% 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{mt3a.tex} 
\versiondate{13.3.08} 
\copyrightdate{1996} 
 
\def\chaptername{Appendix} 
\def\sectionname{Introduction} 
 
\centerline{\bf Appendix to Volume 3} 
 
\medskip 
 
\centerline{\bf Useful Facts} 
 
\medskip 
 
This volume assumes a fairly wide-ranging competence in analysis, a 
solid understanding of elementary set theory and some straightforward 
Boolean algebra.   As in previous volumes, I start with a few pages of 
revision in set theory, but the absolutely essential material is in 
\S3A2, on commutative 
rings, which is the basis of the treatment of Boolean rings in \S311. 
I then give three sections of results in analysis:  topological spaces 
(\S3A3), uniform spaces (\S3A4) and normed spaces (\S3A5).   Finally, I 
add six sentences on group theory (\S3A6). 
 
\discrpage 
 
\frfilename{mt3a1.tex}
\versiondate{31.10.07}
\copyrightdate{1996}

\def\chaptername{Appendix}
\def\sectionname{Set Theory}
\def\Enderton{{\smc Enderton 77}}
\def\Jech{{\smc Jech 03}}
\def\Krivine{{\smc Krivine 71}}
\def\Kunen{{\smc Kunen 80}}

\newsection{3A1}
\cmmnt{
\leader{3A1A}{The axioms of set theory} This treatise is based
on arguments within, or in principle reducible to, `ZFC', meaning
`Zermelo-Fraenkel set theory, including the Axiom of Choice'.   For
discussions of this system, see, for instance, \Krivine, \Jech\ or
\Kunen.   As I remarked in \S2A1, I believe that it is helpful,
as a matter of general principle, to distinguish between results
dependent on the axiom of choice and those which can be proved without
it, or with some relatively weak axiom such as `countable choice'.
(See 134C.   I will go much more deeply into this in Chapter 56 of
Volume 5.)   In Volumes 1 and 2, such a
distinction is useful in appreciating the special features of different
ideas.   In the present volume, however, most of the principal theorems
require something close to the full axiom of choice, and there are few
areas where it seems at present appropriate to work with anything
weaker.   Indeed, at many points we shall approach questions which are,
or may be, undecidable in ZFC;  but with very few exceptions
I postpone discussion of these to
Volume 5.   In particular, I specifically exclude, for the time being,
results dependent on such axioms as the continuum hypothesis.
}%end of comment

\leader{3A1B}{Definition} Let $X$ be a set.   By an {\bf enumeration} of
$X$ I mean a bijection $f:\kappa\to X$ where
$\kappa=\#(X)$\cmmnt{is the initial ordinal equipollent with $X$
(2A1Kb)};  more often than not I shall express
such a function in the form $\langle x_{\xi}\rangle_{\xi<\kappa}$.  In
this case I say that the function $f$, or the family $\langle
x_{\xi}\rangle_{\xi<\kappa}$, {\bf enumerates} $X$.   \cmmnt{You will
see that I am tacitly assuming that $\#(X)$ is
always defined, that is, that the axiom of choice is true.}

\leader{3A1C}{Calculation of cardinalities}\cmmnt{ The following
formulae are basic.}

\spheader 3A1Ca For any sets $X$ and $Y$,
$\#(X\times Y)\le\max(\omega,\#(X),\#(Y))$.
\cmmnt{(\Enderton, p.\ 64;  \Jech,
p.\ 51;    \Krivine, p.\ 33;  \Kunen, 10.13.)}

\spheader 3A1Cb For any $r\in\Bbb N$ and any family
$\langle X_i\rangle_{i\le r}$ of sets,
$\#(\prod_{i=0}^rX_i)\le\max(\omega,\max_{i\le r}\#(X_i))$.
\prooflet{(Induce on $r$.)}

\spheader 3A1Cc For any family $\langle X_i\rangle_{i\in I}$ of sets,
$\#(\bigcup_{i\in I}X_i)\le\max(\omega,\#(I),\sup_{i\in I}\#(X_i))$.
\cmmnt{(\Jech, p.\ 52;  \Krivine, p.\ 33;  \Kunen, 10.21.)}

\spheader 3A1Cd For any set $X$,\cmmnt{ the
set} $[X]^{<\omega}$\cmmnt{ of finite subsets of $X$} has
cardinal at most $\max(\omega,\#(X))$.  \prooflet{(There is a
surjection from $\bigcup_{r\in\Bbb N}X^r$ onto $[X]^{<\omega}$.   For
the notation $[X]^{<\omega}$ see 3A1J below.)}

\leader{3A1D}{Cardinal exponentiation} For a cardinal $\kappa$, I write
$2^{\kappa}$ for $\#(\Cal P\kappa)$.   So
$2^{\omega}=\frak c$, and $\kappa^+\le 2^{\kappa}$ for every $\kappa$.
\cmmnt{(\Enderton, p.\ 132;  {\smc Lipschutz 64}, p.\ 139;  \Jech, p.\ 29;
\Krivine, p.\ 25;  {\smc Halmos 60}, p.\ 93.)}

\leader{3A1E}{Definition}\cmmnt{ The class of infinite initial
ordinals, or cardinals, is a subclass of the class $\text{On}$ of all
ordinals, so is itself well-ordered;  being unbounded, it is a proper
class;  consequently there is a unique increasing enumeration of it as
$\langle\omega_{\xi}\rangle_{\xi\in\text{On}}$.
We have} $\omega_0=\omega$, $\omega_{\xi+1}=\omega_{\xi}^+$ for every
$\xi$\cmmnt{ (compare 2A1Fc)},
$\omega_{\xi}=\bigcup_{\eta<\xi}\omega_{\eta}$
for non-zero limit ordinals $\xi$.  \cmmnt{(\Enderton, pp.\ 213-214;
\Jech, p.\ 30;  \Krivine, p.\ 31.)}

\leader{3A1F}{Cofinal sets (a)} If $P$ is any partially ordered
set\cmmnt{ (definition:  2A1Aa)}, a
subset $Q$ of $P$ is {\bf cofinal} with $P$ if
for every $p\in P$ there is a $q\in Q$ such that $p\le q$.

\spheader 3A1Fb If $P$ is any partially ordered set, the {\bf
cofinality} of $P$, $\cf P$, is the least cardinal of any cofinal
subset of $P$.   \cmmnt{Note that} $\cf P=0$ iff $P=\emptyset$,
and\cmmnt{ that} $\cf P=1$ iff $P$ has
a greatest element.

\spheader 3A1Fc Observe that if $P$ is upwards-directed and $\cf P$ is
finite, then $\cf P$ is either $0$ or $1$\cmmnt{;  for if $Q$ is a
finite, non-empty cofinal set then it has an upper bound, which must be
the greatest  element of $P$}.

\spheader 3A1Fd If $P$ is a totally ordered set of cofinality $\kappa$,
then there is a strictly increasing family $\langle
p_{\xi}\rangle_{\xi<\kappa}$ in $P$ such that $\{p_{\xi}:\xi<\kappa\}$
is cofinal with $P$.   \prooflet{\Prf\ If $\kappa=0$ then
$P=\emptyset$ and this is trivial.    Otherwise, let $Q$ be a cofinal
subset of $P$ with cardinal $\kappa$, and $\{q_{\xi}:\xi<\kappa\}$ an
enumeration of $Q$.   Define $\langle p_{\xi}\rangle_{\xi<\kappa}$
inductively, as follows.   Start with $p_0=q_0$.   Given $\langle
p_{\eta}\rangle_{\eta<\xi}$, where $\xi<\kappa$, then if
$p_{\eta}<q_{\xi}$ for every $\eta<\xi$, take $p_{\xi}=q_{\xi}$;
otherwise, because $\#(\xi)\le\xi<\kappa$, $\{p_{\eta}:\eta<\xi\}$
cannot be cofinal with $P$, so there is a $p_{\xi}\in P$ such that
$p_{\xi}\not\le p_{\eta}$ for every $\eta<\xi$, that is,
$p_{\eta}<p_{\xi}$ for every $\eta<\xi$.   Note that there is some
$\eta<\xi$ such that $q_{\xi}\le p_{\eta}$, so that $q_{\xi}\le
p_{\xi}$.   Continue.

Now $\langle p_{\xi}\rangle_{\xi<\kappa}$ is a strictly increasing
family in $P$ such that $q_{\xi}\le p_{\xi}$ for every $\xi$;   it
follows at once that $\{p_{\xi}:\xi<\kappa\}$ is cofinal with $P$.
\Qed}

\spheader 3A1Fe In particular, for a totally ordered set $P$,
$\cf P=\omega$ iff there is a cofinal strictly increasing sequence in
$P$.

\cmmnt{
\leader{3A1G}{Zorn's Lemma} In Volume 2 I used Zorn's Lemma
only once or twice, giving the arguments in detail.   In the present
volume I feel that continuing in such a manner would often be tedious;
but nevertheless the arguments are not always quite obvious, at least
until you have gained a good deal of experience.   I therefore take a
paragraph to comment on some of the standard forms in which they appear.

The statement of Zorn's Lemma, as quoted in 2A1M, refers to arbitrary
partially ordered sets $P$.   A large proportion of the applications can
in fact be represented more or less naturally by taking $P$ to be a
family $\frak P$ of sets ordered by $\subseteq$;  in such a case, it
will be sufficient to check that (i) $\frak P$ is not empty (ii)
$\bigcup\frak Q\in\frak P$ for every non-empty totally ordered 
$\frak Q\subseteq\frak P$.   
More often than not, this will in fact be true for
all non-empty upwards-directed sets $\frak Q\subseteq\frak P$, and the
line of the argument is sometimes clearer if phrased in this form.

Within this class of partially ordered sets, we can distinguish a
special subclass.   If $A$ is any set and $\perp$ any relation on $A$,
we can consider the collection $\frak P$ of sets $I\subseteq A$ such
that $a\perp b$ for all distinct $a$, $b\in I$.   In this case we need
look no farther before declaring `$\frak P$ has a maximal element';  for
$\emptyset$ necessarily belongs to $\frak P$, and if $\frak Q$ is any
upwards-directed subset of $\frak P$, then $\bigcup\frak Q\in\frak P$.
\prooflet{\Prf\ If $a$, $b$ are distinct elements of $\bigcup\frak Q$,
there are $I_1$, $I_2\in\frak Q$ such that $a\in I_1$, $b\in I_2$;
because $\frak Q$ is upwards-directed, there is an $I\in\frak Q$ such
that $I_1\cup I_2\subseteq I$, so that $a$, $b$ are distinct members of
$I\in\frak P$, and $a\perp b$.\ \QeD}   So $\bigcup\frak Q$ is an upper
bound of $\frak Q$ in $\frak P$;  as $\frak Q$ is arbitrary, $\frak P$
satisfies the conditions of Zorn's Lemma, and must have a maximal
element.

Another important type of partially ordered set in this context is a
family $\Phi$ of functions, ordered by saying that $f\le g$ if $g$ is an
extension of $f$.   In this case, for any non-empty upwards-directed
$\Psi\subseteq\Phi$, we shall have a function $h$ defined by saying that

\Centerline{$\dom h=\bigcup_{f\in\Psi}\dom f$,
\quad $h(x)=f(x)$ whenever $f\in\Psi$, $x\in\dom f$,}

\noindent and the usual attack is to seek to prove that any such $h$
belongs to $\Phi$.

I find that at least once I wish to use Zorn's Lemma `upside down':
that is, I have a non-empty partially ordered set $P$ in which every
non-empty totally ordered subset has a {\it lower} bound.   In this
case, of course, $P$ has a {\it minimal} element.   The point is that
the definition of `partial order' is symmetric, so that $(P,\ge)$ is a
partially ordered set whenever $(P,\le)$ is;  and we can seek to apply
Zorn's Lemma to either.
}%end of comment

\cmmnt{
\leader{3A1H}{Natural numbers and finite ordinals} I remarked
in 2A1De that the first few ordinals

\Centerline{$\emptyset$,
\quad$\{\emptyset\}$,
\quad$\{\emptyset,\{\emptyset\}\}$,
\quad$\{\emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\}\}$,
\quad$\ldots$}

\noindent may be identified with the natural numbers $0,1,2,3,\ldots$;
the idea being that $n=\{0,1,\ldots,\penalty-100{n-}\penalty+100{1}\}$
is a set with $n$ elements.
If we do this, then the set $\Bbb N$ of natural numbers becomes
identified with the first infinite ordinal $\omega$.   This convention
makes it possible to present a number of arguments in a particularly
elegant form.   A typical example is in 344H.   There I wish to describe
an inductive construction for a family
$\langle K_{\sigma}\rangle_{\sigma\in S}$
where $S=\bigcup_{n\in\Bbb N}\{0,1\}^{n}$.   If we think of $n$ as the
set of its predecessors, then $\sigma\in\{0,1\}^n$ becomes a function from
$n$ to $\{0,1\}$;  since the set $n$ has just $n$ members, this
corresponds well to the idea of $\sigma$ as the list of its
$n$ coordinates, except that it would now be natural to list them as
$\sigma(0),\ldots,\sigma(n-1)$ rather than as
$x_1,\ldots,x_n$, which was
the language I favoured in Volume 2.   An extension of $\sigma$ to a member
of $\{0,1\}^{n+1}$ is of the form
$\tau=\sigma^{\smallfrown}\fraction{i}$ where
$\tau(k)=\sigma(k)$ for $k<n$ (`$\tau\restr n=\sigma$') and $\tau(n)=i$.
If $w\in\{0,1\}^{\Bbb N}$, then we can
identify the initial segment $(w(0),w(1),\ldots,w(n-1))$ of its first
$n$ coordinates with the restriction $w\restr n$ of $w$ to the set
$n=\{0,\ldots,n-1\}$.
}%end of comment

\leader{3A1I}{Definitions (a)} If $P$ and $Q$ are
lattices\cmmnt{ (2A1Ad)}, a {\bf lattice homomorphism} from $P$ to $Q$
is a function $f:P\to Q$ such that $f(p\wedge p')=f(p)\wedge f(p')$ and
$f(p\vee p')=f(p)\vee f(p')$ for all $p$, $p'\in P$.   Such a
homomorphism is\cmmnt{ surely}
order-preserving\cmmnt{ (313H)}\prooflet{, for if
$p\le p'$ in $P$ then $f(p')=f(p\vee p')=f(p)\vee f(p')$ and
$f(p)\le f(p')$}.

\spheader 3A1Ib If $P$ is a lattice, a {\bf sublattice} of $P$ is a set
$Q\subseteq P$ such that $p\vee q$ and $p\wedge q$ belong to $Q$ for all
$p$, $q\in Q$.

\spheader 3A1Ic{\bf (i)} A lattice $P$ is {\bf distributive} if

\Centerline{$(p\wedge q)\vee r=(p\vee r)\wedge(q\vee r)$,
\quad$(p\vee q)\wedge r=(p\wedge r)\vee(q\wedge r)$}

\noindent for all $p$, $q$, $r\in P$.

\leaveitout{
if $(p\wedge q)\vee r=(p\vee r)\wedge(q\vee r)\Forall p, q, r$ then
 $(p\wedge r)\vee(q\wedge r)
 =(p\vee q)\wedge(p\vee r)\wedge(r\vee q)\wedge r 
 =(p\vee q)\wedge r\Forall p, q, r$.

$P$ is distributive iff it is isomorphic to a sublattice of some $\Cal PX$
\Prf\Quer\ if $(A,B)\in(\Cal PP)^2$ maximal subject to $A$ an ideal, 
$B$ a filter disjoint from $A$ and $\exists r\in P\setminus(A\cup B)$ 
then there are $p\in A$,
$q\in B$ such that $p\vee r\in B$, $q\wedge r\in A$ but now 
$p\vee q\in B$, $p\vee(q\wedge r)\in A\cap B$ \Bang\Qed
}

\medskip

\quad{\bf (ii)}\dvAnew{2010} In a distributive lattice we have a
{\bf median function} of three variables

\dvro{\Centerline{$\med(p,q,r)
=(p\wedge q)\vee(p\wedge r)\vee(q\wedge r)$.}}
{$$\eqalign{\med(p,q,r)
&=(p\wedge q)\vee(p\wedge r)\vee(q\wedge r)\cr
&=((p\wedge(q\vee r))\vee(q\wedge r)
=(p\vee q)\wedge(p\vee r)\wedge(q\vee r).\cr}$$}

\noindent If $P$ and $Q$ are distributive lattices and $f:P\to Q$ is a
lattice homomorphism, $f(\med(p,q,r))=\med(f(p),f(q),f(r))$ for all $p$,
$q$, $r\in P$.

\medskip

\quad{\bf (iii)}\dvAnew{2012}
If $P$ is a distributive lattice and $I\subseteq P$
is finite, then the sublattice of $P$ generated by $I$ is finite.
\prooflet{\Prf\ If $J=\{\sup I_0:\emptyset\ne I_0\subseteq I\}$ and
$K=\{\inf J_0:\emptyset\ne J_0\subseteq J\}$
then $K$ is a sublattice of $P$.\ \Qed}
%for 611C

\leader{3A1J}{Subsets of given size}\cmmnt{ The following concepts are
used often enough for a special notation to be helpful.}   If $X$ is a
set and $\kappa$ is a cardinal, write

\Centerline{$[X]^{\kappa}=\{A:A\subseteq X,\,\#(A)=\kappa\}$,}

\Centerline{$[X]^{\le\kappa}=\{A:A\subseteq X,\,\#(A)\le\kappa\}$,}

\Centerline{$[X]^{<\kappa}=\{A:A\subseteq X,\,\#(A)<\kappa\}$.}

\cmmnt{\noindent Thus

\Centerline{$[X]^0=[X]^{\le 0}=[X]^{<1}=\{\emptyset\}$,}

\noindent $[X]^2$ is the set of doubleton subsets of $X$,
$[X]^{<\omega}$ is the set of finite subsets of $X$, $[X]^{\le\omega}$
is the set of countable subsets of $X$, and so on.
}%end of comment

\leader{3A1K}{}\cmmnt{ The next result is one of the fundamental
theorems of combinatorics.
In this volume it is used in the proofs of Ornstein's theorem (\S387)
and the Kalton-Roberts theorem (\S392).

\medskip

\noindent}{\bf Hall's Marriage Lemma} Suppose that $X$ and $Y$ are
finite sets and $R\subseteq X\times Y$ is a relation such that
$\#(R[I])\ge\#(I)$ for every $I\subseteq X$.   Then there is an
injective function $f:X\to Y$ such that $(x,f(x))\in R$ for every
$x\in X$.

\cmmnt{\medskip

\noindent{\bf Remark} Recall that $R[I]$ is the set
$\{y:\,\Exists x\in I,\,(x,y)\in R\}$ (1A1Bc).    If we identify a function
with its graph, then `$(x,f(x))\in R$ for every
$x\in X$' becomes `$f\subseteq R$'.
}%end of comment

\proof{{\smc Bollob\'as 79}, p.\ 54, Theorem 7;  {\smc Anderson 87}, 2.2.1;  
{\smc Bose \& Manvel 84}, \S10.2.
}%end of proof of 3A1K

\discrpage

\frfilename{mt3a2.tex}
\versiondate{22.11.07}
\copyrightdate{1995}

\def\chaptername{Appendix}
\def\sectionname{Rings}

\newsection{3A2}

I give a very brief outline of the indispensable parts of the elementary
theory of (commutative) rings.   I assume that you have seen at least a
little group theory.

\leader{3A2A}{Definition} A {\bf ring} is a triple $(R,+,.)$ such that

\inset{$(R,+)$ is an abelian group;  its identity
will\cmmnt{ always} be denoted $0$ or $0_R$;}

\inset{$(R,.)$ is a semigroup, that is, $ab\in R$ for all $a$, $b\in R$
and $a(bc)=(ab)c$ for all $a$, $b$, $c\in R$;}

\inset{$a(b+c)=ab + ac$, $(a+b)c=ac+bc$ for all $a$, $b$, $c\in R$.}

\noindent A {\bf commutative ring} is one in\cmmnt{ which
multiplication is commutative, that is,} $ab=ba$ for all $a$, $b\in R$.

\leader{3A2B}{Elementary facts} Let $R$ be a ring.

\header{3A2Ba}{\bf (a)} $a0=0a=0$ for every $a\in R$.   \prooflet{\Prf\

\Centerline{$a0=a(0+0)=a0+a0$,
\quad$0a=(0+0)a=0a+0a$;}

\noindent because $(R,+)$ is a group, we may subtract $a0$ or $0a$ from
each side of the appropriate equation to see that $0=a0$, $0=0a$.\ \Qed}

\header{3A2Bb}{\bf (b)} $(-a)b=a(-b)=-(ab)$ for all $a$, $b\in R$.
\prooflet{\Prf\

\Centerline{$ab+((-a)b)=(a+(-a))b=0b=0=a0=a(b+(-b))=ab+a(-b)$;}

\noindent subtracting $ab$ from each term, we get
$(-a)b=-(ab)=a(-b)$.\ \Qed}

\leader{3A2C}{Subrings} If $R$ is a ring, a {\bf subring} of $R$ is a
set $S\subseteq R$ such that $0\in S$ and $a+b$, $ab$, $-a$ belong to
$S$ for all $a$, $b\in S$.   In this case $S$, together with the
addition and multiplication induced by those of $R$, is a ring in its
own right.

\leader{3A2D}{Homomorphisms (a)} Let $R$, $S$ be two rings.   A function
$\phi:R\to S$ is a {\bf ring homomorphism} if
$\phi(a+b)=\phi(a)+\phi(b)$ and $\phi(ab)=\phi(a)\phi(b)$ for all $a$,
$b\in R$.      The {\bf kernel} of $\phi$ is $\{a:a\in
R,\,\phi(a)=0_S\}$.

\header{3A2Db}{\bf (b)} Note that if $\phi:R\to S$ is a ring
homomorphism, then it is also a group homomorphism from $(R,+)$ to
$(S,+)$, so that $\phi(0_R)=0_S$ and
$\phi(-a)=-\phi(a)$ for every $a\in R$;   moreover, $\phi[R]$ is a
subring of $S$, and $\phi$ is injective iff its kernel is $\{0_R\}$.

\header{3A2Dc}{\bf (c)} If $R$, $S$ and $T$ are rings, and $\phi:R\to
S$, $\psi:S\to T$ are ring homomorphisms, then $\psi\phi:R\to T$ is a
ring homomorphism\prooflet{, because

\Centerline{$(\psi\phi)(a*b)=\psi(\phi(a*b))=\psi(\phi(a)*\phi(b))
=\psi(\phi(a))*\psi(\phi(b))$}

\noindent for all $a$, $b\in R$, taking $*$ to be either addition or
multiplication}.   If $\phi$ is bijective, then $\phi^{-1}:S\to R$ is a
ring homomorphism\prooflet{, because

\Centerline{$\phi^{-1}(c*d)
=\phi^{-1}(\phi(\phi^{-1}(c))*\phi(\phi^{-1}(d)))
=\phi^{-1}\phi(\phi^{-1}(c)*\phi^{-1}(d))
=\phi^{-1}(c)*\phi^{-1}(d)$}

\noindent for all $c$, $d\in S$, again taking $*$ to be either addition
or multiplication}.

\leader{3A2E}{Ideals (a)} Let $R$ be a ring.   An {\bf ideal} of $R$ is
a subring $I$ of $R$ such that
$ab\in I$ and $ba\in I$ whenever $a\in I$ and $b\in R$.
In this case we write $I\normalsubgroup R$.

\cmmnt{Note that $R$ and $\{0\}$ are always ideals of $R$.}

\header{3A2Eb}{\bf (b)} If $R$ and $S$ are rings and $\phi:R\to S$ is a
ring homomorphism, then the kernel $I$ of $\phi$ is an ideal of $R$.
\prooflet{\Prf\ (i) Because $\phi$ is a group homomorphism, $I$ is a
subgroup of $(R,+)$.   (ii) If $a\in I$, $b\in R$ then

\Centerline{$\phi(ab)=\phi(a)\phi(b)=0_S\phi(b)=0_S$,
\quad$\phi(ba)=\phi(b)\phi(a)=\phi(b)0_S=0_S$}

\noindent so $ab$, $ba\in I$.\ \Qed}

\leader{3A2F}{Quotient rings (a)} Let $R$ be a ring and $I$ an ideal of
$R$.   A {\bf coset} of $I$ is a set of the form $a+I=\{a+x:x\in I\}$
where $a\in R$.   \cmmnt{(Because $+$ is commutative, we do not need
to distinguish between `left cosets' $a+I$ and `right cosets'
$I+a$.)}   Let $R/I$ be the set of cosets of $I$ in $R$.

\header{3A2Fb}{\bf (b)} For $A$, $B\in R/I$, set

\Centerline{$A+B=\{x+y:x\in A,\,y\in B\}$,
\quad$A\cdot B=\{xy+z:x\in A,\,y\in B,\,z\in I\}$.}

\noindent Then $A+B$, $A\cdot B$ both belong to $R/I$;  moreover, if
$A=a+I$
and $B=b+I$, then $A+B=(a+b)+I$ and $A\cdot B=ab+I$.   \prooflet{\Prf\
{(i)}

$$\eqalignno{A+B
&=(a+I)+(b+I)\cr
&=\{(a+x)+(b+y):x,\,y\in I\}\cr
&=\{(a+b)+(x+y):x,\,y\in I\}\cr
\noalign{\noindent (because addition is associative and commutative)}
&\subseteq\{(a+b)+z:z\in I\}=(a+b)+I\cr
\noalign{\noindent (because $I+I\subseteq I$)}
&=\{(a+b)+(z+0):z\in I\}\cr
&\subseteq (a+I)+(b+I)=A+B\cr}$$

\noindent because $0\in I$.   {(ii)}

$$\eqalignno{A\cdot B
&=\{(a+x)(b+y)+z:x,\,y,\,z\in I\}\cr
&=\{ab+(ay+xb+z):x,\,y,\,z\in I\}\cr
&\subseteq\{ab+w:w\in I\}=ab+I\cr
\noalign{\noindent (because $ay$, $xb\in I$ for all $x$, $y\in I$, and
$I$ is closed under addition)}
&=\{(a+0)(b+0)+w:w\in I\}\cr
&\subseteq A\cdot B.\text{  \Qed}\cr}$$
}

\header{3A2Fc}{\bf (c)}\cmmnt{ It is now an elementary exercise to
check that} $(R/I,+,\cdot )$ is a ring, with zero $0+I=I$ and additive
inverses $-(a+I)=(-a)+I$.

\header{3A2Fd}{\bf (d)} Moreover, the map $a\mapsto a+I:R\to R/I$ is a
ring homomorphism.

\header{3A2Fe}{\bf (e)} Note that for $a$, $b\in R$, the following are
equiveridical:  (i) $a\in b+I$;  (ii) $b\in a+I$;  (iii)
$(a+I)\cap(b+I)\ne\emptyset$;  (iv) $a+I=b+I$;  (v) $a-b\in I$.   Thus
the cosets of $I$ are just the equivalence classes in $R$ under the
equivalence relation $a\sim b\iff a+I=b+I$;  accordingly I shall
generally write $a^{\ssbullet}$ for $a+I$\cmmnt{, if there seems no
room for
confusion}.    In particular, the kernel of the canonical map from $R$ to
$R/I$ is just $\{a:a+I=I\}=I=0^{\ssbullet}$.

\header{3A2Ff}{\bf (f)} If $R$ is commutative so is
$R/I$\prooflet{, since

\Centerline{$a^{\ssbullet}b^{\ssbullet}=(ab)^{\ssbullet}
=(ba)^{\ssbullet}=b^{\ssbullet}a^{\ssbullet}$}

\noindent for all $a$, $b\in R$}.

\leader{3A2G}{Factoring homomorphisms through quotient rings:
Proposition} Let $R$ and $S$ be rings, $I$ an ideal of $R$, and
$\phi:R\to S$ a homomorphism such that $I$ is included in the kernel of
$\phi$.   Then we have a ring homomorphism $\pi:R/I\to S$ such that
$\pi(a^{\ssbullet})=\phi(a)$ for every $a\in R$.   $\pi$ is injective
iff $I$ is precisely the kernel of $\phi$.

\proof{ If $a$, $b\in R$ and $a^{\ssbullet}=b^{\ssbullet}$ in $R/I$,
then $a-b\in I$ (3A2Fe), so $\phi(a)-\phi(b)=\phi(a-b)=0$, and
$\phi(a)=\phi(b)$.   This means that the formula offered does indeed
define a function $\pi$ from $R/I$ to $S$.   Now if $a$, $b\in R$ and
$*$ is either multiplication or addition,

\Centerline{$\pi(a^{\ssbullet}*b^{\ssbullet})
=\pi((a*b)^{\ssbullet})
=\phi(a*b)
=\phi(a)*\phi(b)
=\pi(a^{\ssbullet})*\pi(b^{\ssbullet})$.}

\noindent So $\pi$ is a ring homomorphism.

The kernel of $\pi$ is $\{a^{\ssbullet}:\phi(a) = 0\}$, which is $\{0\}$
iff $\phi(a)=0\iff a^{\ssbullet}=0\iff a\in I$.
}%end of proof of 3A2G

\vleader{72pt}{3A2H}{Product rings (a)} Let $\langle R_i\rangle_{i\in I}$ be
any family of rings.   Set $R=\prod_{i\in I}R_i$ and for $a$, $b\in R$
define $a+b$, $ab\in R$ by setting

\Centerline{$(a+b)(i)=a(i)+b(i)$,\quad $(ab)(i)=a(i)b(i)$}

\noindent for every $i\in I$.   \cmmnt{It is easy to check from the
definition in 3A2A that} $R$ is a ring;  its zero is given by the
formula

\Centerline{$0_R(i)=0_{R_i}$ for every $i\in I$,}

\noindent and its additive inverses by the formula

\Centerline{$(-a)(i)=-a(i)$ for every $i\in I$.}

\header{3A2Hb}{\bf (b)} Now let $S$ be any other ring.
Then\cmmnt{ it is easy to see that} a function $\phi:S\to R$ is a ring
homomorphism iff $s\mapsto\phi(s)(i):S\to R_i$ is a ring homomorphism
for every $i\in I$.

\header{3A2Hc}{\bf (c)}\cmmnt{ Note that} $R$ is commutative iff $R_i$
is commutative for every $i$.



\discrpage

\frfilename{mt3a3.tex}
\versiondate{14.12.07}
\copyrightdate{2004}

\def\Bourbaki{{\smc Bourbaki 66}}
%\def\Bushaw{{\smc Bushaw 63}}
%\def\Christenson{{\smc Christenson \& Voxmann 77}}
\def\Dugundji{{\smc Dugundji 66}}
\def\Engelking{{\smc Engelking 89}}
\def\Gaal{{\smc Gaal 64}}
\def\James{{\smc James 87}}
\def\Schubert{{\smc Schubert 68}}

\def\chaptername{Appendix}
\def\sectionname{General topology}

\newsection{3A3}

In \S2A3, I looked at a selection of topics in general topology in some
detail, giving proofs;  the point was that an ordinary elementary course
in the subject would surely go far beyond what we needed there, and at
the same time might omit some of the results I wished to quote.
It seemed therefore worth taking a bit of space to cover the requisite
material, giving readers the option of delaying a proper study of the
subject until a convenient opportunity arose.   In the context of the
present volume, this approach is probably no longer appropriate, since
we need a much greater proportion of the fundamental ideas, and by the
time you have reached familiarity with the topics here you will be
well able to find your way about one of the many excellent textbooks on
the subject.   This time round, therefore, I give most of the results
without proofs (as in \S\S2A1 and 3A1), hoping that some of the
references I
offer will be accessible in all senses.   I do, however, give a full set
of definitions, partly to avoid ambiguity (since even in this relatively
mature subject, there are some awkward divergences remaining in the
usage of different authors), and partly because many of the proofs are
easy enough for even a novice to fill in with a bit of thought, once the
meaning of the words is clear.   \cmmnt{In fact this happens so often
that I will mark with a $*$ those points where a proof needs an idea not
implicit in the preceding work.}

\leader{3A3A}{Taxonomy of topological \dvrocolon{spaces}}\cmmnt{ I
begin with the handful of definitions we need in order to classify the
different types of topological space used in this volume.   A couple
have already been introduced in Volume 2, but I repeat them because the
list would look so odd without them.

\medskip

\noindent}{\bf Definitions} Let $(X,\frak T)$ be a topological space.

\spheader 3A3Aa $X$ is {\bf T$_{\pmb{1}}$} if singleton subsets of $X$ are
closed.

\spheader 3A3Ab $X$ is {\bf Hausdorff} if for any
distinct points $x$,
$y\in X$ there are disjoint open sets $G$, $H\subseteq X$ such that
$x\in G$ and $y\in H$.

\spheader 3A3Ac $X$ is {\bf regular} if whenever $F\subseteq X$ is
closed and $x\in X\setminus F$ there are disjoint open sets $G$,
$H\subseteq X$ such that $x\in G$ and $F\subseteq H$.   \cmmnt{(Note
that in this definition I do not require $X$ to be Hausdorff, following
\James\leaveitout{ and \Christenson} but not \Engelking, \Bourbaki,
\Dugundji,
\Schubert\ or \Gaal\leaveitout{ or \Bushaw}.)}

\spheader 3A3Ad $X$ is {\bf completely regular} if whenever
$F\subseteq X$ is closed and $x\in X\setminus F$ there is a
continuous function
$f:X\to[0,1]$ such that $f(x)=1$ and $f(y)=0$ for every $y\in F$.
\cmmnt{(Note that many authors restrict the phrase `completely
regular' to Hausdorff spaces.)}

\spheader 3A3Ae $X$ is {\bf zero-dimensional} if whenever $G\subseteq X$
is an open set and $x\in G$ then there is an open-and-closed set $H$
such that $x\in H\subseteq G$.

\spheader 3A3Af $X$ is {\bf extremally disconnected} if the closure of
every open set in $X$ is open.

\spheader 3A3Ag $X$ is {\bf compact} if every open cover of $X$
has a finite subcover.

\spheader 3A3Ah $X$ is {\bf locally compact} if for every $x\in X$ there
is a set $K\subseteq X$ such that $x\in\interior K$ and $K$ is
compact\cmmnt{ (in its subspace topology, as defined in 2A3C)}.

\spheader 3A3Ai If every subset of $X$ is open, we call $\frak T$ the
{\bf discrete topology} on $X$.

\leader{3A3B}{Elementary relationships (a)} A completely regular space
is regular.  \prooflet{(\Engelking, p.\ 39;  \Dugundji, p.\ 154;
\Schubert, p.\ 104.)}

\spheader 3A3Bb A locally compact Hausdorff space is completely
regular\cmmnt{, therefore regular}.   \cmmnt{$*$}
\prooflet{(\Engelking,
3.3.1;  \Dugundji, p.\ 238;  \Gaal, p.\ 149\leaveitout{;  Christenson,
p.\ 123}.)}

\spheader 3A3Bc A compact Hausdorff space is locally compact\cmmnt{,
therefore completely regular and regular}.

\spheader 3A3Bd A regular extremally disconnected space is
zero-dimensional.  \prooflet{(\Engelking, 6.2.25.)}

\spheader 3A3Be
Any topology defined by pseudometrics\cmmnt{ (2A3F), in particular
the weak topology of a normed space (2A5I),} is completely
regular\cmmnt{, therefore regular}.  \prooflet{(\Bourbaki, IX.1.5;
\Dugundji, p.\ 200.)}

\spheader 3A3Bf If $X$ is a completely regular Hausdorff
space\cmmnt{ (in particular, if $X$ is (locally) compact and
Hausdorff),} and $x$, $y$ are distinct points in $X$, then there is a
continuous function $f:X\to\Bbb R$ such that $f(x)\ne f(y)$.
\prooflet{(Apply 3A3Ad with $F=\{y\}$, which is closed because $X$ is
Hausdorff.)}

\spheader 3A3Bg An open set in a locally compact Hausdorff space is
locally compact in its subspace topology.
\prooflet{(\Engelking, 3.3.8;  \Bourbaki, I.9.7.)}

\spheader 3A3Bh\dvAnew{2017, 2022} Any subspace of a Hausdorff space is Hausdorff;
any subspace of a completely regular space is completely regular.
\prooflet{(\Engelking, 2.1.6\query).}
% other refs?\query \Cech, 27A.3;  \Kuratowski, \S5.VIII; \Csaszar, 2.5.21.


\leader{3A3C}{Continuous functions} Let $(X,\frak T)$ and $(Y,\frak S)$
be topological spaces.

\spheader 3A3Ca If $f:X\to Y$ is a function and $x\in X$, we say that
$f$ is {\bf continuous at $x$} if $x\in\interior f^{-1}[H]$ whenever
$H\subseteq Y$ is an open set containing $f(x)$.

\spheader 3A3Cb Now a function from $X$ to $Y$ is continuous iff it is
continuous at every point of $X$.  \prooflet{(\Bourbaki, I.2.1;
\Dugundji, p.\ 80;  \Schubert, p.\ 24;  \Gaal, p.\ 183;
\James, p.\ 26\leaveitout{;  \Christenson, p.\ 15}.)}

\spheader 3A3Cc If $f:X\to Y$ is continuous at $x\in X$, and
$A\subseteq X$ is such that $x\in\overline{A}$, then
$f(x)\in\overline{f[A]}$.
\prooflet{(\Bourbaki, I.2.1;  \Schubert, p.\ 23.)}

\spheader 3A3Cd If $f:X\to Y$ is continuous, then
$f[\overline{A}]\subseteq\overline{f[A]}$ for every $A\subseteq X$.
\prooflet{(\Engelking, 1.4.1;  \Bourbaki, I.2.1;  \Dugundji, p.\ 80;
\Schubert, p.\ 24;  \Gaal, p.\ 184;  \James, p.\ 27\leaveitout{;
\Christenson, p.\ 31;  \Bushaw, p.\ 38}.)}

\spheader 3A3Ce A function $f:X\to Y$ is a {\bf homeomorphism} if it is
a continuous bijection and its inverse is also
continuous\cmmnt{;  that is, if $\frak S=\{f[G]:G\in\frak T\}$ and
$\frak T=\{f^{-1}[H]:H\in\frak S\}$}.

\spheader 3A3Cf A function $f:X\to[-\infty,\infty]$ is
{\bf lower semi-continuous} if $\{x:x\in X,\,f(x)>\alpha\}$ is open for
every $\alpha\in\Bbb R$.  \cmmnt{(Cf.\ 225H.)}

\leader{3A3D}{Compact spaces} \cmmnt{Any extended series of
applications of general topology
is likely to involve some new features of compactness.   I start with
the easy bits, continuing from 2A3Nb.

\medskip

}{\bf (a)}\cmmnt{ The first is just a definition of
compactness in terms of closed sets instead of open sets.}    A family
$\Cal F$ of sets has the {\bf finite intersection property} if
$\bigcap\Cal F_0$ is non-empty for every finite
$\Cal F_0\subseteq\Cal F$.   Now a topological space $X$ is compact iff
$\bigcap\Cal F\ne\emptyset$ whenever $\Cal F$ is a family of closed subsets of $X$
with the finite intersection property.   \prooflet{(\Engelking, 3.1.1;
\Bourbaki, I.9.1;  \Dugundji, p.,\ 223;  \Schubert, p.\ 68;  \Gaal, p.\
127\leaveitout{;  \Christenson, p.\ 67;  \Bushaw, p.\ 77}.)}

\spheader 3A3Db\cmmnt{ A marginal generalization of this is the
following.}  Let $X$ be a topological space and $\Cal F$ a family of
closed subsets of $X$ with the finite intersection property.   If
$\Cal F$ contains a compact set then $\bigcap\Cal F\ne\emptyset$.
\prooflet{(Apply (a) to $\{K\cap F:F\in\Cal F\}$ where $K\in\Cal F$ is
compact.)}

\spheader 3A3Dc In a Hausdorff space, compact subsets are closed.
\prooflet{(\Engelking, 3.1.8;  \Bourbaki, I.9.4;  \Dugundji, p.\ 226;
\Schubert, p.\ 70;  \Gaal, p.\ 138;  \James, p.\ 77\leaveitout{;
\Bushaw, p.\ 73}.)}

\spheader 3A3Dd If $X$ is compact, $Y$ is Hausdorff and $\phi:X\to Y$ is
continuous and injective, then $\phi$ is a homeomorphism between $X$ and
$\phi[X]$\cmmnt{ (where $\phi[X]$ is given the subspace topology)}.
\prooflet{(\Engelking, 3.1.13;  \Bourbaki, I.9.4;
\Dugundji, p.\ 226;  \Schubert, p.\ 71;  \Gaal, p.\ 207\leaveitout{;
\Bushaw, p.\ 73}.)}

\spheader 3A3De Let $X$ be a regular topological space and $A$ a subset
of $X$.   Then the following are equiveridical:  (i) $A$ is relatively
compact in $X$\cmmnt{ (that is, $A$ is included in some compact subset
of $X$, as in 2A3Na)};  (ii) $\overline{A}$ is compact;  (iii) every
ultrafilter on $X$ which contains $A$ has a limit in $X$.
%better would be:  every filter containing A has a cluster point
\prooflet{\Prf (ii)$\Rightarrow$(i) is trivial, and
(i)$\Rightarrow$(iii) is a consequence of 2A3R;  neither of these
requires $X$ to be regular.
Now assume (iii) and let $\Cal F$ be an ultrafilter on $X$ containing
$\overline{A}$.   Set

\Centerline{$\Cal H=\{B:B\subseteq X$, there is an open set $G\in\Cal F$
such that $A\cap G\subseteq B\}$.}

\noindent Then $\Cal H$ does not contain $\emptyset$ and
$B_1\cap B_2\in\Cal H$ whenever $B_1$, $B_2\in\Cal H$, so $\Cal H$ is a
filter on $X$,
and it contains $A$.   Let $\Cal H^*\supseteq\Cal H$ be an ultrafilter
(2A1O).   By hypothesis, $\Cal H^*$ has a limit $x$ say.   Because
$A\in\Cal H^*$, $X\setminus\overline{A}$ is an open set not belonging to
$\Cal H^*$, and cannot be a neighbourhood of $x$;  thus $x$ must belong
to $\overline{A}$.   Let $G$ be an open set containing $x$.   Then there
is an open set $H$ such that $x\in H\subseteq\overline{H}\subseteq G$
(this is where I use the hypothesis that $X$ is regular).   Because
$\Cal H^*\to x$, $H\in\Cal H^*$ so $X\setminus\overline{H}$ does not
belong to $\Cal H^*$ and therefore does not belong to $\Cal H$.   But
$X\setminus\overline{H}$ is open, so by the definition of $\Cal H$ it
cannot belong to $\Cal F$.   As $\Cal F$ is an ultrafilter,
$\overline{H}\in\Cal F$ and $G\in\Cal F$.   As $G$ is arbitrary,
$\Cal F\to x$.   As $\Cal F$ is arbitrary, $\overline{A}$ is compact
(2A3R).   Thus (iii)$\Rightarrow$(ii).\ \Qed
}%end of prooflet

\leader{3A3E}{Dense sets} \cmmnt{Recall that a set $D$ in a
topological space $X$ is {\bf dense} if $\overline{D}=X$, and that $X$
is {\bf separable} if it has a countable dense subset (2A3Ud).

\medskip

}{\bf (a)} If $X$ is a topological space, $D\subseteq X$ is
dense and $G\subseteq X$ is dense and open, then $G\cap D$ is dense.
\prooflet{(\Engelking, 1.3.6.)}  Consequently the intersection of
finitely many dense open sets is always dense.

\spheader 3A3Eb If $X$ and $Y$ are topological spaces,
$D\subseteq A\subseteq X$, $D$ is dense in $A$ and
$f:X\to Y$ is a continuous
function, then $f[D]$ is dense in $f[A]$.   \prooflet{(Use 3A3Cd.)}

\vleader{48pt}{3A3F}{Meager sets} Let $X$ be a topological space.

\spheader 3A3Fa A set $A\subseteq X$ is {\bf nowhere dense}
if $\interior\overline{A}=\emptyset$\cmmnt{, that is,
$\interior(X\setminus A)=X\setminus\overline{A}$ is dense, that
is, for every non-empty open set $G$ there is a non-empty open set
$H\subseteq G\setminus A$}.

\spheader 3A3Fb A set $M\subseteq X$ is {\bf meager}
if it is expressible as the union of a sequence of nowhere
dense sets.   A subset of $X$ is {\bf comeager} if its complement is
meager.

\spheader 3A3Fc Any subset of a nowhere dense set is nowhere dense;  the
union of finitely many nowhere dense sets is nowhere dense.
\prooflet{(3A3Ea.)}
%\Bourbaki, IX.5.1;  \Gaal, p.\ 287.

\spheader 3A3Fd Any subset of a meager set is meager;  the union of
countably many meager sets is meager.   \prooflet{(314L.)}

\leader{3A3G}{Baire's theorem for locally compact Hausdorff spaces} Let
$X$ be a locally compact Hausdorff space and $\sequencen{G_n}$ a
sequence of dense open
subsets of $X$.   Then $\bigcap_{n\in\Bbb N}G_n$ is dense.
\cmmnt{$*$} \prooflet{(\Engelking, 3.9.4;  \Bourbaki, IX.5.3;
\Dugundji, p.\ 249;  \Schubert, p.\ 148.)}
Consequently every comeager subset of $X$ is dense.

\leader{3A3H}{Corollary} (a) Let $X$ be a compact Hausdorff space.
Then a non-empty open subset of $X$ cannot be meager.
\prooflet{(\Dugundji, p.\ 250;  \Schubert, p.\ 147.)}

(b) Let $X$ be a non-empty locally compact Hausdorff space.   If
$\sequencen{A_n}$ is a sequence of sets covering $X$, then there is some
$n\in\Bbb N$ such that $\interior\overline{A}_n$ is non-empty.
\prooflet{(\Dugundji, p.\ 250.)}

\leader{3A3I}{Product spaces (a) Definition} Let $\familyiI{X_i}$
be a family of topological spaces, and
$X=\prod_{i\in I}X_i$ their Cartesian product.   We say that a set
$G\subseteq X$ is open for the {\bf product topology} if for every 
$x\in G$ there are a finite $J\subseteq I$ and a family 
$\langle G_j\rangle_{j\in J}$ such that every $G_j$ is an open set in the
corresponding $X_j$ and

\Centerline{$\{y:y\in X,\,y(j)\in G_j$ for every $j\in J\}$}

\noindent contains $x$ and is included in $G$.

\cmmnt{(Of course we must check that this does indeed define a
topology;  see \Engelking, 2.3.1;  \Bourbaki, I.4.1;  \Schubert, p.\
38;  \Gaal, p.\ 144\leaveitout{;  \Christenson, p.\ 162;  \Bushaw, p.\
56}.)}

\spheader 3A3Ib If $\familyiI{X_i}$ is a family of
topological spaces, with product $X$, and $Y$ another topological
space, a function $\phi:Y\to X$ is continuous iff $\pi_i\phi$ is
continuous for every $i\in I$, where $\pi_i(x)=x(i)$ for $x\in X$ and
$i\in I$.   \prooflet{(\Engelking, 2.3.6;  \Bourbaki, I.4.1;  \Dugundji,
p.\ 101;  \Schubert, p.\ 62;  \James, p.\ 31.)}

\spheader 3A3Ic Let $\familyiI{X_i}$ be any family of
non-empty topological spaces, with product $X$.   If $\Cal F$ is a
filter on $X$ and $x\in X$, then $\Cal F\to x$ iff $\pi_i[[\Cal F]]\to
x(i)$ for every $i$, where $\pi_i(y)=y(i)$ for $y\in X$\cmmnt{, and
$\pi_i[[\Cal F]]$ is the image filter on $X_i$ (2A1Ib)}.
\prooflet{(\Bourbaki, I.7.6;  \Schubert, p.\ 61;  \James, p.\ 32.}

\spheader 3A3Id The product of any family of Hausdorff spaces is
Hausdorff.   \prooflet{(\Engelking, 2.3.11;  \Bourbaki, I.8.2;
\Schubert, p.\ 62;  \James, p.\ 87.)}

\spheader 3A3Ie Let $\familyiI{X_i}$ be any family of
topological spaces.   If $D_i$ is a dense subset of $X_i$ for each $i$,
then $\prod_{i\in I}D_i$ is dense in $\prod_{i\in I}X_i$.
\prooflet{(\Engelking, 2.3.5.)}.

\spheader 3A3If Let $\familyiI{X_i}$ be any family of
topological spaces.   If $F_i$ is a closed subset of $X_i$ for each $i$,
then $\prod_{i\in I}F_i$ is closed in $\prod_{i\in I}X_i$.
\prooflet{(\Engelking, 2.3.4;  \Bourbaki, I.4.3.)}

\spheader 3A3Ig Let $\familyiI{(X_i,\frak T_i)}$ be a family of
topological spaces with product $(X,\frak T)$.   Suppose that each
$\frak T_i$ is defined by a family $\Rho_i$ of pseudometrics on
$X_i$\cmmnt{ (2A3F)}.   Then $\frak T$ is defined by the family
$\Rho=\{\tilde\rho_i:i\in I,\rho\in\Rho_i\}$ of pseudometrics on $X$,
where I write $\tilde\rho_i(x,y)=\rho(\pi_i(x),\pi_i(y))$ whenever
$i\in I$, $\rho\in\Rho_i$ and $x$, $y\in X$\cmmnt{, taking $\pi_i$ to
be the coordinate map from $X$ to $X_i$, as in (b)-(c)}.
\prooflet{\Prf\ \cmmnt{(Compare 2A3Tb).}   (i) It is easy to check
that every $\tilde\rho_i$ is a pseudometric on $X$.   Write
$\frak T_{\Rho}$ for the topology generated by $\Rho$.   (ii) If
$x\in G\in\frak T_{\Rho}$, let $\Rho'\subseteq\Rho$ and $\delta>0$ be
such that $\Rho'$
is finite and $\{y:\tau(y,x)\le\delta$ for every $\tau\in\Rho'\}$ is
included in $G$.   Express $\Rho'$ as
$\{\tilde\rho_j:j\in J,\,\rho\in\Rho'_j\}$ where $J\subseteq I$ is
finite and $\Rho'_j\subseteq\Rho_j$ is finite for each $j\in J$.   Set

\Centerline{$G_j=\{t:t\in X_j,\,\rho(t,\pi_j(x))<\delta$ for every
$\rho\in\Rho'_j\}$}

\noindent for every $j\in J$.   Then $G'=\{y:\pi_j(y)\in G_j$ for every
$j\in J\}$ contains $x$, belongs to $\frak T$ and is included in $G$.
As $x$ is arbitrary, $G\in\frak T$;  as $G$ is arbitrary,
$\frak T_{\Rho}\subseteq\frak T$.   (iii) Every $\pi_i$ is
$(\frak T_{\Rho},\frak T_i)$-continuous, by 2A3H;  by (b) above, the
identity
map from $X$ to itself is $(\frak T_{\Rho},\frak T)$-continuous, that
is, $\frak T_{\Rho}\subseteq\frak T$ and $\frak T_{\Rho}=\frak T$, as
claimed.\ \Qed
}%end of prooflet

\spheader 3A3Ih Let $\familyiI{X_i}$ be a family of
topological spaces with product $X$, and $Y$ another topological space.
Then a function $f:X\to Y$ is {\bf separately continuous} if for every
$j\in I$ and $z\in\prod_{i\in I\setminus\{j\}}X_i$ the function
$t\mapsto f(z^{\smallfrown}\fraction{t}):X_j\to Y$ is continuous, where
$z^{\smallfrown}\fraction{t}$ is
the member of $X$ extending $z$ and such that
$(z^{\smallfrown}\fraction{t})(j)=t$.

\leader{3A3J}{Tychonoff's theorem} The product of any family of compact
topological spaces is compact.

\proof{\Engelking, 3.2.4;  \Bourbaki, I.9.5;  \Dugundji, p.\ 224;
\Schubert, p.\ 72;   \Gaal, p.\ 146 and p.\ 272;  \James, p.\
67\leaveitout{; \Christenson, p.\ 164;  \Bushaw, p.\ 79}.
}%end of proof of 3A3J

\leader{3A3K}{The spaces $\{0,1{\delimiter"5267309}^I$, $\BbbR^I$} For any set $I$, we
can think of $\{0,1\}^I$ as the product $\prod_{i\in I}X_i$ where
$X_i=\{0,1\}$ for each $i$.   If we endow each $X_i$ with its discrete
topology, the product topology is the {\bf usual topology} on
$\{0,1\}^I$.   Being a product of Hausdorff spaces, it is Hausdorff;  by
Tychonoff's theorem, it is compact.
A subset $G$ of $\{0,1\}^I$ is open iff for every $x\in G$ there is a
finite $J\subseteq I$ such that
$\{y:y\in\{0,1\}^I,\,y\restr J=x\restr J\}\subseteq G$.

Similarly, the `usual topology' of $\BbbR^I$ is the product topology
when each factor is given its Euclidean topology\cmmnt{ (cf.\
2A3Tc)}.

\leader{3A3L}{Cluster points of filters (a)} Let $X$ be a topological
space and $\Cal F$ a filter on $X$.   A point $x$ of $X$ is a {\bf
cluster point} of $\Cal F$ if $x\in\overline{A}$ for every $A\in\Cal F$.

\spheader 3A3Lb For any topological space $X$, filter $\Cal F$ on $X$
and $x\in X$, $x$ is a cluster point
of $\Cal F$ iff there is a filter $\Cal G\supseteq\Cal F$ such that
$\Cal G\to x$.  \prooflet{(\Engelking, 1.6.8;  \Bourbaki, I.7.2;  \Gaal,
p.\ 260;  \James, p.\ 22\leaveitout{;  \Christenson, p.\ 286}.)}

\leaveitout{
\spheader 3A3Lc For any topological space $X$, filter $\Cal F$ on $X$
and $x\in X$, the following are equiveridical:  (i) $\Cal F\to x$;  (ii)
$x$ is a cluster point of every filter $\Cal G\supseteq\Cal F$;  (iii)
$\Cal G\to x$ for every ultrafilter $\Cal G\supseteq\Cal F$.
\prooflet{\Prf (i)$\Rightarrow$(iii) is elementary, and
(iii)$\Rightarrow$(ii) is a consequence of (b) and the Ultrafilter
Theorem.   Now assume (ii).   \Quer\ If $\Cal F\not\to x$, there is an
open set $G$ containing $x$ and not belonging to $\Cal F$.   Set $\Cal
G=\{B:\,\exists\, F\in\Cal F,\,H\supseteq F\setminus G\}$.   Then $\Cal
G$ is a filter including $\Cal F$ and $X\setminus G$ is a closed set,
belonging to $\Cal G$, and not containing $x$, so $x$ is not a cluster
point of $\Cal G$.\ \Bang\  So (i) is true.   Compare \Dugundji, p.\
214.\ \Qed
}%end of prooflet
}%end of leaveitout

\spheader 3A3Lc If $\sequencen{\alpha_n}$ is a sequence
in $\Bbb R$, $\alpha\in\Bbb R$ and $\lim_{n\to\Cal H}\alpha_n=\alpha$
for every non-principal ultrafilter
$\Cal H$ on $\Bbb N$\cmmnt{ (definition:  2A3Sb)}, then
$\lim_{n\to\infty}\alpha_n=\alpha$.
\prooflet{\Prf\Quer\ If $\sequencen{\alpha_n}\not\to\alpha$, there is
some $\epsilon>0$ such that
$I=\{n:|\alpha_n-\alpha|\ge\epsilon\}$ is infinite.   Now $\Cal
F_0=\{F:F\subseteq\Bbb N,\,I\setminus F$ is finite$\}$ is a filter on
$\Bbb N$, so there is an ultrafilter $\Cal F\supseteq\Cal F_0$.   But
now $\alpha$ cannot be $\lim_{n\to\Cal F}\alpha_n$.\ \Bang\Qed}

\leader{3A3M}{Topology bases (a)} If $X$ is a set and $\Bbb T$ is any
non-empty family of topologies on $X$, $\bigcap\Bbb T$ is a topology on
$X$.   So if $\Cal A$ is any
family of subsets of $X$, the intersection of all the topologies on $X$
including $\Cal A$ is a topology on $X$;  this is the
{\bf topology generated by $\Cal A$}.

\header{3A3Mb}{\bf (b)} If $X$ is a set and $\frak T$ is a topology on
$X$, a {\bf base} for $\frak T$ is a set $\Cal U\subseteq\frak T$ such
that whenever $x\in G\in\frak T$ there is a $U\in\Cal U$ such that
$x\in U\subseteq G$\cmmnt{;  that is, such that
$\frak T=\{\bigcup\Cal G:\Cal G\subseteq\Cal U\}$}.
In this case\cmmnt{, of course,} $\Cal U$ generates $\frak T$.

\header{3A3Mc}{\bf (c)} If $X$ is a set and $\Cal E$ is a family of
subsets of $X$, then $\Cal E$ is a base for a topology on $X$ iff (i)
whenever $E_1$, $E_2\in\Cal E$ and $x\in E_1\cap E_2$ then there is an
$E\in\Cal E$ such that $x\in E\subseteq E_1\cap E_2$ (ii) $\bigcup\Cal
E=X$.  \prooflet{(\Engelking, p.\ 12\leaveitout{;  \Christenson, p.\ 16;
\Bushaw, p.\ 47}.)}

\leader{3A3N}{Uniform convergence (a)} Let $X$ be a set, $(Y,\rho)$ a
metric space and
$\sequencen{f_n}$ a sequence of functions from $X$ to $Y$.   We say
that $\sequencen{f_n}$ converges {\bf uniformly} to a function
$f:X\to Y$ if for every $\epsilon>0$ there is an $n_0\in\Bbb N$ such
that $\rho(f_n(x),f(x))\le\epsilon$ whenever $n\ge n_0$ and $x\in X$.

\spheader 3A3Nb Let $X$ be a topological space and $(Y,\rho)$ a metric
space.   Suppose that $\sequencen{f_n}$ is a
sequence of continuous functions from $X$ to $Y$ converging uniformly
to $f:X\to Y$.   Then $f$ is continuous.  \cmmnt{$*$}
\prooflet{(\Engelking, 1.4.7/4.2.19;  \Gaal, p.\ 202.)}

\leader{3A3O}{One-point compactifications} Let $(X,\frak T)$ be a
locally compact Hausdorff space.   Take any object $x_{\infty}$ not
belonging to $X$ and set $X^*=X\cup\{x_{\infty}\}$.   Let $\frak T^*$ be
the family of those sets $H\subseteq X^*$ such that $H\cap X\in\frak T$
and either $x_{\infty}\notin H$ or $X\setminus H$ is compact (for
$\frak T$).   Then $\frak T^*$ is the unique compact Hausdorff topology on
$X^*$ inducing $\frak T$ as the subspace topology on $X$;
$(X^*,\frak T^*)$ is the {\bf one-point compactification}
of $(X,\frak T)$.   \prooflet{(\Engelking, 3.5.11;
\Bourbaki, I.9.8;  \Dugundji, p.\ 246.)}

\leader{3A3P}{Topologies defined from a sequential convergence:
Proposition}
(a) Let $X$ be a set and $\to^*$ a relation between $X^{\Bbb N}$ and $X$
such that whenever $\sequencen{x_n}\in X^{\Bbb N}$, $x\in X$,
$\sequencen{x_n}\to^*x$ and $\sequencen{x'_n}\in X^{\Bbb N}$ is a
subsequence of $\sequencen{x_n}$ then $\sequencen{x'_n}\to^*x$.
Then there is a unique topology on $X$ for which a set
$F\subseteq X$ is closed iff $x\in F$ whenever $\sequencen{x_n}$ is a
sequence in $F$ and $\sequencen{x_n}\to^*x$.   Moreover, if
$\sequencen{x_n}\to^*x$ then $\sequencen{x_n}$ converges to $x$ for this
topology.

(b) Let $X$ and $Y$ be sets, and suppose that
$\to^*_X\mskip5mu\subseteq X^{\Bbb N}\times X$,
$\to^*_Y\mskip5mu\subseteq Y^{\Bbb N}\times Y$ are relations with the
subsequence property described in (a).   Give $X$ and $Y$ the corresponding
topologies.   If $f:X\to Y$ is a function such that
$\sequencen{f(x_n)}\to^*_Yf(x)$ whenever
$\sequencen{x_n}\to_X^*x$, then $f$ is continuous.

\proof{{\bf (a)(i)} Let $\Cal F$ be the family of those
$F\subseteq X$ which are closed under *-convergence, that is, such that
$x\in F$ whenever $\sequencen{x_n}$ is a sequence in $F$ and
$\sequencen{x_n}\to^*x$.   Of course $\emptyset$ and
$X$ belong to $\Cal F$;  also the intersection of any non-empty subset
of $\Cal F$ belongs to $\Cal F$.
The point is that the union of two members of $\Cal F$ belongs to
$\Cal F$.   \Prf\ Suppose that $F_1$, $F_2\in\Cal F$
and that $\sequencen{x_n}$ is a sequence in $F_1\cup F_2$
*-converging to $x$.   Then there
is a subsequence $\sequencen{x'_n}$ of $\sequencen{x_n}$ which lies
entirely in one of the sets;  say $x'_n\in F_j$ for
every $n$.   By hypothesis, $\sequencen{x'_n}\to^*x$, so
$x\in F_j\subseteq F_1\cup F_2$.   As
$\sequencen{x_n}$ and $x$ are arbitrary, $F_1\cup F_2\in\Cal F$.\ \QeD\
Taking complements, we see that
$\{X\setminus F:F\in\Cal F\}$ is a topology on $X$ for which $\Cal F$ is
the family of closed sets;  and of course
there can be only one such topology.

\medskip

\quad{\bf (ii)} Now suppose that $\sequencen{x_n}\to^*x$.   \Quer\ If
$\sequencen{x_n}$ does not converge topologically to $x$, then there is
an open set $G$ containing $x$ such that $\{n:x_n\notin G\}$ is
infinite, that is, there is a subsequence $\sequencen{x'_n}$ of
$\sequencen{x_n}$ such that $x'_n\notin G$ for every $n$.   Now
$\sequencen{x'_n}\to^*x$;  but $X\setminus G$ is supposed to be closed
under *-convergence, so this is impossible.\ \Bang

\medskip

{\bf (b)} Let $H\subseteq Y$ be open;  set $F=Y\setminus H$ and
let $\sequencen{x_n}$ be a sequence in $f^{-1}[F]$ such that
$\sequencen{x_n}\to^*_Xx$.   Then $\sequencen{f(x_n)}\to^*_Yf(x)$, so
$f(x)\in F$ and $x\in f^{-1}[F]$,   As $\sequencen{x_n}$ and $x$ are
arbitrary, $f^{-1}[F]$ is closed and $f^{-1}[H]$ is open;  as $H$ is
arbitrary, $f$ is continuous.
}%end of proof of 3A3P

\leader{3A3Q}{Miscellaneous definitions} Let $X$ be a topological space.

\spheader 3A3Qa A subset of $X$ is a {\bf zero set} if it is of the
form $f^{-1}[\{0\}]$ for some continuous function $f:X\to\Bbb R$.   A
subset of $X$ is a {\bf cozero set} if its complement is a zero set.
A subset of $X$ is a {\bf G$_{\delta}$ set}
if it is expressible as the intersection of a sequence of open sets.

\spheader 3A3Qb An {\bf isolated point} of $X$ is a point $x\in X$ such
that the singleton set $\{x\}$ is open.

\discrpage

\frfilename{mt3a4.tex}
\versiondate{30.1.08}
\copyrightdate{1996}

\def\chaptername{Appendix}
\def\sectionname{Uniformities}

%\def\ref{\medskip\noindent < \medskip}
\def\Bourbaki{{\smc Bourbaki 66}}
\def\Engelking{{\smc Engelking 89}}
\def\Gaal{{\smc Gaal 64}}
\def\James{{\smc James 87}}
\def\Schubert{{\smc Schubert 68}}

\newsection{3A4}

I continue the work of \S3A3 with some notes on uniformities, so as to
be able to discuss completeness and the extension of uniformly
continuous functions in non-metrizable contexts (3A4F-3A4H).
\cmmnt{As in \S3A3, most of the individual steps are elementary;  I
mark exceptions with a $*$.}%end of comment

\leader{3A4A}{Uniformities (a)} Let $X$ be a set.   A {\bf uniformity}
on $X$ is a filter $\Cal W$ on $X\times X$ such that

\quad(i) $(x,x)\in W$ for every $x\in X$, $W\in\Cal W$;

\quad(ii) for every $W\in\Cal W$,
$W^{-1}=\{(y,x):(x,y)\in W\}\in\Cal W$;

\quad(iii) for every $W\in\Cal W$, there is a $V\in\Cal W$ such that

\Centerline{$V\frsmallcirc V
=\{(x,z):\,\Exists y,\,(x,y)\in V\,\&\,(y,z)\in V\}\subseteq W$.}

\noindent It is convenient to allow the special case $X=\emptyset$,
$\Cal W=\{\emptyset\}$\cmmnt{, even though this is not properly
speaking a filter}.

The pair $(X,\Cal W)$ is now a {\bf uniform space}.

\spheader 3A4Ab If $\Cal W$ is a uniformity on a set $X$, it induces a topology
on $X$, the family of sets $G\subseteq X$ such that
for every $x\in G$ there is a $W\in\Cal W$ such that
$W[\{x\}]\cmmnt{\mskip5mu=\{y:(x,y)\in W\}}$ is included in $G$.
\prooflet{(\Engelking, 8.1.1;  \Bourbaki, II.1.2;  \Gaal, p.\
48;  \Schubert, p.\ 115;  \James, p.\ 101.)}

\spheader 3A4Ac We say that a uniformity is {\bf Hausdorff} if it
induces a Hausdorff topology.

\spheader 3A4Ad If $U$ is a linear topological space, then it has an
associated uniformity

$$\eqalign{\Cal W=\{W:W\subseteq U\times U,\text{ there is an}
&\text{ open set } G\text{ containing }0\cr
&\text{ such that }(u,v)\in W\text{ whenever }u-v\in G\},\cr}$$

\noindent and $\Cal W$ induces the topology of $U$ in the sense of (b) 
above\prooflet{ ({\smc Schaefer 66}, I.1.4)}.

\leader{3A4B}{Uniformities and pseudometrics (a)} If $\Rho$ is a family
of pseudometrics on a set $X$, then the associated uniformity is the
smallest uniformity on $X$ containing all the sets
$W(\rho;\epsilon)=\{(x,y):\rho(x,y)<\epsilon\}$
as $\rho$ runs over $\Rho$, $\epsilon$ over $\ooint{0,\infty}$.
\prooflet{(\Engelking, 8.1.18;  \Bourbaki, IX.1.2.)}

\spheader 3A4Bb If $\Cal W$ is the uniformity defined by a family $\Rho$
of pseudometrics, then the topology induced by $\Cal W$ is the
topology defined from $\Rho$\cmmnt{ (2A3F)}.
\prooflet{({\smc Dugundji 66}, p.\ 203.)}

\spheader 3A4Bc A uniformity $\Cal W$ is {\bf metrizable} if it can be
defined by a single metric.

\spheader 3A4Bd If $U$ is a linear space with a topology defined from a
family $\Tau$ of F-seminorms\cmmnt{ (definition:  2A5B\Latereditions)},
the uniformity defined from the topology\cmmnt{ (3A4Ad)} coincides with
the uniformity defined from the pseudometrics
$\rho_{\tau}(u,v)=\tau(u-v)$ as $\tau$ runs over $\Tau$.   
\prooflet{(Immediate from the definitions.)}

\leader{3A4C}{Uniform continuity (a)} If $(X,\Cal W)$ and $(Y,\Cal V)$
are uniform spaces, a function $\phi:X\to Y$ is {\bf uniformly
continuous} if $\{(x,y):(\phi(x),\phi(y))\in V\}$ belongs to
$\Cal W$ for every $V\in\Cal V$.

\header{3A4Cb}{\bf (b)} The composition of uniformly continuous
functions is uniformly continuous.
\prooflet{(\Bourbaki, II.2.1;  \Schubert, p.\ 118.)}

\spheader 3A4Cc If uniformities $\Cal W$, $\Cal V$ on sets $X$, $Y$ are
defined by
non-empty families $\Rho$, $\Theta$ of pseudometrics, then a function
$\phi:X\to Y$ is uniformly continuous iff for every $\theta\in\Theta$,
$\epsilon>0$ there are $\rho_0,\ldots,\rho_n\in\Rho$ and $\delta>0$ such
that $\theta(\phi(x),\phi(y))\le\epsilon$ whenever $x$, $y\in X$ and
$\max_{i\le n}\rho_i(x,y)\le\delta$.
%3{}25D
\prooflet{(Elementary verification.)}

\leaveitout{If $\Rho$ is upwards-directed then we can say:
$\phi:X\to Y$ is uniformly continuous iff for every $\theta\in\Theta$,
$\epsilon>0$ there are $\rho\in\Rho$ and $\delta>0$ such
that $\theta(\phi(x),\phi(y))\le\epsilon$ whenever $x$, $y\in X$ and
$\rho(x,y)\le\delta$.
\ref
}%end of leaveitout

\spheader 3A4Cd A uniformly continuous function is continuous for the
induced topologies.
\prooflet{(\Bourbaki, II.2.1;  \Schubert, p.\ 118;  \James, p.\ 102.)}

\spheader 3A4Ce Two metrics $\rho$, $\sigma$ on a set $X$ are {\bf
uniformly equivalent} if they give rise to the same uniformity\cmmnt{;  
that is, if for every $\epsilon>0$ there is a $\delta>0$ such that

\Centerline{$\rho(x,y)\le\delta\Rightarrow\sigma(x,y)\le\epsilon$,
\quad$\sigma(x,y)\le\delta\Rightarrow\rho(x,y)\le\epsilon$.}}

\leaveitout{If $\Rho$ and $\Theta$ are two non-empty families of
pseudometrics on
$X$, then they define the same uniformity iff

\inset{for every $\rho\in\Rho$, $\epsilon>0$ there are $\delta>0$,
$\theta_0,\ldots,\theta_n\in\Theta$ such that $\rho(x,y)\le\epsilon$
whenever $\max_{i\le n}\theta_i(x,y)\le\delta$}

\noindent and

\inset{for every $\theta\in\Theta$, $\epsilon>0$ there are $\delta>0$,
$\rho_0,\ldots,\rho_n\in\Rho$ such that $\theta(x,y)\le\epsilon$
whenever $\max_{i\le n}\rho_i(x,y)\le\delta$.}
\ref

If $\Rho$ and $\Theta$ are upwards-directed (2A3Fe), this simplifies to

\inset{for every $\rho\in\Rho$, $\epsilon>0$ there are $\delta>0$,
$\theta\in\Theta$ such that $\rho(x,y)\le\epsilon$ whenever
$\theta(x,y)\le\delta$}

\noindent and

\inset{for every $\theta\in\Theta$, $\epsilon>0$ there are $\delta>0$,
$\rho\in\Rho$ such that $\theta(x,y)\le\epsilon$ whenever
$\rho(x,y)\le\delta$.}
}%end of leaveitout

\spheader 3A4Cf\dvAnew{2013} 
If $U$ and $V$ are linear topological spaces, and
$T:U\to V$ is a continuous linear operator, then $T$ is uniformly
continuous for the uniformities associated with the topologies of
$U$ and $V$\cmmnt{ (3A4Ad)}.   \prooflet{ ({\smc Schaefer 66}, \query).} 

\leader{3A4D}{Subspaces (a)} If $(X,\Cal W)$ is a uniform space and $Y$
is any subset of $X$, then $\Cal W_Y=\{W\cap(Y\times Y):W\in\Cal W\}$ is
a uniformity on $Y$;  it is the {\bf subspace uniformity}.
\prooflet{(\Bourbaki, II.2.4;  \Schubert, p.\ 122.)}

\header{3A4Db}{\bf (b)} If $\Cal W$ defines a topology $\frak T$ on $X$,
then the topology defined by $\Cal W_Y$ is the subspace topology on
$Y$\cmmnt{, as defined in 2A3C}.
\prooflet{(\Schubert, p.\ 122;  \James, p.\ 103.)}

\header{3A4Dc}{\bf (c)} If $\Cal W$ is defined by a family $\Rho$ of
pseudometrics on $X$, then $\Cal W_Y$ is defined by
$\{\rho\,\restr\,Y\times Y:\rho\in\Rho\}$.
\prooflet{(Elementary verification.)}

\leader{3A4E}{Product uniformities (a)} If $(X,\Cal U)$ and $(Y,\Cal V)$
are uniform spaces, the {\bf product uniformity} is the
smallest uniformity $\Cal W$ on $X\times Y$ containing all sets of the
form

\Centerline{$\{((x,y),(x',y')):(x,x')\in U,\,(y,y')\in V\}$}

\noindent as $U$ runs over $\Cal U$ and $V$ over $\Cal V$.
\prooflet{(\Engelking, \S8.2;  \Bourbaki, II.2.6;  \Schubert, p.\ 124;
\James, p.\ 93.)}

\header{3A4Eb}{\bf (b)} If $\Cal U$, $\Cal V$ are defined from families
$\Rho$, $\Theta$ of pseudometrics, then $\Cal W$ will be defined by the
family $\{\tilde\rho:\rho\in\Rho\}\cup\{\bar\theta:\theta\in\Theta\}$,
writing

\Centerline{$\tilde\rho((x,y),(x',y'))=\rho(x,x'),
\quad\bar\theta((x,y),(x',y'))=\theta(y,y')$}

\noindent as in 2A3Tb.
\prooflet{(Elementary verification.)}

\header{3A4Ec}{\bf (c)} If $(X,\Cal U)$, $(Y,\Cal V)$ and $(Z,\Cal W)$
are uniform spaces, a map $\phi:Z\to X\times Y$ is uniformly continuous
iff the coordinate maps $\phi_1:Z\to X$ and $\phi_2:Z\to Y$ are
uniformly continuous.
\prooflet{(\Engelking, 8.2.1;  \Bourbaki, II.2.6;  \Schubert, p.\ 125;
\James, p.\ 93.)}

\leader{3A4F}{Completeness (a)} If $\Cal W$ is a uniformity on a set
$X$, a filter $\Cal F$ on $X$ is {\bf Cauchy} if for every $W\in\Cal W$
there is an $F\in\Cal F$ such that $F\times F\subseteq W$.

Any convergent filter in a uniform space is Cauchy.
\prooflet{(\Bourbaki, II.3.1;  \Gaal, p.\ 276;  \Schubert, p.\ 134;
\James, p.\ 109.)}

\header{3A4Fb}{\bf (b)} A uniform space is {\bf complete} if every
Cauchy filter is convergent.

\header{3A4Fc}{\bf (c)} If $\Cal W$ is defined from a family $\Rho$ of
pseudometrics, then a
filter $\Cal F$ on $X$ is Cauchy iff for every $\rho\in\Rho$ and
$\epsilon>0$ there is an $F\in\Cal F$ such that $\rho(x,y)\le\epsilon$
for all $x$, $y\in F$;  equivalently, for every $\rho\in\Rho$,
$\epsilon>0$ there is an $x\in X$ such that
$U(x;\rho;\epsilon)\in\Cal F$.   \prooflet{(Elementary verification.)}

\spheader 3A4Fd A complete subspace of a Hausdorff uniform space is
closed.
\prooflet{(\Engelking, 8.3.6;  \Bourbaki, II.3.4;  \Schubert, p.\ 135;
\James, p.\ 148.)}   A closed subspace of a complete uniform space is complete
under the subspace uniformity (\query references)\dvAnew{2017}.

\spheader 3A4Fe A metric space is complete iff every Cauchy sequence
converges\cmmnt{ (cf.\ 2A5H)}.
\prooflet{(\Schubert, p.\ 141;  \Gaal, p.\ 276;  \James, p.\ 150.)}

\spheader 3A4Ff If $(X,\rho)$ is a complete metric space, $D\subseteq X$
a dense subset, $(Y,\sigma)$ a metric space and $f:X\to Y$ is an
{\bf isometry} (that is, $\sigma(f(x),f(x'))=\rho(x,x')$ for all $x$,
$x'\in X$), then $f[X]$ is precisely the closure of $f[D]$ in $Y$.
\prooflet{(For $f[X]$ must be complete, and we can use (d).)}

\spheader 3A4Fg If $U$ is a linear space with a linear space topology
and the associated uniformity\cmmnt{ (3A4Ad)}, then a filter $\Cal F$
on $U$ is Cauchy iff for every open set $G$ containing $0$ there is an
$F\in\Cal F$ such that $F-F\subseteq G$\cmmnt{ (cf.\ 2A5F)}.
\prooflet{(Immediate from the definitions.)}

\leader{3A4G}{Extension of uniformly continuous functions:  Theorem} If
$(X,\Cal W)$ is a uniform space, $(Y,\Cal V)$ is a complete uniform
space, $D\subseteq X$ is a dense subset of $X$, and $\phi:D\to Y$ is
uniformly continuous\cmmnt{ (for the subspace uniformity of $D$)},
then there is a uniformly continuous $\hat\phi:X\to Y$ extending $\phi$.
If $Y$ is Hausdorff, the extension is unique.
\cmmnt{$*$} \prooflet{(\Engelking, 8.3.10;  \Bourbaki, II.3.6;  \Gaal,
p.\ 300;  \Schubert, p.\ 137;  \James, p.\ 152.)}

In particular, if $(X,\rho)$ is a metric space, $(Y,\sigma)$ is a
complete metric space, $D\subseteq X$ is a dense subset, and
$\phi:D\to Y$ is an isometry, then there is a unique isometry
$\hat\phi:X\to Y$ extending $\phi$.

\leader{3A4H}{Completions (a) Theorem} If $(X,\Cal W)$ is any Hausdorff
uniform space, then we can find a complete Hausdorff uniform space
$(\hat X,\hat{\Cal W})$ in which $X$ is embedded as a dense
subspace;   moreover, any two such spaces are essentially unique.
\cmmnt{$*$} \prooflet{(\Engelking, 8.3.12;  \Bourbaki, II.3.7;  \Gaal,
p.\ 297 \& p.\ 300;  \Schubert, p.\ 139;  \James, p.\ 156.)}

\spheader 3A4Hb Such a space $(\hat X,\hat{\Cal W})$ is called a
{\bf completion} of $(X,\Cal W)$.   Because it is unique up to
isomorphism as a uniform space, we may call it `the' completion.

\spheader 3A4Hc If $\Cal W$ is the
uniformity defined by a metric $\rho$ on a set $X$, then there is a
unique extension of $\rho$ to a metric $\hat\rho$ on $\hat X$ defining
the uniformity $\hat{\Cal W}$.
\prooflet{(\Bourbaki, IX.1.3.)}

\leader{3A4I}{A note on metric spaces}\cmmnt{ I mention some
elementary facts.}   Let $(X,\rho)$ be a metric
space.   If $x\in X$ and $A\subseteq X$ is non-empty, set

\Centerline{$\rho(x,A)=\inf_{y\in A}\rho(x,y)$.}

\noindent Then $\rho(x,A)=0$ iff $x\in\overline{A}$\cmmnt{ (2A3Kb)}.
If $B\subseteq X$ is another non-empty set, then

\Centerline{$\rho(x,B)\le\rho(x,A)+\sup_{y\in A}\rho(y,B)$.}

\noindent In particular, $\rho(x,\overline{A})=\rho(x,A)$.   If
$\sequencen{A_n}$ is a non-decreasing sequence of non-empty sets with
union $A$, then

\Centerline{$\rho(x,A)=\lim_{n\to\infty}\rho(x,A_n)$.}

\discrpage

\frfilename{mt3a5.tex}
\versiondate{22.5.11}
\copyrightdate{1999}

\def\chaptername{Appendix}
\def\sectionname{Normed spaces}

\def\ref{\discrversionA{\medskip\noindent$<$\medskip}{}}
\def\Bourbaki{{\smc Bourbaki 87}}
\def\DS{{\smc Dunford \& Schwartz 57}}
\def\Lang{{\smc Lang 93}}
\def\Rudin{{\smc Rudin 91}}
\def\Taylor{{\smc Taylor 64}}
\def\Wilansky{{\smc Wilansky 64}}

\newsection{3A5}

I run as quickly as possible over the results, nearly all of them
standard elements of any introductory course in functional analysis,
which I find myself calling on in this volume.   As in the corresponding
section of Volume 2 (\S2A4), a large proportion of these are valid for
both real and complex normed spaces, but as the present volume is almost
exclusively concerned with real linear spaces I leave this unsaid,
except in 3A5M, and if in doubt you may suppose for the time being that
scalars belong
to the field $\Bbb R$.   A couple of the most basic results will be used
in their complex forms in Volume 4.

%interpolate definition of Hamel basis for 376Ya??

\leader{3A5A}{The Hahn-Banach theorem:  analytic forms}\cmmnt{ The
Hahn-Banach theorem is
one of the central ideas of functional analysis, both finite- and
infinite-dimensional, and appears in a remarkable variety of forms.   I
list those formulations which I wish to quote, starting with those which
are more or less `analytic', according to the classification of
\Bourbaki.   Recall that if $U$ is a normed space I write $U^*$ for the
Banach space of bounded linear functionals on $U$.

\medskip

} {\bf (a)} Let $U$ be a linear space and $p:U\to\coint{0,\infty}$ a
functional such that $p(u+v)\le p(u)+p(v)$ and $p(\alpha u)=\alpha p(u)$
whenever $u$, $v\in U$ and $\alpha\ge 0$.   Then for any $u_0\in U$ there is a linear functional
$f:U\to\Bbb R$ such that $f(u_0)=p(u_0)$ and $f(u)\le p(u)$ for every
$u\in U$.
\prooflet{(\Rudin, 3.2;  \DS, II.3.10.)}
%\Bourbaki query
%\Lang query
%\Taylor not there
%cf. 4A4D
%for 3{}91E

\spheader 3A5Ab
Let $U$ be a normed space and $V$ a linear subspace of $U$.   Then for
any $f\in V^*$ there is a $g\in U^*$, extending $f$, with $\|g\|=\|f\|$.
\prooflet{(363R;  \Bourbaki, II.3.2;  \Rudin, 3.3;  \DS, II.3.11;
\Lang, p.\ 69;   \Wilansky, p.\ 66;  \Taylor, 3.7-B \& 4.3-A.)}
%3-63Q\Pedersen, 2.3.3

\spheader 3A5Ac If $U$ is a normed space and $u\in U$ there is an
$f\in U^*$ such that $\|f\|\le 1$ and $f(u)=\|u\|$.
\prooflet{(\Bourbaki, II.3.2;  \Rudin, 3.3;  \DS, II.3.14;  \Wilansky,
p.\ 67;  \Taylor, 3.7-C \& 4.3-B.)}
%3-91F\Pedersen, 2.3.4;

\spheader 3A5Ad If $U$ is a normed space and $V\subseteq U$ is a linear
subspace which is not dense, then there is a non-zero $f\in U^*$ such
that $f(v)=0$ for every $v\in V$.
\prooflet{(\Rudin, 3.5;  \DS, II.3.12;  \Taylor, 4.3-D.)}
%3-72A

\spheader 3A5Ae If $U$ is a normed space, $U^*$ separates the points of
$U$.
\prooflet{(\Rudin, 3.4;  \Lang, p.\ 70;  \DS, II.3.14.)}
%3-68

\leader{3A5B}{Cones (a)}
%\spheader 3A5Ba
Let $U$ be a linear space.  A {\bf convex cone} (with
apex $0$) is a set $C\subseteq U$ such that $\alpha u+\beta v\in C$
whenever $u$, $v\in C$ and $\alpha$, $\beta\ge 0$.   The intersection of
any family of convex cones is a convex cone, so for every subset $A$ of
$U$ there is a smallest convex cone including $A$.

\spheader 3A5Bb Let $U$ be a normed space.   Then the closure of a
convex cone is a convex cone.
\prooflet{(\Bourbaki, II.2.6;  \DS, V.2.1.)}

\leader{3A5C}{Hahn-Banach theorem:  geometric forms (a)} Let $U$ be a
normed space and $C\subseteq U$ a convex set such that $\|u\|\ge 1$ for
every $u\in C$.   Then there is an $f\in U^*$ such that $\|f\|\le 1$ and
$f(u)\ge 1$ for every $u\in C$.   \prooflet{(\DS, V.1.12.)}

\spheader 3A5Cb Let $U$ be a normed space and $C\subseteq U$ a non-empty
convex set such
that $0\notin\overline{C}$.   Then there is an $f\in U^*$ such that
$\inf_{u\in C}f(u)>0$.
\prooflet{(\Bourbaki, II.4.1; \Rudin, 3.4;  \Lang, p.\ 70;
\DS, V.2.12.)}
%3-56

\spheader 3A5Cc Let $U$ be a normed space, $C$ a closed convex subset of
$U$ containing $0$, and $u$ a point of $U\setminus C$.   Then there is
an $f\in U^*$ such that $f(u)>1$ and $f(v)\le 1$ for every $v\in C$.
\prooflet{(Apply (b) to $C-u$ to find a $g\in U^*$ such that
$g(u)<\inf_{v\in C}g(v)$ and now set $f=-\bover1{\alpha}g$ where
$g(u)<\alpha<\inf_{v\in C}g(v)$).}
%for 3-69H

\leader{3A5D}{Separation from finitely-generated cones} Let $U$ be a
linear space over $\Bbb R$ and $u$,
$v_0,\ldots,v_n$ points of $U$ such that $u$ does not belong to the
convex cone generated by $\{v_0,\ldots,v_n\}$.   Then there is a
linear functional $f:U\to\Bbb R$ such that $f(v_i)\ge 0$ for every $i$
and $f(u)<0$.

\proof{{\bf (a)} If $U$ is finite-dimensional this is covered by 
{\smc Gale 60}, p.\ 56.

\medskip

{\bf (b)} For the general case, let $V$ be the linear subspace of $U$
generated by $u$, $v_0,\ldots,v_n$.   Then there is a linear functional
$f_0:V\to\Bbb R$ such that $f_0(u)<0\le f_0(v_i)$ for every $i$.   By
Zorn's Lemma, there is a maximal linear subspace $W\subseteq U$ such
that $W\cap V=\{0\}$.   Now $W+V=U$ (for if $u\notin W+V$, the linear
subspace $W'$ generated by $W\cup\{u\}$ still has trivial intersection
with $V$), so we have an extension of $f_0$ to a linear functional
$f:U\to \Bbb R$ defined
by setting $f(v+w)=f_0(v)$ whenever $v\in V$ and $w\in W$.   Now
$f(u)<0\le\min_{i\le n}f(v_i)$, as required.
}%end of proof of 3A5D
%for 3-51N

\leader{3A5E}{Weak topologies (a)} Let $U$ be any linear space over
$\Bbb R$ and $W$ a subset of the space $U'$ of all linear
functionals from $U$ to $\Bbb R$.   Then I write $\frak T_s(U,W)$ for
the linear space topology
defined by the method of 2A5B from the seminorms
$u\mapsto|f(u)|$ as $f$ runs over $W$.
\cmmnt{ (\Bourbaki, II.6.2;  \Rudin, 3.10;  \DS, V.3.2;  \Taylor,
3.81.)}
%3-76O

\cmmnt{\spheader 3A5Eb I note that the weak topology of a normed space
$U$\cmmnt{ (2A5Ia)} is $\frak T_s(U,U^*)$, while the weak* topology
of $U^*$\cmmnt{ (2A5Ig)} is $\frak T_s(U^*,W)$ where $W$ is the
canonical image of $U$ in $U^{**}$.
\cmmnt{(\Rudin, 3.14.)}
}%end of comment

\spheader 3A5Ec Let $U$ and $V$ be linear spaces over $\Bbb R$ and
$T:U\to V$ a linear operator.   If $W\subseteq U'$ and $Z\subseteq V'$
are such that $gT\in W$ for every $g\in Z$, then $T$ is continuous for
$\frak T_s(U,W)$ and $\frak T_s(V,Z)$.
\cmmnt{(\Bourbaki, II.6.4.)}

\spheader 3A5Ed If $U$ and $V$ are normed spaces and $T:U\to V$ is a
bounded linear operator then we have an {\bf adjoint}
%\cmmnt{ (or {\bf conjugate}, or {\bf dual})}
operator $T':V^*\to U^*$ defined by
saying that $T'g=gT$ for every $g\in V^*$.   $T'$ is linear and is
continuous for the weak* topologies of $U^*$ and $V^*$.
\prooflet{(\Bourbaki, II.6.4;  \DS, \S VI.2;  \Taylor, 4.5.)}
%for 3-67L, 3-93;  \Pedersen, 2.4.12

\spheader 3A5Ee If $U$ is a normed space and $A\subseteq U$ is convex,
then the closure of $A$ for the norm topology is the same as the closure
of $A$ for the weak topology of $U$.   In particular, norm-closed convex
subsets (for instance, norm-closed linear
subspaces) of $U$ are closed for the weak topology.
\prooflet{(\Rudin, 3.12;  \Lang, p.\ 88;  \DS, V.3.13.)}
%for 3-67L, 3-72R

\leader{3A5F}{Weak* topologies:  Theorem} If $U$ is a normed space,
the unit ball of $U^*$ is compact and Hausdorff for the weak*
topology.
\prooflet{(\Rudin, 3.15;  \Lang, p.\ 71;  \DS, V.4.2;
\Taylor, 4.61-A.)}
%for 3-63P\Pedersen, 2.5.2;

\leader{3A5G}{Reflexive spaces (a)} A normed space $U$ is
{\bf reflexive} if every member of $U^{**}$ is of the form
$f\mapsto f(u)$ for some $u\in U$.

\spheader 3A5Gb A normed space is reflexive iff bounded sets are
relatively weakly compact. \prooflet{(\DS, V.4.8;  \Taylor, 4.61-C.)}
%for 3-72

\spheader 3A5Gc If $U$ is a reflexive space, $\sequencen{u_n}$ is a
bounded sequence in $U$ and $\Cal F$ is an ultrafilter on
$\Bbb N$, then $\lim_{n\to\Cal F}u_n$ is defined in $U$ for the weak
topology.
\prooflet{(Use (b) and 2A3Se.)}
%for 3-72A

\leader{3A5H}{(a) Uniform Boundedness Theorem} Let $U$ be a Banach
space, $V$ a normed space, and $A\subseteq\eurm B(U;V)$ a set such that
$\{Tu:T\in A\}$ is bounded in $V$ for every $u\in U$.   Then $A$ is
bounded in $\eurm B(U;V)$.
\prooflet{(\Rudin, 2.6;  \DS, II.3.21;  \Taylor, 4.4-E.)}
%3{}76N 4{}61

\spheader 3A5Hb {\bf Corollary} If $U$ is a
normed space and $A\subseteq U$ is such that $f[A]$ is bounded for every
$f\in U^*$, then $A$ is bounded.
\prooflet{(\Wilansky, p.\ 117;  \Taylor, 4.4-AS.)}
%3{}76
Consequently any relatively weakly compact set in $U$ is bounded.
\prooflet{(\Rudin, 3.18.)}
%3{}56

\leader{*3A5I}{Strong operator topologies}
If $U$ and $V$ are normed spaces,
the {\bf strong operator topology} on $\eurm B(U;V)$ is that defined by
the seminorms $T\mapsto\|Tu\|$ as $u$ runs over $U$.
If $U$ is a Banach space, $V$ is a normed space and
$A\subseteq\eurm B(U;V)$,
then $A$ is relatively compact for the strong operator topology
iff $\{Tu:T\in A\}$ is relatively compact in $V$ for every $u\in U$.
\prooflet{(Put 3A5Ha and 2A3R together.)}

\leader{3A5J}{Completions} Let $U$ be a normed space.

\spheader 3A5Ja\cmmnt{ Recall that} $U$ has a metric $\rho$ associated 
with the norm\cmmnt{ (2A4Bb)}, and\cmmnt{ that} the topology defined by 
$\rho$ is a
linear space topology\cmmnt{ (2A5D, 2A5B)}.   This topology defines a
uniformity $\Cal W$\cmmnt{ (3A4Ad)} which is also the uniformity
defined by $\rho$\cmmnt{ (3A4Bd)}.   The norm itself is a uniformly
continuous function from $U$ to $\Bbb R$\prooflet{ (because
$|\|u\|-\|v\||\le\|u-v\|$ for all $u$, $v\in U$)}.

\spheader 3A5Jb Let $(\hat U,\hat\Cal W)$ be the uniform space
completion of $(U,\Cal W)$\cmmnt{ (3A4H)}.   Then addition and scalar
multiplication and the norm extend uniquely to make $\hat U$ a Banach
space.  \prooflet{({\smc Schaefer 66}, I.1.5;  \Lang, p.\ 78.)}

\spheader 3A5Jc If $U$ and $V$ are Banach spaces with dense linear
subspaces $U_0$ and $V_0$, then any norm-preserving isomorphism between
$U_0$ and $V_0$ extends uniquely to a norm-preserving isomorphism
between $U$ and $V$\prooflet{ (use 3A4G)}.
%for 3-63I

\leader{3A5K}{Normed algebras} If $U$ is a normed
algebra\cmmnt{ (2A4J)}, its multiplication, regarded as a function
from $U\times U$ to $U$, is continuous.
\prooflet{(\Wilansky, p.\ 259.)}
%for 3-63I

\leader{3A5L}{Compact operators} 
Let $U$ and $V$ be Banach spaces.

\spheader 3A5La A linear operator $T:U\to V$ is
{\bf compact}
%\cmmnt{ or {\bf completely continuous}}
if $\{Tu:\|u\|\le 1\}$ is relatively compact in $V$ for the topology
defined by the norm of $V$.

\spheader 3A5Lb A linear operator $T:U\to V$ is {\bf weakly compact} if
$\{Tu:\|u\|\le 1\}$ is relatively weakly compact in $V$.
Of course compact operators are weakly compact;
\cmmnt{because weakly compact sets must be norm-bounded (3A5Hb),}
weakly compact operators are bounded.

\leader{3A5M}{Hilbert spaces}\cmmnt{ I mentioned the phrases `inner
product space', `Hilbert space' briefly in 244N and 244P, without
explanation, as I did not there rely on any of the abstract theory of
these spaces.   For the main result of \S396 we need one of their
fundamental properties, so I now skim over the definitions.

\medskip

} {\bf (a)} An {\bf inner product space} is a linear space $U$ over
$\RoverC$ together with an operator
$\innerprod{\,\,}{\,\,}:U\times U\to\RoverC$ such that

\Centerline{$\innerprod{u_1+u_2}{v}
=\innerprod{u_1}{v}+\innerprod{u_2}{v}$,
\quad$\innerprod{\alpha u}{v}=\alpha \innerprod{u}{v}$,
\quad$\innerprod{u}{v}=\overline{\innerprod{v}{u}}$}

\noindent (the complex conjugate of $\innerprod{v}{u}$),

\Centerline{$\innerprod{u}{u}\ge 0$,
\quad$u=0$ whenever $\innerprod{u}{u}=0$}

\noindent for all $u$, $u_1$, $u_2$, $v\in U$ and $\alpha \in\RoverC$.

\spheader 3A5Mb\dvArevised{2022} 
If $U$ is any inner product space, we have a norm on $U$
defined by setting $\|u\|=\sqrt{\innerprod{u}{u}}$ for every $u\in U$, and
$|\innerprod{u}{v}|\le\|u\|\|v\|$ for all $u$, $v\in U$.
\prooflet{(\Taylor, 3.2-B.\query)}

\spheader 3A5Mc A {\bf Hilbert space} is an inner product space which
is\cmmnt{ a Banach space under the norm of (b) above, that is, is}
complete in the metric defined from its norm.

\spheader 3A5Md If $U$ is a Hilbert space, $C\subseteq U$ is a non-empty
closed convex set, and $u\in U$, then there is a unique $v\in C$ such
that $\|u-v\|=\inf_{w\in C}\|u-w\|$.  \prooflet{(\Taylor, 4.81-A;  compare
244Yn.)}

\spheader 3A5Me\dvAnew{2022} If $U$ is an inner product space, 
$C\subseteq U$ is a 
convex set, $u$, $u'\in U$ and $v$, $v'\in C$ are such that
that $\|u-v\|=\inf_{w\in C}\|u-w\|$ and $\|u'-v'\|=\inf_{w\in C}\|u'-w\|$,
then $\|v'-v\|\le\|u'-u\|$.   \prooflet{\Prf\ If $v'=v$ this is trivial.   
Otherwise,
set $\gamma=\|v'-v\|$ and $e=\bover1{\gamma}(v'-v)$.   If
$0<\alpha\le\gamma$, $v+\alpha e\in C$ so

\Centerline{$\|u-v\|^2\le\|u-v-\alpha e\|^2
=\|u-v\|^2-2\alpha\Real\innerprod{u-v}{e}+\alpha^2$}

\noindent and $2\Real\innerprod{u-v}{e}\le\alpha$;  it follows that
$\Real\innerprod{u-v}{e}\le 0$.   Similarly, $\Real\innerprod{u'-v'}{e}\ge 0$, 
and

\Centerline{$\|v'-v\|=\innerprod{v'-v}{e}\le\Real\innerprod{u'-u}{e}
\le\|u'-u\|\|e\|=\|u'-u\|$}

\noindent as claimed.\ \Qed}
%check Taylor query





\leader{*3A5N}{Bounded sets in linear topological
spaces}\dvAnew{2009}\cmmnt{ There is a
point in \S377 %377E
where a concept from the general theory of linear topological spaces
helps an idea to flow more freely.}
Let $U$ be a linear topological space over $\RoverC$.

\spheader 3A5Na\dvAformerly{4{}A4Bf} A set
$A\subseteq U$ is {\bf bounded} if for every neighbourhood $G$ of $0$ 
there is an $n\in\Bbb N$ such that $A\subseteq nG$.

\spheader 3A5Nb If $A\subseteq U$ is bounded, then

\quad(i) every subset of $A$ is bounded;

\quad(ii) the closure of $A$ is bounded;

\quad(iii) $\alpha A$ is bounded for every $\alpha\in\RoverC$;

\quad(iv) $A\cup B$ and $A+B$ are bounded for every bounded $B\subseteq U$;

\quad(v) if $V$ is another linear topological space, 
and $T:U\to V$ is a continuous linear operator, then $T[A]$ is bounded.

\spheader 3A5Nc If $A\subseteq U$ is relatively compact, it is bounded.

\spheader 3A5Nd If $U$ is a normed space, and $A\subseteq U$, then the
following are equiveridical:

\quad(i) $A$ is bounded in the sense of (a) above for the norm topology of
$U$;

\quad(ii) $A$ is bounded in the sense of 2A4Bc\cmmnt{, that is,
$\{\|u\|:u\in A\}$ is bounded above in $\Bbb R$};

\quad(iii) $A$ is bounded for the weak topology of $U$.

\proof{{\smc Schaefer 66}, \S I.5;
{\smc K\"othe 69}, \S15.6.   For (d-iii), use 3A5Hb.}

\discrpage

\frfilename{mt3a6.tex} 
\versiondate{6.8.08} 
\copyrightdate{1996} 
      
\def\chaptername{Appendix} 
\def\sectionname{Group Theory} 
      
\newsection{3A6} 
      
For Chapter 38 we need four definitions and two results from elementary 
abstract group theory. 
      
\leader{3A6A}{Definition} If $G$ is a group, \cmmnt{I will say that} an 
element $g$ of $G$ is an {\bf involution} if its order is 
2\cmmnt{, that is, $g^2=e$, the identity of $G$, but $g\ne e$}. 
      
\leader{3A6B}{Definition} If $G$ is a group, the set $\Aut G$ of 
{\bf automorphisms} of $G$ (that is, bijective homomorphisms from $G$ to 
itself) is a group.   For $g\in G$ define $\hat g:G\to G$ by writing 
$\hat g(h)=ghg^{-1}$ for every $h\in G$;  then $\hat g\in\Aut G$, and 
the map $g\mapsto\hat g$ is a homomorphism from $G$ onto a normal subgroup 
$J$ of $\Aut G$\cmmnt{ ({\smc Rotman 84}, p.\ 130)}. 
We call $J$ the group of {\bf inner automorphisms} 
of $G$.   Members of $(\Aut G)\setminus J$ are called {\bf outer automorphisms}. 
      
\leader{3A6C}{Normal subgroups} For any group $G$, the family of 
normal subgroups of $G$, ordered by $\subseteq$, 
is a Dedekind complete lattice, with $H\vee K=HK$ and 
$H\wedge K=H\cap K$.   \cmmnt{({\smc Davey \& Priestley 90}, 
2.8 \& 2.19.)} 
%for 3{8}2 

\discrversionA{\leader{[dvA]3A6?}{An outer automorphism for $S_6$}\dvAnew{2017}
{\bf (a)} 
Suppose that
$H$ is a subgroup of $A_6$ which is isomorphic to $A_5$.   Then $H$ has just
six conjugates in $A_6$.   \Prf\ $H$ has $5\cdot 4\cdot 3\cdot 2\cdot 1/5=24$
elements of order 5, so has $24/(5-1)=6$ Sylow 5-subgroups;  similarly
$A_6$ has $6\cdot 5\cdot 4\cdot 3\cdot 2/5=144$
elements of order 5 and $36$ Sylow 5-subgroups.   Looking at the conjugacy
action of $A_6$ on $\Cal PA_6$, the stabilizer of $H$ includes $H$, so the
orbit of $H$ has at most $\#(A_6)/\#(H)=6$ members.   On the other hand, every
conjugate of $H$ has just 24 Sylow 5-subgroups and all the Sylow 5-subgroups of
$A_6$ are conjugate in $A_6$, so every Sylow 5-subgroup of $A_6$ belongs to some
conjugate of $H$, and $H$ has at least $144/24=6$ conjugates.\ \Qed

Note that it follows that $H$ is its own normalizer, because whenever $G$ is a
finite group acting on a set $Z$,
$\#(\text{stabilizer of }z)\cdot\#(\text{orbit of }z)=\#(G)$ for every
$z\in Z$.

\spheader 3A6?b There is a subgroup $H$ of $A_6$, isomorphic to $A_5$, such that
every element of order 3 in $H$ is a product of two disjoint 3-cycles in $A_6$.
\Prf\ The isometry group of a regular icosahedron $X$ is isomorphic to $A_5$.
Consider its action on the six-element set $D$ of long diagonals of $X$.  
Because $A_5$ is simple, this action is either trivial or faithful;  surely it
is not trivial, so it is faithful.   In $A_5$ there are $5\cdot 4\cdot 3/3=20$
elements of order $3$, corresponding to 10 Sylow 3-subgroups;  now $X$ has
$20/2=10$ pair of opposite faces, and for each such pair we see we have a 
Sylow 3-subgroup corresponding to rotations around the axis through the centres
of the pair of faces.   So these are precisely the Sylow 3-subgroups of the
group $H_0$ of permutations of $D$ corresponding to the action of $A_5$ on $D$.
Looking at such rotations, we see that they permute cyclically the 
three diagonals
through the vertices of the faces involved, and also the three diagonals through
the opposite vertices faces linked to those.   So they correspond to products of
disjoint 3-cycles in $D$.   Identifying $D$ with $\{0,\ldots,5\}$, we get a
subgroup $H$ of $A_6$ with the required properties.\ \Qed

\spheader 3A6?c There is an automorphism $g:A_6\to A_6$ such that $g(\rho)$ is a
product of two disjoint 3-cycles whenever $\rho\in A_6$ is a 3-cycle.
\Prf\ Take $H$ from (b), and let $\Cal K$ be the set of conjugates of
$H$ in $A_6$;  by (a), $\#(\Cal K)=6$.   Then the conjugacy action of $A_6$ on
$\Cal K$ is non-trivial, therefore faithful;  write $f$ for the corresponding
injective homomorphism from $A_6$ to $S_{\Cal K}$.   If $\rho\in A_6$ has order
3, then $f(\rho)$ has order $3$, so is an even permutation of $\Cal K$;   since
every member of $A_6$ is expressible as a product of 3-cycles, every member of
$f[A_6]$ is an even permutation;  because the number of even permutations of
$\Cal K$ is equal to $\#(A_6)$, $f[A_6]$ is precisely the set $A_{\Cal K}$
of even
permutations of $\Cal K$.   Observe next that if $\rho\in A_6$ is a 3-cycle, it
cannot belong to any member of $\Cal K$, because if $K\in\Cal K$ then every
element of order 3 in $K$ is a product of two disjoint 3-cycles.   But this
means that $\rho K\rho^{-1}\ne K$ (see last sentence in (a)).   Thus
$f(\rho)\in A_{\Cal K}$ is an element of order 3 with no fixed point in 
$\Cal K$, and must be a product of two disjoint 3-cycles.   Accordingly any
bijection between $\Cal K$ and $6$ will convert $f$ into an automorphism of
$A_6$ with the required property.\ \Qed

\spheader 3A6?d There is an outer automorphism of $S_6$.  \query
}{}      
\frnewpage 
      
      
      
      
      
      
      
\ifresultsonly\else
\frfilename{mt3aconc.tex}
\versiondate{5.3.08}
\copyrightdate{2008}

\gdef\topparagraph{}
\gdef\bottomparagraph{}
\gdef\newparagraph{}

\def\chaptername{Appendix}
\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

\leader{3{}A3P}{Definitions} This paragraph, referred to in the 2003 
edition of Volume 4, is now 3A3Q.
% zero set, cozero set, G$_{\delta}$ set, isolated point     

\leader{3{}A5K}{Compact operators} This paragraph, referred to in the 2003
and 2006 editions of Volume 4, is now 3A5L.

\leader{3{}A5L}{Inner product spaces} This paragraph, referred to in the 
2003 and 2006 editions of Volume 4, is now 3A5M.

\fi

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

% except for references in ...X., ...Y., ...Z. give locations in form
%  342L  rather than  342Lb
%  \S511 {\it notes\/}, \S512 {\it intro.}]

\gdef\topparagraph{}
\gdef\bottomparagraph{}
\gdef\newparagraph{}
\gdef\chaptername{References}
\gdef\sectionname{References}
\gdef\headlinesectionname{References}

\centerline{\bf References for Volume 3}

\medskip

%A

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

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

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

\medskip
%B

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

Balcar B., Jech T.\ \& Paz\'ak T.\ [05]
`Complete ccc Boolean algebras, the order sequential topology,
and a problem of von Neumann',
Bull.\ London Math.\ Soc.\ 37 (2005) 885-898.
\cmmnt{[393L, 393Q.]}
% arXiv.math.LO/0312473

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\medskip
%C

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

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

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

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

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

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

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

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

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

\medskip
%D

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

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

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

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

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

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

\medskip
%E

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

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

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

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

\medskip
%F

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

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

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

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

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

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

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

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

\medskip
%G

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

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

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

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

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

\medskip%H

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

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

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

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

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

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

\medskip %I

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

\medskip %J

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

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

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

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

Judah H.\ [93] (ed.) {\it Proceedings of the Bar-Ilan Conference on Set
Theory and the Reals, 1991.} Amer.\ Math.\ Soc.\ (Israel Mathematical
Conference Proceedings 6), 1993.

\medskip %K

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

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

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

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

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

Kelley J.L.\ [59] `Measures on Boolean algebras', Pacific J.\ Math.\ 9
(1959) 1165-1177.  \cmmnt{[\S391 {\it notes\/}.]}

\leaveitout{
Kelley J.L. \& Namioka I.\ [76] {\it Linear Topological Spaces.}
Springer, 1976.
}

\leaveitout{
King J.L.F.\ [00] `The generic transformation has roots of all orders',
Colloquium Mathematicum 84/85  (2000) 521-547.
  % if  \frak A  is the Lebesgue measure algebra
  % a comeager set in the Polish group
  % \Aut_{\bar\mu}\frak A   has roots of all orders
}

Kolmogorov A.N. [58] `New metric invariants of transitive dynamical
systems and automorphisms of Lebesgue spaces', Dokl.\ Akad.\ Nauk SSSR
119 (1958) 861-864.  \cmmnt{[385P.]}

K\"olzow D.\ \& Maharam-Stone D.\ [82] (eds.) {\it Measure Theory
Oberwolfach 1981.}   Springer, 1982 (Lecture Notes in Math.\ 945).

Koppelberg S.\ [89] {\it General Theory of Boolean
Algebras\/}, vol.\ 1 of {\smc Monk 89}.
\cmmnt{[Chap.\ 31 {\it intro\/}., \S332 {\it notes\/}.]}

K\"othe G.\ [69] {\it Topological Vector Spaces I.}   Springer, 1969.
\cmmnt{[\S356 {\it notes\/}, 3A5N.]}

Kranz P.\ \& Labuda I.\ [93] (eds.) {\it Proceedings of the Orlicz
Memorial Conference, 1991\/}, unpublished manuscript, available from
first editor ({\tt mmkranz{\@}olemiss.edu}).

Krengel, U.\ [63] `"Uber den Absolutbetrag stetiger linearer
Operatoren und seine Anwendung auf ergodische Zerlegungen', Math.\
Scand.\ 13 (1963) 151-187.   \cmmnt{[371Xb.]}

Krieger W.\ [76] `On ergodic flows and the isomorphism of factors',
Math.\ Ann.\ 223 (1976) 19-70.   [\S388 {\it notes}.]
   %Dye's theorem in general form, von-Neumann-algebra version

Krivine J.-L.\ [71] {\it Introduction to Axiomatic Set Theory.}
D.Reidel, 1971.  \cmmnt{[\S3A1.]}

Kullback S.\ [67]
`A lower bound for discrimination information in terms of variation',
IEEE Trans.\ on Information Theory 13 (1967) 126-127.
\cmmnt{[386G.]}
%Wendt S IN5 EL24 T IT

Kunen K.\ [80] {\it Set Theory.}  North-Holland, 1980.
\cmmnt{[\S3A1.]}

Kwapien S.\ [73] `On the form of a linear operator on the space of
all measurable functions', Bull.\ Acad.\ Polon.\ Sci.\ 21 (1973)
951-954.  \cmmnt{[\S375 {\it notes\/}.]}

\medskip %L

Lang S.\ [93] {\it Real and Functional Analysis.}  Springer, 1993.
\cmmnt{[\S3A5.]}

Liapounoff A.A.\ [1940] `Sur les fonctions-vecteurs compl\`etement
additives', Bull.\ Acad.\ Sci.\ URSS (Izvestia Akad.\ Nauk SSSR) 4 (1940)
465-478.
\cmmnt{[326H.]}

Lindenstrauss J.\ \& Tzafriri L.\ [79] {\it Classical Banach Spaces II.}
Springer, 1979, reprinted in {\smc Lindenstrauss \& Tzafriri 96}.
\cmmnt{[\S354 {\it notes\/}, 374Xj.]}

Lindenstrauss J.\ \& Tzafriri L.\ [96] {\it Classical Banach Spaces I \& II.}  Springer, 1996.

Lipschutz S.\ [64] {\it Set Theory and Related Topics.}  McGraw-Hill,
1964 (Schaum's Outline Series).  \cmmnt{[3A1D.]}

Luxemburg W.A.J.\ [67a] `Is every integral normal?',
Bull.\ Amer.\ Math.\ Soc.\ 73 (1967) 685-688.   \cmmnt{[363S.]}

Luxemburg W.A.J.\ [67b] `Rearrangement-invariant Banach function spaces',
pp.\ 83-144 in {\smc Coleman \& Ribenboim 67}.
\cmmnt{[\S374 {\it notes}.]}

Luxemburg W.A.J.\ \& Zaanen A.C.\ [71] {\it Riesz Spaces
I.} North-Holland, 1971.  \cmmnt{[Chap.\ 35 {\it intro\/}.]}

\medskip %M

Macheras N.D., Musia\l\ K.\ \& Strauss W.\ [99] `On products of
admissible liftings and densities',
J.\ for Analysis and its Applications 18 (1999) 651-668. \cmmnt{[346G.]}

Macheras N.D.\ \& Strauss W.\ [95] `Products of lower densities', J.\
for Analysis and its Applications 14 (1995) 25-32.  \cmmnt{[346Xf.]}

Macheras N.D.\  \& Strauss W.\ [96a] `On products of almost strong
liftings', J.\ Australian Math.\ Soc.\ (A) 60 (1996) 1-23.
\cmmnt{[346Yc.]}

Macheras N.D.\ \& Strauss W.\ [96b] `The product lifting for arbitrary products of complete
probability spaces', Atti Sem.\ Math.\ Fis.\ Univ.\ Modena 44 (1996) 485-496.
\cmmnt{[346H, 346Yd.]}

Maharam D.\ [1942] `On homogeneous measure algebras', Proc.\ Nat.\ Acad.\
Sci.\ U.S.A.\ 28 (1942) 108-111.  \cmmnt{[331F, 332B.]}

Maharam D.\ [1947] `An algebraic characterization of measure algebras',
Ann.\ Math.\ 48 (1947) 154-167.
\cmmnt{[393J.]}

Maharam D.\ [58] `On a theorem of von Neumann', Proc.\ Amer.\ Math.\
Soc.\ 9 (1958) 987-994.
\cmmnt{[\S341 {\it notes\/}, \S346 {\it notes\/}.]}

Marczewski E.\ [53] `On compact measures',
Fund.\ Math.\ 40 (1953) 113-124.  \cmmnt{[342A.]}

McCune W.\ [97] `Solution of the Robbins problem',
J.\ Automated Reasoning 19 (1997) 263-276.
\cmmnt{[311Yc.]}

Miller B.D.\ [04] PhD Thesis, University of California, Berkeley, 2004.
%\query
\cmmnt{[382Xc, \S382 {\it notes\/}.]}

Monk J.D.\ [89] (ed.) {\it Handbook of Boolean Algebra.}
North-Holland, 1989.

\medskip %N

Nadkarni M.G.\ [90] `On the existence of a finite invariant measure',
Proc.\ Indian Acad.\ Sci., Math.\ Sci.\ 100 (1990) 203-220.
\cmmnt{[\S395 {\it notes\/}.]}

\medskip %O

Ornstein D.S.\ [74] {\it Ergodic Theory, Randomness and Dynamical Systems.}  Yale U.P., 1974.
\cmmnt{[\S387 {\it notes\/}.]}

Ornstein D.S.\ \& Shields P.C.\ [73] `An uncountable family of
$K$-automorphisms', Advances in Math.\ 10 (1973) 63-88.
\cmmnt{[\S382 {\it notes\/}.]}

\medskip%P

%Pedersen G.K.\ [89] {\it Analysis Now.}  Springer, 1989.

Perovi\'c \v{Z}.\ \& Veli\v{c}kovi\'c B.\ [18] `Ranks of Maharam algebras',
Advances in Math.\ 330 (2018) 253-279.
\cmmnt{[394A.]}

Petersen K.\ [83] {\it Ergodic Theory.}   Cambridge U.P., 1983.
\cmmnt{[328Xa, 385C, \S385 {\it notes\/}, 386E.]}

\medskip %Q%R

Roberts J.W.\ [93] `Maharam's problem', in {\smc Kranz \& Labuda 93}.
\cmmnt{[\S394 {\it notes\/}.]}

Rotman J.J.\ [84] `An Introduction to the Theory of Groups', Allyn \&
Bacon, 1984.
\cmmnt{[\S384 {\it notes\/}, 3A6B.]}

Rudin W. [91] {\it Functional Analysis.}  McGraw-Hill, 1991.
\cmmnt{[\S3A5.]}

Ryzhikov V.V.\ [93] `Factorization of an automorphism of a complete Boolean algebra into a product of three involutions', Mat.\ Zametki (=Math.\ Notes of Russian Acad.\ Sci.) 54 (1993) 79-84.
\cmmnt{[\S382 {\it notes\/}.]}

\medskip %S

Sazonov V.V.\ [66] `On perfect measures', A.M.S.\ Translations (2) 48
(1966) 229-254.   \cmmnt{[\S342 {\it notes\/}.]}

Schaefer H.H.\ [66] {\it Topological Vector Spaces.}  MacMillan, 1966;  reprinted with corrections Springer, 1971.
\cmmnt{[3A4A, 3A5J, 3A5N.]}
%Store A	QA 611.S3    6640 0159 9

Schaefer H.H.\ [74] {\it Banach Lattices and Positive Operators.}
Springer, 1974.   \cmmnt{[Chap.\ 35 {\it intro\/}., \S354 {\it notes\/}.]}

Schubert H.\ [68] {\it Topology.}  Allyn \& Bacon, 1968.
\cmmnt{[\S3A3, \S3A4.]}

Shelah S.\ [98] `The lifting problem with the full ideal',
J.\ Applied Analysis 4 (1998) 1-17.   \cmmnt{[341L.]}

Sikorski R.\ [64] {\it Boolean Algebras.}  Springer, 1964.
\cmmnt{[Chap.\ 31 {\it intro\/}.]}

Sina\v\i\ Ya.G.\ [59] `The notion of entropy of a dynamical system',
Dokl.\ Akad.\ Nauk SSSR 125 (1959) 768-771.  \cmmnt{[385P.]}

Sina\v\i\ Ya.G.\ [62] `Weak isomorphism of transformations with an invariant measure', Soviet Math.\ 3 (1962) 1725-1729.
\cmmnt{[387E.]}
% ({\cyr Dokl.\ Akad.\ Nauk SSSR 147 (1962) 797-800}).

Smorodinsky M.\ [71] {\it Ergodic Theory, Entropy.}
Springer, 1971 (Lecture Notes in Math., 214).
\cmmnt{[\S387 {\it notes\/}.]}

\v St\v ep\'anek P.\ \& Rubin M.\ [89] `Homogeneous Boolean algebras',
pp.\ 679-715 in {\smc Monk 89}.   \cmmnt{[382S, \S382 {\it notes\/}.]}

\medskip %T

Talagrand M.\ [82a] `Closed convex hull of set of measurable functions,
Riemann-\vthsp measurable functions and measurability of translations',
Ann.\ Institut Fourier (Grenoble) 32 (1982) 39-69.
\cmmnt{[\S346 {\it notes\/}.]}

Talagrand M.\ [82b] `La pathologie des rel\grv{e}vements invariants',
Proc.\ Amer.\ Math.\ Soc.\ 84 (1982) 379-382.
\cmmnt{[345F.]}

Talagrand M.\ [84] {\it Pettis integral and measure theory.}
Mem.\ Amer.\ Math.\ Soc.\  307 (1984).
\cmmnt{[\S346 {\it notes\/}.]}

%Talagrand M.\ [06] `Maharam's problem', C.R.Acad.Sci.\ Paris 342 (2006)
%501-503.

Talagrand M.\ [08] `Maharam's problem', Annals of Math.\ 168 (2008)
981-1009.
\cmmnt{[\S394.]}

Taylor A.E.\ [64] {\it Introduction to Functional Analysis.}
Wiley, 1964.
\cmmnt{[\S3A5.]}

Todor\v{c}evi\'c S.\ [04]
`A problem of von Neumann and Maharam about algebras supporting
continuous submeasures',
Fund.\ Math.\ 183 (2004) 169-183.
\cmmnt{[393S.]}

Truss J.K.\ [89] `Infinite permutation groups I:  products of conjugacy
classes',
J.\ Algebra 120 (1989) 454-493.
\cmmnt{[\S382 {\it notes\/}.]}

\medskip %U%V

Vladimirov D.A.\ [02] {\it Boolean algebras in analysis.}
Kluwer, 2002 (Math.\ and its Appl.\ 540).
\cmmnt{[367L.]}

Vulikh B.C.\ [67] {\it Introduction to the Theory of Partially Ordered
Vector Spaces.}   Wolters-Noordhoff, 1967.
\cmmnt{[\S364 {\it notes\/}.]}

\medskip %W

Wagon S. [85] {\it The Banach-Tarski Paradox.}   Cambridge U.P., 1985.
\cmmnt{[\S395 {\it notes\/}.]}

Wilansky A. [64] {\it Functional Analysis.}  Blaisdell, 1964.
\cmmnt{[\S3A5.]}

\medskip %X%Y%Z

Zaanen A.C.\ [83] {\it Riesz Spaces II.}  North-Holland, 1983.
\cmmnt{[Chap.\ 35 {\it intro.\/}, 376K, \S376 {\it notes\/}.]}


\frnewpage

}%end of references


%---------------------------------------------------------------%

\end



