% 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{mt1a.tex}
\versiondate{3.1.04}
\copyrightdate{1996}

\def\chaptername{Appendix}
\def\sectionname{Introduction}

\gdef\topparagraph{}
\gdef\bottomparagraph{Appendix to Vol.\ 1 {\it intro.}}

\centerline{\bf Appendix to Volume 1}

\medskip

\centerline{\bf Useful Facts}

\medskip

Each volume of this treatise will have an appendix, containing very
brief accounts of material which many readers will have met before but
some may not, and which is relevant to some topic dealt with in the
volume.   For this first volume the appendix is short, partly because
the volume itself is short, but mostly because the required basic
knowledge of analysis is so fundamental that it must be done properly
from a regular textbook or in a regular course.   However I do set out a
few details that might be omitted from some first courses in analysis,
describing some not-quite-standard notation and the elementary theory of
countable sets (\S1A1), open and closed sets in Euclidean space (\S1A2)
and upper and lower limits of sequences and functions (\S1A3).

\discrpage

\frfilename{mt1a1.tex}
\versiondate{5.11.03}
\copyrightdate{1996}

\def\chaptername{Appendix}
\def\sectionname{Set theory}

\newsection{1A1}

In 111E-111F I briefly discussed `countable' sets.   The approach
there was along what seemed to be the shortest path to the facts
immediately needed, and it is perhaps right that I should here indicate
a more conventional route.   I take the opportunity to list some
notation which I find convenient but is not universally employed.

\leader{1A1A}{Square bracket notations}\cmmnt{ I use square brackets
$[$ and $]$ in a variety of ways;  the context will I hope always make
it clear what interpretation is expected.

\header{1A1Aa}}{\bf (a)} For $a$, $b\in\Bbb R$, I write

\Centerline{$[a,b]=\{x:a\le x\le b\}$,
\quad$\ooint{a,b}=\{x:a<x<b\}$,}

\Centerline{$\coint{a,b}=\{x:a\le x<b\}$,
\quad$\ocint{a,b}=\{x:a<x\le b\}$.}

\noindent It is natural, when these formulae appear, to jump to the
conclusion that $a<b$;  but just occasionally it is useful to interpret
them when
$b\le a$, in which case\cmmnt{ I follow the formulae above literally,
so that}

\Centerline{$[a,a]=\{a\}$,
\quad$\ooint{a,a}=\coint{a,a}=\ocint{a,a}=\emptyset$,}

\Centerline{$[a,b]=\ooint{a,b}=\coint{a,b}=\ocint{a,b}=\emptyset$ if
$b<a$.}

\spheader 1A1Ab\cmmnt{ We can interpret the formulae with infinite $a$
or $b$;
for example,}

\Centerline{$\ooint{-\infty,b}=\{x:x<b\}$,
\quad$\ooint{a,\infty}=\{x:a<x\}$,
\quad$\ooint{-\infty,\infty}=\Bbb R$,}

\Centerline{$\coint{a,\infty}=\{x:x\ge a\}$,
\quad$\ocint{-\infty,b}=\{x:x\le b\}$,}

\cmmnt{\noindent and even}

\Centerline{$[0,\infty]=\{x:x\in\Bbb R,\,x\ge 0\}\cup\{\infty\}$,
\quad$[-\infty,\infty]=\Bbb R\cup\{-\infty,\infty\}$.}

\cmmnt{\spheader 1A1Ac With some circumspection -- since further
choices have to be made, which it is safer to set out explicitly
when the occasion arises -- we
can use similar formulae for `intervals' in multidimensional space
$\BbbR^r$;  see, for instance, 115A or 136D;  and even in general
partially
ordered sets, though these will not be important to us before Volume 3.
}

\cmmnt{\spheader 1A1Ad Perhaps I owe you an explanation for my choice
of $\ooint{a,b}$, $\coint{a,b}$ in favour of $(a,b)$, $[a,b)$, which are
both commoner and more pleasing to the eye.   In the first instance it
is simply because the formula

\Centerline{$(1,2)\in\ooint{0,2}\times\ooint{1,3}$}

\noindent makes better sense than its translation.   Generally, it leads
to a slightly better balance in the number of appearances of $($ and
$[$, even allowing for the further uses of $[\ldots]$ which I am about
to specify.
}

\leader{1A1B}{Direct and inverse images}\cmmnt{ I now describe an
entirely different use of square brackets, belonging to abstract set
theory rather than to the theory of the real number system.

\header{1A1Ba}}{\bf (a)} If $f$ is a function and $A$ is a set, I write

\Centerline{$f[A]=\{f(x):x\in A\cap\dom f\}$}

\noindent for the {\bf direct image} of $A$ under $f$.   \cmmnt{Note
that while
$A$ will often be a subset of the domain of $f$, this is not assumed.}

\spheader 1A1Bb If $f$ is a function and $B$ is a set, I write

\Centerline{$f^{-1}[B]=\{x:x\in\dom f,\,f(x)\in B\}$}

\noindent for the {\bf inverse image} of $B$ under $f$.   \cmmnt{This
time, it
is important to note that there is no presumption that $f$ is injective,
or that $f^{-1}$ is a function;  the formula $f^{-1}[\,\,]$ is being
given a meaning independent of any meaning of
the expression $f^{-1}$.   But it is easy to see that when $f$ is
injective, so that we have a true inverse function $f^{-1}$ (defined on
the set of values of $f$, $f[\dom f]$), then $f^{-1}[B]$, as defined
here, agrees with its interpretation under (a).}

\spheader 1A1Bc Now suppose that $R$ is a relation, that is, a set of
ordered pairs, and $A$, $B$ are sets.   Then\cmmnt{ I write}

\Centerline{$R[A]=\{y:\,\exists\,x\in A$ such that $(x,y)\in R\}$,}

\Centerline{$R^{-1}[B]=\{x:\,\exists\,y\in B$ such that $(x,y)\in R\}$.}

\cmmnt{\noindent If we write

\Centerline{$R^{-1}=\{(y,x):(x,y)\in R\}$,}

\noindent then we have an alternative interpretation of $R^{-1}[B]$
which agrees with the one just given.   Moreover, if $R$ is the graph of
a function $f$, that is, if for every $x$ there is at most one $y$ such
that $(x,y)\in R$, then the formulae here agree with those of
(a)-(b) above.
}%end of comment

\cmmnt{\spheader 1A1Bd (The following is addressed exclusively to
readers who have been taught to distinguish between the words `set' and
`class'.)   I have used the word `set' more than once above.   But that
was purely for euphony.   The same formulae can be used with arbitrary
classes, though in some set theories the expressions involved may not be
recognised as `terms' in the technical sense.
}%end of comment

\cmmnt{
\leader{1A1C}{Countable sets} In 111Fa I defined `countable set' as
follows:  a set $K$ is countable if either it is empty or there is a
surjective function from $\Bbb N$ to $K$.   A commoner formulation is to
say that a set $K$ is countable iff either it is finite or there is a
bijection between $\Bbb N$ and $K$.   So I should check at once
that these two formulations agree.
}%end of comment

\leader{1A1D}{Proposition} Let $K$ be a set.   Then the following are
equiveridical:

(i) either $K$ is empty or there is a surjection from $\Bbb N$ onto $K$;

(ii) either $K$ is finite or there is a bijection between $\Bbb N$ and
$K$;

(iii) there is an injection from $K$ to $\Bbb N$.

\proof{{\bf (a)(i)$\Rightarrow$(iii)} Assume (i).   If $K$ is
empty, then the empty function is an injection from $K$ to $\Bbb N$.
Otherwise, there is a surjection $\phi:\Bbb N\to K$.
Now, for each $k\in K$, set

\Centerline{$\psi(k)=\min\{n:n\in\Bbb N$, $\phi(n)=k\}$;}

\noindent this is always well-defined because $\phi$ is surjective, so
that $\{n:\phi(n)=k\}$ is never empty, and must have a least member.
Because $\phi\psi(k)=k$ for every $k$, $\psi$ must be injective, so is
the required injection from $K$ to $\Bbb N$.

\medskip

{\bf (b)(iii)$\Rightarrow$(ii)} Assume (iii);  let $\psi:K\to\Bbb N$ be
an injection, and set $A=\psi[K]\subseteq\Bbb N$.   Then $\psi$ is a
bijection between $K$ and $A$.   If $K$ is finite, then of course (ii)
is satisfied.   Otherwise, $A$ must also be infinite.   Define
$\phi:A\to\Bbb N$ by setting

\Centerline{$\phi(m)=\#(\{i:i\in A,\,i<m\})$,}

\noindent the number of elements of $A$ less than $m$, for each
$m\in A$;  thus $\phi(m)$ is the position of $m$ if the elements of $A$ are listed from the bottom, starting at $0$ for the least element of $A$.   
Then $\phi:A\to\Bbb N$ is a bijection, because $A$
is infinite, and $\phi\psi:K\to\Bbb N$ is a bijection.

\medskip

{\bf (c)(ii)$\Rightarrow$(i)} If $K$ is empty, surely it satisfies (i).
If $K$ is finite and not empty, list its members as $k_0,\ldots,k_n$;
now set $\phi(i)=k_i$ for $i\le n$, $k_0$ for $i>n$ to get a surjection
$\phi:\Bbb N\to K$.   If $K$ is infinite, there is a bijection from
$\Bbb N$ to $K$, which is of course also a surjection from $\Bbb N$ to
$K$.   So (i) is true in all cases.
}%end of proof of 1A1D

\cmmnt{\medskip

\noindent{\bf Remark} I referred to the `empty function' in the proof
above.   This is the function with domain $\emptyset$;  having said
this, any, or no, rule for calculating the function will have the same
effect, since it will never be applied.   By examining your feelings
about this construction you can learn something about your basic
attitude to mathematics.   You may feel that it is an artificial
irrelevance, or you may feel that it is as necessary as the number $0$.
Both are entirely legitimate feelings, and the fully rounded
mathematician alternates between them;  but I have to say that I myself
tend to the latter more often than the former, and that when I say
`function' in this treatise the empty function will generally be in the
back of my mind as a possibility.
}%end of comment

\leader{1A1E}{Properties of countable sets}\cmmnt{ Let me recapitulate
the basic properties of countable sets:

\medskip

}{\bf (a)} If $K$ is countable and $\phi:K\to L$ is a
surjection, then $L$ is countable.   \prooflet{\Prf\ If $K$ is empty
then so is
$L$.   Otherwise there is a surjection $\psi:\Bbb N\to K$, so $\phi\psi$
is a surjection from $\Bbb N$ onto $L$, and $L$ is countable. \Qed}

\header{1A1Eb}{\bf (b)} If $K$ is countable and $\phi:L\to K$ is an
injection, then $L$ is countable. \prooflet{\Prf\ By 1A1D(iii), there is
an injection $\psi:K\to\Bbb N$;  now $\psi\phi:L\to\Bbb N$ is injective, so $L$ is countable.  \Qed}

\header{1A1Ec}{\bf (c)} In particular, any subset of a countable set is
countable\cmmnt{ (as in 111F(b-i))}.

\header{1A1Ed}{\bf (d)} The Cartesian product of finitely many countable
sets is countable\cmmnt{ (111Fb(iii)-(iv))}.

\header{1A1Ee}{\bf (e)} $\Bbb Z$ is countable.   \prooflet{\Prf\ The map
$(m,n)\mapsto m-n:\Bbb N\times\Bbb N\to\Bbb Z$ is surjective.  \Qed}

\header{1A1Ef}{\bf (f)} $\Bbb Q$ is countable.   \prooflet{\Prf\ The map
$(m,n)\mapsto \bover{m}{n+1}:\Bbb Z\times\Bbb N\to\Bbb Q$ is surjective.
\Qed}

\leader{1A1F}{}\cmmnt{ Another fundamental property is worth
distinguishing from these, as it relies on a slightly deeper argument.

\medskip

\noindent}{\bf Theorem} If $\Cal K$ is a countable collection of
countable sets, then

\Centerline{$\bigcup\Cal K=\{x:\,\exists\,K\in\Cal K,\, x\in K\}$}

\noindent is countable.

\proof{ Set

\Centerline{$\Cal K'=\Cal K\setminus\{\emptyset\}
=\{K:K\in\Cal K,\,K\ne\emptyset\}$;}

\noindent then $\Cal K'\subseteq\Cal K$, so is countable, and
$\bigcup\Cal K'=\bigcup\Cal K$.   If $\Cal K'=\emptyset$, then

\Centerline{$\bigcup\Cal K=\bigcup\Cal K'=\emptyset$}

\noindent is surely countable.   Otherwise, let
$m\mapsto K_m:\Bbb N\to\Cal K'$ be a surjection.   For each
$m\in\Bbb N$, $K_m$ is a
non-empty countable set, so there is a surjection
$n\mapsto k_{mn}:\Bbb N\to K_m$.   Now
$(m,n)\mapsto k_{mn}:\Bbb N\times\Bbb N\to\bigcup\Cal K$ is a surjection (if $k\in\bigcup\Cal K$, there is a $K\in\Cal K'$
such that $k\in K$;  there is an $m\in\Bbb N$ such that $K=K_m$;  there
is an $n\in\Bbb N$ such that $k=k_{mn}$).   So $\bigcup\Cal K$ is
countable, as required.
}%end of proof of 1A1F

\cmmnt{
\leader{*1A1G}{Remark} I divide this result from the
`elementary' facts in 1A1E partly because it uses a different principle
of argument
from any necessary for the earlier work.   In the middle of the proof I
wrote `so there is a surjection $n\mapsto k_{mn}:\Bbb N\to K_m$'.
That there is a surjection from $\Bbb N$ onto $K_m$ does indeed follow
from the immediately preceding statement `$K_m$ is a non-empty
countable set'.   The sleight of hand lies in immediately naming such a
surjection as `$n\mapsto k_{mn}$'.   There may of course be many
surjections from $\Bbb N$ to $K_m$ -- as a rule, indeed, there will be
uncountably many -- and what I am in effect doing here is picking
arbitrarily on one of them.   The choice has to be arbitrary, because I
am working in a totally abstract context, and while in any particular
application of this theorem there may be some natural surjection to use,
I have no way of forecasting what approach, if any, might offer a
criterion for distinguishing a particular function here.   Now it has
been a basic method of mathematical argument, from Euclid's time at
least, that we are willing to give a name to an object, a `general
point' or an `arbitrary number', without specifying exactly which
object we are naming.   But here I am picking out simultaneously
infinitely many objects, all arbitrary members of certain sets.   This
is a use of the {\bf Axiom of Choice}.

I do not recall ever having had a student criticise an argument in the
form of that in 1A1F on the grounds that it uses a new, and possibly
illegitimate, principle;  I am sure that it never occurred to me that
anything exceptionable was being done in these cases, until someone
pointed it out.   If you find that discussions of this kind are
irrelevant to your own mathematical interests, you can certainly pass
them by, at least until you reach Volume 5.   Mathematical systems have
been studied in which the axiom of choice is false;  they are of great
interest but so far remain peripheral to the
subject.   Systems in which the axiom of choice is so false that the
union of countably many countable sets is sometimes uncountable have a
character all of their own, and in particular the theory of Lebesgue
measure is transformed;  I will come to this
possibility in Chapter 56 of Volume 5.

For a brief comment on other ways of using the axiom of choice, see
134C.
}%end of comment

\leader{1A1H}{Some uncountable sets}\cmmnt{ Of course not all sets are
countable.   In 114G/115G I remark that all countable subsets of
Euclidean space are negligible for Lebesgue measure;  consequently, any
set which is not negligible -- for instance,
any non-trivial interval -- must be uncountable.
But perhaps it will be helpful if I offer here
elementary arguments to show that $\Bbb R$ and $\Cal P\Bbb N$ are not
countable.

\medskip

}{\bf (a)} There is no surjection from $\Bbb N$ onto
$\Bbb R$.   \prooflet{\Prf\ Let $n\mapsto a_n:\Bbb N\to\Bbb R$ be any
function.   For
each $n\in\Bbb N$, express $a_n$ in decimal form as

\Centerline{$a_n=k_n + 0\cdot\epsilon_{n1}\epsilon_{n2}\ldots
=k_n+\sum_{i=1}^{\infty}10^{-i}\epsilon_{ni}$,}

\noindent where $k_n\in\Bbb Z$ is the greatest integer not greater than
$a_n$, and each $\epsilon_{ni}$ is an integer between $0$ and $9$;  for
definiteness, if $a_n$ happens to be an exact decimal, use the
terminating expansion, so that the $\epsilon_{ni}$ are eventually $0$
rather than eventually $9$.

Now define $\epsilon_i$, for $i\ge 1$, by saying that

$$\eqalign{\epsilon_i&=6\text{ if }\epsilon_{ii}<6,\cr
&=5\text{ if }\epsilon_{ii}\ge 6.\cr}$$

\noindent Consider $a=k_0+1+\sum_{i=1}^{\infty}10^{-i}\epsilon_i$, so
that $a=k_0+1+0\cdot\epsilon_1\epsilon_2\ldots$ in decimal form.   I
claim
that  $a\ne a_n$ for every $n$.   Of course $a\ne a_0$ because
$a_0<k_0+1\le a$.   If $n\ge 1$, then $\epsilon_n\ne\epsilon_{nn}$;
because no $\epsilon_i$ is either $0$ or $9$, there is no alternative
decimal expansion of $a$, so the expansion
$a_n=k_n+0\cdot\epsilon_{n1}\epsilon_{n2}\ldots$ cannot represent $a$,
and
$a\ne a_n$.

Thus I have constructed a real number which is not in the list
$a_0,a_1,\ldots$.   As $\sequencen{a_n}$ is arbitrary, there is no
surjection from $\Bbb N$ onto $\Bbb R$.  \Qed}

Thus $\Bbb R$ is uncountable.

\header{1A1Hb}{\bf (b)} There is no surjection from $\Bbb N$ onto its
power set $\Cal P\Bbb N$.   \prooflet{\Prf\ Let
$n\mapsto A_n:\Bbb N\to\Cal P\Bbb N$ be any function.   Set

\Centerline{$A=\{n:n\in\Bbb N,\,n\notin A_n\}$.}

\noindent If $n\in\Bbb N$, then

\inset{{\it either} $n\in A_n$, in which case $n\notin A$,}

\inset{{\it or} $n\notin A_n$, in which case $n\in A$.}

\noindent Thus in both cases we have $n\in A\symmdiff A_n$, so that
$A\ne A_n$.   As $n$ is arbitrary, $A\notin\{A_n:n\in\Bbb N\}$ and
$n\mapsto A_n$ is not a surjection.   As $\sequencen{A_n}$ is arbitrary,
there is no surjection from $\Bbb N$ onto $\Cal P\Bbb N$.   \Qed}

Thus $\Cal P\Bbb N$ is\cmmnt{ also} uncountable.

\cmmnt{
\leader{1A1I}{Remark} In fact it is the case that there is a
bijection between $\Bbb R$ and $\Cal P\Bbb N$ (2A1Ha);   so that
the uncountability of
both can be established by just one of the arguments above.
}

\leader{1A1J}{Notation}\cmmnt{ For definiteness, I remark here that} I
will say that a family $\Cal A$ of sets is a {\bf partition} of a set $X$
whenever $\Cal A$ is a disjoint cover of $X$, that
is, $X=\bigcup\Cal A$
and $A\cap A'=\emptyset$ for all distinct $A$, $A'\in\Cal A$\cmmnt{;
in particular, the empty set may or may not belong to $\Cal A$}.
Similarly, an indexed family $\familyiI{A_i}$ is a {\bf partition}
partition of $X$ if
$\bigcup_{i\in I}A_i=X$ and $A_i\cap A_j=\emptyset$ for all distinct
$i$, $j\in I$\cmmnt{;  again, one or more of the $A_i$ may be empty}.

\exercises{}

\cmmnt{\Notesheader{1A1} The ideas of
1A1C-1A1I %1A1C 1A1D 1A1E 1A1F 1A1H 1A1I
are essentially due to G.F.Cantor.   These concepts are
fundamental to modern set theory, and indeed to very large parts of
modern pure mathematics.   The notes above hardly begin to suggest the
extraordinary fertility of these ideas, which need a book of their own
for their proper expression;   my only aim here has been to try to make
sense of those tiny parts of the subject which are needed in the present
volume.    In later volumes I will present results which call on
substantially more advanced ideas, which I will discuss in appendices to
those volumes.
}%end of notes

\discrpage

\frfilename{mt1a2.tex}
\versiondate{21.11.03}
\copyrightdate{1994}

\def\chaptername{Appendix}
\def\sectionname{Open and closed sets in $\BbbR^r$}

\newsection{1A2}

\def\headlinesectionname{Open and closed sets in $\eightBbb R^r$}

In 111G I gave the definition of an open set in
$\Bbb R$ or $\BbbR^r$, and in 121D I used, in passing, some of the
basic properties of these sets;  perhaps it will be helpful if I set out a tiny part of the elementary theory.

\leader{1A2A}{Open sets} Recall that a set $G\subseteq\Bbb R$ is {\bf
open} if for every $x\in G$
there is a $\delta>0$ such that $\ooint{x-\delta,x+\delta}\subseteq G$;
similarly, a set $G\subseteq\BbbR^r$ is {\bf open} if for every 
$x\in G$ there is a $\delta>0$ such that $U(x,\delta)\subseteq G$, where
$U(x,\delta)=\{y:\|y-x\|<\delta\}$, writing
$\|z\|$ for $\sqrt{\zeta_1^2+\ldots+\zeta_r^2}$ if
$z=(\zeta_1,\ldots,\zeta_r)$.   \cmmnt{Henceforth I give the arguments
for general $r$;  if you are at present interested only in the
one-dimensional case, you should
have no difficulty in reading them as if $r=1$ throughout.}

\vleader{36pt}{1A2B}{The family of all open sets} 
Let $\frak T$ be the family of open sets of $\BbbR^r$.   \cmmnt{Then
$\frak T$ has the following properties.}

(a) $\emptyset\in\frak T$\cmmnt{, that is, the empty set is open}.
\prooflet{\Prf\ Because the definition of `$\emptyset$
is open' begins with `for every $x\in\emptyset$, $\ldots$', it must
be vacuously satisfied by the empty set.\ \Qed}

(b) $\BbbR^r\in\frak T$\cmmnt{, that is, the whole space under
consideration is an open set}.   \prooflet{\Prf\ 
$U(x,1)\subseteq\BbbR^r$ for every $x\in\BbbR^r$.\ \Qed}

(c) If $G$, $H\in\frak T$ then $G\cap H\in\frak T$\cmmnt{;  that is,
the intersection of two open sets is always an open set}.
\prooflet{\Prf\
Let $x\in G\cap H$.   Then there are $\delta_1$, $\delta_2>0$ such that
$U(x,\delta_1)\subseteq G$ and $U(x,\delta_2)\subseteq H$.   Set
$\delta=\min(\delta_1,\delta_2)>0$;  then

\Centerline{$U(x,\delta)=\{y:\|y-x\|<\min(\delta_1,\delta_2)\}
=U(x,\delta_1)\cap U(x,\delta_2)\subseteq G\cap H$.}

\noindent   As $x$ is
arbitrary, $G\cap H$ is open.\ \Qed}

(d) If $\Cal G\subseteq\frak T$, then

\Centerline{$\bigcup\Cal G=\{x:\,\exists\, G\in\Cal G,\,x\in G\}
\in\frak T$\dvro{.}{;}}

\noindent\cmmnt{that is, the union of any family of open sets is
open.}   \prooflet{\Prf\ Let $x\in\bigcup\Cal G$.   Then there is a
$G\in\Cal G$ such that
$x\in G$.   Because $G\in\frak T$, there is a $\delta>0$ such that


\Centerline{$U(x,\delta)\subseteq G\subseteq\bigcup\Cal G$.}

\noindent   As $x$ is arbitrary,
$\bigcup\Cal G\in\frak T$.\ \Qed}

\leader{1A2C}{Cauchy's inequality:  Proposition}  For all
$x$, $y\in\BbbR^r$,
$\|x+y\|\le\|x\|+\|y\|$.

\proof{ Express $x$ as $(\xi_1,\ldots,\xi_r)$, $y$ as
$(\eta_1,\ldots,\eta_r)$;
set $\alpha=\|x\|$, $\beta=\|y\|$.   Then both $\alpha$ and $\beta$
are non-negative.   If $\alpha=0$ then $\sum_{j=1}^r\xi_j^2=0$ so every
$\xi_j=0$ and $x=\tbf{0}$, so $\|x+y\|=\|y\|=\|x\|+\|y\|$;  if
$\beta=0$, then $y=\tbf{0}$ and $\|x+y\|=\|x\|=\|x\|+\|y\|$.
Otherwise, consider

$$\eqalign{\alpha\beta\|x+y\|^2
&\le\alpha\beta\|x+y\|^2+\|\alpha y-\beta x\|^2\cr
&=\alpha\beta\sum_{j=1}^r(\xi_j+\eta_j)^2
+\sum_{j=1}^r(\alpha\eta_j-\beta\xi_j)^2\cr
&=\sum_{j=1}^r\alpha\beta\xi_j^2+\alpha\beta\eta_j^2
+\alpha^2\eta_j^2+\beta^2\xi_j^2\cr
&=\alpha^3\beta+\alpha\beta^3+\alpha^2\beta^2+\beta^2\alpha^2\cr
&=\alpha\beta(\alpha+\beta)^2
=\alpha\beta(\|x\|+\|y\|)^2.\cr}$$

\noindent Dividing both sides by $\alpha\beta$ and taking square roots
we have the result.
}%end of proof of 1A2C

\leader{1A2D}{Corollary} $U(x,\delta)$\cmmnt{, as defined in 1A2A,} is
open, for every $x\in\BbbR^r$ and $\delta>0$.

\proof{ If
$y\in U(x,\delta)$, then $\eta=\delta-\|y-x\|>0$.   Now if $z\in
U(y,\eta)$,

\Centerline{$\|z-x\|=\|(z-y)+(y-x)\|
\le\|z-y\|+\|y-x\|<\eta+\|y-x\|=\delta$,}

\noindent and $z\in U(x,\delta)$;  thus
$U(y,\eta)\subseteq U(x,\delta)$.   As $y$ is arbitrary,
$U(x,\delta)$ is open.
}%end of proof of 1A2D

\leader{1A2E}{Closed sets:  Definition} A set $F\subseteq\BbbR^r$ is
{\bf closed} if
$\BbbR^r\setminus F$ is open.  \cmmnt{({\it Warning!} `Most'
subsets of
$\BbbR^r$ are neither open nor closed;  two subsets of $\BbbR^r$,
viz., $\emptyset$ and $\BbbR^r$, are both open and closed.)
Corresponding to
the list in 1A2B, we have the following properties of the family $\Cal
F$ of closed subsets of $\BbbR^r$.}

\leader{1A2F}{Proposition} Let $\Cal F$ be the family of closed subsets of $\BbbR^r$.

 (a) $\emptyset\in\Cal F$\prooflet{ (because
$\BbbR^r\in\frak T$)}.

(b) $\BbbR^r\in\Cal F$\prooflet{ (because $\emptyset\in\frak T$)}.

(c) If $E$, $F\in\Cal F$ then $E\cup F\in\Cal F$\dvro{.}{, because

\Centerline{$\BbbR^r\setminus(E\cup F)
=(\BbbR^r\setminus E)\cap(\BbbR^r\setminus F)\in\frak T$.}}

(d) If $\Cal E\subseteq\Cal F$ is a {\it non-empty} family of
closed sets, then

\Centerline{$\bigcap\Cal E
=\{x:x\in F\Forall F\in\Cal E\}\cmmnt{\mskip5mu =\BbbR^r\setminus\bigcupop_{F\in\Cal E}(\BbbR^r\setminus F)}\in\Cal F$.}

\cmmnt{\medskip

\noindent{\bf Remark} In (d), we need to assume that 
$\Cal E\ne\emptyset$ to ensure that $\bigcap\Cal E\subseteq\BbbR^r$.
}%end of comment

\leader{1A2G}{}\cmmnt{ Corresponding to 1A2D, we have the following
fact:

\medskip

\noindent}{\bf Lemma} If $x\in\BbbR^r$ and $\delta\ge 0$ then
$B(x,\delta)=\{y:\|y-x\|\le\delta\}$ is closed.

\proof{ Set $G=\BbbR^r\setminus B(x,\delta)$.   If $y\in G$, then
$\eta=\|y-x\|-\delta>0$;   if $z\in U(y,\eta)$, then

\Centerline{$\delta+\eta=\|y-x\|\le\|y-z\|+\|z-x\|<\eta+\|z-x\|$,}

\noindent so $\|z-x\|>\delta$ and $z\in G$.   So 
$U(y,\eta)\subseteq G$.   As $y$ is arbitrary, $G$ is open and $B(x,\delta)$ is closed.
}%end of proof of 1A2G

\exercises{}

\discrpage


\frfilename{mt1a3.tex}
\versiondate{18.12.03}
\copyrightdate{1994}

\def\chaptername{Appendix}
\def\sectionname{Lim sups and lim infs}

\newsection{1A3}

It occurs to me that not every foundation
course in real analysis has time to deal with the concepts $\limsup$ and
$\liminf$.

\leader{1A3A}{Definition (a)} For a real sequence $\sequencen{a_n}$,
write

\Centerline{$\limsup_{n\to\infty}a_n=\lim_{n\to\infty}\sup_{m\ge n}a_m
=\inf_{n\in\Bbb N}\sup_{m\ge n}a_m$,}

\Centerline{$\liminf_{n\to\infty}a_n=\lim_{n\to\infty}\inf_{m\ge n}a_m
=\sup_{n\in\Bbb N}\inf_{m\ge n}a_m$;}

\noindent  if we
allow the values $\pm \infty$, both for suprema and infima and for
limits\cmmnt{ (see 112Ba)}, these will always be 
defined\cmmnt{, because the sequences

\Centerline{$\sequencen{\sup_{m\ge n}a_m}$,
\quad$\sequencen{\inf_{m\ge n}a_m}$}

\noindent are monotonic}.


\cmmnt{\header{1A3Ab}{\bf (b)} Explicitly:


\Centerline{$\limsup_{n\to\infty}a_n=\infty\,\iff\,\{a_n:n\in\Bbb N\}$
is unbounded above,}

\Centerline{$\limsup_{n\to\infty}a_n=-\infty\,\iff\,\lim_{n\to\infty}a_n
=-\infty$,}

\noindent that is, if and only if for every $a\in\Bbb R$ there is an
$n_0\in\Bbb N$ such that $a_n\le a$ for every $n\ge n_0$;

\Centerline{$\liminf_{n\to\infty}a_n=-\infty\, \iff\,\{a_n:n\in\Bbb N\}$
is unbounded
below,}

\Centerline{$\liminf_{n\to\infty}a_n=\infty\, \iff
\,\lim_{n\to\infty}a_n
=\infty$,}

\noindent that is, if and only if for every $a\in\Bbb R$ there is an
$n_0\in\Bbb N$ such that $a_n\ge a$ for every $n\ge n_0$.
}%end of comment

\header{1A3Ac}{\bf (c)} For \cmmnt{finite $a\in\Bbb R$, we have

\inset{$\limsup_{n\to\infty}a_n=a$ iff (i) for
every $\epsilon>0$ there is an $n_0\in\Bbb N$ such that $a_n\le
a+\epsilon$ for every $n\ge n_0$ (ii) for every $\epsilon>0$,
$n_0\in\Bbb N$ there is an $n\ge n_0$ such that $a_n\ge a-\epsilon$,}

\noindent while

\inset{$\liminf_{n\to\infty}a_n=a$ iff (i) for
every $\epsilon>0$ there is an $n_0\in\Bbb N$ such that $a_n\ge
a-\epsilon$ for every $n\ge n_0$ (ii) for every $\epsilon>0$,
$n_0\in\Bbb N$ there is an $n\ge n_0$ such that $a_n\le a+\epsilon$.}

\noindent Generally, for }%end of comment
$u\in[-\infty,\infty]$, we can say that

\inset{$\limsup_{n\to\infty}a_n=u$ iff (i) for every $v>u$ (if any)
there is an
$n_0\in\Bbb N$ such that $a_n\le v$ for every $n\ge n_0$ (ii) for every
$v<u$, $n_0\in\Bbb N$ there is an $n\ge n_0$ such that $a_n\ge v$,}

\inset{$\liminf_{n\to\infty}a_n=u$ iff (i) for every $v<u$ there is an
$n_0\in\Bbb N$ such that $a_n\ge v$ for every $n\ge n_0$ (ii) for every
$v>u$, $n_0\in\Bbb N$ there is an $n\ge n_0$ such that $a_n\le v$.}


\leader{1A3B}{}\cmmnt{ We have the following basic results.

\medskip

\noindent}{\bf Proposition} For any sequences $\sequencen{a_n}$,
$\sequencen{b_n}$ in $\Bbb R$,

(a) $\liminf_{n\to\infty}a_n\le\limsup_{n\to\infty}a_n$,

(b) $\lim_{n\to\infty}a_n=u\in[-\infty,\infty]$ iff
$\limsup_{n\to\infty}a_n=\liminf_{n\to\infty}a_n=u$,

(c) $\liminf_{n\to\infty}a_n=-\limsup_{n\to\infty}(-a_n)$,

(d) $\limsup_{n\to\infty}(a_n+b_n)
\le\limsup_{n\to\infty}a_n+\limsup_{n\to\infty}b_n$,

(e) $\liminf_{n\to\infty}(a_n+b_n)
\ge\liminf_{n\to\infty}a_n+\liminf_{n\to\infty}b_n$,

(f) $\limsup_{n\to\infty}ca_n=c\limsup_{n\to\infty}a_n$ if $c\ge 0$,

(g) $\liminf_{n\to\infty}ca_n=c\liminf_{n\to\infty}a_n$ if $c\ge 0$,

\noindent with the proviso in (d) and (e) that we must be able to
interpret the right-hand-side of the inequality according to the rules in 
135A, while in (f) and (g) we take $0\cdot\infty=0\cdot(-\infty)=0$.

\proof{{\bf (a)} $\sup_{m\ge n}a_m\ge\inf_{m\ge n}a_m$ for every
$n$, so

\Centerline{$\limsup_{n\to\infty}a_n=\lim_{n\to\infty}\sup_{m\ge
n}a_m\ge\lim_{n\to\infty}\inf_{m\ge n}a_m=\liminf_{n\to\infty}a_n$.}

\medskip

{\bf (b)} Using the last description of $\limsup_{n\to\infty}$ and
$\liminf_{n\to\infty}$ in 1A3Ac, and a corresponding description of
$\lim_{n\to\infty}$, we have

$$\eqalign{\lim_{n\to\infty}&a_n=u\cr
&\iff\text{ for every }v>u\text{ there is an }n_1\in\Bbb N
    \text{ such that }a_n\le v\text{ for every }n\ge n_1\cr
&\qquad\text{ and for every }v<u\text{ there is an }n_2\in\Bbb N
    \text{ such that }a_n\ge v\text{ for every }n\ge n_2\cr
&\iff\text{ for every }v>u\text{ there is an }n_1\in\Bbb N
    \text{ such that }a_n\le v\text{ for every }n\ge n_1\cr
&\qquad\text{ and for every }v<u,\,n_0\in\Bbb N
    \text{ there is an }n\ge n_0\text{ such that }a_n\ge v\cr
&\qquad\text{ and for every }v<u\text{ there is an }n_2\in\Bbb N
    \text{ such that }a_n\ge v\text{ for every }n\ge n_2\cr
&\qquad\text{ and for every }v>u,\,n_0\in\Bbb N
    \text{ there is an }n\ge n_0\text{ such that }a_n\le v\cr
&\iff\limsup_{n\to\infty}a_n=\liminf_{n\to\infty}a_n=u.\cr}$$

\medskip

{\bf (c)} This is just a matter of turning the formulae upside down:

$$\eqalign{\liminf_{n\to\infty}a_n
&=\sup_{n\in\Bbb N}\inf_{\lower 0.4ex\hbox{$\scriptstyle m\ge n$}}a_m
=\sup_{n\in\Bbb N}(-\sup_{m\ge n}(-a_m))\cr
&=-\inf_{\lower 0.4ex\hbox{$\scriptstyle n\in\Bbb N$}}
     \sup_{m\ge n}(-a_m)
=-\limsup_{n\to\infty}(-a_n).\cr}$$

\medskip

{\bf (d)} If $v>\limsup_{n\to\infty}a_n+\limsup_{n\to\infty}b_n$, there
are $v_1$, $v_2$ such that $v_1>\limsup_{n\to\infty}a_n$,
$v_2>\limsup_{n\to\infty}b_n$ and $v_1+v_2=v$.   Now there are $n_1$,
$n_2\in\Bbb N$ such that $\sup_{m\ge n_1}a_n\le v_1$ and 
$\sup_{m\ge n_2}b_n\le v_2$;  so that

$$\eqalign{\sup_{m\ge\max(n_1,n_2)}a_m+b_m
&\le\sup_{m\ge\max(n_1,n_2)}a_m+\sup_{m\ge\max(n_1,n_2)}b_m\cr
&\le\sup_{m\ge n_1}a_m+\sup_{m\ge n_2}b_m
\le v_1+v_2
= v.\cr}$$

\noindent As $v$ is arbitrary,

\Centerline{$\limsup_{n\to\infty}a_n+b_n=\inf_{n\in\Bbb N}\sup_{m\ge
n}a_m+b_m\le\limsup_{n\to\infty}a_n+\limsup_{n\to\infty}b_n$.}

\medskip

{\bf (e)} Putting (c) and (d) together,

$$\eqalign{\liminf_{n\to\infty}a_n+b_n
&=-\limsup_{n\to\infty}(-a_n)+(-b_n)\cr
&\ge-\limsup_{n\to\infty}(-a_n)-\limsup_{n\to\infty}(-b_n)
=\liminf_{n\to\infty}a_n+\liminf_{n\to\infty}b_n.\cr}$$

\medskip

{\bf (f)} Because $c\ge 0$,

$$\eqalign{\limsup_{n\to\infty}ca_n
&=\inf_{n\in\Bbb N}\sup_{m\ge n}ca_m
=\inf_{n\in\Bbb N}c\sup_{m\ge n}a_m\cr
&=c\inf_{n\in\Bbb N}\sup_{m\ge n}a_m
=c\limsup_{n\to\infty}a_n.\cr}$$

\medskip

{\bf (g)} Finally,

\Centerline{$\liminf_{n\to\infty}ca_n
=-\limsup_{n\to\infty}c(-a_n)
=-c\limsup_{n\to\infty}(-a_n)
=c\liminf_{n\to\infty}a_n$.}
}%end of proof of 1A3B

\cmmnt{
\leader{1A3C}{Remark} Of course the familiar results that
$\lim_{n\to\infty}a_n+b_n=\lim_{n\to\infty}a_n+\lim_{n\to\infty}b_n$,
$\lim_{n\to\infty}ca_n=c\lim_{n\to\infty}a_n$ are immediate corollaries
of 1A3B.
}%end of comment

\leader{*1A3D}{Other expressions of the same idea} 
The concepts of $\limsup$ and
$\liminf$ may be applied in any context in which we can consider the
limit of a real-valued function.   For instance, if $f$ is a real-valued
function defined (at least) on a punctured interval of the form
$\{x:0<|c-x|\le\epsilon\}$ where $c\in\Bbb R$ and $\epsilon>0$, then

\Centerline{$\limsup_{t\to c}f(t)
=\lim_{\delta\downarrow 0}\sup_{0<|t-c|\le\delta}f(t)
=\inf_{0<\delta\le\epsilon}\sup_{0<|t-c|\le\delta}f(t)$,}

\Centerline{$\liminf_{t\to c}f(t)
=\lim_{\delta\downarrow 0}\inf_{0<|t-c|\le\delta}f(t)
=\sup_{0<\delta\le\epsilon}\inf_{0<|t-c|\le\delta}f(t)$,}

\noindent allowing $\infty$ and $-\infty$ whenever they seem called for.
Or if $f$ is defined on the half-open interval
$\ocint{c,c+\epsilon}$, we can say

\Centerline{$\limsup_{t\downarrow c}f(t)
=\lim_{\delta\downarrow 0}\sup_{c<t\le c+\delta}f(t)
=\inf_{0<\delta\le\epsilon}\sup_{c<t\le c+\delta}f(t)$,}

\Centerline{$\liminf_{t\downarrow c}f(t)
=\lim_{\delta\downarrow 0}\inf_{c<t\le c+\delta}f(t)
=\sup_{0<\delta\le\epsilon}\inf_{c<t\le c+\delta}f(t)$.}

\noindent Similarly, if $f$ is defined on $\coint{M,\infty}$ for some
$M\in\Bbb R$, we have

\Centerline{$\limsup_{t\to\infty}f(t)
=\lim_{a\to\infty}\sup_{t\ge a}f(t)
=\inf_{a\ge M}\sup_{t\ge a}f(t)$,}

\Centerline{$\liminf_{t\to\infty}f(t)
=\lim_{a\to\infty}\inf_{t\ge a}f(t)
=\sup_{a\ge M}\inf_{t\ge a}f(t)$.}

\cmmnt{\noindent A further extension of the idea is 
examined briefly in 2A3S in Volume 2.}

\frnewpage

\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.

\frnewpage

}%end of references

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

\end



