,
with applications to problems in image segmentation',
Ann.\ Univ.\ Ferrara, Sez.\ VII (N.S.) 35 (1989) 187-213.
\cmmnt{[484B.]}

Tao T.\ [07] `A correspondence principle between (hyper)graph theory and
probability theory, and the (hyper)graph removal lemma',
J.\ d'Analyse Math.\ 103 (2007) 1-45'
%({\tt http://front.math.ucdavis.edu/\penalty-100math.CO/{\penalty-100}0602037}).
\cmmnt{[459I, \S497.]}

Tao T.\ [l08] {\it Topics in Ergodic Theory,} lecture notes.
{\tt http://en.wordpress.com/tag/254a-ergodic-theory}, 2008.
\cmmnt{[494F.]}

Taylor A.E.\ [64] {\it Introduction to Functional Analysis.}  Wiley,
1964.
\cmmnt{[\S4A4.]}

Taylor S.J.\ [53] `The Hausdorff $\alpha$-dimensional measure of Brownian
paths in $n$-space',
Proc.\ Cambridge Philosophical Soc.\ 49 (1953) 31-39.
\cmmnt{[477L.]}

Tops{\o}e F.\ [70a] `Compactness in spaces of measures', Studia Math.\ 36
(1970) 195-222.
\cmmnt{[413J, 416K.]}

Tops{\o}e F.\ [70b] {\it Topology and Measure.}  Springer, 1970 (Lecture
Notes in Mathematics 133).
\cmmnt{[437J.]}
%used in chap 41

T\"ornquist A.\ [11] `On the pointwise implementation of near-actions',
Trans.\ Amer.\ Math.\ Soc.\ 363 (2011) 4929-4944.
%({\tt http://arxiv.org/abs/0909.3869}).
\cmmnt{[425D, 425Ya.]}

\medskip%U

Ulam S.\ [1930] `Zur Masstheorie in der allgemeinen
Mengenlehre', Fund.\ Math.\ 16 (1930) 140-150.
\cmmnt{[419G, 438C.]}

Uspenski\v\i\ V.V.\ [88] `Why compact groups are dyadic', pp.\ 601-610
in {\smc Frol\'\i k 88}.
\cmmnt{[4A5T.]}

\medskip%V

Vasershtein L.N.\ [69] `Markov processes over denumerable products of
spaces describing large systems of automata', Problems of Information
Transmission 5 (1969) 47-52.
\cmmnt{[457L.]}

Veech W.A.\ [71] `Some questions of uniform distribution', Annals of Math.\ (2) 94 (1971) 125-138.
\cmmnt{[491H.]}

Veech W.A.\ [77] `Topological dynamics', Bull.\ Amer.\ Math.\ Soc.\ 83 (1977) 775-830.
\cmmnt{[493H.]}

{\cyr Vinokurov V.G.} \& {\cyr Ma{kh}kamov B.M.\ [73] `O prostranstva{kh}
s sover{sh}enno\u{i}, no ne kompaktno\u{i} mero\u{i}',
Nau{ch}ye zapiski Ta{sh}ken{t}skogo instituta narodnogo
{kh}oz{ya}\u{i}stva 71 (1973) 97-103.}
\cmmnt{[451U.]}
% `Ta{sh}ken{t}skogo instituta' taken from ref in Fund Math

\medskip%W

Wage M.L.\ [80] `Products of Radon spaces', Russian Math. Surveys 35:3
(1980) 185-187.
\cmmnt{[\S434 {\it notes\/}.]}

Wagon S.\ [85] {\it The Banach-Tarski Paradox.}  Cambridge U.P., 1985.
\cmmnt{[449Yj.]}

Weil A.\ [1940] {\it L'int\'egration dans les groupes topologiques et ses
applications,} Hermann, 1940 (Actualit\'es Scientifiques et Industrielles 869).
\cmmnt{[445N.]}

\indexheader{Wheeler}
Wheeler R.F.\ [83] `A survey of Baire measures and strict topologies',
Expositiones Math.\ 1 (1983) 97-190.
  % printer's error "2" for "1" at head of paper refers to part, not
%volume
\cmmnt{[\S435 {\it notes\/}, \S437 {\it notes\/}.]}

%\medskip%X\medskip%Y\medskip%Z

%Zindulka O.\ [p98] `Killing residual measures', preprint, 1998.

\frnewpage
}%end of references


%---------------------------------------------------------------%

\end



