% 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{mt12.tex}
\versiondate{7.4.05}
\copyrightdate{1994}

\def\chaptername{Integration}

\newchapter{12}

If you look along the appropriate shelf of your college's library, you
will see that the words `measure' and `integration' go together
like Siamese twins.   The linkage is both more complex and more intimate
than any simple explanation can describe.   But if we say that one of
the concepts on which integration is based is that of `area under a
curve', then it is clear that any method of determining `areas' ought
to correspond to a method of integrating functions;  and this has from
the beginning been an essential part of the Lebesgue theory.   For a
literal description of the integral of a non-negative function in terms
of the area of its ordinate set, I think it best to wait until Chapter
25 in Volume 2.
In the present chapter I seek to give a concise description of the
standard integral of a real-valued function on a general measure space,
with the half-dozen most important theorems concerning this integral.

The construction bristles with technical difficulties at every step, and
you will find it easy to understand why it was not done before
1901.   What may be less clear is why it was ever done at all.   So
perhaps you should immediately read the statements of
123A-123D %123A 123B 123C 123D
below.
It is the case (some of the details will appear, rather late, in \S436
in Volume 4)
that any theory of integration powerful enough to have theorems of this
kind must essentially encompass all the ideas of this chapter, 
and nearly all the ideas of the last.

\discrpage
\frfilename{mt121.tex}
\versiondate{21.12.03}
\copyrightdate{1994}
     
\def\chaptername{Integration}
\def\sectionname{Measurable functions}
     
\newsection{121}
     
In this section, I take a step back to develop ideas relating to
$\sigma$-algebras of sets, following \S111;  there will be no mention
of `measures' here, except in the exercises.   The aim is to
establish the concept of `measurable function' (121C) and a variety of 
associated techniques.
The best single example of a $\sigma$-algebra to bear in mind when
reading this chapter is probably the $\sigma$-algebra of Borel subsets of
$\Bbb R$\cmmnt{ (111G)};  the 
$\sigma$-algebra of Lebesgue measurable subsets of 
$\Bbb R$\cmmnt{ (114E)} is a good second.
     
Throughout the exposition here (starting with 121A) I seek to deal with
functions which are not defined on the whole of the space $X$ under
consideration.   I believe that there are compelling reasons for facing
up to such functions at an early stage\cmmnt{ (see 121G)};  but undeniably
they add to the technical difficulties, and it would be fair to read
through the chapter once with the mental reservation that all functions
are taken to be defined everywhere, before returning to deal with the
general case.
     
     
\leader{121A}{Lemma} Let $X$ be a set and $\Sigma$ a $\sigma$-algebra of
subsets of $X$.   Let $D$ be any subset of $X$ and write
     
\Centerline{$\Sigma_D=\{E\cap D:E\in\Sigma\}$.}
     
\noindent Then $\Sigma_D$ is a $\sigma$-algebra of subsets of $D$.
     
\proof{{\bf (i)} $\emptyset=\emptyset\cap D\in\Sigma_D$ because
$\emptyset\in\Sigma$.
     
\medskip
     
{\bf (ii)} If $F\in\Sigma_D$, there is an $E\in\Sigma$ such that
$F=E\cap D$;  now $D\setminus F=(X\setminus E)\cap D\in\Sigma_D$ because
$X\setminus E\in\Sigma$.
     
\medskip
     
{\bf (iii)} If $\langle F_n\rangle_{n\in\Bbb N}$ is any sequence in
$\Sigma_D$, then for each $n\in\Bbb N$ we may choose an $E_n\in\Sigma$
such that $F_n=E_n\cap D$;  now
$\bigcup_{n\in\Bbb N}F_n=(\bigcup_{n\in\Bbb N}E_n)\cap D\in\Sigma_D$ because
$\bigcup_{n\in\Bbb N}E_n\in\Sigma$.
}%end of proof of 121A
     
\medskip
     
\noindent{\bf Notation} I will call $\Sigma_D$ the {\bf subspace
$\sigma$-algebra} of subsets of $D$, and I will say that its members are
{\bf relatively measurable} in $D$.   
$\Sigma_D$ is also sometimes called the {\bf trace} of $\Sigma$ on $D$.
     
\leader{121B}{Proposition} Let $X$ be a set, $\Sigma$ a $\sigma$-algebra
of subsets of $X$, and $D$ a subset of $X$.   Write $\Sigma_D$ for the
subspace $\sigma$-algebra of subsets of $D$.   Then for any function
$f:D\to\Bbb R$ the following assertions are equiveridical\cmmnt{, that is, if one of them is true so are all the others}:
     
\quad(i) $\{x:f(x)<a\}\in\Sigma_D$ for every $a\in\Bbb R$;
     
\quad(ii) $\{x:f(x)\le a\}\in\Sigma_D$ for every $a\in\Bbb R$;
     
\quad(iii) $\{x:f(x)>a\}\in\Sigma_D$ for every $a\in\Bbb R$;
     
\quad(iv) $\{x:f(x)\ge a\}\in\Sigma_D$ for every $a\in\Bbb R$.
     
\proof{{\bf (i)$\Rightarrow$(ii)} Assume (i), and let
$a\in\Bbb R$.   Then
     
\Centerline{$\{x:f(x)\le a\}
=\bigcap_{n\in\Bbb N}\{x:f(x)<a+2^{-n}\}\in\Sigma_D$}
     
\noindent because $\{x:f(x)<a+2^{-n}\}\in\Sigma_D$ for every $n$ and
$\Sigma_D$ is closed under countable intersections (111Dd).
Because $a$ is arbitrary, (ii) is true.
     
\wheader{121B}{4}{2}{2}{72pt}
     
{\bf (ii)$\Rightarrow$(iii)} Assume (ii), and let $a\in\Bbb R$.   Then
     
\Centerline{$\{x:f(x)>a\}=D\setminus\{x:f(x)\le a\}\in\Sigma_D$}
     
\noindent because $\{x:f(x)\le a\}\in\Sigma_D$ and $\Sigma_D$ is closed
under complementation.   Because $a$ is arbitrary, (iii) is
true.
     
\medskip
     
{\bf (iii)$\Rightarrow$(iv)} Assume (iii), and let $a\in\Bbb R$.   Then
     
\Centerline{$\{x:f(x)\ge a\}
=\bigcap_{n\in\Bbb N}\{x:f(x)>a-2^{-n}\}\in\Sigma_D$}
     
\noindent because $\{x:f(x)>a-2^{-n}\}\in\Sigma_D$ for every $n$ and
$\Sigma_D$ is closed under countable intersections.
Because $a$ is arbitrary, (iv) is true.
     
\medskip
     
{\bf (iv)$\Rightarrow$(i)} Assume (iv), and let $a\in\Bbb R$.   Then
     
\Centerline{$\{x:f(x)<a\}=D\setminus\{x:f(x)\ge a\}\in\Sigma_D$}
     
\noindent because $\{x:f(x)\ge a\}\in\Sigma_D$ and $\Sigma_D$ is closed
under complementation.   Because $a$ is arbitrary, (i) is
true.
}%end of proof of 121B
     
\leader{121C}{Definition} Let $X$ be a set, $\Sigma$ a $\sigma$-algebra
of subsets of $X$, and $D$ a subset of $X$.   A function $f:D\to\Bbb R$
is called {\bf measurable} (or {\bf $\Sigma$-measurable}) if it
satisfies any, or equivalently all, of the conditions (i)-(iv) of 121B.
     
If $X$ is $\Bbb R$ or $\BbbR^r$, and $\Sigma$ is its Borel
$\sigma$-algebra\cmmnt{ (111G)}, a $\Sigma$-measurable function is
called {\bf Borel measurable}.   If $X$ is $\Bbb R$ or $\BbbR^r$, and 
$\Sigma$ is
the $\sigma$-algebra of Lebesgue measurable sets\cmmnt{ (114E, 115E)},
a $\Sigma$-measurable function is called {\bf Lebesgue measurable}.
     
\cmmnt{\medskip
     
\noindent{\bf Remark} Naturally the principal case here is when $D=X$.
However, partially-defined functions are so common, and so important, in
analysis (consider, for instance, the real function $\ln\sin$) that it
seems worth while, from the beginning, to establish techniques for
handling them efficiently.
     
Many authors develop a theory of `extended real numbers' at this
point, working with $[-\infty,\infty]=\Bbb R\cup\{-\infty,\infty\}$, and
defining measurability for functions taking values in this set.   I
outline such a theory in \S135 below.
}%end of comment
     
\leader{121D}{Proposition} Let $X$ be $\BbbR^r$ for some $r\ge 1$, $D$
a subset of $X$, and $g:D\to\Bbb R$ a function.
     
(a) If $g$ is Borel measurable it is Lebesgue measurable.
     
(b) If $g$ is continuous it is Borel measurable.
     
(c) If $r=1$ and $g$ is monotonic it is Borel measurable.
     
\proof{{\bf (a)} This is immediate from the definitions in 121C,
if we recall that the Borel $\sigma$-algebra is included in the Lebesgue
$\sigma$-algebra (114G, 115G).
     
\medskip
     
{\bf (b)} Take $a\in\Bbb R$.   Set
     
\Centerline{$\Cal G=\{G:G\subseteq\BbbR^r\text{ is
open},\,g(x)<a\Forall x\in G\cap D\}$,}
     
\Centerline{$G_0=\bigcup\Cal G
=\{x:\exists\enskip G\in\Cal G,\,x\in G\}$.}
     
\noindent Then $G_0$ is a union of open sets, therefore open (1A2Bd).
Next,
     
\Centerline{$\{x:g(x)<a\}=G_0\cap D$.}
     
\noindent\Prf\ {\bf (i)} If $g(x)<a$, then (because $g$ is
continuous) there is a $\delta>0$ such that $|g(y)-g(x)|<a-g(x)$
whenever $y\in D$ and $\|y-x\|<\delta$.   But
$\{y:\|y-x\|<\delta\}$ is open (1A2D),  so belongs to $\Cal G$ and is
included in
$G_0$, and $x\in G_0\cap D$.   {\bf (ii)} If $x\in
G_0\cap D$, then there is a $G\in\Cal G$ such that $x\in G$;  now
$g(y)<a$ for every $y\in G\cap D$, so, in particular, $g(x)<a$.\ \Qed
     
Finally, $G_0$, being open, is a Borel set.   As $a$ is
arbitrary, $g$ is Borel measurable.
     
\medskip
     
{\bf (c)} Suppose first that $g$ is non-decreasing.   Let $a\in\Bbb R$
and write $E=\{x:g(x)<a\}$.  If $E=D$ or $E=\emptyset$ then of course it
is the intersection of $D$ with a Borel set.   Otherwise, $E$ is
non-empty and bounded above in $\Bbb R$, so has a supremum $c\in\Bbb R$.
Now $E$ must be either $D\cap\ooint{-\infty,c}$ or
$D\cap\ocint{-\infty,c}$, according to whether $c\in E$ or not, and in
either case is the intersection of $D$ with a Borel set (see 114G).
     
Similarly, if $g$ is non-increasing, $\{x:g(x)>a\}$ will again be the
intersection of $D$ with either $\emptyset$ or $\Bbb R$ or
$\ocint{-\infty,c}$ or $\ooint{-\infty,c}$ for some $c$.   So in this
case 121B(iii) will be satisfied.
     
\medskip
     
\noindent{\bf Remark} I see that in part (b) of the above proof I use
some basic facts about open sets in $\BbbR^r$.   These are covered in
detail in \S1A2.   If they are new to you it would probably be sensible
to rehearse the arguments with $r=1$, so that $D\subseteq\Bbb R$, before
embracing the general case.
}%end of proof of 121D
     
     
\vleader{108pt}{121E}{Theorem} 
Let $X$ be a set and $\Sigma$ a $\sigma$-algebra
of subsets of $X$.   Let $f$ and $g$ be real-valued functions defined on
domains $\dom f$, $\dom g\subseteq X$.
     
(a) If $f$ is constant it is
measurable.
     
(b) If $f$ and $g$ are measurable, so is $f+g$, where
$(f+g)(x)=f(x)+g(x)$ for $x\in
\dom f\cap\dom g$.
     
(c) If $f$ is measurable and $c\in\Bbb R$, then $cf$ is measurable,
where
$(cf)(x)=c\cdot f(x)$ for $x\in \dom f$.
     
(d) If $f$ and $g$ are measurable, so is $f\times g$, where $(f\times
g)(x)=f(x)\times g(x)$ for $x\in \dom f\cap\dom g$.
     
(e) If $f$ and $g$ are measurable, so is $f/g$, where
$(f/g)(x)=f(x)/g(x)$ when
$x\in \dom f\cap\dom g$ and $g(x)\ne 0$.
     
(f) If $f$ is measurable and $E\subseteq\Bbb R$ is a Borel set, then
there is an
$F\in\Sigma$ such that $f^{-1}[E]=\{x:f(x)\in E\}$ is equal to $F\cap
\dom f$.
     
(g) If $f$ is measurable and $h$ is  a Borel measurable function from a
subset $\dom h$
of $\Bbb R$ to $\Bbb R$, then $hf$ is measurable, where
$(hf)(x)=h(f(x))$ for $x\in\dom(hf)=\{y:y\in\dom f,\,f(y)\in\dom h\}$.
     
(h) If $f$ is measurable and $A$ is any set, then $f\restr A$ is
measurable, where $\dom(f\restr A)=A\cap\dom f$ and
$(f\restr A)(x)=f(x)$ for $x\in A\cap\dom f$.
     
\proof{ For any $D\subseteq X$ write $\Sigma_D$ for the subspace
$\sigma$-algebra of subsets of $D$.
     
\medskip
     
{\bf (a)} If $f(x)=c$ for every $x\in \dom f$, then
$\{x:f(x)<a\}=\dom f$ if $c<a$, $\emptyset$ otherwise, and in either
case
belongs to $\Sigma_{\dom f}$.
     
\medskip
     
{\bf (b)} Write $D=\dom(f+g)=\dom f\cap\dom g$.   If $a\in\Bbb R$ then
set $K=\{(q,q'):q,\,q'\in\Bbb Q,\,q+q'\le
a\}$.   Then $K$ is a subset of $\Bbb Q\times\Bbb Q$, so is countable
(111Fb, 1A1E).      For $q\in\Bbb Q$ choose sets $F_q$,
$G_q\in\Sigma$ such that
     
\Centerline{$\{x:f(x)<q\}=F_q\cap\dom f$,
\quad$\{x:g(x)<q\}=G_q\cap\dom g$.}
     
\noindent For each $(q,q')\in K$, the set
     
\Centerline{$E_{qq'}
=\{x:f(x)<q,\,g(x)<q'\}=F_q\cap G_{q'}\cap D$}
     
\noindent belongs to $\Sigma_D$.
Finally, if $(f+g)(x)<a$, then we can find
$q\in\ooint{f(x),a-g(x)}$, $q'\in\ocint{g(x),a-q}$, so that
$(q,q')\in K$ and
$x\in E_{qq'}$;  while if $(q,q')\in K$ and $x\in E_{qq'}$, then
$(f+g)(x)<q+q'\le a$.   Thus
     
\Centerline{$\{x:(f+g)(x)<a\}=\bigcup_{(q,q')\in K}E_{qq'}\in\Sigma_D$}
     
\noindent by 111Fa.   As $a$ is arbitrary, $f+g$ is measurable.
     
\medskip
     
{\bf (c)} Write $D=\dom f$.   Let $a\in\Bbb R$.   If $c>0$, then
     
\Centerline{$\{x:cf(x)<a\}=\{x:f(x)<\Bover{a}{c}\}\in\Sigma_D$.}
     
\noindent If $c<0$, then
     
\Centerline{$\{x:cf(x)<a\}=\{x:f(x)>\Bover{a}{c}\}\in\Sigma_D$.}
     
\noindent While if $c=0$, then $\{x:cf(x)<a\}$ is either $D$ or
$\emptyset$, as in (a) above, so belongs to $\Sigma_D$.   As $a$ is
arbitrary, $cf$ is measurable.
     
\medskip
     
{\bf (d)} Write $D=\dom(f\times g)=\dom f\cap\dom g$.   Let
$a\in\Bbb R$.   Let $K$ be
     
\Centerline{$\{(q_1,q_2,q_3,q_4):q_1,\ldots,q_4\in\Bbb Q,\,uv<a
\text{ whenever }u\in\ooint{q_1,q_2},\,v\in\ooint{q_3,q_4}\}$.}
     
\noindent Then $K$ is countable.   For $q\in\Bbb Q$ choose sets $F_q$,
$F'_q$, $G_q$, $G'_q\in\Sigma$ such that
     
\Centerline{$\{x:f(x)<q\}=F_q\cap\dom f$,
\quad$\{x:f(x)>q\}=F'_q\cap\dom f$,}
     
\Centerline{$\{x:g(x)<q\}=G_q\cap\dom g$,
\quad$\{x:g(x)>q\}=G'_q\cap\dom g$.}
     
\noindent  For $(q_1,q_2,q_3,q_4)\in K$ set
     
$$\eqalign{E_{q_1q_2q_3q_4}
&=\{x:f(x)\in\ooint{q_1,q_2},\,g(x)\in\ooint{q_3,q_4}\}\cr
&=D\cap F'_{q_1}\cap F_{q_2}\cap G'_{q_3}\cap G_{q_4}
\in\Sigma_D;\cr}$$
     
\noindent then $E=\bigcup_{(q_1,q_2,q_3,q_4)\in
K}E_{q_1q_2q_3q_4}\in\Sigma_D$.
     
Now $E=\{x:(f\times g)(x)<a\}$.   \Prf\ {\bf (i)} If $(f\times g)(x)<a$,
set
$u=f(x)$, $v=g(x)$.   Set
     
\Centerline{$\eta=\min(1,\Bover{a-uv}{1+|u|+|v|})>0$.}
     
\noindent   Take
$q_1,\ldots,q_4\in\Bbb Q$ such that
     
\Centerline{$u-\eta\le q_1<u<q_2\le u+\eta$,
\quad$v-\eta\le q_3<v<q_4\le v+\eta$.}
     
\noindent If $u'\in\ooint{q_1,q_2}$,
$v'\in\ooint{q_3,q_4}$, then $|u'-u|<\eta$ and $|v'-v|<\eta$, so
     
$$\eqalign{u'v'-uv
&=(u'-u)(v'-v)+(u'-u)v+u(v'-v)\cr
&<\eta^2+\eta|v|+|u|\eta
\le\eta(1+|u|+|v|)
\le a-uv,\cr}$$
     
\noindent and $u'v'<a$.   Accordingly $(q_1,q_2,q_3,q_4)\in K$.
Also $x\in E_{q_1q_2q_3q_4}$, so $x\in E$.   Thus $\{x:(f\times
g)(x)<a\}\subseteq E$.   {\bf (ii)} On the other hand, if $x\in E$,
there are
$q_1,\ldots,q_4$ such that $(q_1,q_2,q_3,q_4)\in K$ and $x\in
E_{q_1q_2q_3q_4}$, so that $f(x)\in\ooint{q_1,q_2}$ and
$g(x)\in\ooint{q_3,q_4}$ and $f(x)g(x)<a$.   So
$E\subseteq\{x:(f\times g)(x)<a\}$.\ \Qed
     
Thus $\{x:(f\times g)(x)<a\}\in\Sigma_D$.   As $a$ is arbitrary,
$f\times g$ is measurable.
     
\medskip
     
{\bf (e)} In view of (d), it will be enough to show that $1/g$ is
measurable.   Now if $a>0$,
$\{x:1/g(x)<a\}=\{x:g(x)>1/a\}\cup\{x:g(x)<0\}$;  if $a<0$, then
$\{x:1/g(x)<a\}=\{x:1/a<g(x)<0\}$;  and if $a=0$, then
$\{x:1/g(x)<a\}=\{x:g(x)<0\}$.   And all of these belong to
$\Sigma_{\dom 1/g}$.
     
\medskip
     
{\bf (f)} Write $D=\dom f$ and consider the set
     
\Centerline{$\Tau=\{E:E\subseteq\Bbb R,\,f^{-1}[E]\in\Sigma_D\}$.}
     
\noindent Then $\Tau$ is a $\sigma$-algebra of subsets of $\Bbb R$.
\Prf\ {(i)} $f^{-1}[\emptyset]=\emptyset\in\Sigma_D$, so
$\emptyset\in\Tau$.   {(ii)} If $E\in\Tau$, then
$f^{-1}[\Bbb R\setminus E]=D\setminus f^{-1}[E]\in\Sigma_D$ so
$\Bbb R\setminus E\in\Tau$.
{(iii)} If $\langle E_n\rangle_{n\in\Bbb N}$ is a sequence in
$\Tau$, then
$f^{-1}[\bigcup_{n\in\Bbb N}E_n]
=\bigcup_{n\in\Bbb N}f^{-1}[E_n]\in\Sigma_D$ because $\Sigma_D$ is a
$\sigma$-algebra, so $\bigcup_{n\in\Bbb N}E_n\in\Tau$.\ \Qed
     
Next, $\Tau$ contains all sets of the form $H_a=\ooint{-\infty,a}$ for
$a\in\Bbb R$, by the definition of measurability of $f$.   The result
follows by arguments already used in 114G above.
First, all open subsets of $\Bbb R$
belong to $\Tau$.   \Prf\ Let $G\subseteq\Bbb R$ be open.   Let
$K\subseteq\Bbb Q\times\Bbb Q$ be the set of pairs $(q,q')$ of rational
numbers such that $\coint{q,q'}\subseteq G$.    $K$
is countable.   Also, every $\coint{q,q'}$ belongs to $\Tau$, being
$H_{q'}\setminus H_q$.   So
$G'=\bigcup_{(q,q')\in K}\coint{q,q'}\in\Tau$.
     
By the definition of $K$, $G'\subseteq G$.   On the other hand, if $x\in
G$, there is a $\delta>0$
such that $\ooint{x-\delta,x+\delta}\subseteq G$.
Now there are rational numbers $q\in\ocint{x-\delta,x}$ and
$q'\in\ocint{x,x+\delta}$, so that $(q,q')\in K$ and $x\in
\coint{q,q'}\subseteq G'$.
As $x$ is arbitrary, $G=G'$ and $G\in\Tau$.\ \Qed
     
Finally, $\Tau$ is a
$\sigma$-algebra of subsets of $\Bbb R$ including the family of open
sets, so must contain every Borel set, by the definition of Borel set
(111G).
     
\medskip
     
{\bf (g)} If $a\in\Bbb R$, then $\{y:h(y)<a\}$ is of the form 
$E\cap\dom h$, where $E$ is a Borel subset of $\Bbb R$.   
Next, $f^{-1}[E]$ is of
the form $F\cap\dom f$, where $F\in\Sigma$, by (f) above.   So
     
\Centerline{$\{x:(hf)(x)<a\}=F\cap\dom hf\in\Sigma_{\dom hf}$.}
     
\noindent   As $a$ is
arbitrary, $hf$ is measurable.
     
\medskip
     
{\bf (h)} The point is that $\Sigma_{A\cap\dom f}=\{E\cap
A:E\in\Sigma_{\dom f}\}$.   So if $a\in\Bbb R$,
     
\Centerline{$\{x:(f\restr
A)(x)<a\}=A\cap\{x:f(x)<a\}\in\Sigma_{\dom(f\restr A)}$.}
}%end of proof of 121E
     
\cmmnt{\medskip
     
\noindent{\bf Remarks} Of course part (c) of this theorem is just a
matter of putting (a) and (d) together, while (e) is a consequence of
(d), (g) and the fact that continuous functions are Borel measurable
(121Db).
     
I hope you will recognise the technique in the proof of part (d) as a
version of arguments which may be used to prove that the limit of a
product is the product of the limits, or that the product of continuous
functions is continuous.   In fact (b) and (d) here, as well as the
theorems on sums and products of limits, are consequences of the fact
that addition and multiplication are continuous functions.   In
121K I give a general result which may be used to exploit such
facts.
     
Really, part (f) here is the essence of the concept of `measurable'
real-valued function.   The point of the definition in 121B-121C is that
the Borel $\sigma$-algebra of $\Bbb R$ can be generated by any of the
families $\{\ooint{-\infty,a}:a\in\Bbb R\}$,
$\{\ocint{-\infty,a}:a\in\Bbb R\},\ldots$.   (See 121Yc(ii).)
There are many routes covering this territory in rather
fewer words than I have used, at the cost of slightly greater
abstraction.
}%end of comment
     
     
\leader{121F}{Theorem} Let $X$ be a set and $\Sigma$ a $\sigma$-algebra
of
subsets of $X$.   Let $\langle f_n\rangle_{n\in\Bbb N}$ be a sequence of
$\Sigma$-measurable real-valued functions with domains included in $X$.
     
(a) Define a function $\lim_{n\to\infty}f_n$ by writing
     
\Centerline{$(\lim_{n\to\infty}f_n)(x)=\lim_{n\to\infty}f_n(x)$}
     
\noindent for all those
$x\in\bigcup_{n\in\Bbb N}\bigcap_{m\ge n}\dom f_m$ for which the limit exists in $\Bbb R$.   Then $\lim_{n\to\infty}f_n$ is
$\Sigma$-measurable.
     
     
(b) Define a function $\sup_{n\in\Bbb N}f_n$ by writing
     
\Centerline{$(\sup_{n\in\Bbb N}f_n)(x)=\sup_{n\in\Bbb N}f_n(x)$}
     
\noindent for all those $x\in\bigcap_{n\in\Bbb N}\dom f_n$ for which the
supremum exists in $\Bbb R$.   Then $\sup_{n\in\Bbb N}f_n$ is
$\Sigma$-measurable.
     
(c) Define a function $\inf_{n\in\Bbb N}f_n$ by writing
     
\Centerline{$(\inf_{n\in\Bbb N}f_n)(x)=\inf_{n\in\Bbb N}f_n(x)$}
     
\noindent for all those $x\in\bigcap_{n\in\Bbb N}\dom f_n$ for which the
infimum exists in $\Bbb R$.   Then $\inf_{n\in\Bbb N}f_n$ is
$\Sigma$-measurable.
     
(d) Define a function $\limsup_{n\to\infty}f_n$ by writing
     
\Centerline{$(\limsup_{n\to\infty}f_n)(x)=\limsup_{n\to\infty}f_n(x)$}
     
\noindent for all those 
$x\in\bigcup_{n\in\Bbb N}\bigcap_{m\ge n}\dom f_m$ for which the 
$\limsup$ exists in $\Bbb R$.   Then $\limsup_{n\to\infty}f_n$ is
$\Sigma$-measurable.
     
(e) Define a function $\liminf_{n\to\infty}f_n$ by writing
     
\Centerline{$(\liminf_{n\to\infty}f_n)(x)=\liminf_{n\to\infty}f_n(x)$}
     
\noindent for all those $x\in\bigcup_{n\in\Bbb N}\bigcap_{m\ge n}\dom
f_m$ for which the $\liminf$ exists in $\Bbb R$.   Then
$\liminf_{n\in\Bbb N}f_n$ is
$\Sigma$-measurable.
     
\proof{ For $n\in\Bbb N$, $a\in\Bbb R$ choose
$H_n(a)\in\Sigma$ such that $\{x:f_n(x)\le a\}=H_n(a)\cap \dom f_n$.
The proofs are now a matter of observing the following facts:
     
\medskip
     
{\bf (a)}
$\{x:(\lim_{n\to\infty}f_n)(x)\le a\}
=\dom(\lim_{n\to\infty}f_n)
  \cap\bigcap_{k\in\Bbb N}\bigcup_{n\in\Bbb N}
    \bigcap_{m\ge n}H_m(a+2^{-k})$;
     
\medskip
     
     
{\bf (b)} $\{x:(\sup_{n\in\Bbb N}f_n)(x)\le a\}=\dom(\sup_{n\in\Bbb
N}f_n)\cap\bigcap_{n\in\Bbb N}H_n(a)$;
     
\medskip
     
{\bf (c)} $\inf_{n\in\Bbb N}f_n=-\sup_{n\in\Bbb N}(-f_n)$;
     
\medskip
     
{\bf (d)} $\limsup_{n\to\infty}f_n
=\lim_{n\to\infty}\sup_{m\in\Bbb N}f_{m+n}$;
     
\medskip
     
{\bf (e)} $\liminf_{n\to\infty}f_n=-\limsup_{n\to\infty}(-f_n)$.
}%end of proof of 121F
     
     
\cmmnt{
\leader{121G}{Remarks} It is at this point that we first
encounter clearly the problem of functions which are not defined
everywhere.  (The quotient $f/g$ of 121Ee also need not be defined
everywhere on the
common domain of $f$ and $g$, but it is less important and more
easily dealt with.)   The whole point of the theory of measure and
integration, since Lebesgue, is that we can deal with limits of
sequences of functions;  and the set on which $\lim_{n\to\infty}f_n(x)$
exists can be decidedly irregular, even for apparently well-behaved
functions $f_n$.   (If you have encountered the theory of Fourier
series, then an appropriate example to think of is the sequence of
partial sums $f_n(x)={1\over 2}a_0+\sum_{k=1}^n(a_k\cos kx+b_k\sin kx)$
of a Fourier series in which $\sum_{k=1}^{\infty}|a_k|+|b_k|=\infty$, so
that the series is not uniformly absolutely summable, but may be
conditionally summable at certain points.)
     
I have tried to make it clear what domains I mean
to attach to the functions $\sup_{n\in\Bbb N}f_n$,
$\lim_{n\to\infty}f_n$, etc.   The guiding principle is that they should
be the set of all $x\in X$ for which the defining formulae
$\sup_{n\in\Bbb N}f_n(x)$, $\lim_{n\to\infty}f_n(x)$ can be interpreted
as real numbers.   (As I noted in 121C, I am for the time being avoiding
`$\infty$' as a value of a function, though it gives little
difficulty, and some formulae are more naturally interpreted by allowing
it.)   But in the case of $\lim$, $\limsup$, $\liminf$ it should be
noted that I am using the restrictive definition, that
$\lim_{n\to\infty}a_n$ can be regarded as existing only when there is
some $n\in\Bbb N$ such that $a_m$ is defined for every $m\ge n$.   There
are occasions when it would be more natural to admit the limit when we
know only that $a_m$ is defined for infinitely many $m$;  but such a
convention could make 121Fa false, unless care was taken.
     
As in 111E-111F, we can use the ideas of parts (b), (c) here to discuss
functions of the form $\sup_{k\in K}f_k$, $\inf_{k\in K}f_k$ for any
family $\langle f_k\rangle_{k\in K}$ of measurable functions indexed by
a non-empty countable set $K$.
     
In this theorem and the last, the functions $f$, $g$, $f_n$ have been
permitted to have arbitrary domains, and consequently there is nothing
that can be said about the domains of the constructed functions.
However, it is of course the case that if the original functions have
measurable domains, so do the functions constructed from them by the
rules I propose.   I spell out the details in the next proposition.
}%end of comment
     
\leader{121H}{Proposition}  Let $X$ be a set and $\Sigma$ a
$\sigma$-algebra
of subsets of $X$;  let $f$, $g$ and $f_n$, for $n\in\Bbb N$, be
$\Sigma$-measurable real-valued functions whose domains belong to
$\Sigma$.   Then all the functions
     
\Centerline{$f+g$, \quad$f\times g$, \quad$f/g$,}
     
\Centerline{$\sup_{n\in\Bbb N}f_n$, \quad$\inf_{n\in\Bbb N}f_n$,
\quad$\lim_{n\to\infty}f_n$, \quad$\limsup_{n\to\infty}f_n$,
\quad$\liminf_{n\to\infty}f_n$}
     
\noindent have domains belonging to $\Sigma$.   Moreover, if $h$ is a
Borel measurable real-valued function defined on a Borel subset of
$\Bbb R$, then $\dom hf\in\Sigma$.
     
\proof{ For the first two, we have $\dom(f+g)=\dom(f\times
g)=\dom f\cap \dom g$.   Next, if $E$ is a Borel subset of $\Bbb R$,
there is an $H\in\Sigma$ such that
$f^{-1}[E]=H\cap\dom f$;  so
$f^{-1}[E]\in\Sigma$.   Thus
     
\Centerline{$\dom hf=f^{-1}[\dom h]\in\Sigma$.}
     
\noindent Setting $h(a)=1/a$ for $a\in\Bbb R\setminus\{0\}$, we see that
$\dom(1/f)\in\Sigma$.   ($\dom h=\Bbb R\setminus\{0\}$ is a Borel set
because it is open.)   Similarly, $\dom(1/g)$ and $\dom(f/g)=\dom
f\cap\dom(1/g)$ belong to $\Sigma$.
     
Now for the infinite combinations.   Set
$H_n(a)=\{x:x\in\dom f_n,\,f_n(x)<a\}$ for
$n\in\Bbb N$, $a\in\Bbb R$;  as just explained, every $H_n(a)$
belongs to $\Sigma$.   Now
     
\Centerline{$\dom(\sup_{n\in\Bbb N}f_n)
=\bigcup_{m\in\Bbb N}\bigcap_{n\in\Bbb N}H_n(m)\in\Sigma$.}
     
\noindent Next, $|f_m-f_n|$ is
measurable, with domain in $\Sigma$, for all $m$, $n\in\Bbb N$ (applying
the results above to $-f_n=-1\cdot f_n$, $f_m-f_n=f_m+(-f_n)$ and
$|f_m-f_n|=|\,\,|\circ(f_m-f_n)$), so
     
\Centerline{$G_{mnk}
=\{x:x\in\dom f_m\cap\dom f_n,\,|f_m(x)-f_n(x)|\le 2^{-k}\}\in\Sigma$}
     
\noindent for all $m$, $n$, $k\in\Bbb N$.   Accordingly
     
\Centerline{$\dom(\lim_{n\to\infty}f_n)
=\{x:\Exists n,\,\langle f_m(x)\rangle_{m\ge n}$ is Cauchy$\}
=\bigcap_{k\in\Bbb N}\bigcup_{n\in\Bbb N}\bigcap_{m\ge n}G_{mnk}
\in\Sigma$.}
     
Manipulating the above results as in (c), (d) and (e) of the proof of
121F, we easily complete the proof.
}%end of proof of 121H
     
\cmmnt{\medskip
     
\noindent{\bf Remark} Note the use of the General Principle of
Convergence in the proof above.   I am not sure whether this will strike
you as `natural', and there are alternative methods;  but the formula
     
\Centerline{$\{x:\lim_{n\to\infty}f_n(x)$ exists in $\Bbb R\}
=\{x:\sequencen{f_n(x)}$ is Cauchy$\}$}
     
\noindent is one worth storing in your long-term memory.
}%end of comment
     
\leader{*121I}{}\cmmnt{ I end this section with two results which can be safely passed by on first reading, but which you will need at some point
to master if you wish to go farther into measure theory than the present
chapter, as both are essential parts of the concept of `measurable
function'.
     
\medskip
     
\noindent}{\bf Proposition} Let $X$ be a set and $\Sigma$ a
$\sigma$-algebra of subsets of $X$.   Let $D$ be a subset of $X$ and
$f:D\to\Bbb R$ a function.   Then $f$ is measurable iff there is a
measurable function $h:X\to\Bbb R$ extending $f$.
     
\proof{{\bf (a)} If $h:X\to\Bbb R$ is measurable and $f=h\restr D$, then
$f$ is measurable by 121Eh.
     
\medskip
     
{\bf (b)} Now suppose that $f$ is measurable.
     
\medskip
     
\quad{\bf (i)} For each
$q\in\Bbb Q$, the set $D_q=\{x:x\in D,\,f(x)\le q\}$ belongs to the subspace $\sigma$-algebra $\Sigma_D$, that is, there is an $E_q\in\Sigma$ such that $D_q=E_q\cap D$.   Set
     
\Centerline{$F=X\setminus\bigcup_{q\in\Bbb Q}E_q$,}
     
\Centerline{$G=\bigcap_{n\in\Bbb N}\bigcup_{q\in\Bbb Q,q\le -n}E_q$;}
     
\noindent then both $F$ and $G$ belong to $\Sigma$, and are disjoint
from $D$.   \Prf\ If $x\in D$, there is a $q\in\Bbb Q$ such that
$f(x)\le q$, so that $x\in E_q$ and $x\notin F$.   Also there is an
$n\in\Bbb N$ such that $f(x)>-n$, so that $x\notin E_{q'}$ for $q'\le-n$
and $x\notin G$.\ \Qed
     
Set $H=X\setminus(F\cup G)\in\Sigma$.   For $x\in H$,
     
\Centerline{$\{q:q\in\Bbb Q,\,x\in E_q\}$}
     
\noindent is non-empty and bounded below, so we may set
     
\Centerline{$h(x)=\inf\{q:x\in E_q\}$;}
     
\noindent for $x\in F\cup G$, set $h(x)=0$.   This defines
$h:X\to\Bbb R$.
     
\medskip
     
\quad{\bf (ii)} $h(x)=f(x)$ for $x\in D$.   \Prf\ As remarked above, $x\in H$.   If $q\in\Bbb Q$ and $x\in E_q$, then $f(x)\le q$;  consequently $h(x)\ge f(x)$.   On the other
hand, given $\epsilon>0$, there is a
$q\in\Bbb Q\cap[f(x),f(x)+\epsilon]$, and now $x\in E_q$, so
$h(x)\le q\le f(x)+\epsilon$;  as $\epsilon$ is arbitrary,
$h(x)\le f(x)$.\ \Qed
     
\medskip
     
\quad{\bf (iii)} $h$ is measurable.   \Prf\ If $a>0$ then
     
\Centerline{$\{x:h(x)<a\}=(H\cap\bigcup_{q<a}E_q)\cup(F\cup
G)\in\Sigma$,}
     
\noindent while if $a\le 0$
     
\Centerline{$\{x:h(x)<a\}=H\cap\bigcup_{q<a}E_q\in\Sigma$.   \Qed}
     
\noindent This completes the proof.
}%end of proof of 121I
     
\leader{*121J}{}\cmmnt{ The next proposition may illuminate 121E, as well as
being indispensable for the work of Volume 2.   I start with a useful
description of the Borel sets of $\BbbR^r$.
     
\medskip
     
\noindent}{\bf Lemma} Let $r\ge 1$ be an integer, and write $\Cal J$ for
the family of subsets of $\BbbR^r$ of the form
$\{x:\xi_i\le\alpha\}$ where $i\le r$, $\alpha\in\Bbb R$, writing
$x=(\xi_1,\ldots,\xi_r)$, as in \S115.   Then the
$\sigma$-algebra of subsets of $\BbbR^r$ generated by $\Cal J$ is
precisely the $\sigma$-algebra $\Cal B$ of Borel subsets of $\BbbR^r$.
     
\proof{{\bf (a)}   All the sets in $\Cal J$ are closed, so must
belong to $\Cal B$;  writing $\Sigma$ for the $\sigma$-algebra
generated by $\Cal J$, we must have $\Sigma\subseteq\Cal B$.
     
\medskip
     
     
{\bf (b)} The next step is to observe that all half-open
intervals of the form
     
\Centerline{$\ocint{a,b}
=\{x:\alpha_i<\xi_i\le\beta_i\Forall i\le r\}$}
     
\noindent belong to $\Sigma$;  this is because
     
\Centerline{$\ocint{a,b}=\bigcap_{i\le
r}(\{x:\xi_i\le\beta_i\}\setminus\{x:\xi_i\le\alpha_i\})$.}
     
\noindent     It follows that all open sets belong to $\Sigma$.
\Prf\ (Compare the proof of 121Ef.)  Let $G\subseteq\BbbR^r$ be an open
set.   Let $\Cal I$ be the set
of all intervals of the form
$\ocint{q,q'}$ which are included in $G$, where $q$, $q'\in\BbbQ^r$.
Then $\Cal
I$ is a countable subset of $\Sigma$, so (because $\Sigma$ is a
$\sigma$-algebra) $\bigcup\Cal I\in\Sigma$.   By the definition of
$\Cal I$, $\bigcup\Cal I\subseteq G$.   But also, if $x\in G$, there is a $\delta>0$ such that the open ball $U(x,\delta)$ with centre $x$ and radius $\delta$ is included in $G$ (1A2A).
Now, for each $i\le r$, we can find rational numbers $\alpha_i$,
$\beta_i$ such that
     
\Centerline{$\xi_i-\Bover{\delta}r
\le\alpha_i<\xi_i\le\beta_i<\xi_i+\Bover{\delta}r$,}
     
\noindent so that
     
\Centerline{$x\in\ocint{a,b}\subseteq U(x,\delta)\subseteq G$}
     
\noindent and $x\in\ocint{a,b}\in\Cal I$.   Thus $x\in\bigcup\Cal I$.
As $x$ is arbitrary, $G\subseteq\bigcup\Cal I$ and $G=\bigcup\Cal
I\in\Sigma$.\ \Qed
     
\medskip
     
{\bf (c)} Thus $\Sigma$ is a $\sigma$-algebra of sets containing
every open set, and must include $\Cal B$, the smallest such
$\sigma$-algebra.
}%end of proof of 121J
     
\cmmnt{\medskip
     
\noindent{\bf Remark} Compare the proof of 115G.
}%end of comment
     
\leader{*121K}{Proposition} Let $X$ be a set and $\Sigma$ a
$\sigma$-algebra of subsets of $X$.   Let $r\ge 1$ be an integer, and
$f_1,\ldots,f_r$ measurable functions defined on subsets of $X$.
Set $D=\bigcap_{i\le r}\dom f_i$ and for $x\in D$ set
$f(x)=(f_1(x),\ldots,f_r(x))\in\BbbR^r$.   Then
     
(a) for any Borel set $E\subseteq\BbbR^r$, $f^{-1}[E]$ belongs to the subspace $\sigma$-algebra $\Sigma_D$;
     
(b) if $h$ is a Borel measurable function from a subset $\dom h$ of
$\BbbR^r$ to $\Bbb R$, then the composition $hf$ is measurable.
     
\proof{{\bf (a)(i)} Consider the set
     
\Centerline{$\Tau=\{E:E\subseteq\BbbR^r,\,f^{-1}[E]\in\Sigma_D\}$.}
     
\noindent Then $\Tau$ is a $\sigma$-algebra of subsets of $\BbbR^r$.
\Prf\ (Compare 121Ef.) {\bf ($\pmb{\alpha}$)}
$f^{-1}[\emptyset]=\emptyset\in\Sigma_D$, so
$\emptyset\in\Tau$.   {\bf ($\pmb{\beta}$)} If $E\in\Tau$,
then $f^{-1}[\BbbR^r\setminus
E]=D\setminus f^{-1}[E]\in\Sigma_D$ so $\BbbR^r\setminus E\in\Tau$.
{\bf ($\pmb{\gamma}$)} If $\langle E_n\rangle_{n\in\Bbb N}$ is a
sequence in $\Tau$, then
$f^{-1}[\bigcup_{n\in\Bbb N}E_n]
=\bigcup_{n\in\Bbb N}f^{-1}[E_n]\in\Sigma_D$ because $\Sigma_D$ is a
$\sigma$-algebra, so $\bigcup_{n\in\Bbb N}E_n\in\Tau$.\ \Qed
     
\medskip
     
\quad{\bf (ii)}
Next, for any $i\le r$ and $\alpha\in\Bbb R$, $J=\{x:\xi_i\le\alpha\}$
belongs to $\Tau$, because
     
\Centerline{$f^{-1}[J]
=\{x:x\in D,\, f_i(x)\le \alpha\}\in\Sigma_D$.}
     
\noindent So $\Tau$ includes the family $\Cal J$ of 121J and therefore
includes the $\sigma$-algebra $\Cal B$ generated by $\Cal J$, that is,
contains every Borel subset of $\BbbR^r$.
     
\medskip
     
{\bf (b)} Now the rest follows by the argument of 121Eg.   If
$a\in\Bbb R$, then $\{y:y\in\dom h,\,h(y)<a\}$ is of the form $E\cap\dom h$, where
$E$ is a
Borel subset of $\BbbR^r$, so $\{x:x\in\dom(hf),\,(hf)(x)<a\}
=f^{-1}[E]\cap\dom(hf)$ belongs to
$\Sigma_{\dom hf}$.
}%end of proof of 121K
     
\exercises{
\leader{121X}{Basic exercises $\pmb{>}$(a)}
%\spheader 121Xa
Let $X$ be a set, $\Sigma$ a $\sigma$-algebra of
subsets of $X$, and $D\subseteq X$.   Let $\sequencen{D_n}$ be a
partition of $D$ into relatively measurable sets and $\sequencen{f_n}$
a sequence of measurable real-valued functions such that
$D_n\subseteq\dom f_n$ for each $n$.   Define $f:D\to\Bbb R$ by setting
$f(x)=f_n(x)$ whenever $n\in\Bbb N$, $x\in D_n$.   Show that $f$ is
measurable.
%121C
     
\spheader 121Xb Let $X$ be a set and $\Sigma$ a
$\sigma$-algebra of subsets of $X$.   If $f$ and $g$ are measurable
real-valued functions defined on subsets of $X$, show that $f^+$, $f^-$,
$f\wedge g$ and $f\vee g$ are measurable, where
     
\Centerline{$f^+(x)=\max(f(x),0)$ for $x\in\dom f$,}
     
\Centerline{$f^-(x)=\max(-f(x),0)$ for $x\in\dom f$,}
     
\Centerline{$(f\vee g)(x)=\max(f(x),g(x))$ for $x\in\dom f\cap\dom g$,}
     
\Centerline{$(f\wedge g)(x)=\min(f(x),g(x))$ for
$x\in\dom f\cap\dom g$.}
%121E
     
\sqheader 121Xc Let $(X,\Sigma,\mu)$ be a measure space.   Write
$\eusm L^0$ for the set of real-valued functions $f$ such that
($\alpha$) $\dom f$ is a conegligible subset of $X$ ($\beta$) there is a
conegligible set $E\subseteq X$ such that $f\restr E$ is measurable.
(i) Show that the set $E$ of clause ($\beta$) in the last sentence may
be taken to belong to $\Sigma$ and be included in $\dom f$.   (ii) Show
that if $f$, $g\in\eusm L^0$ and $c\in\Bbb R$, then $f+g$, $cf$,
$f\times g$, $|f|$, $f^+$, $f^-$, $f\wedge g$, $f\vee g$
all belong to $\eusm L^0$.   (iii) Show that if $f$, $g\in\eusm L^0$ and
$g\ne 0$ a.e.\ then $f/g\in\eusm L^0$.   (iv) Show that if
$\sequencen{f_n}$ is a sequence in $\eusm L^0$ then the functions
     
\Centerline{$\lim_{n\to\infty}f_n$,\quad$\sup_{n\in\Bbb N}f_n$,
\quad$\inf_{n\in\Bbb N}f_n$,\quad $\limsup_{n\to\infty}f_n$,
\quad$\liminf_{n\to\infty}f_n$}
     
\noindent belong to $\eusm L^0$ whenever they are defined almost
everywhere as real-valued functions.   (v) Show that if $f\in\eusm L^0$
and $h:\Bbb R\to\Bbb R$ is Borel measurable then $hf\in\eusm L^0$.
%121F
     
\sqheader 121Xd Consider the following four families of subsets
of $\Bbb R$:
     
\Centerline{$\Cal A_1=\{\ooint{-\infty,a}:a\in\Bbb R\}$,
\quad$\Cal A_2=\{\ocint{-\infty,a}:a\in\Bbb R\}$,}
     
\Centerline{$\Cal A_3=\{\ooint{a,\infty}:a\in\Bbb R\}$,
\quad$\Cal A_4=\{\coint{a,\infty}:a\in\Bbb R\}$.}
     
\noindent Show that for each $j$ the $\sigma$-algebra of subsets of
$\Bbb R$ generated by $\Cal A_j$ is the $\sigma$-algebra of Borel sets.
%121J
     
\spheader 121Xe Let $D$ be any subset of $\BbbR^r$, where $r\ge 1$.
Write $\frak T_D$ for the set $\{G\cap D:G\subseteq\BbbR^r$ is
open$\}$.  (i) Show that $\frak T_D$ satisfies the properties of open
sets listed in 1A2B.   (ii) Let $\Cal B$ be the 
$\sigma$-algebra of Borel sets
in $\BbbR^r$, and $\Cal B(D)$ the subspace $\sigma$-algebra on $D$.
Show that $\Cal B(D)$ is just the $\sigma$-algebra of subsets of $D$
generated by $\frak T_D$.   \Hint{($\alpha$) observe that
$\frak T_D\subseteq\Cal B(D)$ ($\beta$) consider 
$\{E:E\subseteq\BbbR^r,\,E\cap D$ 
belongs to the $\sigma$-algebra generated by $\frak T_D\}$.}
%121K
     
\spheader 121Xf Let $(X,\Sigma,\mu)$ be a measure space and
define $\eusm L^0$ as in 121Xc.   Show that if $f_1,\ldots,f_r$ belong
to $\eusm L^0$ and $h:\BbbR^r\to\Bbb R$ is Borel measurable then
$h(f_1,\ldots,f_r)$ belongs to $\eusm L^0$.
%121Xc 121K
     
\leader{121Y}{Further exercises (a)}
%\spheader 121Ya
Let $X$ and $Y$ be sets, $\Sigma$ a $\sigma$-algebra of subsets of
$X$, $\phi:X\to Y$ a function and $g$ a real-valued function defined on a
subset of $Y$.   Set
$\Tau=\{F:F\subseteq Y,\,\phi^{-1}[F]\in\Sigma\}$;   then $\Tau$ is a
$\sigma$-algebra of subsets of $Y$ (see 111Xc).   (i) Show that if $g$ is
$\Tau$-measurable then $g\phi$ is $\Sigma$-measurable.   
(ii) Give an example in which $g\phi$ is
$\Sigma$-measurable but $g$ is not $\Tau$-measurable.
(iii) Show that if
$g\phi$ is $\Sigma$-measurable and {\it either} $\phi$ is injective 
{\it or} $\dom(g\phi)\in\Sigma$ {\it or} $\phi[X]\subseteq\dom g$, 
then $g$ is $\Tau$-measurable.\footnote{I am grateful to P.Wallace Thompson
for pointing out the error in the original version of this exercise.}   
%121C
     
\spheader 121Yb Let $X$ and $Y$ be sets, $\Tau$ a
$\sigma$-algebra of subsets of $Y$ and $\phi:X\to Y$ a function.   Set
$\Sigma=\{\phi^{-1}[F]:F\in\Tau\}$, as in 111Xd.   Show that a
function $f:X\to\Bbb R$ is $\Sigma$-measurable iff there is a
$\Tau$-measurable function $g:Y\to\Bbb R$ such that $f=g\phi$.
%121I
     
\spheader 121Yc Let $X$ and $Y$ be sets and $\Sigma$, $\Tau$
$\sigma$-algebras of subsets of $X$, $Y$ respectively.   I say that a
function $\phi:X\to Y$ is $(\Sigma,\Tau)${\bf-measurable} if
$\phi^{-1}[F]\in\Sigma$ for every $F\in\Tau$.   (i) Show that if
$\Sigma$, $\Tau$, $\Upsilon$ are $\sigma$-algebras of subsets of $X$,
$Y$, $Z$ respectively, and $\phi:X\to Y$ is $(\Sigma,\Tau)$-measurable,
$\psi:Y\to Z$ is $(\Tau,\Upsilon)$-measurable, then $\psi\phi:X\to Z$ is
$(\Sigma,\Upsilon)$-measurable.   
(ii) Suppose that $\Cal A\subseteq\Tau$ is such that $\Tau$ is the
$\sigma$-algebra of subsets of $Y$ generated by $\Cal A$ (111Gb).   Show
that $\phi:X\to Y$ is $(\Sigma,\Tau)$-measurable iff
$\phi^{-1}[A]\in\Sigma$ for every $A\in\Cal A$.
(iii) For $r\ge 1$, write
$\Cal B_r$ for the
$\sigma$-algebra of Borel subsets of $\BbbR^r$.   Show that if $X$ is
any set and $\Sigma$ is a $\sigma$-algebra of subsets of $X$, then a
function $f:X\to\BbbR^r$ is $(\Sigma,\Cal B_r)$-measurable iff
$\pi_if:X\to\Bbb R$ is $(\Sigma,\Cal B_1)$-measurable for every
$i\le r$, writing $\pi_i(x)=\xi_i$ for $i\le r$,
$x=(\xi_1,\ldots,\xi_r)\in\BbbR^r$.    (iv) Rewrite these ideas
for partially-defined functions.
%121+
     
\spheader 121Yd Let $X$ be a set and $\Sigma$ a $\sigma$-algebra
of subsets of $X$.   For $r\ge 1$, $D\subseteq X$ say that a function
$\phi:D\to\BbbR^r$ is {\bf measurable} if $\phi^{-1}[G]$ is relatively
measurable in $D$ for every open set $G\subseteq\BbbR^r$.   If
$X=\BbbR^s$ and $\Sigma$ is the 
$\sigma$-algebra $\Cal B_s$ of Borel subsets of $\BbbR^s$, 
say that $\phi$ is {\bf Borel measurable}.
(i) Show that $\phi$ is measurable in this sense iff all its
coordinate functions $\phi_i:D\to\Bbb R$ are measurable in the sense of
121C, taking $\phi(x)=(\phi_i(x),\ldots,\phi_r(x))$ for $x\in D$.   (In
particular, this definition agrees with 121C when $r=1$.)
(ii) Show that $\phi:D\to\BbbR^r$ is measurable iff it is
$(\Sigma,\Cal B_r)$-measurable in the sense of 121Yc.
(iii) Show that if $\phi:D\to\BbbR^r$ is measurable and
$\psi:E\to\BbbR^s$ is Borel measurable, where $E\subseteq\BbbR^r$,
then $\psi\phi:\phi^{-1}[E]\to\BbbR^s$ is measurable.
(iv) Show that any continuous function from a subset of
$\BbbR^s$ to $\BbbR^r$ is Borel measurable.
%121Yc
     
\spheader 121Ye Let $X$ be a set and $\theta$ an outer measure
on $X$;  let $\mu$ be the measure defined from $\theta$ by
\Caratheodory's method, and $\Sigma$ its domain.   Suppose that
$f:X\to\Bbb R$ is a function such that
     
\Centerline{$\theta\{x:x\in A,\,f(x)\le a\}
  +\theta\{x:x\in A,\,f(x)\ge b\}\le\theta A$}
     
\noindent whenever $A\subseteq X$ and $a<b$ in $\Bbb R$.   Show that $f$
is $\Sigma$-measurable.   ({\it Hint\/}: suppose that $a\in\Bbb R$ and
$\theta A<\infty$.   Set
     
\Centerline{$B_k=\{x:x\in A,\,
 a+\Bover1{2k+2}\le f(x)\le a+\Bover1{2k+1}\}$,}
     
\Centerline{$B'_k=\{x:x\in A,\,
  a+\Bover1{2k+3}\le f(x)\le a+\Bover1{2k+2}\}$}
     
\noindent for $k\in\Bbb N$.   Show that 
$\sum_{k=0}^{\infty}\theta B_k\le\theta A$, 
and check a similar result for $B'_k$.   Hence show that
     
\Centerline{$\theta\{x:x\in A,\,f(x)>a\}
=\lim_{k\to\infty}\theta\{x:x\in A,\,f(x)\ge a+\Bover1k\}$.)}
%121+
}%end of exercises
     
\endnotes{
\Notesheader{121} I find myself offering no fewer than three definitions
of `measurable function', in 121C, 121Yc and 121Yd.   It is in fact the
last which is probably the most important and the best guide to further
ideas.   Nevertheless, the overwhelming majority of applications refer
to real-valued functions, and the four equivalent conditions of 121B are
the most natural and most convenient to use.   The fact that they all
coincide with the condition of 121Yd corresponds to the fact that they
are all of the form
     
\Centerline{$f^{-1}[E]\in\Sigma_D$ for every $E\in\Cal A$}
     
\noindent where $\Cal A$ is a family of subsets of $\Bbb R$ generating
the Borel $\sigma$-algebra (121Xd, 121Yc(ii)).
     
The class of measurable functions may well be the widest you have yet
seen, not counting the family of all real-valued functions;  all easily
describable functions between subsets of $\Bbb R$ are measurable.   Not
only is the space of measurable functions closed under addition and
multiplication and composition with continuous functions (121E), but any
natural operation acting on a sequence of measurable functions will
produce a measurable function (121F, 121Xb, 121Xa).   It is {\it not}
however the case that the composition of two Lebesgue measurable
functions from $\Bbb R$ to itself is always Lebesgue measurable;  I
offer a counter-example in 134Ib.   The point here is that a function is
called `measurable' if it is $(\Sigma,\Cal B)$-measurable, in the
language of 121Yc, where $\Cal B$ is the 
$\sigma$-algebra of Borel sets.   Such a
function can well fail to be $(\Sigma,\Sigma)$-measurable, if $\Sigma$
properly includes $\Cal B$, so the natural argument for compositions
(121Yc(i)) fails.   Nevertheless, for reasons which I will hint at in
\S134, non-Lebesgue-measurable functions are hard to come by, and only
in the most rarefied kinds of real analysis do they appear in any
natural way.   You may therefore approach the question of whether a
particular function is Lebesgue measurable with reasonable confidence
that it is, and that the proof is merely a challenge to your technique.
     
You will observe that the results of 121E are mostly covered by
121I-121K, which also include large parts of 114G and 115G;  and that
121Kb is covered by 121Yd(iii).   You can count yourself as having
mastered this part of the subject when you find my exposition tediously
repetitive.   Of course, in order to deduce 121Ed from 121K, for
instance, you have to know that multiplication, regarded as a function
from $\BbbR^2$ to $\Bbb R$, is continuous, therefore Borel measurable;
the proof of this is embedded in the proof I give of 121Ed (look at the
formula for $\eta$ half way through).
}%end of notes
     
\discrpage
     
     
\frfilename{mt122.tex}
\versiondate{4.1.04}
\copyrightdate{1994}

\def\chaptername{Integration}
\def\sectionname{Definition of the integral}

\newsection{122}

I set out the definition of ordinary integration for real-valued
functions defined on an arbitrary measure space, with its most basic
properties.

\leader{122A}{Definitions}  Let $(X,\Sigma,\mu)$ be a measure space.

\header{122Aa}{\bf (a)} For any set $A\subseteq X$, I write $\chi A$
for
the {\bf indicator function} or {\bf characteristic function} of $A$, 
the function
from $X$ to $\{0,1\}$ given by setting $\chi A(x)=1$ if $x\in A$, $0$ if
$x\in X\setminus A$.   \cmmnt{(Of course this notation depends on it
being understood which is the `universal' set $X$ under consideration;
perhaps I should call it the `indicator function of $A$ as a
subset of $X$'.)}
\cmmnt{Observe that} $\chi A$ is $\Sigma$-measurable\cmmnt{, in
the
sense of 121C above,} iff $A\in\Sigma$\cmmnt{ (because
$A=\{x:\chi A(x)>0\}$)}.

\header{122Ab}{\bf (b)} Now a {\bf simple} function on $X$ is a
function
of the form
$\sum_{i=0}^n a_i\chi E_i$, where $E_0,\ldots,E_n$ are measurable
sets of finite measure and $a_0,\ldots, a_n$ belong to
$\Bbb R$.   \cmmnt{{\bf Warning!} Some authors allow arbitrary sets
$E_i$, so that a simple function on $X$ is any function taking only
finitely many values.}

\leader{122B}{Lemma} Let $(X,\Sigma,\mu)$ be a measure space.

(a) Every simple function on $X$ is measurable.

(b) If $f$, $g:X\to\Bbb R$ are simple functions, so is $f+g$.

(c) If $f:X\to \Bbb R$ is a simple function and $c\in\Bbb R$, then
$cf:X\to\Bbb R$ is a simple function.

(d) The constant zero function is simple.

\proof{ (a) comes from the facts that $\chi E$ is
measurable for measurable $E$, and that sums and scalar multiples of
measurable functions are measurable (121Eb-121Ec).  (b)-(d) are
trivial.
}%end of proof of 122B

\vleader{42pt}{122C}{Lemma} Let $(X,\Sigma,\mu)$ be a measure space.

(a) If $E_0,\ldots,E_n$ are measurable sets of finite measure,
there are disjoint measurable sets $G_0,\ldots,G_m$ of finite measure
such that each $E_i$ is expressible as a union of some of the $G_j$.

(b) If $f:X\to\Bbb R$ is a simple function, it is expressible in the
form $\sum_{j=0}^mb_j\chi G_j$ where $G_0,\ldots,G_m$ are disjoint
measurable sets of finite measure.

(c) If $E_0,\ldots,E_n$ are measurable sets of finite measure, and
$a_0,\ldots,a_n\in\Bbb R$ are such that $\sum_{i=0}^na_i\chi E_i(x)\ge
0$ for every $x\in X$, then $\sum_{i=0}^na_i\mu E_i\ge 0$.

\proof{{\bf (a)} Set $m=2^{n+1}-2$, and enumerate the
non-empty subsets of $\{0,\ldots,n\}$ as $I_0,\ldots,I_m$.   For each
$j\le m$, set

\Centerline{$G_j
=\bigcap_{i\in I_j}E_i\setminus\bigcup_{i\le n,i\notin I_j}E_i$.}

\noindent   Then every $G_j$ is a measurable set, being obtained from
finitely many measurable sets by the operations $\cup$, $\cap$ and
$\setminus$, and has finite measure, because $I_j\ne\emptyset$ and
$G_j\subseteq E_i$ if $i\in I_j$.   Moreover, the $G_j$ are disjoint,
for if $i\in I_j\setminus I_k$ then $G_j\subseteq E_i$ and $G_k\cap
E_i=\emptyset$.   Finally, if $k\le n$ and $x\in E_k$, there is a
$j\le
m$ such that $I_j=\{i:i\le n,\,x\in E_i\}$, and in this case $x\in
G_j\subseteq E_k$;  thus  $E_k$ is the union of those $G_j$ which it
includes.

\medskip

{\bf (b)} Express $f$ as $\sum_{i=0}^na_i\chi E_i$ where
$E_0,\ldots,E_n$ are measurable sets of finite measure and
$a_0,\ldots,a_n$ are real numbers.   Let $G_0,\ldots,G_m$ be disjoint
measurable sets of finite measure such that every $E_i$ is expressible
as a union of appropriate $G_j$.   Set $c_{ij}=1$ if $G_j\subseteq
E_i$,
$0$ otherwise, so that, because the $G_j$ are disjoint, $\chi
E_i=\sum_{j=0}^mc_{ij}\chi G_j$ for each $i$.      Then

\Centerline{$f=\sum_{i=0}^na_i\chi E_i
=\sum_{i=0}^n\sum_{j=0}^ma_ic_{ij}\chi G_j
=\sum_{j=0}^mb_j\chi G_j$,}

\noindent setting $b_j=\sum_{i=0}^na_ic_{ij}$ for each $j\le m$.

\medskip

{\bf (c)} Set $f=\sum_{i=0}^na_i\chi E_i$, and take $G_j$, $c_{ij}$,
$b_j$ as in  (b).   Then $b_j\mu G_j\ge 0$ for every $j$.   \Prf\ If
$G_j=\emptyset$, this is trivial.  Otherwise, let $x\in G_j$;  then

\Centerline{$0\le f(x)=\sum_{i=0}^nb_i\chi G_i(x)=b_j\chi
G_j(x)=b_j$,}

\noindent so again $b_j\mu G_j\ge 0$.\ \Qed\    Next, because the
$G_j$
are disjoint,

\Centerline{$\mu E_i=\sum_{j=0}^mc_{ij}\mu G_j$}

\noindent for each $i$, so

\Centerline{$\sum_{i=0}^na_i\mu E_i
=\sum_{i=0}^n\sum_{j=0}^ma_ic_{ij}\mu G_j
=\sum_{j=0}^mb_j\mu G_j
\ge 0$,}

\noindent as required.
}%end of proof of 122C

\leader{122D}{Corollary} Let $(X,\Sigma,\mu)$ be a measure space.   If

\Centerline{$\sum_{i=0}^m a_i\chi E_i
=\sum_{j=0}^n b_j\chi F_j$,}

\noindent where all the $E_i$ and $F_j$ are
measurable sets of finite measure and the $ a_i$, $ b_j$ are real
numbers, then

\Centerline{$\sum_{i=0}^m a_i\mu E_i=\sum_{j=0}^n b_j\mu F_j$.}

\proof{ Apply 122Cc to
$\sum_{i=0}^m a_i\chi E_i+\sum_{j=0}^n(-b_j)\chi F_j$ to see that
$\sum_{i=0}^ma_i\mu E_i-\sum_{j=0}^nb_j\mu F_j\ge 0$;   now reverse
the
roles of the two sums to get the opposite inequality.
}%end of proof of 122D


\leader{122E}{Definition} Let $(X,\Sigma,\mu)$ be a measure space.
Then we
may define the {\bf integral} $\int f$ of $f$, for simple functions
$f:X\to\Bbb R$, by saying that
$\int f=\sum_{i=0}^m a_i\mu E_i$ whenever $f=\sum_{i=0}^m a_i\chi
E_i$ and every $E_i$ is a measurable set of finite measure\cmmnt{;
122D promises us that it won't matter which representation of $f$ we pick
on for the calculation}.

\leader{122F}{Proposition} Let $(X,\Sigma,\mu)$ be a measure space.

(a) If $f$, $g:X\to\Bbb R$ are simple functions, then $f+g$ is a
simple
function and $\int f+g=\int f+\int g$.

(b) If $f$ is a simple function and $c\in\Bbb R$, then $cf$ is a
simple
function and $\int cf=c\int f$.

(c)  If $f$, $g$ are simple functions and $f(x)\le g(x)$ for every
$x\in
X$, then $\int f\le \int g$.

\proof{ (a) and (b) are immediate from the formula given
for $\int f$ in 122E.   As for (c), observe that $g-f$ is a
non-negative
simple function, so that $\int g-f\ge 0$, by 122Cc;  but this means
that
$\int g-\int f\ge 0$.
}%end of proof of 122F


\vleader{72pt}{122G}{Lemma} Let $(X,\Sigma,\mu)$ be a measure space.
If $\langle f_n\rangle_{n\in\Bbb N}$ is a sequence of simple functions
which is non-decreasing (in the sense that $f_n(x)\le f_{n+1}(x)$ for
every $n\in\Bbb N$, $x\in X$) and $f$ is a simple function such that
$f(x)\le\sup_{n\in\Bbb N}f_n(x)$ for almost every $x\in X$ (allowing
$\sup_{n\in\Bbb N}f_n(x)=\infty$ in this formula), then
$\int f\le\sup_{n\in\Bbb N}\int f_n$.

\proof{  Note that $f-f_0$ is a simple function, so
$H=\{x:(f-f_0)(x)\ne 0\}$ is a finite union of sets of finite measure,
and $\mu H<\infty$;  also $f-f_0$ is bounded, so there is an $M\ge 0$
such that $(f-f_0)(x)\le M$ for every $x\in X$.

Let $\epsilon>0$.   For each $n\in\Bbb N$, set
$H_n=\{x:(f-f_n)(x)\ge\epsilon\}$.   Then each $H_n$ is measurable (by
121E), and $\langle H_n\rangle_{n\in\Bbb N}$ is a non-increasing
sequence of sets with intersection

\Centerline{$\bigcap_{n\in\Bbb N}H_n
=\{x:f(x)\ge\epsilon+\sup_{n\in\Bbb N}f_n(x)\}
\subseteq\{x:f(x)>\sup_{n\in\Bbb N}f_n(x)\}$.}

\noindent Because $f(x)\le\sup_{n\in\Bbb N}f_n(x)$ for almost every
$x$,
$\{x:f(x)>\sup_{n\in\Bbb N}f_n(x)\}$ and $\bigcap_{n\in\Bbb N}H_n$ are
negligible.   Also $\mu H_0<\infty$, because
$H_0\subseteq H$.
Consequently

\Centerline{$\lim_{n\to\infty}\mu H_n=\mu(\bigcap_{n\in\Bbb N}H_n)=0$}

\noindent (112Cf).   Let $n$ be so large that $\mu H_n\le\epsilon$.

Consider the simple function $g=f_n+\epsilon\chi H+M\chi H_n$.   Then
$f\le g$, so

\Centerline{$\int f\le\int g=\int f_n+\epsilon\mu H+M\mu H_n
\le\int f_n+\epsilon(M+\mu H)$.}

\noindent As $\epsilon$ is arbitrary,
$\int f\le\sup_{n\in\Bbb N}\int f_n$.
}%end of proof of 122G


\leader{122H}{Definition} Let $(X,\Sigma,\mu)$ be a measure space.
For the rest of this section, I will write $U$ for the set of
functions $f$ such that

\quad(i) the domain of $f$ is a conegligible subset of $X$ and
$f(x)\in\coint{0,\infty}$ for each $x\in\dom f$,

\quad(ii) there is a non-decreasing sequence $\sequencen{f_n}$ of
non-negative
simple functions such that $\sup_{n\in\Bbb N}\int f_n<\infty$ and
$\lim_{n\to\infty}f_n(x)\penalty-100=f(x)$ for almost
every $x\in X$.

\leader{122I}{Lemma} If $f$ and $\langle f_n\rangle_{n\in\Bbb N}$ are
as
in 122H, then

\Centerline{$\sup_{n\in\Bbb N}\int f_n
=\sup\{\int g:g$ is a simple function and $g\leae f\}$.}

\proof{ Of course

\Centerline{$\sup_{n\in\Bbb N}\int f_n
\le \sup\{\int g:g$ is a simple function and $g\leae f\}$}

\noindent because $f_n\leae f$  for each $n$.   On the other hand, if
$g$ is a simple function and $g\leae f$, then
$g(x)\le\sup_{n\in\Bbb N}f_n(x)$ for almost every $x$, so
$\int g\le\sup_{n\in\Bbb N}\int f_n$ by 122G.   Thus

\Centerline{$\sup_{n\in\Bbb N}\int f_n
\ge \sup\{\int g:g$ is a simple function and $g\leae f\}$,}

\noindent as required.
}%end of proof of 122I


\leader{122J}{Lemma} Let $(X,\Sigma,\mu)$ be a measure
space\cmmnt{, and define $U$ as in 122H}.

(a) If $f$ is a function defined on a conegligible subset of $X$ and
taking values in $\coint{0,\infty}$, then $f\in U$ iff
there is a conegligible measurable set $E\subseteq\dom f$ such that

\inset{($\alpha$)
$f\restr E$ is measurable,

($\beta$) for every $\epsilon>0$,
$\mu\{x:x\in E,\,f(x)\ge\epsilon\}<\infty$,

($\gamma$)
$\sup\{\int g:g$ is a simple function, $g\leae f\}<\infty$.}

(b) Suppose that $f\in U$ and that $h$ is a function defined on a
conegligible subset of $X$ and taking values in $\coint{0,\infty}$.
Suppose that $h\leae f$ and there is a conegligible
$F\subseteq X$ such that $h\restr F$ is measurable.   Then $h\in U$.

\proof{{\bf (a)(i)} Suppose that $f\in U$.   Then there is a
non-decreasing sequence $\langle f_n\rangle_{n\in\Bbb N}$ of
non-negative
simple functions such that $f\eae\lim_{n\to\infty}f_n$ and
$\sup_{n\in\Bbb N}\int f_n=c<\infty$.   The set
$\{x:f(x)=\lim_{n\to\infty}f_n(x)\}$ is conegligible, so includes a
measurable conegligible set $E$ say.   Now
$f\restr E=(\lim_{n\to\infty}f_n)\restr E$ is measurable, by 121Fa
and 121Eh; thus ($\alpha$) is satisfied.   Next, given $\epsilon>0$,
set $H_n=\{x:x\in E,\,f_n(x)\ge{1\over 2}\epsilon\}$;  then
$f_n\ge{1\over 2}\epsilon\chi H_n$, so

\Centerline{$\Bover12\epsilon\mu H_n
=\int\bover12\epsilon\chi H_n\le\int f_n\le c$,}

\noindent for each $n$.   Now $\langle H_n\rangle_{n\in\Bbb N}$ is
non-decreasing, so

\Centerline{$\mu(\bigcup_{n\in\Bbb N}H_n)
=\sup_{n\in\Bbb N}\mu H_n\le 2c/\epsilon$,}

\noindent by 112Ce.   Accordingly

\Centerline{$\mu\{x:x\in E,\,f(x)\ge\epsilon\}
\le\mu(\bigcup_{n\in\Bbb N}H_n)\le 2c/\epsilon<\infty$.}

\noindent As $\epsilon$ is arbitrary, ($\beta$) is satisfied.
Finally, ($\gamma$) is satisfied by 122I.

\medskip

\quad{\bf (ii)} Now suppose that the conditions ($\alpha$)-($\gamma$)
are satisfied.   Take an appropriate conegligible $E\in\Sigma$, and
for
each $n\in\Bbb N$ define $f_n:X\to\Bbb R$ by setting

$$\eqalign{f_n(x)&=2^{-n}k\text{ if }x\in E,\, 0\le k<4^n,\,
2^{-n}k\le f(x)<2^{-n}(k+1),\cr
&=0\text{ if }x\in X\setminus E,\cr
&=2^n\text{ if }x\in E\text{ and }f(x)\ge 2^n.\cr}$$

\noindent Then $f_n$ is a non-negative simple function, being
expressible as

\Centerline{$f_n
=\sum_{k=1}^{4^n}2^{-n}\chi\{x:x\in E,\,f(x)\ge 2^{-n}k\}$;}

\noindent all the sets $\{x:x\in E,\,f(x)\ge 2^{-n}k\}$ being
measurable
(because $f\restr E$ is measurable) and of finite measure, by
($\beta$).
Also it is easy to see that $\langle f_n\rangle_{n\in\Bbb N}$ is a
non-decreasing sequence which converges to $f$ at every point of $E$,
so
that $f\eae\lim_{n\to\infty}f_n$.   Finally,

\Centerline{$\lim_{n\to\infty}\int f_n
=\sup_{n\in\Bbb N}\int f_n\le\sup\{\int g:g\le f\text{ is simple}\}
<\infty$,}

\noindent by ($\gamma$).   So $f\in U$.

\medskip

{\bf (b)} Let $E$ be a set as in (a).   The sets $E$, $F$ and
$\{x:h(x)\le f(x)\}$ are all conegligible, so there is a conegligible
measurable set $E'$ included in their intersection.   Now
$E'\subseteq\dom h$, $h\restr E'$ is measurable,

\Centerline{$\mu\{x:x\in E',\,h(x)\ge\epsilon\}
\le\mu\{x:x\in E,\,f(x)\ge\epsilon\}<\infty$}

\noindent for
every $\epsilon>0$, and

\Centerline{$\sup\{\int g:g\text{ is simple},\, g\leae h\}
\le\sup\{\int g:g\text{ is simple},\,g\leae f\}<\infty$.}

\noindent So $h\in U$.
}%end of proof of 122J

\leader{122K}{Definition} Let $(X,\Sigma,\mu)$ be a measure
space\cmmnt{, and define $U$ as in 122H}.   For $f\in U$, set

\Centerline{$\int f
=\sup\{\int g:g$ is a simple function and $g\leae f\}$.}

\cmmnt{\noindent By 122I, we see that
$\int f=\lim_{n\to\infty}\int f_n$ whenever $\sequencen{f_n}$ is a
non-decreasing sequence
of simple functions converging to $f$ almost everywhere;  in
particular,
if $f\in U$ is itself a simple function, then $\int f$, as defined
here,
agrees with the original definition of $\int f$ in 122E, since we may
take $f_n=f$ for every $n$.}



\leader{122L}{Lemma} Let $(X,\Sigma,\mu)$ be a measure space.

(a) If $f$, $g\in U$ then $f+g\in U$ and $\int f+g=\int f+\int g$.

(b) If $f\in U$ and $c\ge 0$ then $cf\in U$ and $\int cf=c\int f$.

(c) If $f$, $g\in U$ and $f\leae g$ then $\int f\le \int g$.

(d) If $f\in U$ and $g$ is a function with domain a conegligible
subset
of $X$, taking values in $\coint{0,\infty}$, and equal to $f$ almost
everywhere, then $g\in U$ and $\int g=\int f$.

(e) If $f_1$, $g_1$, $f_2$, $g_2\in U$ and $f_1-f_2=g_1-g_2$, then
$\int
f_1-\int f_2=\int g_1-\int g_2$.

\proof{{\bf (a)} We know that there are non-decreasing sequences
$\langle f_n\rangle_{n\in\Bbb N}$, $\langle g_n\rangle_{n\in\Bbb N}$
of
non-negative simple functions such that $f\eae\lim_{n\to\infty}f_n$,
$g\eae\lim_{n\to\infty}g_n$, $\sup_{n\in\Bbb N}\int f_n<\infty$ and
$\sup_{n\in\Bbb N}\int g_n<\infty$.   Now $\langle
f_n+g_n\rangle_{n\in\Bbb N}$ is a non-decreasing sequence of simple
functions converging to $f+g$ a.e., and

\Centerline{$\sup_{n\in\Bbb N}\int f_n+g_n
=\lim_{n\to\infty}\int f_n+g_n
=\lim_{n\to\infty}\int f_n+\lim_{n\to\infty}\int g_n=\int f+\int g$.}

\noindent   Accordingly $f+g\in U$, and also, as remarked in 122K,

\Centerline{$\int f+g=\lim_{n\to\infty}\int f_n+g_n=\int f+\int g$.}

\medskip

{\bf (b)} We know that there is a non-decreasing sequence
$\langle f_n\rangle_{n\in\Bbb N}$ of
non-negative simple functions such that $f\eae\lim_{n\to\infty}f_n$
and $\sup_{n\in\Bbb N}\int f_n<\infty$.   Now
$\langle cf_n\rangle_{n\in\Bbb N}$ is a non-decreasing sequence of
simple functions converging to $cf$ a.e., and

\Centerline{$\sup_{n\in\Bbb N}\int cf_n
=\lim_{n\to\infty}\int cf_n=c\lim_{n\to\infty}\int f_n=c\int f$.}

\noindent Accordingly $cf\in U$, and also, as remarked in 122K,

\Centerline{$\int cf=\lim_{n\to\infty}\int cf_n=c\int f$.}

\medskip

{\bf (c)} This is obvious from 122K.

\medskip

{\bf (d)} If $\langle f_n\rangle_{n\in\Bbb N}$ is a non-decreasing
sequence of simple functions converging to $f$ a.e., then it also
converges to $g$ a.e.;  so $g\in U$ and

\Centerline{$\int g=\lim_{n\to\infty}\int f_n=\int f$.}

\medskip

{\bf (e)} By (a), $f_1+g_2$ and $f_2+g_1$ both belong to $U$.   Also,
they are equal at any point at which all four functions are defined,
which is almost everywhere.   So

\Centerline{$\int f_1+\int g_2=\int f_1+g_2
=\int f_2+g_1=\int f_2+\int g_1$,}

\noindent using (a) and (d).   Shifting $\int g_2$ and $\int f_2$
across the equation, we have the result.
}%end of proof of 122L


\leader{122M}{Definition} Let $(X,\Sigma,\mu)$ be a measure space.
\cmmnt{Define $U$ as in 122H.}   A real-valued function $f$ is {\bf
integrable}, or {\bf integrable over
$X$}, or {\bf $\mu$-integrable over $X$}, if it is expressible as
$f_1-f_2$ with $f_1$, $f_2\in U$, and in this case its {\bf integral}
is

\Centerline{$\int f=\int f_1-\int f_2$.}

\cmmnt{
\leader{122N}{Remarks (a)} We see from 122Le that the
integral
$\int f$ is uniquely defined  by the
formula in 122M.   Secondly, if $f\in U$, then $f=f-\tbf{0}$ is
integrable, and the
integral here agrees with that defined in 122K.   Finally, if $f$ is a
simple function, then it can be expressed as $f_1-f_2$ where $f_1$,
$f_2$ are non-negative simple functions (if $f=\sum_{i=0}^na_i\chi
E_i$, where each $E_i$ is measurable and of finite measure, set

\Centerline{$f_1=\sum_{i=0}^na_i^+\chi E_i$,
\quad$f_2=\sum_{i=0}^na_i^-\chi E_i$,}

\noindent writing $a_i^+=\max(a_i,0)$, $a_i^-=\max(-a_i,0)$);  so that

\Centerline{$\int f=\int f_1-\int f_2=$$\sum_{i=0}^na_i\mu E_i$,}

\noindent and the definition of 122M is consistent with the definition
of 122E.


\header{122Nb}{\bf (b)} Alternative notations which I will use for
$\int
f$ are $\int_Xf$,
$\int fd\mu$, $\int f(x)\mu(dx)$, $\int f(x)dx$, $\int_Xf(x)\mu(dx)$,
etc., according to which aspects of the context seem due for emphasis.

When $\mu$ is Lebesgue measure on $\Bbb R$ or $\BbbR^r$ we say that
$\int f$ is the {\bf Lebesgue integral} of $f$, and that $f$ is {\bf
Lebesgue integrable} if this is defined.

\header{122Nc}{\bf (c)} Note that when I say, in 122M, that `$f$ can
be
expressed as $f_1-f_2$', I mean
to interpret $f_1-f_2$ according to the rules set out in 121E, so that
$\dom f$ must be $\dom(f_1-f_2)=\dom f_1\cap\dom f_2$, and is surely
conegligible.
}%end of comment

\leader{122O}{Theorem} Let $(X,\Sigma,\mu)$ be a measure space.

(a) If $f$ and $g$ are integrable over $X$
then $f+g$ is integrable and $\int f+g=\int f+\int g$.

(b) If $f$ is integrable over $X$ and $c\in\Bbb R$ then $cf$
is integrable and $\int cf=c\int f$.

(c) If $f$ is integrable over $X$ and $f\ge 0$ a.e.\ then
$\int f\ge 0$.

(d) If $f$ and $g$ are integrable over $X$ and $f\leae g$
then $\int f\le\int g$.

\proof{{\bf (a)} Express $f$ as $f_1-f_2$ and $g$ as $g_1-g_2$
where $f_1$, $f_2$, $g_1$ and $g_2$ belong to $U$, as defined in 122H.
Then $f+g=(f_1+g_1)-(f_2+g_2)$ is integrable because $U$ is closed
under addition (122La), and

\Centerline{$\int f+g=\int f_1+g_1-\int f_2+g_2
=\int f_1+\int g_1-\int f_2-\int g_2=\int f+\int g$.}

\medskip

{\bf (b)} Express $f$ as $f_1-f_2$ where $f_1$, $f_2$ belong to $U$.
If $c\ge 0$ then $cf=cf_1-cf_2$ is integrable because $U$ is closed
under multiplication by non-negative scalars (122Lb), and

\Centerline{$\int cf=\int cf_1-\int cf_2=c\int f_1-c\int f_2=c\int
f$.}

\noindent   If $c=-1$ then $-f=f_2-f_1$ is integrable and

\Centerline{$\int (-f)=\int f_2-\int f_1=-\int f$.}

\noindent    Putting
these together we get the result for $c<0$.

\medskip

{\bf (c)} Express $f$ as $f_1-f_2$ where $f_1$, $f_2\in U$.   Then
$f_2\leae f_1$, so $\int f_2\le\int f_1$ (122Lc), and $\int f\ge 0$.

\medskip

{\bf (d)} Apply (c) to $g-f$.
}%end of proof of 122O


\leader{122P}{Theorem} Let $(X,\Sigma,\mu)$ be a measure space and
$f$ a
real-valued function defined on a conegligible subset of $X$.   Then
the following are equiveridical:

\quad(i)  $f$ is integrable;

\quad(ii) $|f|\in U$\cmmnt{, as defined in 122H,} and there is a
conegligible set $E\subseteq X$ such that $f\restr E$ is measurable;

\quad(iii) there are a $g\in U$ and a
conegligible set $E\subseteq X$  such that $|f|\leae g$
and $f\restr E$ is measurable.

\proof{{\bf (i)$\Rightarrow$(iii)} Suppose that $f$ is
integrable.   Let $f_1$,
$f_2\in U$ be such that $f=f_1-f_2$.   Then there are conegligible
sets $E_1$, $E_2$ such that $f_1\restr E_1$ and $f_2\restr
E_2$ are measurable;  set $E=E_1\cap E_2$, so that $E$ also is a
conegligible set.   Now $f\restr E=f_1\restr E-f_2\restr E$
is measurable.   Next, $f_1+f_2\in U$ (122La) and $|f|(x)\le
f_1(x)+f_2(x)$ for every $x\in\dom f$, so we may take $g=f_1+f_2$.

\medskip

{\bf (iii)$\Rightarrow$(ii)} If $f\restr E$ is measurable, so is
$|f|\restr E=|f\restr E|$ (121Eg);  so if $g\in U$ and
$|f|\leae g$, then $|f|\in U$ by 122Jb.

\medskip

{\bf (ii)$\Rightarrow$(i)} Suppose that $f$ satisfies the conditions
of
(ii).   Set
$f^+={1\over 2}(|f|+f)$ and $f^-={1\over 2}(|f|-f)$.    Of course
$|f|\restr E$, $f^+\restr E$ and $f^-\restr E$ are all measurable.
Also $0\le f^+(x)\le |f|(x)$ and $0\le f^-(x)\le |f|(x)$ for every
$x\in\dom f$, while $|f|\in U$ by hypothesis, so $f^+$ and $f^-$
belong
to $U$ by 122Jb.  Finally, $f=f^+-f^-$, so $f$ is integrable.
}%end of proof of 122P

\leader{122Q}{Remark} The condition `there is a conegligible set $E$
such that $f\restr E$ is measurable' recurs so often that I think it
worth having a phrase for it;  I will call such functions {\bf
virtually
measurable}, or {\bf $\mu$-virtually measurable} if it seems necessary
to specify the measure.

\leader{122R}{Corollary} Let $(X,\Sigma,\mu)$ be a measure space.

(a) A non-negative real-valued function, defined on a subset of $X$,
is integrable iff it belongs to $U$\cmmnt{, as defined in 122H}.

(b) If $f$ is integrable over $X$ and $h$ is a real-valued function,
defined on a conegligible subset of $X$ and equal to $f$ almost
everywhere, then $h$ is integrable, with $\int h=\int f$.

(c) If $f$ is integrable over $X$, $f\ge 0$ a.e.\ and $\int f\le 0$,
then $f=0$ a.e.

(d) If $f$ and $g$ are integrable over $X$, $f\leae g$ and
$\int g\le\int f$, then $f\eae g$.

(e) If $f$ is integrable over $X$, so is $|f|$, and
$|\int f|\le\int|f|$.

\proof{{\bf (a)} If $f$ is integrable then $f=|f|\in U$, by
122P(ii).   If $f\in U$ then $f=f-\tbf{0}$ is integrable, by 122M.

\medskip

{\bf (b)} Let $E$, $F$ be conegligible sets such that $f\restr E$ is
measurable and $h\restr F=f\restr F$;  then $E\cap F$ is conegligible
and $h\restr(E\cap F)=(f\restr E)\restr F$ is measurable.   Next,
there
is a $g\in U$ such that $|f|\leae g$, and of course $|h|\leae g$.
So $h$ is integrable by 122P(iii).   By 122Od, applied to $f$ and $h$
and then to $h$ and $f$, $\int h=\int f$.

\medskip

{\bf (c)} \Quer\ Suppose, if possible, otherwise.   Let $E\subseteq X$
be a conegligible set such that $f\restr E$ is measurable (122P(ii)),
and $E'\subseteq E\cap\dom f$ a conegligible measurable set.   Then
$F=\{x:x\in E',\,f(x)>0\}$ must be non-negligible.   Set
$F_k=\{x:x\in
E',\,f(x)\ge 2^{-k}\}$ for each $k\in\Bbb N$, so that
$F=\bigcup_{k\in\Bbb N}F_k$ and there is a $k$ such that $\mu F_k>0$.
But consider $g=2^{-k}\chi F_k$.   Because $f\ge 0$ a.e.\ and
$f\ge 2^{-k}$ on $F_k$, $f\geae g$, so that

\Centerline{$ 0<2^{-k}\mu F_k=\int g\le\int f$,}

\noindent by 122Od;  which is impossible.  \Bang

\medskip

{\bf (d)} Apply (c) to $g-f$.

\medskip

{\bf (e)} By (i)$\Rightarrow$(ii) of 122P, $|f|$ is integrable.   Now
$f^+={1\over 2}(|f|+f)$ and $f^-={1\over 2}(|f|-f)$ are both
integrable
and non-negative, so have non-negative integrals, and

\Centerline{$|\int f|=|\int f^+-\int f^-|\le\int f^++\int
f^-=\int|f|$.}
}%end of proof of 122R

\exercises{
\leader{122X}{Basic exercises (a)} Let $(X,\Sigma,\mu)$ be a measure
space.   (i)
Show that if $f:X\to\Bbb R$ is simple so is $|f|$, setting
$|f|(x)=|f(x)|$ for
$x\in\dom f=X$.  (ii) Show that if $f$, $g:X\to\Bbb R$ are simple
functions so are $f\vee g$ and $f\wedge g$, as defined in 121Xb.
%122D

\sqheader 122Xb Let $(X,\Sigma,\mu)$ be a measure space and $f$
a real-valued function which is integrable over $X$.   Show that for
every $\epsilon>0$ there is a simple function $g:X\to\Bbb R$ such that
$\int|f-g|\le\epsilon$.   \Hint{consider non-negative $f$ first.}
%122O

\spheader 122Xc Let $(X,\Sigma,\mu)$ be a measure space, and
write $\eusm L^1$ for the set of all real-valued functions which are
integrable over $X$.   Let $\Phi\subseteq\eusm L^1$ be such that

\inset{(i) $\chi E\in\Phi$ whenever $E\in\Sigma$ and $\mu E<\infty$;

(ii) $f+g\in\Phi$ for all $f$, $g\in\Phi$;

(iii) $cf\in\Phi$ whenever $c\in\Bbb R$, $f\in\Phi$;

(iv) $f\in\Phi$ whenever $f\in\eusm L^1$ is such that there is a
non-decreasing sequence $\sequencen{f_n}$ in $\Phi$ for which
$\lim_{n\to\infty}f_n=f$ almost everywhere.}

\noindent Show that $\Phi=\eusm L^1$.
%122O

\sqheader 122Xd Let $\mu$ be counting measure on $\Bbb N$ (112Bd).
Show that a function $f:\Bbb N\to\Bbb R$ (that is, a sequence
$\sequencen{f(n)}$) is $\mu$-integrable iff it is absolutely summable,
and in this case

\Centerline{$\int fd\mu=\int_{\Bbb N}f(n)\mu(dn)
=\sum_{n=0}^{\infty}f(n)$.}
%122O

\sqheader 122Xe Let $(X,\Sigma,\mu)$ be a measure space and $f$, $g$
two
virtually measurable real-valued functions defined on subsets of $X$.
(i) Show that $f+g$, $f\times g$ and $f/g$, defined as in 121E, are
all
virtually measurable.   (ii) Show that if $h$ is a Borel
measurable real-valued function defined on any subset of $\Bbb R$,
then
the composition $hf$ is virtually measurable.
%122Q

\sqheader 122Xf Let $(X,\Sigma,\mu)$ be a measure space and
$\sequencen{f_n}$ a sequence of virtually measurable real-valued
functions defined on subsets of $X$.   Show that
$\lim_{n\to\infty}f_n$,
$\sup_{n\in\Bbb N}f_n$, $\inf_{n\in\Bbb N}f_n$,
$\limsup_{n\to\infty}f_n$
and $\liminf_{n\to\infty}f_n$, defined as in 121F, are virtually
measurable.
%122Q

\sqheader 122Xg Let $(X,\Sigma,\mu)$ be a measure space and $f$,
$g$ real-valued functions which are integrable over $X$.   Show that
$f\wedge g$ and $f\vee g$, as defined in 121Xb, are integrable.
%122R

\sqheader 122Xh Let $(X,\Sigma,\mu)$ be a measure space, $f$ a
real-valued function which is integrable over $X$, and $g$ a bounded
real-valued virtually measurable function defined on a conegligible
subset of $X$.    Show that $f\times g$, defined as in 121Ed, is
integrable.
%122R

%maybe better in 122Y?
\spheader 122Xi Let $X$ be a set, $\Sigma$ a $\sigma$-algebra of
subsets
of $X$, and $\mu_1$, $\mu_2$ two measures with domain $\Sigma$.   Set
$\mu E=\mu_1E+\mu_2E$ for $E\in\Sigma$.
Show that for any
real-valued function $f$ defined on a subset of $X$,
$\int fd\mu=\int fd\mu_1+\int fd\mu_2$ in the sense that if one side
is defined as a real
number so is the other, and they are then equal.   \Hint{($\alpha$)
Check that a subset of $X$ is $\mu$-conegligible iff it is
$\mu_i$-conegligible for
both $i$.  ($\beta$) Check the result for simple functions $f$.
($\gamma$) Now consider general non-negative $f$.}

\leader{122Y}{Further exercises (a)} Let $(X,\Sigma,\mu)$ be a
`complete' measure space, that is, one
in which all negligible sets are measurable (see, for instance,
113Xa). Show that if
$f$ is a virtually measurable real-valued function defined on a subset
of $X$, then $f$ is measurable.   Use this fact to find appropriate
simplifications of 122J and 122P for such spaces $(X,\Sigma,\mu)$.
%122P

\spheader 122Yb Write $\eusm L^1$ for the set of all Lebesgue
integrable real-valued functions on
$\Bbb R$.   Let $\Phi\subseteq\eusm L^1$ be such that

\inset{(i) $\chi I\in\Phi$ whenever $I$ is a bounded half-open
interval
in $\Bbb R$;

(ii) $f+g\in\Phi$ for all $f$, $g\in\Phi$;

(iii) $cf\in\Phi$ whenever $c\in\Bbb R$, $f\in\Phi$;

(iv) $f\in\Phi$ whenever $f\in\eusm L^1$ is such that there is a
non-decreasing sequence $\sequencen{f_n}$ in $\Phi$ for which
$\lim_{n\to\infty}f_n=f$ almost everywhere.}

\noindent Show that $\Phi=\eusm L^1$.   \Hint{show that ($\alpha$)
$\chi E\in\Phi$ whenever $E$ is expressible as the union of finitely
many half-open intervals ($\beta$) $\chi E\in\Phi$ whenever $E$ has
finite measure and is expressible as the union of a sequence of
half-open
intervals ($\gamma$) $\chi E\in\Phi$ whenever $E$ is measurable and
has
finite measure.}
%122O, 122Xc

\spheader 122Yc Let $X$ be any set, and let $\mu$ be counting
measure on $X$.   Let $f:X\to\Bbb R$ be a function;  set
$f^+(x)=\max(0,f(x))$, $f^-(x)=\max(0,-f(x))$ for $x\in X$.   Show
that the following are equiveridical:  (i) $\int fd\mu$ exists in $\Bbb R$,
and is equal to $s$;  (ii) for every $\epsilon>0$ there is a finite
$K\subseteq X$ such that $|s-\sum_{i\in I}f(i)|\le\epsilon$ whenever
$I\subseteq X$ is
a finite set including $K$ (iii) $\sum_{x\in X}f^+(x)$ and
$\sum_{x\in X}f^-(x)$, defined as in 112Bd, are finite, and
$s=\sum_{x\in X}f^+(x)-\sum_{x\in X}f^-(x)$.
%122Xd, 122O

\spheader 122Yd Let $(X,\Sigma,\mu)$ be a measure space.   Let
us say that a function $g:X\to\Bbb R$ is {\bf quasi-simple} if it is
expressible as $\sum_{i=0}^{\infty}a_i\chi G_i$, where
$\sequence{i}{G_i}$ is a partition of $X$ into measurable sets,
$\sequence{i}{a_i}$ is a sequence in $\Bbb R$, and
$\sum_{i=0}^{\infty}|a_i|\mu G_i<\infty$, counting $0\cdot\infty$ as
$0$, so that there can be $G_i$ of infinite measure provided that the
corresponding $a_i$ are zero.

\quad(i) Show that if $g$ and $h$ are quasi-simple functions so are
$g+h$, $|g|$ and $cg$, for any $c\in \Bbb R$.   \Hint{for $g+h$ you
will need 111F(b-ii) or its equivalent.}

\quad(ii) Show from first principles (I mean, without using anything
later than 122F in this chapter) that if
$g=\sum_{i=0}^{\infty}a_i\chi G_i$ and
$h=\sum_{i=0}^{\infty}b_i\chi H_i$ are quasi-simple
functions, and  $g\leae h$, then
$\sum_{i=0}^{\infty}a_i\mu G_i\le\sum_{i=0}^{\infty}b_i\mu H_i$.

\quad{(iii)} Hence show that we have a functional $I$ defined by
saying that $I(g)=\sum_{i=0}^{\infty}a_i\mu G_i$ whenever $g$ is a
quasi-simple
function represented as $\sum_{i=0}^{\infty}a_i\chi G_i$.

\quad{(iv)} Show that if $g$ and $h$ are quasi-simple functions and
$c\in\Bbb R$,
then $I(g+h)=I(g)+I(h)$ and $I(cg)=cI(g)$, and that $I(g)\le I(h)$ if
$g\leae h$.

\quad{(v)} Show that if $g$ is a quasi-simple function then $g$ is
integrable and $\int g=I(g)$.   (I do now allow you to use
122G-122R.)

\quad{(vi)} Show that a real-valued function $f$, defined on a
conegligible subset of $X$, is integrable iff for every $\epsilon>0$
there are quasi-simple functions $g$, $h$ such that $g\leae f\leae h$
and $I(h)-I(g)\le\epsilon$.
%122R

\spheader 122Ye Let $\mu$ be Lebesgue measure on $\Bbb R$.   Let
us say (for this exercise only) that a real-valued function $g$ with
$\dom
g\subseteq\Bbb R$ is `pseudo-simple' if it is expressible as
$\sum_{i=0}^{\infty}a_i\chi J_i$, where $\sequence{i}{J_i}$ is a
sequence
of bounded half-open intervals ({\it not} necessarily disjoint) and
$\sum_{i=0}^{\infty}|a_i|\mu J_i<\infty$.    (Interpret the infinite
sum
$\sum_{i=0}^{\infty}a_i\chi J_i$ as in 121F, so that

\Centerline{$\dom(\sum_{i=0}^{\infty}a_i\chi J_i)
=\{x:\lim_{n\to\infty}\sum_{i=0}^na_i(\chi J_i)(x)
  \text{ exists in }\Bbb R\}$.)}

\quad(i) Show that if $g$, $h$ are pseudo-simple functions so are
$g+h$
and $cg$, for any $c\in \Bbb R$.

\quad(ii) Show that if $g$ is a pseudo-simple function then $g$ is
integrable.

\quad(iii) Show that a real-valued function $f$, defined on a
conegligible subset of $\Bbb R$, is integrable iff for every
$\epsilon>0$
there are pseudo-simple functions $g$, $h$ such that $g\leae f\leae h$
and $\int h-g\,d\mu\le\epsilon$.   \Hint{Take $\Phi$ to be the set
of integrable functions with this property, and show that it satisfies
the
conditions of 122Yb.}
%122Yd, 122R

\spheader 122Yf Repeat 122Yb and 122Ye for Lebesgue measure on
$\BbbR^r$,
where $r>1$.
%122Yd, 122Yb, 122Ye, 122R

\spheader 122Yg Let $(X,\Sigma,\mu)$ be a measure space, 
and assume that there is at least one partition of $X$ into infinitely
many non-empty measurable sets.   Let $f:X\to\Bbb R$ be a function, 
and $a\in\Bbb R$.   Show that the following are equiveridical:

\quad (i) $f$ is integrable, with $\int f=a$;

\quad (ii) for every $\epsilon>0$ there is a partition
$\sequencen{E_n}$
of $X$ into non-empty measurable sets such that

\Centerline{$\sum_{n=0}^{\infty}|f(t_n)|\mu E_n<\infty$,
\quad$|a-\sum_{n=0}^{\infty}f(t_n)\mu E_n|\le\epsilon$}

\noindent whenever $\sequencen{t_n}$ is a sequence such that
$t_n\in E_n\cap\dom f$ for every $n$.   (As usual, take
$0\cdot\infty=0$ in these formulae.)  \Hint{use 122Yd.}
%122Yd, 122R

\spheader 122Yh Find a re-formulation of (g) which covers the case
of measure spaces which can {\it not} be partitioned into sequences of
non-empty measurable sets.
%122Yg, 122Yd, 122R

\spheader 122Yi Let $X$ be a set, $\Sigma$ a $\sigma$-algebra of
subsets
of $X$, and $\sequencen{\mu_n}$ a sequence of measures with domain
$\Sigma$.   Set $\mu E=\sum_{n=0}^{\infty}\mu_nE$ for $E\in\Sigma$.
(i) Show that $\mu$ is a measure.
(ii) Show that for any real-valued function $f$ defined on a subset of $X$,
$f$ is $\mu$-integrable iff it is $\mu_n$-integrable for every $n$ and
$\sum_{n=0}^{\infty}\int|f|d\mu_n$ is finite, and that then
$\int fd\mu=\sum_{n=0}^{\infty}\int fd\mu_n$.
%122Xi

\spheader 122Yj Let $X$ be a set, $\Sigma$ a $\sigma$-algebra of
subsets
of $X$, and $\langle\mu_i\rangle_{i\in I}$ a family of measures with
domain $\Sigma$.
Set $\mu E=\sum_{i\in I}\mu_iE$ for $E\in\Sigma$.
(i) Show that $\mu$ is a measure.
(ii) Show that for any $\Sigma$-measurable function $f:X\to\Bbb R$,
$f$ is $\mu$-integrable iff it is $\mu_i$-integrable for every $i$ and
$\sum_{i\in I}\int|f|d\mu_i$ is finite.
%122Yi, 122Xi
}%end of exercises

\endnotes{
\Notesheader{122} Just as in \S121, some extra
technical problems are caused by my insistence on trying to integrate
(i) functions which are not defined on the whole of the measure space
under consideration (ii) functions which are not, strictly speaking,
measurable, but are only measurable on some conegligible set.   There
is
nothing in the present section to justify either of these
elaborations.
In the next section, however, we shall be looking at the limits of
sequences of functions, and these limits need not be defined at every
point;  and the examples in which the limits are not everywhere
defined are not in any sense pathological, but are central to the most
important applications of the theory.

The question of integrating not-quite-measurable functions is more
disputable.   I will discuss this point further after formally
introducing
`complete' measure spaces in Chapter 21.   For the moment, I will
say only that I think it is worth taking the trouble to have a
formalisation which integrates as many functions as is reasonably
possible;  the original point of the Lebesgue integral being, in part,
that it enables us to integrate more functions than its predecessors.

The definition of `integration' here proceeds in three
distinguishable stages:  (i) integration of simple functions
(122A-122G);  (ii) integration of non-negative functions (122H-122L);
(iii) integration of general real-valued functions (122M-122R).   I
have
taken each stage slowly, passing to non-negative integrable functions
only when I have a full set of the requisite lemmas on simple
functions,
for instance.   There are, of course, innumerable alternative routes;
see, for instance, 122Yd, which offers a definition using two steps
rather than three.   I
prefer the longer, gentler climb partly because (to my eye) it gives a
clearer view of the ideas and partly because it corresponds to an
almost
canonical method of proving properties of integrable functions:  we
prove them first for simple functions, then for non-negative
integrable
functions, then for general integrable functions.   (The hint I give
for
122Yb conforms to this philosophy.   See also 122Xc;  but I
do not give this as a formally expressed theorem, because the exact
order of proof varies from case to case, and I think it is best
remembered as a method of attack rather than as a specific result to
apply.)

You have a right to feel that this section has been singularly
abstract,
and gives very little idea of which of your favourite functions are
likely to be integrable, let alone what the integrals are.   I hope
that
Chapter 13 will provide some help in this direction, though I have to
say that for really useful methods for calculating integrals we must
wait for Chapters 22, 25 and 26 in the next volume.    If you want to
know the true centre of the arguments of this section, I would myself
locate it in 122G, 122H and 122K.   The point is that the ideas of
122A-122F apply to a much wider class of structures $(X,\Sigma,\mu)$,
because they involve only operations on finitely many members of
$\Sigma$ at a time;  there is no mention of sequences of sets.   The
key
that makes all the rest possible is 122G, which is founded on 112Cf.
And after 122G-122K, the rest of the section, although by no means
elementary, really is no more than a careful series of checks to
ensure
that the functional defined in 122K behaves as we expect it to.

Many of the results of this section (including the key one, 122G) will
be superseded by stronger results in the following section.   But I
should remark on Lemma 122Ja, which will periodically recur as a most
useful criterion for integrability of non-negative functions (see
122Ra).

There is another point about the standard integral as defined here.   It
is an `absolute' integral, meaning that if $f$ is integrable so is
$|f|$ (122P).   This means that although the Lebesgue integral extends
the `proper' Riemann integral (see 134K below), there are functions
with
finite `improper' Riemann integrals which are not Lebesgue integrable;
a typical example is $f(x)=\bover{\sin x}x$, where
$\lim_{a\to\infty}\int_0^af$ exists in $\Bbb R$, while
$\lim_{a\to\infty}\int_0^a|f|=\infty$, so that $f$ is not integrable,
in the sense defined here, over the whole interval $\ooint{0,\infty}$.
(For full proofs of these assertions, see 283D and 282Xm in Volume
2.)   If you have encountered the theory of `absolutely' and
`conditionally'
summable series, you will be aware that the latter can exhibit very
confusing behaviour, and will appreciate that restricting the notion of
`integrable' to mean `absolutely integrable' is a great convenience.

Indeed, it is more than just a convenience;  it is necessary to make the
definition work at the level of abstraction used in this chapter.
Consider the example of counting measure $\mu$ on $\Bbb N$ (112Bd,
122Xd).   The structure $(\Bbb N,\Cal P\Bbb N,\mu)$ is invariant under
permutations;
that is, $\mu(\pi[A])=\mu A$ for every $A\subseteq\Bbb N$ and every
permutation $\pi:\Bbb N\to\Bbb N$.   Consequently, {\it any} definition
of integration which depends only on the structure
$(\Bbb N,\Cal P\Bbb N,\mu)$ must also be invariant under
permutations, that is,

\Centerline{$\int f(\pi(n))\mu(dn)=\int f(n)\mu(dn)$}

\noindent for any integrable function $f$ and any permutation $\pi$.
But of course (as I hope you have been told) a series
$\sequencen{f(n)}$ such that
$\sum_{n=0}^{\infty}f(\pi(n))=\sum_{n=0}^{\infty}f(n)\in\Bbb R$ for
any permutation $\pi$ must be absolutely summable.   Thus if we are to
define an integral on an abstract measure space $(X,\Sigma,\mu)$ in
terms depending only on $\Sigma$ and $\mu$, we are nearly inevitably
forced to define an absolute integral.

Naturally there are important contexts in which this restriction is an
embarrassment, and in which some kind of `improper' integral seems
appropriate.   A typical one is the theory of Fourier transforms, in
which we find ourselves looking at $\lim_{a\to\infty}\int_{-a}^af$ in
place of $\int_{-\infty}^{\infty}f$ (see \S283).   A vast number of
more or less abstract forms of improper integral have been proposed;
 many are interesting and some are important;  but none rivals the
`standard' integral as described in this chapter.   (For an attempt at a
systematic examination of a particular class of such improper
integrals, see Chapter 48 in Volume 4.)

Much less work has been done on the integration of non-measurable
functions -- to speak more exactly, of functions which are not equal
almost everywhere to a measurable integrable function.   I am sure that
this is simply because there are too few important problems to show us
which way to turn.   In 134C below I mention the question of whether
there is {\it any} non-measurable real-valued function on $\Bbb R$.
The standard answer is `yes', but no such function can possibly arise
as a result of any ordinary construction.    Consequently the
majority of
questions concerning non-measurable functions appear in very special
contexts, and so far I have seen none which gives any useful hint of
what a generally appropriate extension of the notion of
`integrability' might be.
}%end of notes

\discrpage

\frfilename{mt123.tex}
\versiondate{18.11.04}
\copyrightdate{1994}

\def\chaptername{Integration}
\def\sectionname{The convergence theorems}

\newsection{123}

The great labour we have gone through so far has not yet been justified
by any theorems powerful enough to make it worth while.   We come now to
the heart of the modern theory of integration,  the `convergence
theorems', describing conditions under which we can integrate the
limit of a sequence of integrable functions.

\leader{123A}{B.Levi's theorem} Let $(X,\Sigma,\mu)$ be a measure space
and $\langle f_n\rangle_{n\in\Bbb N}$ a sequence of real-valued
functions, all integrable over $X$, such that (i) $f_n\leae f_{n+1}$
for every $n\in\Bbb N$ (ii)
$\sup_{n\in\Bbb N}\int f_n<\infty$.   Then $f=\lim_{n\to\infty}f_n$ is
integrable, and $\int f=\lim_{n\to\infty}\int f_n$.

\medskip

\noindent{\bf Remarks}\cmmnt{ I ought to repeat at once the
conventions I am
following here.   Each of the functions $f_n$ is taken to be defined on
a conegligible set $\dom f_n\subseteq X$, as in 122Nc, and the limit
function $f$ is taken to have domain

\Centerline{$\{x:x\in\bigcup_{n\in\Bbb N}\bigcap_{m\ge n}\dom f_m$,
$\lim_{n\to\infty}f_n(x)$ is defined in $\Bbb R\}$,}

\noindent as in 121Fa.}  \dvro{The}{You would miss no important idea if
you supposed
that every $f_n$ was defined everywhere on $X$;  but the} statement
`$f$ is integrable' includes the assertion `$f$ is defined, as a
real number, almost everywhere'\cmmnt{, and
this is an essential part of the theorem}.

\proof{{\bf (a)} Let us first deal with the case in which
$f_0=0$ a.e.
Write $c=\sup_{n\in\Bbb N}\int f_n=\lim_{n\to\infty}\int f_n$ (noting
that, by 122Od, $\sequencen{\int f_n}$ is a
non-decreasing sequence).

\medskip

\quad{\bf (i)} All the sets $\dom f_n$, $\{x:f_0(x)=0\}$, $\{x:f_n(x)
\le f_{n+1}(x)\}$ are conegligible, so their intersection $F$ also is.
For each $n\in\Bbb N$ there is a conegligible set $E_n$ such that
$f_n\restr E_n$ is measurable (122P);  let $E^*$ be a measurable
conegligible
set included in the conegligible set $F\cap\bigcap_{n\in\Bbb N}E_n$.

\medskip

\quad{\bf (ii)} For $a>0$ and $n\in\Bbb N$ set
$H_n(a)=\{x:x\in E^*$, $f_n(x)\ge a\}$;
then $H_n(a)$ is measurable because $f_n\restr E_n$ is
measurable and $E^*$ is a measurable subset of $E_n$.   Also $a\chi
H_n(a)
\le f_n$ everywhere on $E^*$, so

\Centerline{$a\mu H_n(a)=\int a\chi H_n(a)\le\int f_n\le c$.}

\noindent Because $f_n(x)\le f_{n+1}(x)$ for every $x\in E^*$,
$H_n(a)\subseteq H_{n+1}(a)$ for every $n\in\Bbb N$, and writing
$H(a)=\bigcup_{n\in\Bbb N}H_n(a)$, we have

\Centerline{$\mu H(a)=\lim_{n\to\infty}\mu H_n(a)\le\Bover{c}{a}$}

\noindent (112Ce).   In particular, $\mu H(a)<\infty$ for every $a$.
Furthermore,

\Centerline{$\mu(\bigcap_{k\ge 1}H(k))\le\inf_{k\ge 1}\mu H(k)
\le\inf_{k\ge 1}\Bover{c}{k}=0$.}

\noindent  Set $E=E^*\setminus\bigcap_{k\ge 1}H(k)$;  then $E$ is
conegligible.

\medskip

\quad{\bf (iii)} If $x\in E$, there is some $k$ such that
$x\notin H(k)$, that is, $x\notin\bigcup_{n\in\Bbb N}H_n(k)$, that is,
$f_n(x)<k$
for every $n$;  moreover, $\langle f_n(x)\rangle_{n\in\Bbb N}$ is a
non-decreasing sequence, so
$f(x)=\lim_{n\to\infty}f_n(x)=\sup_{n\in\Bbb N}f_n(x)$ is defined in
$\Bbb R$.   Thus the limit function $f$ is
defined almost everywhere.
Because every $f_n\restr E$ is measurable (121Eh), so is
$f\restr E=\lim_{n\to\infty}f_n\restr E$ (121Fa).
If $\epsilon>0$ then $\{x:x\in E,\,f(x)\ge\epsilon\}$ is included in
$H({1\over 2}\epsilon)$, so has finite measure.

\medskip

\quad{\bf (iv)}
Now suppose that $g$ is a simple function and that $g\leae f$.   As
in the proof of 122G, $H=\{x:g(x)\ne 0\}$ has finite measure, and
$g$ is bounded above by $M$ say.

Let $\epsilon>0$.   For each $n\in\Bbb N$ set
$G_n=\{x:x\in E,\,(g-f_n) (x)\ge\epsilon\}$.
Then each $G_n$ is measurable, and $\langle G_n \rangle_{n\in\Bbb N}$
is a non-increasing sequence with intersection

\Centerline{$\{x:x\in E,\,g(x)\ge\epsilon +\sup_{n\in\Bbb N}f_n(x)\}
\subseteq\{x:g(x)>f(x)\}$,}

\noindent which is negligible.
Also $\mu G_0<\infty$ because $G_0\subseteq H$.
Consequently $\lim_{n\to\infty}\mu G_n=0$ (112Cf).
Let $n$ be such that $\mu G_n\le\epsilon$.   Then, for any $x\in E$,

\Centerline{$g(x)\le f_n(x)+\epsilon\chi H(x)+M\chi G_n(x)$,}

\noindent so

\Centerline{$g\leae f_n+M\chi G_n+\epsilon\chi H$}

\noindent and

\Centerline{$\int g\le\int f_n+M\mu G_n+\epsilon\mu H
\le c+\epsilon(M+\mu H)$.\footnote{I am grateful to P.Wallace Thompson for
noticing a fault at this stage in previous editions.}}

\noindent As $\epsilon$ is arbitrary, $\int g\le c$.

\medskip

\quad{\bf (v)} Accordingly, $f\restr E$ (which is non-negative)
satisfies the conditions of Lemma 122Ja,
and is integrable.   Moreover, its integral is at
most $c$, by Definition 122K.   Because $f\eae f\restr E$, $f$ also is
integrable, with the same integral (122Rb).   On the other hand,
$f\geae f_n$ for
each $n$, so $\int f\ge\sup_{n\in\Bbb N}\int f_n=c$, by 122Od.

This completes the proof when $f_0=0$ a.e.

\medskip

{\bf (b)} For the general case, consider the sequence
$\langle f'_n\rangle_{n\in\Bbb N}=\langle f_n-f_0\rangle_{n\in\Bbb N}$.
By (a),
$f'=\lim_{n\to\infty}f'_n$ is integrable, and $\int f'=\lim_{n\to\infty}
\int f'_n$;  now $\lim_{n\to\infty}f_n\eae f'+f_0$, so is
integrable, with integral $\int f'+\int f_0=\lim_{n\to\infty}\int f_n$.

\medskip

\noindent{\bf Remark} You may have observed, without surprise, that the
argument of (a-iv) in the proof here repeats that used for the special
case 122G.
}%end of proof of 123A

\leader{123B}{Fatou's Lemma} Let $(X,\Sigma,\mu)$ be a measure space,
and $\langle f_n\rangle_{n\in\Bbb N}$ a sequence of real-valued
functions, all integrable over $X$.   Suppose that every $f_n$ is
non-negative a.e., and
that $\liminf_{n\to\infty}\int f_n<\infty$.   Then 
$\liminf_{n\to\infty}f_n$ is integrable, and
$\int\liminf_{n\to\infty}f_n\le\liminf_{n\to\infty}\int f_n$.

\cmmnt{\medskip

\noindent{\bf Remark} Once again, this theorem includes the assertion
that $\liminf_{n\to\infty}f_n(x)$ is defined in $\Bbb R$ for almost
every $x\in X$.}

\proof{ Set $c=\liminf_{n\to\infty}\int f_n$ and
$f=\liminf_{n\to\infty}f_n$.
For each $n\in\Bbb N$, let $E_n$ be a conegligible
set such that $f_n'=f_n\restr E_n$ is measurable and non-negative.   Set
$g_n=\inf_{m\ge n}f'_m$;  then each $g_n$ is measurable (121Fc),
non-negative and defined on the conegligible set $\bigcap_{m\ge n}E_m$,
and $g_n\leae f_n$;  by 122Re and 122Ra, $|f_n|$ belongs to $U$, as defined
in 122H, while $|g_n|\leae|f_n|$, so $g_n$ is integrable (122P) with
$\int g_n\le\inf_{m\ge n}\int f_m\le c$.
Next, $g_n(x)\le g_{n+1}(x)$ for every
$x\in\dom g_n$, so $\langle g_n\rangle_{n\in\Bbb N}$ satisfies the
conditions of B.Levi's theorem (123A), and $g=\lim_{n\to\infty}g_n$
is integrable, with $\int g=\lim_{n\to\infty}\int g_n\le c$.
Finally, because every $f'_n$ is equal to $f_n$ almost everywhere,
$g=\liminf_{n\to\infty}f'_n\eae f$, and $\int f$ exists, equal to
$\int g\le c$.
}%end of proof of 123B

\leader{123C}{Lebesgue's Dominated Convergence Theorem} Let
$(X,\Sigma,\mu)$
be a measure space and $\langle f_n\rangle_{n\in\Bbb N}$ a sequence of
real-valued functions, all integrable over $X$, such that
$f(x)=\lim_{n\to\infty}f_n(x)$
exists in $\Bbb R$ for almost every $x\in X$.   Suppose moreover that
there is an
integrable function $g$ such that $|f_n|\leae g$ for every $n$.
Then $f$ is integrable, and $\lim_{n\to\infty}\int f_n$ exists and is
equal to $\int f$.

\proof{ Consider $\tilde f_n=f_n+g$ for each $n\in\Bbb N$.
Then $0\le\tilde f_n\le 2g$ a.e.\ for each $n$, so
$\tilde c=\liminf_{n\to\infty}\int \tilde f_n$
exists in $\Bbb R$, and $\tilde f=\liminf_{n\to\infty}\tilde f_n$ is
integrable,
with $\int \tilde f\le \tilde c$, by Fatou's Lemma (123B).
But observe that $f\eae\tilde f-g$, since $f(x)=\tilde f(x)-g(x)$ at
least whenever $f(x)$ and $g(x)$ are both defined, so $f$
is integrable, with

\Centerline{$\int f=\int \tilde f-\int g
\le\liminf_{n\to\infty}\int \tilde f_n-\int g
=\liminf_{n\to\infty}\int f_n$.}

\noindent Similarly, considering $\langle -f_n\rangle_{n\in\Bbb N}$,

\Centerline{$\int(-f)\le\liminf_{n\to\infty}\int(-f_n)$,}

\noindent that is,

\Centerline{$\int f\ge\limsup_{n\to\infty}\int f_n$.}

\noindent So $\lim_{n\to\infty}\int f_n$ exists and is equal to
$\int f$.
}%end of proof of 123C

\cmmnt{\medskip

\noindent{\bf Remark} We have at last reached the point where the
technical problems associated with partially-defined functions
are reducing, or rather, are being covered efficiently by the
conventions I am using concerning the
interpretation of such formulae as `$\limsup$'.
}%end of comment

\leader{123D}{}\cmmnt{ To try to show the power of these theorems, I
give a result here which is one of the standard applications of the
theory.

\medskip

\noindent}{\bf Corollary} Let $(X,\Sigma,\mu)$ be a measure space and
$\ooint{a,b}$ a non-empty open interval in $\Bbb R$.   Let
$f:X\times\ooint{a,b}\to\Bbb R$ be a function such that

\inset{(i) the integral $F(t)=\int f(x,t)dx$ is defined for every
$t\in\ooint{a,b}$;}

\inset{(ii) the partial derivative $\pd{f}{t}$ of $f$ with respect to
the
second variable is defined everywhere in $X\times\ooint{a,b}$;}

\inset{(iii) there is an integrable function $g:X\to\coint{0,\infty}$
such that $|\pd{f}{t}(x,t)|\le g(x)$ for every $x\in X$ and
$t\in\ooint{a,b}$.}

\noindent Then the derivative $F'(t)$ and the integral
$\biggerint\pd{f}{t}(x,t)dx$ exist for every $t\in\ooint{a,b}$, 
and are equal.

\proof{{\bf (a)} Let $t$ be any point of $\ooint{a,b}$, and
$\sequencen{t_n}$ any sequence in $\ooint{a,b}\setminus\{t\}$ converging
to $t$.   Consider

\Centerline{$\Bover{F(t_n)-F(t)}{t_n-t}
=\int\Bover{f(x,t_n)-f(x,t)}{t_n-t}\mu(dx)$}

\noindent for each $n$.   (This step uses 122O.)   If we set

\Centerline{$f_n(x)=\Bover{f(x,t_n)-f(x,t)}{t_n-t}$,}

\noindent for $x\in X$, then we see from the Mean Value Theorem that
there is a $\tau$ (depending, of course, on both $n$ and $x$), lying between $t_n$ and $t$, such that $f_n(x)=\pd{f}{t}(x,\tau)$, so that
$|f_n(x)|\le g(x)$.   At the same time, $\lim_{n\to\infty}f_n(x)=\pd{f}{t}(x,t)$ for
every $x$.   So Lebesgue's Dominated Convergence Theorem tells us that
$\biggerint\pd{f}{t}(x,t)dx$ exists and is equal to

\Centerline{$\lim_{n\to\infty}\int f_n(x)dx
=\lim_{n\to\infty}\Bover{F(t_n)-F(t)}{t_n-t}$.}

\wheader{123D}{6}{2}{2}{12pt}

{\bf (b)} Because $\sequencen{t_n}$ is arbitrary,

\Centerline{$\lim_{s\to t}\Bover{F(s)-F(t)}{s-t}
=\int\Pd{f}{t}(x,t)dx$,}

\noindent as claimed.
}%end of proof of 123D

\cmmnt{\medskip

\noindent{\bf Remark} In the next volume
I offer a variation on this theorem, with
both hypotheses and conclusion weakened (252Ye).
}%end of comment

\exercises{
\leader{123X}{Basic exercises $\pmb{>}$(a)}
%\spheader 123Xa
Let $(X,\Sigma,\mu)$ be a measure space, and $\sequencen{f_n}$ a
sequence of real-valued functions, all integrable over $X$, such that
$\sum_{n=0}^{\infty}\int|f_n|$ is finite.   Show that
$f(x)=\sum_{n=0}^{\infty}f_n(x)$ is defined in $\Bbb R$ for almost every
$x\in X$, and that $\int f=\sum_{n=0}^{\infty}\int f_n$.
\Hint{consider first the case in which every $f_n$ is non-negative.}

\spheader 123Xb Let $(X,\Sigma,\mu)$ be a measure
space.   Suppose that $T$ is any subset of $\Bbb R$, and
$\langle f_t\rangle_{t\in T}$ a family of functions,
all integrable over $X$, such that, for any $t\in T$,

\Centerline{$f_t(x)=\lim_{s\in T,s\to t}f_s(x)$}

\noindent for almost every $x\in X$.    Suppose moreover that there is
an integrable function $g$ such that $|f_t|\leae g$ for every $t\in T$.
Show that $t\mapsto\int f_t:T\to\Bbb R$ is continuous.

\sqheader 123Xc Let $f$ be a real-valued function defined
everywhere
on $\coint{0,\infty}$, endowed with Lebesgue measure.   Its (real)
{\bf Laplace transform} is the function $F$ defined by

\Centerline{$F(s)=\int_0^{\infty}e^{-sx}f(x)dx$}

\noindent for all those real numbers $s$ for which the integral is
defined.

\quad(i) Show that if $s\in\dom F$ and $s'\ge s$ then
$s'\in\dom F$ (because $e^{-s'x}e^{sx}\le 1$ for all $x$).   (How do you
know that $x\mapsto e^{-s'x}e^{sx}$ is measurable?)

\quad(ii) Show that $F$ is differentiable on the interior of its domain.
\Hint{note that if $a_0\in\dom F$ and $a_0<a<b$ then there is some
$M$ such that
$xe^{-sx}|f(x)|\le Me^{-a_0x}|f(x)|$ whenever $x\in\coint{0,\infty}$,
$s\in[a,b]$.}

\quad(iii) Show that if $F$ is defined anywhere then
$\lim_{s\to\infty}F(s)=0$.   \Hint{use Lebesgue's Dominated
Convergence Theorem to show that $\lim_{n\to\infty}F(s_n)=0$ whenever
$\lim_{n\to\infty}s_n=\infty$.}

\quad(iv) Show that if $f$, $g$ have Laplace transforms $F$, $G$ then
the Laplace transform of $f+g$ is $F+G$, at least on $\dom F\cap \dom
G$.

\spheader 123Xd Let $(X,\Sigma,\mu)$ be a measure space
and $\langle f_n\rangle_{n\in\Bbb N}$ a sequence of
real-valued functions, all integrable over $X$, such that
there is an
integrable function $g$ such that $|f_n|\leae g$ for every $n$.
Show that $\limsup_{n\to\infty}f_n$ is integrable and that
$\int\limsup_{n\to\infty}f_n\ge\limsup_{n\to\infty}\int f_n$.
%123C

\leader{123Y}{Further exercises (a)}
%\spheader 123Ya also in \S235
Let $(X,\Sigma,\mu)$ be a measure
space, $Y$ any set and
$\phi:X\to Y$ any function;  let $\mu\phi^{-1}$ be the image measure on
$Y$ (112Xf).   Show that if $h:Y\to\Bbb R$ is 
$\mu\phi^{-1}$-integrable then $h\phi$ is $\mu$-integrable, and the
integrals are then equal.

\header{123Yb}{\bf (b)} Explain how to adapt 123Xc to the case in which
$f$ is undefined on a negligible subset of $\Bbb R$.

\header{123Yc}{\bf (c)} Let $(X,\Sigma,\mu)$ be a measure space and
$a<b$
in $\Bbb R$.   Let $f:X\times\ooint{a,b}\to\coint{0,\infty}$ be a
function
such that $\int f(x,t)dx$ is defined for every $t\in\ooint{a,b}$ and
$t\mapsto f(x,t)$ is continuous for every $x\in X$.   Suppose that
$c\in\ooint{a,b}$ is such that $\liminf_{t\to c}\int f(x,t)dx<\infty$.
Show that $\int\liminf_{t\to c}f(x,t)dx$ is defined and less than or
equal to $\liminf_{t\to c}\int f(x,t)dx$.

\header{123Yd}{\bf (d)} Show that there is a function
$f:\BbbR^2\to\{0,1\}$ such that (i) the Lebesgue integral
$\int f(x,t)dx$ is defined and equal to $1$ for every $t\ne 0$ (ii) the
function $x\mapsto\liminf_{t\to 0}f(x,t)$ is not
Lebesgue measurable.   ({\it
Remark\/}:  you will of course have to start your construction from a
non-measurable subset of $\Bbb R$;  see 134B for such a set.)

\spheader 123Ye Let $(Y,\Tau,\nu)$ be a measure space.  Let $X$ be a
set, $\Sigma$ a $\sigma$-algebra of subsets of $X$, and
$\langle\mu_y\rangle_{y\in Y}$ a family of measures on $X$ such that
$\mu_yX$ is finite for every $y$ and $\mu E=\int\mu_yE\,\nu(dy)$ is
defined for every $E\in\Sigma$.   (i) Show that
$\mu:\Sigma\to\coint{0,\infty}$ is
a measure.   (ii) Show that if $f:X\to\coint{0,\infty}$ is a
$\Sigma$-measurable function, then $f$ is
$\mu$-integrable iff it is $\mu_y$-integrable for almost every $y\in Y$
and $\int\bigl(\int fd\mu_y\bigr)\nu(dy)$ is defined, and that this is
then $\int fd\mu$.

\spheader 123Yf Let $(X,\Sigma,\mu)$ be a measure space, and
$\sequencen{f_n}$ a sequence of virtually measurable real-valued
functions all defined almost everywhere in $X$.   Suppose that
$\sum_{n=0}^{\infty}\int|f_n(x)-1|\mu(dx)<\infty$.   Show that
$\prod_{n=0}^{\infty}f_n(x)$ is defined in $\Bbb R$ for almost every
$x\in X$.
%123A

}%end of exercises

\endnotes{
\Notesheader{123} I hope that 123D and its special case 123Xc will help
you to believe that the theory here has useful applications.

All the theorems of this section can be thought of as `exchange of
limit' theorems, setting out conditions under which

$$\lim_{n\to\infty}\int f_n=\int\lim_{n\to\infty}f_n,$$

\noindent or

$$\Bover{\partial}{\partial t}{\int}f\,dx=\int\Bover{\partial
f}{\partial
t}dx.$$

\noindent Even for functions which are accessible to much more primitive
methods of integration (e.g., the Riemann integral), theorems of this
type can involve laborious validation of inequalities.   The power of
Lebesgue's integral is that it gives general theorems which cover a
reasonable proportion of the important cases which arise in practice.
(I have to admit, however, that nothing is more typical of applied
analysis than its need for special results which are related to, but not
derivable
from, the standard general theorems.)   For instance, in 123Xc, the fact
that the range of integration is the unbounded interval
$\coint{0,\infty}$
adds no difficulty.   Of course this is connected with the fact that we
consider only integrals of functions with integrable absolute values.

The limits used in 123A-123C are all limits of sequences;  it is of
course part of the essence of measure theory that we expect to be able
to handle
countable families of sets or functions, but that anything larger is
alarming.   Nevertheless, there are many contexts in which we can take
other types of limit.   I describe some in 123D, 123Xb and 123Xc(iii).
The point is that in such limits as $\lim_{t\to u}\phi(t)$, where
$u\in[-\infty,\infty]$, we shall have $\lim_{t\to u}\phi(t)=a$ iff
$\lim_{n\to\infty}\phi(t_n)=a$ whenever $\sequencen{t_n}$ converges to
$u$;  so that when seeking a limit $\lim_{t\to u}\int f_t$, for some
family $\langle f_t\rangle_{t\in T}$ of functions, it will be sufficient
if we can find $\lim_{n\to\infty}\int f_{t_n}$ for enough sequences
$\sequencen{t_n}$.   This type of argument will be effective for any of
the standard limits $\lim_{t\uparrow a}$, $\lim_{t\downarrow a}$,
$\lim_{t\to a}$, $\lim_{t\to\infty}$, $\lim_{t\to-\infty}$ of basic
calculus, and can be used in conjunction either with B.Levi's theorem or
with Lebesgue's theorem.   I should perhaps remark that a difficulty
arises with a similar extension of Fatou's lemma (123Yc-123Yd).
}%end of notes

\frnewpage
\frfilename{mt12conc.tex}
\versiondate{21.12.03}
\copyrightdate{2003}
     
\def\chaptername{Concordance}
\def\sectionname{Concordance}
     
\centerline{\bf Concordance for Chapter 12}
     
\bigskip
     
I list here the section and paragraph numbers which have (to my knowledge) appeared in print in references to this chapter, and which have since been changed.
     
\medskip

\leaveitout{\leader{1{}21A}{Subspace $\sigma$-algebras} This has been referred
to in {\smc Okada \& Ricker 19}.}

\leader{1{}21Yb}{$(\Sigma,\Tau)$-measurable functions} Exercise 1{}21Yb 
in the 2000 and 2001 editions, referred to in the 2001 and 2003 editions 
of Volume 2, has been moved to 121Yc.     

\leaveitout{\leader{1{}22A}{Simple functions} 
referred to in {\smc Okada \& Ricker 19}.

\leader{1{}22H, 1{}22I, 1{}22K}{Non-negative integrable functions} 
referred to in {\smc Okada \& Ricker 19}.

\leader{1{}22M, 1{}22O, 1{}22P, 1{}22R}{Integrable functions} 
referred to in {\smc Okada \& Ricker 19}.


\leader{1{}22Q}{Virtually measurable functions} 
referred to in {\smc Okada \& Ricker 19}.
}

\discrpage
     
\references{
\frfilename{mt1r.tex}
\versiondate{31.5.03}

% except for references in ...X., ...Y., ...Z. give locations in form
%  342L  rather than  342Lb

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

\centerline{\bf References for Volume 1}

\medskip

In addition to those (very few) works which I have mentioned in the
course of this volume, I list some of the books from which I myself
learnt measure theory, as a mark of grateful respect, and to
give you an opportunity to sample alternative approaches.

\medskip

Bartle R.G.\ [66] {\it The Elements of Integration.}  Wiley, 1966.

Cohn D.L.\ [80] {\it Measure Theory.}  Birkh\"auser, 1980.

Facenda J.A. \& Freniche F.J.\ [02]
{\it Integraci\'on de funciones de varias variables.}
Ediciones Pir\'amide, 2002.

Halmos P.R.\ [50] {\it Measure Theory.}   Van Nostrand, 1950.

Lebesgue H.\ [1901] `Sur une g\'en\'eralisation de l'int\'egrale
d\'efinie', C.R.Acad.Sci.\ (Paris) 132 (1901) 1025-1027.
Reprinted in {\smc Lebesgue 72}, vol.\ 1.

\leaveitout{Lebesgue H.\ [1902] {\it Int\'egrale, Longueur, Aire,}  th\`ese pr\'esent\'ee \`a la Facult\'e de Sciences de Paris,  Annali di Matematica Pura Appl.\ 7 (1902).   Reprinted in {\smc
Lebesgue 72}, vol.\ 1.}

Lebesgue H.\ [1904] {\it Le\c{c}ons sur l'int\'egration et la recherche
des fonctions primitives.} Gauthier-Villars, 1904.
Reprinted in {\smc Lebesgue 72}, vol.\ 2.
\cmmnt{[Vol.\ 1 {\it intro.}]}

Lebesgue H.\ [1918] `Remarques sur les th\'eories de la mesure et de l'int\'egration', Ann.\ Sci.\ Ecole Norm.\ Sup.\ 35 (1918) 191-250.
Reprinted in {\smc Lebesgue 72}, vol.\ 2.
\cmmnt{[Vol.\ 1 {\it intro.}]}

Lebesgue H.\ [72] {\it Oeuvres Scientifiques.}   L'Enseignement Math\'ematique, Institut de Math\'ematiques, Univ.\ de Gen\`eve, 1972.

Munroe M.E.\ [53] {\it Introduction to Measure and Integration.}
Addison-Wesley, 1953.

Royden H.L.\ [63] {\it Real Analysis.}  Macmillan, 1963.

Solovay R.M.\ [70] `A model of set theory in which every set of reals
is Lebesgue measurable', Annals of Math.\ 92 (1970) 1-56.   \cmmnt{[134C.]}

Tao T.\ [11] {\it An introduction to measure theory.}   Amer.\ Math.\ Soc.,
2011;   {\tt terrytao.files.wordpress.com/2011/01/measure-book1.pdf}.

Widom H.\ [69] {\it Lectures on Measure and Integration.}  Van Nostrand Reinhold, 1969.

Williamson J.H.\ [62] {\it Lebesgue Integration.}  Holt, Rinehart \& Winston, 1962.

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