% 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\Sous{\Cal S\text{ous}}  %used in vol 6
\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{mt27.tex} 
\versiondate{26.8.13} 
\copyrightdate{1996} 
 
\loadeusm 
\def\hatcalBnu{\hat{\Cal B}_{\nu}} 
 
\def\chaptername{Probability theory} 
\def\sectionname{Introduction} 
 
\newchapter{27} 
 
Lebesgue created his theory of integration in response to a 
number of problems in real analysis, and all his life seems to have 
thought of it as a tool for use in geometry and calculus  
({\smc Lebesgue 72}, vols.\ 1 and 2).   Remarkably, it 
turned out, when suitably adapted, to provide a solid foundation for 
probability theory.   The development of this approach is generally 
associated with the name of Kolmogorov.   It has so come to 
dominate modern abstract probability theory that many authors ignore all 
other methods.   I do not propose to commit myself to any view on 
whether 
$\sigma$-additive measures are the only way to give a rigorous 
foundation to probability theory, or whether they are adequate to deal 
with all probabilistic ideas;  there are some serious philosophical 
questions here, since probability theory, at least in its applied 
aspects, seeks to help us to understand the material world outside mathematics. 
But from my position as a measure theorist, it is 
incontrovertible that probability theory is among the central 
applications of the concepts and theorems of measure theory, and is one 
of the most vital sources of new ideas;  and that every measure theorist 
must be alert to the intuitions which probabilistic methods can provide. 
 
I have written the preceding paragraph in terms suggesting that  
`probability theory' is somehow distinguishable from the rest of measure 
theory;  this is another point on which I should prefer not to put 
forward any opinion as definitive.   But undoubtedly there is a 
distinction, rather deeper than the elementary point that probability 
deals (almost) exclusively with spaces of measure $1$. 
M.Lo\`eve argues persuasively ({\smc Lo\`eve 77}, $\S$10.2) 
that the essence of 
probability theory is the artificial nature of the probability spaces 
themselves.   In measure theory, when we wish to integrate a function, 
we usually feel that we have a proper function with a domain and 
values.   In probability theory, when we take the expectation of a 
random variable, the variable is an `observable' or `the result 
of an experiment';  we are generally uncertain, or ignorant, or 
indifferent concerning the factors underlying the variable.   Let me 
give an example from the theorems below.   In the proof of the Central 
Limit Theorem (274F), I find that I need an auxiliary list 
$Z_0,\ldots,Z_n$ of random variables, independent of each other and of 
the original sequence $X_0,\ldots,X_n$.   I create such a sequence by 
taking a product space $\Omega\times\Omega'$, and writing 
$X'_i(\omega,\omega')=X_i(\omega)$, while the $Z_i$ are functions of 
$\omega'$.   Now the difference between the $X_i$ and the $X'_i$ is of a 
type which a well-trained analyst would ordinarily take seriously.   We 
do not think that the function $x\mapsto x^2:[0,1]\to[0,1]$ is the 
same thing as the function $(x_1,x_2)\mapsto x_1^2:[0,1]^2\to[0,1]$.   But a 
probabilist is likely to feel that it is positively pedantic to start 
writing $X'_i$ instead of $X_i$.   He did not believe in the space 
$\Omega$ in the first place, and if it turns out to be inadequate for 
his intuition he enlarges it without a qualm.   Lo\`eve calls 
probability 
spaces `fictions', `inventions of the imagination' in Larousse's 
words;  they are necessary in the models Kolmogorov has taught us 
to use, but we have a vast amount of freedom in choosing them, and in 
their essence they are nothing so definite as a set with points. 
 
A probability space, therefore, is somehow a more shadowy entity in 
probability theory than it is in measure theory.   The important objects 
in probability theory are random variables and distributions, 
particularly joint distributions.   In this volume I shall deal 
exclusively with random variables which can be thought of as 
taking values in some power of $\Bbb R$;  but this is not the central 
point.   What is vital is that somehow the {\it codomain}, the potential 
set of values, of a random variable, is much better defined than its 
{\it domain}.   Consequently our attention is focused not on any 
features of the artificial space which it is convenient to use as the 
underlying probability space -- I write `underlying', though 
it is the most superficial and easily changed aspect of the model 
-- but on the distribution on the codomain induced by the random 
variable.   Thus the Central Limit Theorem, which speaks only of 
distributions, is actually more important in applied probability than 
the Strong Law of Large Numbers, which claims to tell us what a 
long-term average will almost certainly be. 
 
W.Feller ({\smc Feller 66}) goes even farther than Lo\`eve, and as 
far as 
possible works entirely with distributions, setting up machinery which 
enables him to go for long stretches without mentioning probability 
spaces at all.   I make no attempt to emulate him.   But the approach is 
instructive and faithful to the essence of the subject. 
 
Probability theory includes more mathematics than can easily be 
encompassed in a lifetime, and I have selected for this introductory 
chapter the two limit theorems I have already mentioned, the Strong Law 
of Large Numbers and the Central Limit Theorem,  
together with some material on martingales (\S\S275-276).    
They illustrate not 
only the special character of probability theory -- so that you will be 
able to form your own judgement on the remarks above -- but also some of  
its chief contributions to `pure' measure theory, the concepts of  
`independence' and `conditional expectation'. 
 
 
\discrpage 
 
\frfilename{mt271.tex}
\versiondate{11.12.08}
\copyrightdate{1995}

\def\chaptername{Probability theory}
\def\sectionname{Distributions}

\newsection{271}

I start this chapter with a discussion of `probability
distributions', the probability measures on $\BbbR^n$ defined by
families $(X_1,\ldots,X_n)$ of random variables.   I give the basic
results describing the circumstances under which two distributions are
equal (271G),  integration with respect to a distribution (271E), and
probability density functions (271H-271K).   %271H 271I 271J 271K

\medskip

\leader{271A}{Notation }\cmmnt{I have just spent some paragraphs on an
attempt to describe the essential difference between probability theory
and measure theory.   But there is a quicker test by which you may
discover whether your author is a measure theorist or a probabilist:
open any page, and look for the phrases `measurable function' and
`random variable', and the formulae `$\int fd\mu$' and `$\Expn(X)$'.
The first member of each pair will enable you to diagnose `measure' and
the second `probability', with little danger of error.   So far in
this treatise I have firmly used measure theorists' terminology, with a
few individual quirks.   But in a chapter on probability theory I find
that measure-theoretic notation, while perfectly adequate in a formal
sense, does such violence to the familiar formulations as to render them
unnatural.   Moreover, you must surely at some point -- if you have not
already done so -- become familiar with probabilists' language.   So in
this chapter I will make a substantial step in that
direction.   Happily, I think that this can be done without setting up
any direct conflicts, so that I shall be able, in later volumes, to call
upon this work in whichever notation then seems appropriate, without
needing to re-formulate it.

\medskip

}{\bf (a)}\dvro{ Let}{ So let} $(\Omega,\Sigma,\mu)$ be a
probability space.   \cmmnt{I take
the opportunity given by a new phrase to make a technical move.}   A
{\bf real-valued random variable}
on $\Omega$ will be a member of
$\eusm L^0(\mu)$\cmmnt{, as defined in 241A;  that is, a real-valued
function $X$ defined on a conegligible subset of $\Omega$ such that $X$
is measurable with respect to the completion $\hat\mu$ of $\mu$, or, if
you prefer,
such that $X\restr E$ is $\Sigma$-measurable for some conegligible set
$E\subseteq\Omega$}.\cmmnt{\footnote{For an account of how this
terminology became standard, see
{\tt
http://www.dartmouth.edu/$\sim$chance/\penalty-100Doob/\penalty-100conversation.html}.}}
% http://www.dartmouth.edu/~chance/Doob/conversation.html  % Apr09

\header{271Ab}{\bf (b)} If $X$ is a real-valued random variable on a
probability space
$(\Omega,\Sigma,\mu)$, write $\Expn(X)=\int X\,d\mu$ if this is defined in
$[-\infty,\infty]$\cmmnt{ in the
sense of Chapter 12 and \S133}.
In this case I will call $\Expn(X)$ the {\bf mean} or {\bf expectation}
of $X$.   Thus we may say that
`$X$ {\bf has a finite expectation}' in place of `$X$ is
integrable'.   \cmmnt{133A says that
`$\Expn(X+Y)=\Expn(X)+\Expn(Y)$ whenever $\Expn(X)$
and $\Expn(Y)$ and their sum are defined in $[-\infty,\infty]$', and
122P becomes `a real-valued
random variable $X$ has a finite expectation iff $\Expn(|X|)<\infty$'.}

\header{271Ac}{\bf (c)} If $X$ is a real-valued random variable with
finite expectation, the {\bf variance} of $X$ is

\Centerline{$\Var(X)=\Expn(X-\Expn(X))^2
=\Expn(X^2-2\Expn(X)X+\Expn(X)^2)=\Expn(X^2)-(\Expn(X))^2$\dvro{.}{}}

\cmmnt{\noindent (Note that this formula
shows that $\Expn(X)^2\le\Expn(X^2)$;  compare 244Xd(i).)   $\Var(X)$
is finite iff $\Expn(X^2)<\infty$, that is, iff $X\in\eusm L^2(\mu)$
(244A).   In particular, $X+Y$ and $cX$ have finite variance whenever
$X$ and $Y$ do and $c\in\Bbb R$.}


\header{271Ad}{\bf (d)} I shall allow myself to use such formulae as

\Centerline{$\Pr(X>a)$,\quad $\Pr(X-\epsilon\le Y\le X+\delta)$,}

\noindent where $X$ and $Y$ are random variables on the same probability
space $(\Omega,\Sigma,\mu)$, to mean respectively

\Centerline{$\hat\mu\{\omega:\omega\in\dom X,\,X(\omega)>a\}$,}

\Centerline{$\hat\mu\{\omega:\omega\in\dom X\cap\dom Y,\,
X(\omega)-\epsilon\le Y(\omega)\le X(\omega)+\delta\}$,}

\noindent writing $\hat\mu$ for the completion of $\mu$\cmmnt{ as
usual}.   \cmmnt{There
are two points to note here.   First, $\Pr$ depends on $\hat\mu$, not
on $\mu$;  in effect, the notation automatically directs us to complete
the probability space $(\Omega,\Sigma,\mu)$.   I could, of course,
equally well write

\Centerline{$\Pr(X^2+Y^2>1)
=\mu^*\{\omega:\omega\in\dom X\cap\dom Y,\,
X(\omega)^2+Y(\omega)^2>1\}$,}

\noindent taking $\mu^*$ to be the outer measure on $\Omega$ associated
with $\mu$ (132B).   Secondly,} I will use this notation {\it only for
predicates corresponding to Borel measurable sets};  that is to
say, I shall write

\Centerline{$\Pr(\psi(X_1,\ldots,X_n))
=\hat\mu\{\omega:\omega\in\bigcap_{i\le n}\dom X_i,\,
\psi(X_1(\omega),\ldots,X_n(\omega))\}$}

\noindent only when the set

\Centerline{$\{(\alpha_1,\ldots,\alpha_n):
\psi(\alpha_1,\ldots,\alpha_n)\}$}

\noindent is a Borel set in $\BbbR^n$.    \cmmnt{Part of the reason
for this restriction will appear in the next few paragraphs;
$\Pr(\psi(X_1,\ldots,X_n))$ must be something calculable from knowledge
of the joint distribution of $X_1,\ldots,X_n$, as defined in 271C.
In fact we can safely extend the idea to `universally measurable'
predicates $\psi$, to be discussed in Volume 4.   But it could happen
that $\mu$ gave a measure to a set of the form
$\{\omega:X(\omega)\in A\}$ for some exceedingly irregular set $A$, and
in such a case it would be prudent to regard this as an accidental
pathology of the probability space, and to treat it in a rather
different way.

(I see that I have rather glibly assumed that the formula above defines
$\Pr(\psi(X_1,\ldots,X_n))$ for every Borel predicate $\psi$.   This is
a consequence of 271Bb below.)
}

\leader{271B}{Theorem} Let $(\Omega,\Sigma,\mu)$ be a probability space,
and $X_1,\ldots,X_n$ real-valued random variables on $\Omega$.    Set
$\pmb{X}(\omega)=(X_1(\omega),\ldots,X_n(\omega))$ for
$\omega\in\bigcap_{i\le n}\dom X_i$.

(a) There is a unique Radon measure $\nu$ on $\BbbR^n$ such that

\Centerline{$\nu\ocint{-\infty,a}=\Pr(X_i\le\alpha_i$
for every $i\le n)$}

\noindent whenever $a=(\alpha_1,\ldots,\alpha_n)
\in\BbbR^n$\cmmnt{, writing $\ocint{-\infty,a}$ for
$\prod_{i\le n}\ocint{-\infty,\alpha_i}$};

(b) $\nu\BbbR^n=1$ and $\nu E=\hat\mu(\pmb{X}^{-1}[E])$ whenever $\nu E$
is defined, where $\hat\mu$ is the
completion of $\mu$;  in particular, $\nu E=\Pr((X_1,\ldots,X_n)\in E)$
for every Borel set $E\subseteq\BbbR^n$.

\proof{ Let $\hat\Sigma$ be the domain of $\hat\mu$, and set
$D=\bigcap_{i\le n}\dom X_i=\dom\pmb{X}$;  then $D$ is conegligible, so
belongs to $\hat\Sigma$.   Let $\hat\mu_D=\hat\mu\restrp\Cal PD$ be the
subspace measure on $D$ (131B, 214B), and $\nu_0$ the image measure
$\hat\mu_D\pmb{X}^{-1}$ (234D);  let $\Tau$ be the domain of $\nu_0$.

Write $\Cal B$ for the algebra of Borel sets in $\BbbR^n$.  Then
$\Cal B\subseteq\Tau$.   \Prf\ For $i\le n$, $\alpha\in\Bbb R$ set
$F_{i\alpha}=\{x:x\in\BbbR^n,\,\xi_i\le\alpha\}$,
$H_{i\alpha}=\{\omega:\omega\in\dom X_i,\,X_i(\omega)\le\alpha\}$.
$X_i$ is $\hat\Sigma$-measurable and its domain is in $\hat\Sigma$, so
$H_{i\alpha}\in\hat\Sigma$, and
$\pmb{X}^{-1}[F_{i\alpha}]=D\cap H_{i\alpha}$ is $\hat\mu_D$-measurable.
Thus $F_{i\alpha}\in\Tau$.
As $\Tau$ is a $\sigma$-algebra of subsets of $\BbbR^n$,
$\Cal B\subseteq\Tau$ (121J).\ \Qed

Accordingly $\nu_0\restr\Cal B$ is a measure on $\BbbR^n$ with domain
$\Cal B$;  of course $\nu_0\BbbR^n=\hat\mu D=1$.   By 256C, the
completion $\nu$ of $\nu_0\restr\Cal B$ is a Radon measure on
$\BbbR^n$, and $\nu\BbbR^n=\nu_0\BbbR^n=1$.

For $E\in\Cal B$,

\Centerline{$\nu E=\nu_0E=\hat\mu_D\pmb{X}^{-1}[E]
=\hat\mu\pmb{X}^{-1}[E]
=\Pr((X_1,\ldots,X_n)\in E)$.}

\noindent More generally, if $E\in\dom\nu$, then there are Borel sets
$E'$, $E''$ such that $E'\subseteq E\subseteq E''$ and
$\nu(E''\setminus E')=0$, so that
$\pmb{X}^{-1}[E']\subseteq\pmb{X}^{-1}[E]\subseteq\pmb{X}^{-1}[E'']$ and
$\hat\mu(\pmb{X}^{-1}[E'']\setminus\pmb{X}^{-1}[E'])=0$.   This means
that $\pmb{X}^{-1}[E]\in\hat\Sigma$ and

\Centerline{$\hat\mu\pmb{X}^{-1}[E]=\hat\mu\pmb{X}^{-1}[E']=\nu E'=\nu
E$.}

As for the uniqueness of $\nu$, if $\nuprime$ is any Radon measure on
$\Bbb
R^n$ such that
$\nuprime\ocint{-\infty,a}=\Pr(X_i\le\alpha_i\Forall i\le n)$ for every
$a\in\BbbR^n$, then surely

\Centerline{$\nuprime\BbbR^n
=\lim_{k\to\infty}\nuprime\ocint{-\infty,k\tbf{1}}
=\lim_{k\to\infty}\nu\ocint{-\infty,k\tbf{1}}=1=\nu\BbbR^n$.}

\noindent Also $\Cal I=\{\ocint{-\infty,a}:a\in\BbbR^n\}$ is closed
under finite intersections, and $\nu$ and $\nuprime$ agree on $\Cal I$.
By the Monotone Class Theorem (or rather, its corollary 136C), $\nu$ and
$\nuprime$ agree on the $\sigma$-algebra generated by $\Cal I$, which is
$\Cal B$ (121J), and are identical (256D).
}%end of proof of 271B

\leader{271C}{Definition} Let $(\Omega,\Sigma,\mu)$ be a probability
space and $X_1,\ldots,X_n$ real-valued random variables on $\Omega$.
By the ({\bf joint}) {\bf distribution} or {\bf law} $\nu_{\pmb{X}}$ of
the family $\pmb{X}=(X_1,\ldots,X_n)$ I shall mean the Radon probability
measure $\nu$ of 271B.   \cmmnt{If we think of $\pmb{X}$ as a function
from $\bigcap_{i\le n}\dom X_i$ to $\BbbR^n$, then
$\nu_{\pmb{X}}E=\Pr(\pmb{X}\in E)$ for every Borel set
$E\subseteq\BbbR^n$.}

\leader{271D}{Remarks }\cmmnt{{\bf (a)} The choice of the Radon
probability
measure $\nu_{\pmb{X}}$ as `the' distribution of $\pmb{X}$, with the
insistence that `Radon measures' should be complete, is of course
somewhat arbitrary.   Apart from the general principle that one should
always complete measures, these conventions fit better with some of the
work in Volume 4 and with such results as 272G below.

\spheader 271Db
Observe that in order to speak of the distribution of a family
$\pmb{X}=(X_1,\ldots,X_n)$ of random variables, it is essential that all
the $X_i$ should be based on the same probability space.

\spheader 271Dc I see that the language I have chosen
allows the
$X_i$ to have different domains, so that the family $(X_1,\ldots,X_n)$
may not be exactly identifiable with the corresponding function from
$\bigcap_{i\le n}\dom X_i$ to $\BbbR^n$.   I hope however that using
the same symbol $\pmb{X}$ for both will cause no confusion.

\spheader 271Dd It is not useful to think of the whole image
measure
$\nu_0=\hat\mu_D\pmb{X}^{-1}$ in the proof of 271B as the distribution
of $\pmb{X}$, unless it happens to be equal to $\nu=\nu_{\pmb{X}}$.
The `distribution' of a random variable is exactly that aspect of it
which can be divorced from any consideration of the underlying space
$(\Omega,\Sigma,\mu)$, and the point of such results as 271K and 272G is
that distributions can be calculated from each other, without going back
to the relatively fluid and uncertain model of a random variable in
terms of a function on a probability space.

\medskip

}{\bf (e)} %271De
If $\pmb{X}=(X_1,\ldots,X_n)$ and
$\pmb{Y}=(Y_1,\ldots,Y_n)$ are such that $X_i\eae Y_i$ for each $i$,
then\cmmnt{

\Centerline{$\{\omega:\omega\in\bigcap_{i\le n}\dom X_i,
  \,X_i(\omega)\le\alpha_i\Forall i\le n\}
\symmdiff\{\omega:\omega\in\bigcap_{i\le n}\dom Y_i,
  \,Y_i(\omega)\le\alpha_i\Forall i\le n\}$}

\noindent is negligible, so

$$\eqalign{\Pr(X_i\le\alpha_i\Forall i\le n)
&=\hat\mu\{\omega:\omega\in\bigcap_{i\le n}\dom X_i,
  \,X_i(\omega)\le\alpha_i\Forall i\le n\}\cr
&=\Pr(Y_i\le\alpha_i\Forall i\le n)\cr}$$

\noindent for all $\alpha_0,\ldots,\alpha_n\in\Bbb R$,
and} $\nu_{\pmb{X}}=\nu_{\pmb{Y}}$.   \cmmnt{This means that we can,
if we wish, think of a distribution as a measure $\nu_{\pmb{u}}$ where
$\pmb{u}=(u_0,\ldots,u_n)$ is a finite sequence in $L^0(\mu)$.   In the
present chapter I shall not emphasize this approach, but it will always
be at the back of my mind.
}%end of comment

\leader{271E}{Measurable functions of random variables:  Proposition}
Let $\pmb{X}=(X_1,\ldots,X_n)$ be a family of random variables
\cmmnt{(as always in such a context, I mean them all to be on the
same probability space $(\Omega,\Sigma,\mu)$)};  write $\Tau_{\pmb{X}}$
for the domain of the distribution $\nu_{\pmb{X}}$, and let $h$
be a $\Tau_{\pmb{X}}$-measurable real-valued function defined
$\nu_{\pmb{X}}$-a.e.\ on $\BbbR^n$.   Then we have a random
variable $Y=h(X_1,\ldots,X_n)$ defined by setting

\Centerline{$h(X_1,\ldots,X_n)(\omega)
=h(X_1(\omega),\ldots,X_n(\omega))$ for every
$\omega\in\pmb{X}^{-1}[\dom h]$.}

\noindent  The distribution $\nu_Y$ of $Y$ is the measure
on $\Bbb R$ defined by the formula

\Centerline{$\nu_YF=\nu_{\pmb{X}}h^{-1}[F]$}

\noindent for just those sets $F\subseteq\Bbb R$ such that
$h^{-1}[F]\in\Tau_{\pmb{X}}$.   Also

\Centerline{$\Expn(Y)=\int h\,d\nu_{\pmb{X}}$}

\noindent in the sense that if one of these exists in
$[-\infty,\infty]$, so does the other, and they are then equal.

\proof{{\bf (a)(i)} Once again, write $(\Omega,\hat\Sigma,\hat\mu)$ for the
completion of $(\Omega,\Sigma,\mu)$.   Since

\Centerline{$\Omega\setminus\dom Y
\subseteq\bigcup_{i\le n}(\Omega\setminus\dom X_i)
  \cup\pmb{X}^{-1}[\BbbR^n\setminus\dom h]$}

\noindent is negligible (using 271Bb), $\dom Y$ is
conegligible.   If $a\in\Bbb R$, then

\Centerline{$E=\{x:x\in\dom h,\,h(x)\le a\}\in\Tau_{\pmb{X}},$}

\noindent so

\Centerline{$\{\omega:\omega\in\Omega,\,Y(\omega)\le a\}
=\pmb{X}^{-1}[E]\in\hat\Sigma$.}

\noindent As $a$ is arbitrary, $Y$ is
$\hat\Sigma$-measurable, and is a random variable.

\medskip

\quad{\bf (ii)} Let $\tilde h:\BbbR^n\to\Bbb R$ be any extension of $h$
to the whole of $\BbbR^n$.   Then $\tilde h$ is
$\Tau_{\pmb{X}}$-measurable, so the ordinary image measure
$\nu_{\pmb{X}}\tilde h^{-1}$, defined on
$\{F:\tilde h^{-1}[F]\in\dom\nu_{\pmb{X}}\}$, is a Radon probability
measure on $\Bbb R$ (256G).   But for any $A\subseteq\Bbb R$,

\Centerline{$\tilde h^{-1}[A]\symmdiff h^{-1}[A]
\subseteq\BbbR^n\setminus\dom h$}

\noindent is $\nu_{\pmb{X}}$-negligible, so
$\nu_{\pmb{X}}h^{-1}[F]=\nu_{\pmb{X}}\tilde h^{-1}[F]$ if either is
defined.

If $F\subseteq\Bbb R$ is a Borel set, then

\Centerline{$\nu_YF=\hat\mu\{\omega:Y(\omega)\in F\}
=\hat\mu(\pmb{X}^{-1}[h^{-1}[F]])=\nu_{\pmb{X}}(h^{-1}[F])$.}

\noindent So $\nu_Y$ and $\nu_{\pmb{X}}\tilde h^{-1}$ agree on the Borel
sets and are equal (256D again).

\medskip

{\bf (b)} Now apply Theorem 235E to the measures $\hat\mu$ and
$\nu_{\pmb{X}}$ and the function $\phi=\pmb{X}$.   We have

\Centerline{$\int\chi(\pmb{X}^{-1}[F])d\hat\mu
=\hat\mu(\pmb{X}^{-1}[F])=\nu_{\pmb{X}}F$}

\noindent for every $F\in\Tau_{\pmb{X}}$, by 271Bb.   Because $h$ is
$\nu_{\pmb{X}}$-virtually measurable and defined $\nu_{\pmb{X}}$-a.e.,
235Eb tells us that

\Centerline{$\int h(\pmb{X})d\mu=\int h(\pmb{X})d\hat\mu
=\int h\,d\nu_{\pmb{X}}$}

\noindent whenever either side is defined in $[-\infty,\infty]$,
which is exactly the result we need.
}%end of proof of 271E

\vleader{72pt}{271F}{Corollary} If $X$ is a single random variable with
distribution $\nu_{X}$, then

\Centerline{$\Expn(X)=\int_{-\infty}^{\infty}x\,\nu_X(dx)$}

\noindent if either is defined in $[-\infty,\infty]$.   Similarly

\Centerline{$\Expn(X^2)=\int_{-\infty}^{\infty} x^2\,\nu_X(dx)$}

\noindent (whatever $X$ may be).   If $X$, $Y$ are two random
variables\cmmnt{ (on the same
probability space!)} then we have

\Centerline{$\Expn(X\times Y)=\int xy\,\nu_{(X,Y)}d(x,y)$}

\noindent if either side is defined in $[-\infty,\infty]$.

\medskip

\noindent{\bf Remark}\dvAnew{2013} If $\nu$ is the distribution of a
real-valued random variable, that is, a Radon probability measure
on $\Bbb R$, I will say that the {\bf expectation} $\Expn(\nu)$ of $\nu$ is
$\int_{-\infty}^{\infty}x\,\nu(dx)$ if this is defined;  if $\nu$ has
finite expectation, then its {\bf variance} $\Var(\nu)$ will be
$\int x^2\,\nu(dx)-(\Expn(\nu))^2$.   Thus if $X$ is a real-valued random
variable with distribution $\nu_X$, $\Expn(X)=\Expn(\nu_X)$ and
$\Var(X)=\Var(\nu_X)$ whenever these are defined.

\leader{271G}{Distribution functions (a)} If $X$ is a real-valued random
variable, its {\bf distribution
function} is the function $F_X:\Bbb R\to[0,1]$ defined by setting

\Centerline{$F_X(a)=\Pr(X\le a)=\nu_X\ocint{-\infty,a}$}

\noindent for every $a\in\Bbb R$.    ({\bf Warning!}  some authors
prefer $F_X(a)=\Pr(X<a)$.)   \cmmnt{Observe that $F_X$ is
non-decreasing, that $\lim_{a\to-\infty}F_X(a)=0$, that
$\lim_{a\to\infty}F_X(a)=1$ and that $\lim_{x\downarrow a}F_X(x)=F_X(a)$
for every $a\in\Bbb R$.    By 271Ba,} $X$ and $Y$ have the same
distribution iff $F_X=F_Y$.

\spheader 271Gb
If $X_1,\ldots,X_n$ are real-valued random variables on the same
probability space, their {\bf (joint) distribution function} is the
function $F_{\pmb{X}}:\BbbR^n\to[0,1]$ defined by writing

\Centerline{$F_{\pmb{X}}(a)=\Pr(X_i\le\alpha_i\Forall i\le n)$}

\noindent whenever $a=(\alpha_1,\ldots,\alpha_n)\in\BbbR^n$.   If
$\pmb{X}$ and $\pmb{Y}$ have the same distribution function, they have
the same distribution\cmmnt{, by the $n$-dimensional version of
271B}.

\leader{271H}{Densities} Let $\pmb{X}=(X_1,\ldots,X_n)$ be a family of
random
variables, all defined on the same probability space.   A {\bf density
function}
for $(X_1,\ldots,X_n)$ is a Radon-Nikod\'ym derivative, with respect to
Lebesgue measure, for the distribution
$\nu_{\pmb{X}}$\cmmnt{;  that is, a
non-negative function $f$, integrable with
respect to Lebesgue measure $\mu_L$ on $\BbbR^n$, such that

\Centerline{$\int_Efd\mu_L=\nu_{\pmb{X}}E=\Pr(\pmb{X}\in E)$}

\noindent for every Borel set $E\subseteq\BbbR^n$ (256J) -- if there is
such a function, of course}.

\leader{271I}{Proposition} Let $\pmb{X}=(X_1,\ldots,X_n)$ be a family of
random variables, all defined on the same probability space.   Write
$\mu_L$ for Lebesgue measure on $\BbbR^n$.

(a) There is a density function for $\pmb{X}$ iff $\Pr(\pmb{X}\in E)=0$
for every Borel set $E$ such that $\mu_LE=0$.

(b) A non-negative Lebesgue integrable function $f$ is a density
function for $\pmb{X}$ iff
$\int_{\ocint{-\infty,a}}fd\mu_L=\Pr(\pmb{X}\in\ocint{-\infty,a})$ for
every $a\in\BbbR^n$.

(c) Suppose that $f$ is a density function for $\pmb{X}$, and
$G=\{x:f(x)>0\}$.
Then if $h$ is a Lebesgue measurable real-valued function defined almost
everywhere in $G$,

\Centerline{$\Expn(h(\pmb{X}))=\int h\,d\nu_{\pmb{X}}
=\int h\times fd\mu_L$}

\noindent if any of the three integrals is defined in
$[-\infty,\infty]$, interpreting $(h\times f)(x)$ as $0$ if $f(x)=0$ and
$x\notin\dom h$.

\proof{{\bf (a)} Apply 256J to the Radon probability measure
$\nu_{\pmb{X}}$.

\medskip

{\bf (b)} Of course the condition is necessary.   If
it is satisfied, then (by B.Levi's theorem)

\Centerline{$\int fd\mu_L
=\lim_{k\to\infty}\int_{\ocint{-\infty,k\pmb{1}}}fd\mu_L
=\lim_{k\to\infty}\nu_{\pmb{X}}\ocint{-\infty,k\pmb{1}}
=1$.}

\noindent So we have a Radon probability measure $\nu$ defined by
writing

\Centerline{$\nu E=\int_Efd\mu_L$}

\noindent whenever $E\cap\{x:f(x)>0\}$ is Lebesgue measurable (256E).
We are supposing that
$\nu\ocint{-\infty,a}=\nu_{\pmb{X}}\ocint{-\infty,a}$ for every
$a\in\BbbR^n$;  by 271Ba, as usual, $\nu=\nu_{\pmb{X}}$, so

\Centerline{$\int_Efd\mu_L=\nu E=\nu_{\pmb{X}}E
=\Pr(\pmb{X}\in E)$}

\noindent for every Borel set $E\subseteq\BbbR^n$, and $f$ is a density
function for $\pmb{X}$.

\medskip

{\bf (c)} By 256E, $\nu_{\pmb{X}}$ is the indefinite-integral measure
over $\mu$ associated with $f$.   So, writing
$G=\{x:f(x)>0\}$, we have

\Centerline{$\int h\,d\nu_{\pmb{X}}=\int h\times fd\mu_L$}

\noindent whenever either is defined in $[-\infty,\infty]$ (235K).
By 234La, $h$ is
$\Tau_{\pmb{X}}$-measurable and defined $\nu_{\pmb{X}}$-almost
everywhere, where $\Tau_{\pmb{X}}=\dom\nu_{\pmb{X}}$, so
$\Expn(h(\pmb{X}))=\int h\,d\nu_{\pmb{X}}$ by 271E.
}%end of proof of 271I

\leader{271J}{}\cmmnt{ The machinery developed in \S263 is sufficient
to give a very general result on the densities of random variables of
the form $\phi(\pmb{X})$, as follows.

\medskip

\noindent}{\bf Theorem} Let $\pmb{X}=(X_1,\ldots,X_n)$ be a family of
random variables, and $D\subseteq\BbbR^n$ a Borel set such that
$\Pr(\pmb{X}\in D)=1$.   Let $\phi:D\to\BbbR^n$ be a
function which is differentiable relative to its domain everywhere
in $D$;  for $x\in D$, let $T(x)$ be a derivative of $\phi$ at $x$, and
set $J(x)=|\det T(x)|$.   Suppose that $J(x)\ne 0$ for each $x\in D$,
and that $\pmb{X}$ has a density function $f$;  and suppose moreover
that $\sequence{k}{D_k}$ is a disjoint sequence of Borel sets, with
union $D$, such that $\phi_k=\phi\restr D_k$ is injective for every $k$.
Then $\phi(\pmb{X})$ has a density function $g=\sum_{k=0}^{\infty}g_k$
where

$$\eqalignno{g_k(y)&=\Bover{f(\phi_k^{-1}(y))}{J(\phi_k^{-1}(y))}
\text{ for }y\in\phi[D_k\cap\dom f],\cr
&=0\text{ for }y\in\BbbR^n\setminus\phi[D_k].\cr}$$

\proof{ By 262Ia, $\phi$ is continuous, therefore Borel measurable, so
$\phi(\pmb{X})$ is a random variable.

For the moment, fix $k\in\Bbb N$ and a Borel set $F\subseteq\BbbR^n$.
By 263D(iii), $\phi[D_k]$ is
measurable, and by 263D(ii) $\phi[D_k\setminus\dom f]$ is negligible.
The function $g_k$ is such that $f(x)=J(x)g_k(\phi(x))$ for
every $x\in D_k\cap\dom f$, so by 263D(v) we have

$$\eqalign{\int_Fg_k\,d\mu
&=\int_{\phi[D_k]} g_k\times\chi F\,d\mu
=\int_{D_k} J(x)g_k(\phi(x))\chi F(\phi(x))\mu(dx)\cr
&=\int_{D_k\cap\phi^{-1}[F]}fd\mu
=\Pr(\pmb{X}\in D_k\cap\phi^{-1}[F]).\cr}$$

\noindent (The integral
$\int_{\phi[D_k]} g_k\times\chi F$ is defined because
$\int_{D_k}J\times(g_k\times\chi F)\phi$ is defined, and the integral
$\int
g_k\times\chi F$ is defined because $\phi[D_k]$ is measurable and $g$ is
zero off $\phi[D_k]$.)

Now sum over $k$.   Every $g_k$ is non-negative, so by B.Levi's theorem
(123A, 123Xa)

$$\eqalign{\int_Fg\,d\mu
&=\sum_{k=0}^{\infty}\int_Fg_k\,d\mu
=\sum_{k=0}^{\infty}\Pr(\pmb{X}\in D_k\cap\phi^{-1}[F])\cr
&=\Pr(\pmb{X}\in\phi^{-1}[F])
=\Pr(\phi(\pmb{X})\in F).\cr}$$

\noindent As $F$ is arbitrary, $g$ is a density function for
$\phi(\pmb{X})$, as
claimed.
}%end of proof of 271J

\leader{271K}{}\cmmnt{ The application of the last theorem to ordinary
transformations is sometimes indirect, so I give an example.

\medskip

\noindent}{\bf Proposition} Let $X$, $Y$ be two random variables with a
joint density function $f$.   Then $X\times Y$ has a density function
$h$, where

\Centerline{$h(u)
=\int_{-\infty}^{\infty}\Bover1{|v|}f(\bover{u}{v},v)dv$}

\noindent whenever this is defined in $\Bbb R$.

\proof{ Set $\phi(x,y)=(xy,y)$ for $x$, $y\in\BbbR^2$.   Then $\phi$ is
differentiable, with derivative $T(x,y)=\Matrix{y&x\\0&1}$, so
$J(x,y)=|\det T(x,y)|=|y|$.   Set $D=\{(x,y):y\ne 0\}$;  then $D$ is a
conegligible Borel set in $\BbbR^2$ and $\phi\restr D$ is injective.
Now $\phi[D]=D$ and $\phi^{-1}(u,v)=(\bover{u}{v},v)$ for $v\ne 0$.   So
$\phi(X,Y)=(X\times Y,Y)$ has a density function $g$, where

\Centerline{$g(u,v)=\Bover{f(u/v,v)}{|v|}$ if $v\ne 0$.}

To find a density function for $X\times Y$, we calculate

\Centerline{$\Pr(X\times Y\le a)
=\int_{\ocint{-\infty,a}\times\Bbb R}g
=\int_{-\infty}^a\int_{-\infty}^{\infty}g(u,v)dv\,du
=\int_{-\infty}^ah$}

\noindent by Fubini's theorem (252B, 252C).   In particular, $h$ is
defined
and finite almost everywhere;  and by 271Ib it is a density function for
$X\times Y$.
}%end of proof of 271K

\leader{*271L}{}\cmmnt{ When a random variable is presented as the
limit of a sequence of random variables the following can be very
useful.

\medskip

\noindent}{\bf Proposition} Let $\sequencen{X_n}$ be a sequence of
real-valued random variables converging in measure to a random variable
$X$\cmmnt{ (definition:  245A)}.   Writing $F_{X_n}$, $F_X$ for the
distribution functions of $X_n$, $X$ respectively,

\Centerline{$F_X(a)=\inf_{b>a}\liminf_{n\to\infty}F_{X_n}(b)
=\inf_{b>a}\limsup_{n\to\infty}F_{X_n}(b)$}

\noindent for every $a\in\Bbb R$.

\proof{ Set $\gamma=\inf_{b>a}\liminf_{n\to\infty}F_{X_n}(b)$,
$\gamma'=\inf_{b>a}\limsup_{n\to\infty}F_{X_n}(b)$.

\medskip

{\bf (a)} $F_X(a)\le\gamma$.   \Prf\ Take any $b>a$ and $\epsilon>0$.
Then there is an $n_0\in\Bbb N$ such that
$\Pr(|X_n-X|\ge b-a)\le\epsilon$ for every $n\ge n_0$ (245F).   Now, for
$n\ge n_0$,

\Centerline{$F_X(a)
=\Pr(X\le a)
\le\Pr(X_n\le b)+\Pr(X_n-X\ge b-a)
\le F_{X_n}(b)+\epsilon$.}

\noindent So $F_X(a)\le\liminf_{n\to\infty}F_{X_n}(b)+\epsilon$;  as
$\epsilon$ is arbitrary, $F_X(a)\le\liminf_{n\to\infty}F_{X_n}(b)$;  as
$b$ is arbitrary, $F_X(a)\le\gamma$.\ \Qed

\medskip

{\bf (b)} $\gamma'\le F_X(a)$.   \Prf\ Let $\epsilon>0$.   Then there is
a $\delta>0$ such that $F_X(a+2\delta)\le F_X(a)+\epsilon$ (271Ga).
Next, there is an $n_0\in\Bbb N$ such that
$\Pr(|X_n-X|\ge\delta)\le\epsilon$ for every $n\ge n_0$.
In this case, for $n\ge n_0$,

$$\eqalign{F_{X_n}(a+\delta)
&=\Pr(X_n\le a+\delta)
\le\Pr(X\le a+2\delta)+\Pr(X-X_n\ge\delta)\cr
&\le F_X(a+2\delta)+\epsilon
\le F_X(a)+2\epsilon.\cr}$$

\noindent Accordingly

\Centerline{$\gamma'\le\limsup_{n\to\infty}F_{X_n}(a+\delta)
\le F_X(a)+2\epsilon$.}

\noindent As $\epsilon$ is arbitrary, $\gamma'\le F_X(a)$.\ \Qed

\medskip

{\bf (c)} Since of course $\gamma\le\gamma'$, we must have
$F_X(a)=\gamma=\gamma'$, as claimed.
}%end of proof of 271L

\exercises{
\leader{271X}{Basic exercises $\pmb{>}$(a)}
%\spheader 271Xa
Let $X$ be a real-valued random variable with finite expectation, and
$\epsilon>0$.   Show that
$\Pr(|X-\Expn(X)|\ge\epsilon)\le\Bover1{\epsilon^2}\Var(X)$.
(This is {\bf Chebyshev's inequality}.)
%271A

\sqheader 271Xb Let $F:\Bbb R\to[0,1]$ be a non-decreasing
function such that
(i) $\lim_{a\to-\infty}F(a)=0$ (ii) $\lim_{a\to\infty}F(a)=1$ (iii)
$\lim_{x\downarrow a}F(x)=F(a)$ for every $a\in\Bbb R$.   Show that
there is a unique Radon probability measure $\nu$ in $\Bbb R$ such that
$F(a)=\nu\ocint{-\infty,a}$ for every
$a\in\Bbb R$.   \Hint{114Xa.}   Hence show that $F$
is the distribution function of some random variable.
%271G

\sqheader 271Xc Let $X$ be a real-valued random variable with a density
function $f$.   (i) Show
that $|X|$ has a density function $g_1$ where $g_1(x)=f(x)+f(-x)$
whenever $x\ge 0$
and $f(x)$, $f(-x)$ are both defined, $0$ otherwise.   (ii) Show that
$X^2$ has a density function $g_2$ where
$g_2(x)=(f(\sqrt{x})+f(-\sqrt{x}))/2\sqrt{x}$ whenever $x>0$ and this is
defined, $0$ for other $x$.   (iii) Show that if $\Pr(X=0)=0$ then $1/X$
has a density function $g_3$ where $g_3(x)=\bover1{x^2}f(\bover1x)$
whenever this is
defined.   (iv) Show that if $\Pr(X<0)=0$ then $\sqrt{X}$ has a density
function
$g_4$ where $g_4(x)=2xf(x^2)$ if $x\ge 0$ and $f(x^2)$ is defined, $0$
otherwise.
%271J

\sqheader 271Xd Let $X$ and $Y$ be random variables with a joint density
function $f:\BbbR^2\to\Bbb R$.   Show that $X+Y$ has a density function
$h$ where $h(u)=\int f(u-v,v)dv$ for almost every $u$.
%271K

\spheader 271Xe Let $X$, $Y$ be random variables with a joint density
function
$f:\BbbR^2\to\Bbb R$.   Show that $X/Y$ has a density function $h$ where
$h(u)=\int|v|f(uv,v)dv$ for almost every $u$.
%271K

\spheader 271Xf Devise an alternative proof of 271K by using Fubini's
theorem and one-dimensional substitutions to show that

\Centerline{$\int_a^b\int_{-\infty}^{\infty}
\Bover1{|v|}f(\bover{u}{v},v)dv\,du
=\int_{\{(u,v):a\le uv\le b\}}f$}

\noindent whenever $a\le b$ in $\Bbb R$.
%271K

\leader{271Y}{Further exercises (a)}
%\spheader 271Ya
Let $\frak T$ be the topology of
$\BbbR^{\Bbb N}$ and $\Cal B$ the $\sigma$-algebra of Borel sets
(256Yf).   (i) Let $\Cal I$ be the family of sets of the form

\Centerline{$\{x:x\in\BbbR^{\Bbb N},\,
x(i)\le \alpha_i\Forall i\le n\}$,}

\noindent where $n\in\Bbb N$ and $\alpha_i\in\Bbb R$
for each $i\le n$.   Show that $\Cal B$ is the smallest family of
subsets of $\BbbR^{\Bbb N}$ such that ($\alpha$)
$\Cal I\subseteq\Cal B$
($\beta$) $B\setminus A\in\Cal B$ whenever $A$, $B\in\Cal B$ and
$A\subseteq B$ ($\gamma$) $\bigcup_{k\in\Bbb N}A_k\in\Cal B$ for every
non-decreasing sequence $\sequence{k}{A_k}$ in $\Cal B$.   (ii) Show
that if $\mu$, $\mu'$ are two totally finite measures defined on
$\BbbR^{\Bbb N}$, and
$\mu F$ and $\mu' F$ are defined and equal for every $F\in\Cal I$, then
$\mu E$ and $\mu' E$ are defined and equal for every $E\in\Cal B$.
(iii) Show that if $\Omega$ is a set and $\Sigma$ a $\sigma$-algebra
of subsets of $\Omega$ and $X:\Omega\to\BbbR^{\Bbb N}$ is a function,
then $X^{-1}[E]\in\Sigma$ for every $E\in\Cal B$ iff $\pi_iX$ is
$\Sigma$-measurable for every $i\in \Bbb N$, where $\pi_i(x)=x(i)$ for
each $x\in\BbbR^{\Bbb N}$, $i\in \Bbb N$.   (iv) Show that if
$\pmb{X}=\sequence{i}{X_i}$ is a sequence of real-valued random
variables on a probability space $(\Omega,\Sigma,\mu)$, then there is a
unique probability measure $\nu_{\pmb{X}}^{\Cal B}$, with domain
$\Cal B$, such that
$\nu_{\pmb{X}}^{\Cal B}\{x:x(i)\le\alpha_i\Forall i\le n\}
=\Pr(X_i\le\alpha_i\Forall i\le n)$ for every
$\alpha_0,\ldots,\alpha_n\in\Bbb R$.    (v) Under the conditions of
(iv), show that there is a unique Radon measure $\nu_{\pmb{X}}$ on
$\BbbR^{\Bbb N}$ (in the sense of 256Yf) such that
$\nu_{\pmb{X}}\{x:x(i)\le\alpha_i\Forall i\le n\}
=\Pr(X_i\le\alpha_i\Forall i\le n)$ for every
$\alpha_0,\ldots,\alpha_n\in\Bbb R$.

\header{271Yb}{\bf (b)} Let $F:\BbbR^2\to[0,1]$ be a function.   Show
that the following are equiveridical:  (i) $F$ is the distribution
function
of some pair $(X_1,X_2)$ of random variables (ii) there is a probability
measure $\nu$ on $\BbbR^2$ such that $\nu\ocint{-\infty,a}=F(a)$ for
every $a\in\BbbR^2$ (iii)($\alpha$)
$F(\alpha_1,\alpha_2)+F(\beta_1,\beta_2)
\ge F(\alpha_1,\beta_2)+F(\alpha_2,\beta_1)$ whenever
$\alpha_1\le\beta_1$
and $\alpha_2\le\beta_2$ ($\beta$) $F(\alpha_1,\alpha_2)
=\lim_{\xi_1\downarrow\alpha_1,\xi_2\downarrow\alpha_2}F(\xi_1,\xi_2)$
for every $\alpha_1$, $\alpha_2$ ($\gamma$)
$\lim_{\alpha\to-\infty}F(\alpha,\beta)
=\lim_{\alpha\to-\infty}F(\beta,\alpha)=0$ for all $\beta$
($\delta$) $\lim_{\alpha\to\infty}F(\alpha,\alpha)=1$.
\Hint{for non-empty half-open intervals $\ocint{a,b}$, set
$\lambda\ocint{a,b}=F(\alpha_1,\alpha_2)+F(\beta_1,\beta_2)
  -F(\alpha_1,\beta_2)-F(\alpha_2,\beta_1)$, and
continue as in 115B-115F.}

\header{271Yc}{\bf (c)} Generalize (b) to higher dimensions, finding a
suitable formula to stand in place of that in (iii-$\alpha$) of (b).

\spheader 271Yd Let $(\Omega,\Sigma,\mu)$ be a probability space and
$\Cal F$ a filter on $\eusm L^0(\mu)$ converging to
$X_0\in\eusm L^0(\mu)$ for the topology of convergence in measure.
Show that, writing $F_X$ for the distribution function of
$X\in\eusm L^0(\mu)$,

\Centerline{$F_{X_0}(a)=\inf_{b>a}\liminf_{X\to\Cal F}F_X(b)
=\inf_{b>a}\limsup_{X\to\Cal F}F_X(b)$}

\noindent for every $a\in\Bbb R$.
%271L

\spheader 271Ye\dvAnew{2009.} Let $X$, $Y$ be non-negative random variables
with the same distribution, and $h:\coint{0,\infty}\to\coint{0,\infty}$ a
non-decreasing function.   Show that
$\Expn(X\times hY)\le\Expn(Y\times hY)$.
\Hint{in the language of 252Yo, $(Y\times hY)^*=Y^*\times(hY)^*$.}
}%end of exercises

\endnotes{
\Notesheader{271} Most of this section seems to have been taken up with
technicalities.   This is perhaps unsurprising in view of the fact that
it is devoted to the relationship between a vector random variable
$\pmb{X}$ and the associated distribution $\nu_{\pmb{X}}$, and this
necessarily leads us into the minefield which I attempted to chart in
\S235.   Indeed, I call on results from \S235 twice;  once in 271E,
with a $\phi(\omega)=\pmb{X}(\omega)$ and $J(\omega)=1$, and once in
271I, with $\phi(x)=x$ and $J(x)=f(x)$.

Distribution functions of one-dimensional random variables are easily
characterized (271Xb);  in higher dimensions we have to work harder
(271Yb-271Yc).   Distributions, rather than distribution functions, can
be described for infinite sequences of random variables (271Ya);
indeed, these ideas can be extended to uncountable families, but this
requires proper topological measure theory, and belongs in Volume 4.

The statement of 271J is elaborate, not to say cumbersome.  The point is
that many of the most important transformations $\phi$ are not
themselves injective, but can easily be dissected into injective
fragments (see, for instance, 271Xc and 263Xd).   The point of 271K is
that we frequently wish to apply the ideas here to
transformations which are singular, and indeed change the dimension of
the random variable.   I have not given the theorems which make such
applications routine and suggest rather that you seek out tricks such as
that used in the proof of 271K, which in any case are necessary if you
want amenable formulae.   Of course other methods are available (271Xf).
}%end of notes

\discrpage

\frfilename{mt272.tex}
\versiondate{3.4.09}
\copyrightdate{2000}

\def\hatcalBnu{\hat\Cal B_{\nu}}

\def\chaptername{Probability theory}
\def\sectionname{Independence}

\newsection{272}

I introduce the concept of `independence' for families of events,
$\sigma$-algebras and random variables.   The first part of the section,
down to 272G, amounts to an analysis of the elementary relationships
between the three manifestations of the idea.   In 272G I give the
fundamental result that the joint distribution of a (finite) independent
family of random variables is just the product of the individual
distributions.   Further expressions of the connexion between
independence and product measures are in 272J, 272M and 272N.
I give a
version of the zero-one law (272O), and I end the section with a
group of basic results from probability theory concerning sums and
products of independent random variables (272R-272W).
%272R 272S 272T 272U 272V 272W

\vleader{48pt}{272A}{Definitions} Let $(\Omega,\Sigma,\mu)$ be a 
probability space.

\header{272Aa}{\bf (a)} A family $\langle E_i\rangle_{i\in I}$ in
$\Sigma$ is {\bf (stochastically) independent} if

\Centerline{$\mu(E_{i_1}\cap E_{i_2}\cap\ldots\cap E_{i_n})
=\prod_{j=1}^n\mu E_{i_j}$}

\noindent whenever $i_1,\ldots,i_n$ are distinct members of $I$.

\header{272Ab}{\bf (b)} A family $\langle\Sigma_i\rangle_{i\in I}$ of
$\sigma$-subalgebras of $\Sigma$ is {\bf (stochastically) independent}
if

\Centerline{$\mu(E_{1}\cap E_{2}\cap\ldots\cap E_{n})
=\prod_{j=1}^n\mu E_{j}$}

\noindent whenever $i_1,\ldots,i_n$ are distinct members of $I$ and
$E_j\in\Sigma_{i_j}$ for every $j\le n$.

\header{272Ac}{\bf (c)} A family $\langle X_i\rangle_{i\in I}$ of
real-valued random variables on $\Omega$ is {\bf (stochastically)
independent} if

\Centerline{$\Pr(X_{i_j}\le\alpha_j\text{ for every }j\le n)
=\prod_{j=1}^n\Pr(X_{i_j}\le \alpha_j)$}

\noindent whenever $i_1,\ldots,i_n$ are distinct members of $I$ and
$\alpha_1,\ldots,\alpha_n\in\Bbb R$.

\cmmnt{
\leader{272B}{Remarks (a)} This is perhaps the central
contribution of
probability theory to measure theory, and as such deserves the most
careful scrutiny.   The idea of `independence' comes from outside
mathematics altogether, in the notion of events which have independent
causes.  I suppose that 272G and 272M are the results below which most
clearly show the measure-theoretic aspects of the concept.   It is not
an accident that both involve product measures;  one of the wonders of
measure theory is the fact that the same technical devices are used in
establishing the probability theory of stochastic independence and the
geometry of multi-dimensional volume.

\header{272Bb}
{\bf (b)} In the following paragraphs I will try to describe some
relationships between the three notions of independence just defined.
But it is worth noting at once the fact that, in all three cases, a
family is independent iff all its finite subfamilies are independent.
Consequently any subfamily of an independent family is independent.
Another elementary fact which is immediate from the definitions is that
if $\langle \Sigma_i\rangle_{i\in I}$ is an independent family of
$\sigma$-algebras, and $\Sigma'_i$ is a $\sigma$-subalgebra of
$\Sigma_i$ for each $i$, then $\langle\Sigma'_i\rangle_{i\in I}$ is an
independent family.

\header{272Bc}
{\bf (c)} A useful reformulation of 272Ab is the following:  A family
$\langle\Sigma_i\rangle_{i\in I}$ of $\sigma$-subalgebras of $\Sigma$ is
independent iff

\Centerline{$\mu(\bigcap_{i\in I}E_i)=\prod_{i\in I}\mu E_i$}

\noindent whenever $E_i\in\Sigma_i$ for every $i$ and
$\{i:E_i\ne\Omega\}$ is finite.   (Here I follow the convention of 254F,
saying that for a family $\langle\alpha_i\rangle_{i\in I}$ in $[0,1]$ we
take $\prod_{i\in I}\alpha_i=1$ if $I=\emptyset$, and otherwise it is to
be $\inf_{J\subseteq I,J\text{ is finite}}\prod_{i\in J}\alpha_j$.)

\header{272Bd}
{\bf (d)} In 272Aa-b I speak of sets $E_i\in\Sigma$ and algebras
$\Sigma_i\subseteq\Sigma$.   In fact (272Ac already gives a hint of
this) we shall more often than not be concerned with $\hat\Sigma$
rather than with $\Sigma$, if there is a difference, where
$(\Omega,\hat\Sigma,\hat\mu)$ is the completion of
$(\Omega,\Sigma,\mu)$.
}%end of comment

\leader{272C}{The $\sigma$-subalgebra defined by a random
variable}\cmmnt{ To relate
272Ab to 272Ac we need the following notion.}   Let
$(\Omega,\Sigma,\mu)$
be a probability space and $X$ a real-valued random variable defined on
$\Omega$.   Write $\Cal B$ for the
$\sigma$-algebra of Borel subsets of $\Bbb R$, and $\Sigma_X$ for

\Centerline{$\{X^{-1}[F]:F\in\Cal B\}
\cup\{(\Omega\setminus\dom X)\cup X^{-1}[F]:F\in\Cal B\}$.}

\noindent Then $\Sigma_X$ is a $\sigma$-algebra of subsets of $\Omega$.
\prooflet{\Prf\

\Centerline{$\emptyset=X^{-1}[\emptyset]\in\Sigma_X$;}

\noindent if $F\in\Cal B$ then

\Centerline{$\Omega\setminus X^{-1}[F]
=(\Omega\setminus\dom X)\cup X^{-1}[\Bbb R\setminus F]\in\Sigma_X$,}

\Centerline{$\Omega\setminus((\Omega\setminus\dom X)\cup X^{-1}[F])
=X^{-1}[\Bbb R\setminus F]\in\Sigma_X$;}

\noindent if $\sequence{k}{F_k}$ is any sequence in $\Cal B$ then

\Centerline{$\bigcup_{k\in\Bbb N}X^{-1}[F_k]=X^{-1}[\bigcup_{k\in\Bbb
N}F_k]$,}

\noindent so

\Centerline{$\bigcup_{k\in\Bbb N}X^{-1}[F_k]$,
\quad $(\Omega\setminus\dom X)\cup\bigcup_{k\in\Bbb N}X^{-1}[F_k]$}

\noindent belong to $\Sigma_X$.   \Qed}

\cmmnt{Evidently }$\Sigma_X$ is the smallest $\sigma$-algebra of
subsets of $\Omega$, containing $\dom X$, for which $X$ is measurable.
\cmmnt{Also }$\Sigma_X$ is a subalgebra of
$\hat\Sigma$, where $\hat\Sigma$ is the domain of the completion of
$\mu$\cmmnt{ (271Aa)}.

\cmmnt{Now we have the following result.}

\leader{272D}{Proposition} Let $(\Omega,\Sigma,\mu)$ be a probability
space and $\langle X_i\rangle_{i\in I}$ a family of real-valued random
variables on $\Omega$.   For each $i\in I$, let $\Sigma_i$ be the
$\sigma$-algebra defined by $X_i$\cmmnt{, as in 272C}.   Then the
following are equiveridical:

(i) $\langle X_i\rangle_{i\in I}$ is independent;

(ii) whenever $i_1,\ldots,i_n$ are distinct members of $I$ and
$F_1,\ldots,F_n$ are Borel subsets of $\Bbb R$, then

\Centerline{$\Pr(X_{i_j}\in F_j\text{ for every }j\le n)
=\prod_{j=1}^n\Pr(X_{i_j}\in F_j)$;}

(iii) whenever $\langle F_i\rangle_{i\in I}$ is a family of Borel
subsets of $\Bbb R$, and $\{i:F_i\ne\Bbb R\}$ is finite, then

\Centerline{$\hat\mu\bigl(
\bigcap_{i\in I}(X_i^{-1}[F_i]\cup(\Omega\setminus\dom X_i))\bigr)
=\prod_{i\in I}\Pr(X_i\in F_i)$,}

\noindent where $\hat\mu$ is the completion of $\mu$;

(iv) $\langle\Sigma_i\rangle_{i\in I}$ is independent with respect to
$\hat\mu$.

\proof{{\bf (a)(i)$\Rightarrow$(ii)} Write
$\pmb{X}=(X_{i_1},\ldots,X_{i_n})$.   Write $\nu_{\pmb{X}}$ for the
joint distribution of $\pmb{X}$, and for each $j\le n$ write $\nu_j$ for
the distribution of $X_{i_j}$;  let  $\nu$ be the product of
$\nu_1,\ldots,\nu_n$ as described in 254A-254C.   (I
wrote \S254 out as for infinite products.   If you are interested
only in finite products of probability spaces, which are adequate for
our needs in this paragraph, I recommend reading \S\S251-252 with the
mental proviso that all measures are probabilities, and then \S254
with the proviso that the set $I$ is finite.)   By 256K, $\nu$ is a
Radon measure on $\BbbR^n$.   (This is an induction on $n$, relying on
254N for assurance that we can regard $\nu$ as the repeated product
$(\ldots((\nu_1\times\nu_2)\times\nu_3)\times\ldots\nu_{n-1})
\times\nu_n$.)   Then for any
$a=(\alpha_1,\ldots,\alpha_n)\in\BbbR^n$, we have

$$\eqalignno{\nu\ocint{-\infty,a}
&=\nu\bigl(\prod_{j=1}^n\ocint{-\infty,\alpha_j}\bigr)
=\prod_{j=1}^n\nu_{j}\ocint{-\infty,\alpha_j}\cr
\noalign{\noindent (using 254Fb)}
&=\prod_{j=1}^n\Pr(X_{i_j}\le\alpha_j)
=\Pr(X_{i_j}\le\alpha_j\text{ for every }j\le n)\cr
\noalign{\noindent (using the condition (i))}
&=\nu_{\pmb{X}}\ocint{-\infty,a}.\cr}$$

\noindent By the uniqueness assertion in 271Ba, $\nu=\nu_{\pmb{X}}$.
In particular, if $F_1,\ldots,F_n$ are Borel subsets of $\Bbb R$,

$$\eqalign{\Pr(X_{i_j}\in F_j\text{ for every }j\le n)
&=\Pr(\pmb{X}\in\prod_{j\le n}F_j)
=\nu_{\pmb{X}}(\prod_{j\le n}F_j)\cr
&=\nu(\prod_{j\le n}F_j)
=\prod_{j=1}^n\nu_{{j}}F_j
=\prod_{j=1}^n\Pr(X_{i_j}\in F_j),\cr}$$

\noindent as required.


\medskip

{\bf (b)(ii)$\Rightarrow$(i)} is trivial, if we recall that all sets
$\ocint{-\infty,\alpha}$ are Borel sets, so that the definition of
independence given in 272Ac is just a special case of (ii).

\medskip

{\bf (c)(ii)$\Rightarrow$(iv)} Assume (ii), and suppose that
$i_1,\ldots,i_n$ are distinct members of $I$ and $E_j\in\Sigma_{{i_j}}$
for each $j\le n$.    For each $j$, set $E'_j=E_j\cap\dom X_{i_j}$, so
that $E'_j$ may be expressed as $X_{i_j}^{-1}[F_j]$ for some Borel set
$F_j\subseteq\Bbb R$.   Then $\hat\mu(E_j\setminus E'_j)=0$ for each
$j$, so

$$\eqalignno{\hat\mu(\bigcap_{1\le j\le n}E_j)
&=\hat\mu(\bigcap_{1\le j\le n}E'_j)
=\Pr(X_{i_1}\in F_1,\ldots,X_{i_n}\in F_n)\cr
&=\prod_{j=1}^n\Pr(X_{i_j}\in F_j)\cr
\noalign{\noindent (using (ii))}
&=\prod_{i=1}^n\hat\mu E_j.\cr}$$

\noindent As $E_1,\ldots,E_k$ are arbitrary,
$\langle\Sigma_{i}\rangle_{i\in I}$ is independent.

\medskip

{\bf (d)(iv)$\Rightarrow$(ii)}
Now suppose that $\langle\Sigma_{i}\rangle_{i\in I}$ is independent.
If $i_1,\ldots,i_n$ are distinct members of $I$ and $F_1,\ldots,F_n$ are
Borel sets in $\Bbb R$, then $X_{i_j}^{-1}[F_j]\in\Sigma_{i_j}$ for each
$j$, so

$$\eqalignno{\Pr(X_{i_1}\in F_1,\ldots,X_{i_n}\in F_n)
&=\hat\mu(\bigcap_{1\le j\le n}X_{i_j}^{-1}[F_j])\cr
&=\prod_{i=1}^n\hat\mu X_{i_j}^{-1}[F_j]
=\prod_{j=1}^n\Pr(X_{i_j}\in F_j)\cr.}$$

\medskip

%272D
{\bf (e)} Finally, observe that (iii) is nothing but a re-formulation of
(ii), because if $F_i=\Bbb R$ then $\Pr(X_i\in F_i)=1$ and
$X_i^{-1}[F_i]\cup(\Omega\setminus\dom X_i)=\Omega$.
}%end of proof of 272D

\leader{272E}{Corollary} Let $\langle X_i\rangle_{i\in I}$ be an
independent family of real-valued random variables, and
$\langle h_i\rangle_{i\in I}$ any family of Borel measurable functions
from $\Bbb R$ to $\Bbb R$.   Then $\langle h_i(X_i)\rangle_{i\in I}$ is
independent.

\proof{ Writing $\Sigma_i$ for the $\sigma$-algebra defined by $X_i$,
$\Sigma'_i$ for the $\sigma$-algebra defined by $h(X_i)$, $h(X_i)$ is
$\Sigma_i$-measurable (121Eg) so $\Sigma'_i\subseteq\Sigma_i$ for every
$i$ and $\langle\Sigma'_i\rangle_{i\in I}$ is independent, as in 272Bb.
}%end of proof of 272E

\leader{272F}{}\cmmnt{ Similarly, we can relate the definition in
272Aa to the others.

\medskip

\noindent}{\bf Proposition} Let $(\Omega,\Sigma,\mu)$ be a probability
space, and $\langle E_i\rangle_{i\in I}$ a family in $\Sigma$.   Set
$\Sigma_i=\{\emptyset,E_i,\Omega\setminus E_i,\Omega\}$, the
($\sigma$-\nobreak)algebra of subsets of $\Omega$ generated by $E_i$, and
$X_i=\chi E_i$, the indicator function of $E_i$.   Then the
following are equiveridical:

(i) $\langle E_i\rangle_{i\in I}$ is independent;

(ii) $\langle \Sigma_i\rangle_{i\in I}$ is independent;

(iii) $\langle X_i\rangle_{i\in I}$ is independent.

\proof{{\bf (i)$\Rightarrow$(iii)} If $i_1,\ldots,i_n$ are distinct
members of $I$ and $\alpha_1,\ldots,\alpha_n\in\Bbb R$, then for each
$j\le n$ the set $G_j=\{\omega:X_{i_j}(\omega)\le\alpha_j\}$ is either
$E_{i_j}$ or $\emptyset$ or $\Omega$.   If any $G_j$ is empty, then

\Centerline{$\Pr(X_{i_j}\le\alpha_j\text{ for every}j\le
n\}=0=\prod_{j=1}^n\Pr(X_{i_j}\le\alpha_j)$.}

\noindent Otherwise, set $K=\{j:G_j=E_{i_j}\}$;  then

$$\eqalign{\Pr(X_{i_j}\le\alpha_j\text{ for every}j\le n\}
&=\mu(\bigcap_{j\le n}G_j)
=\mu(\bigcap_{j\in K}E_{i_j})\cr
&=\prod_{j\in K}\mu E_{i_j}
=\prod_{j=1}^n\Pr(X_{i_j}\le\alpha_j).\cr}$$

\noindent As $i_1,\ldots,i_n$ and $\alpha_1,\ldots,\alpha_n$ are
arbitrary, $\langle X_i\rangle_{i\in I}$ is independent.

\medskip

{\bf (iii)$\Rightarrow$(ii)} follows from (i)$\Rightarrow$(iii) of 272D,
because $\Sigma_i$ is the $\sigma$-algebra defined by $X_i$.

\medskip

{\bf (ii)$\Rightarrow$(i)} is trivial, because $E_i\in\Sigma_i$ for each
$i$.
}%end of proof of 272F

\cmmnt{\medskip

\noindent{\bf Remark} You will I hope feel that while the theory of
product measures might be appropriate to 272D, it is surely rather heavy
machinery to use on what ought to be a simple combinatorial problem like
(iii)$\Rightarrow$(ii) of this proposition.   I suggest that you
construct an `elementary' proof, and examine which of the ideas of
the theory of product measures (and the Monotone Class Theorem, 136B)
are actually needed here.
}%end of comment

\leader{272G}{Distributions of independent
random \dvrocolon{variables}}\cmmnt{ I have
not tried to describe the `joint distribution' of an infinite family of
random variables.   (Indications of how to deal with a countable family
are offered in 271Ya and 272Yb.   For uncountable families I will wait
until \S454 in Volume 4.)   As, however, the independence of a
family of random variables is determined by the behaviour of finite
subfamilies, we can approach it through the following proposition.

\medskip

\noindent}{\bf Theorem} Let $\pmb{X}=(X_1,\ldots,X_n)$ be a finite
family of real-valued random variables on a probability space.   Let
$\nu_{\pmb{X}}$ be the corresponding distribution on $\BbbR^n$.   Then
the following are equiveridical:

(i) $X_1,\ldots,X_n$ are independent;

(ii) $\nu_{\pmb{X}}$ can be expressed as a product of $n$ probability
measures $\nu_1,\ldots,\nu_n$, one for each factor $\Bbb R$ of $\Bbb
R^n$;

(iii)  $\nu_{\pmb{X}}$ is the product measure of $\nu_{X_1},\ldots,
\nu_{X_n}$, writing $\nu_{X_i}$ for the distribution of the random
variable $X_i$.

\proof{{\bf (a)(i)$\Rightarrow$(iii)} In the proof of
(i)$\Rightarrow$(ii) of 272D above I showed that
$\nu_{\pmb{X}}$ is the product $\nu$ of
$\nu_{X_1},\ldots,\nu_{X_n}$.

\medskip

{\bf (b)(iii)$\Rightarrow$(ii)} is trivial.

\medskip

{\bf (c)(ii)$\Rightarrow$(i)} Suppose that $\nu_{\pmb{X}}$ is
expressible as a product $\nu_1\times\ldots\times\nu_n$.   Take
$a=(\alpha_1,\ldots,\alpha_n)$ in $\BbbR^n$.  Then

\Centerline{$\Pr(X_i\le\alpha_i\Forall i\le n)
=\Pr(\pmb{X}\in\ocint{-\infty,a})
=\nu_{\pmb{X}}(\ocint{-\infty,a})
=\prod_{i=1}^n\nu_i\ocint{-\infty,\alpha_i}$.}

\noindent On the other hand, setting
$F_i=\{(\xi_1,\ldots,\xi_n):\xi_i\le\alpha_i\}$, we must have

\Centerline{$\nu_i\ocint{-\infty,\alpha_i}
=\nu_{\pmb{X}}F_i=\Pr(\pmb{X}\in F_i)
=\Pr(X_i\le\alpha_i)$}

\noindent for each $i$.   So we get

\Centerline{$\Pr(X_i\le\alpha_i\text{ for every }i\le n)
=\prod_{i=1}^n\Pr(X_i\le\alpha_i)$,}

\noindent as required.
}%end of proof of 272G

\leader{272H}{Corollary} Suppose that $\langle X_i\rangle_{i\in I}$ is
an independent family of real-valued random variables on a probability
space $(\Omega,\Sigma,\mu)$, and that for each $i\in I$ we are given
another real-valued random variable $Y_i$ on $\Omega$ such that
$Y_i\eae X_i$.   Then $\langle Y_i\rangle_{i\in I}$ is independent.

\proof{ For every distinct $i_1,\ldots,i_n\in I$, if we set
$\pmb{X}=(X_{i_1},\ldots,X_{i_n})$ and
$\pmb{Y}=(Y_{i_1},\ldots,Y_{i_n})$, then $\pmb{X}\eae\pmb{Y}$, so
$\nu_{\pmb{X}}$, $\nu_{\pmb{Y}}$ are equal (271De).   By 272G,
$Y_{i_1},\ldots,Y_{i_n}$ must be independent because
$X_{i_1},\ldots,X_{i_n}$ are.   As $i_1,\ldots,i_n$ are arbitrary, the
whole family $\langle Y_i\rangle_{i\in I}$ is independent.
}%end of proof of 272H

\cmmnt{\medskip

\noindent{\bf Remark} It follows that we may speak of independent
families in the space $L^0(\mu)$ of equivalence classes of
random variables (241C), saying that
$\langle X^{\ssbullet}_i\rangle_{i\in I}$ is independent iff
$\langle X_i\rangle_{i\in I}$ is.
}%end of comment

\leader{272I}{Corollary} Suppose that $X_1,\ldots,X_n$ are independent
real-valued random variables with density functions
$f_1,\ldots,f_n$\cmmnt{ (271H)}.
Then $\pmb{X}=(X_1,\ldots,X_n)$ has a density function $f$ given by
setting $f(x)=\prod_{i=1}^nf_i(\xi_i)$ whenever
$x=(\xi_1,\ldots,\xi_n)\in\prod_{i\le n}\dom(f_i)\subseteq\BbbR^n$.

\proof{ For $n=2$ this is covered by 253I;  the general case follows by
induction on $n$.
}%end of proof of 272I

\leader{272J}{}\cmmnt{ The most important theorems of the subject
refer to
independent families of random variables, rather than independent
families of $\sigma$-algebras.   The value of the concept of independent
$\sigma$-algebras lies in such results as the following.

\medskip

\noindent}{\bf Proposition} Let $(\Omega,\Sigma,\mu)$ be a complete
probability space, and $\langle\Sigma_i\rangle_{i\in I}$ a
family of $\sigma$-subalgebras of $\Sigma$.   For each $i\in I$ let
$\mu_i$ be the restriction of $\mu$ to $\Sigma_i$, and let
$(\Omega^I,\Lambda,\lambda)$ be the product probability space of the
family $\langle(\Omega,\Sigma_i,\mu_i)\rangle_{i\in I}$.   Define
$\phi:\Omega\to\Omega^I$ by setting $\phi(\omega)(i)=\omega$ whenever
$\omega\in\Omega$ and $i\in I$.   Then $\phi$ is \imp\ iff
$\langle\Sigma_i\rangle_{i\in I}$ is independent.

\proof{ This is virtually a restatement of 254Fb and 254G.   (i) If
$\phi$ is \imp, $i_1,\ldots,i_n\in I$ are distinct and
$E_j\in\Sigma_{i_j}$ for each $j$, then
$\bigcap_{j\le n}E_{i_j}=\phi^{-1}[\{x:x(i_j)\in E_j$ for every
$j\le n\}]$, so that

\Centerline{$\mu(\bigcap_{j\le n}E_{i_j})
=\lambda\{x:x(i_j)\in E_j\text{ for every }j\le n\}
=\prod_{j=1}^n\mu_{i_j}E_{i_j}
=\prod_{j=1}^n\mu E_{i_j}$.}

\noindent (ii) If $\langle\Sigma_i\rangle_{i\in I}$ is independent,
$E_i\in\Sigma_i$
for every $i\in I$ and $\{i:E_i\ne\Omega\}$ is finite, then

\Centerline{$\mu\phi^{-1}[\prod_{i\in I}E_i]
=\mu(\bigcap_{i\in I}E_i)
=\prod_{i\in I}\mu E_i
=\prod_{i\in I}\mu_iE_i$.}

\noindent So
the conditions of 254G are satisfied and $\mu\phi^{-1}[W]=\lambda W$ for
every $W\in\Lambda$.
}%end of proof of 272J

\leader{272K}{Proposition} Let $(\Omega,\Sigma,\mu)$ be a probability
space and $\langle\Sigma_i\rangle_{i\in I}$ an independent family of
$\sigma$-subalgebras of $\Sigma$.   Let
$\langle J(s)\rangle_{s\in S}$ be a disjoint family of subsets of $I$,
and for each $s\in S$ let
$\tilde\Sigma_s$ be the $\sigma$-algebra of subsets of $\Omega$
generated by $\bigcup_{i\in J(s)}\Sigma_i$.   Then
$\langle\tilde\Sigma_s\rangle_{s\in S}$ is independent.

\proof{ Let $(\Omega,\hat\Sigma,\hat\mu)$ be the completion of
$(\Omega,\Sigma,\mu)$.   On $\Omega^I$ let $\lambda$ be the product of
the measures $\mu\restr\Sigma_i$, and let $\phi:\Omega\to\Omega^I$ be
the diagonal map, as in 272J.   $\phi$ is \imp\ for $\hat\mu$ and
$\lambda$, by 272J.

We can identify $\lambda$ with the product of
$\langle\lambda_s\rangle_{s\in S}$, where for each $s\in S\,\,\lambda_s$
is the product of $\langle\mu\restr\Sigma_i\rangle_{i\in J(s)}$
(254N).  For $s\in S$, let $\Lambda_s$ be the domain of $\lambda_s$, and
set $\pi_s(x)=x\restr J(s)$ for $x\in\Omega^I$, so that $\pi_s$ is \imp\
for $\lambda$ and $\lambda_s$ (254Oa), and $\phi_s=\pi_s\phi$ is \imp\
for $\hat\mu$ and $\lambda_s$;  of course $\phi_s$ is the diagonal map
from $\Omega$ to $\Omega^{J(s)}$.   Set
$\Sigma_s^*=\{\phi_s^{-1}[H]:H\in\Lambda_s\}$.   Then $\Sigma_s^*$ is a
$\sigma$-subalgebra of $\hat\Sigma$, and
$\Sigma_s^*\supseteq\tilde\Sigma_s$, because

\Centerline{$E=\phi_s^{-1}[\{x:x(i)\in E\}]\in\Sigma^*_s$}

\noindent whenever $i\in J(s)$a dn $E\in\Sigma_i$.

Now suppose that $s_1,\ldots,s_n\in S$ are distinct and that
$E_j\in\tilde\Sigma_{s_j}$ for each $j$.   Then $E_j\in\Sigma_{s_j}^*$,
so there are $H_j\in\Lambda_{s_j}$ such that $E_j=\phi_{s_j}^{-1}[H_j]$
for each $j$.   Set

\Centerline{$W=\{x:x\in\Omega^I,\,x\restr J(s_j)\in H_j$ for every $j\le
n\}$.}

\noindent Because we can identify $\lambda$ with the product of the
$\lambda_s$, we have

\Centerline{$\lambda W=\prod_{j=1}^n\lambda_{s_j}H_j
=\prod_{j=1}^n\hat\mu(\phi_{s_j}^{-1}[H_j])
=\prod_{j=1}^n\hat\mu E_j=\prod_{j=1}^n\mu E_j$.}

\noindent On the other hand, $\phi^{-1}[W]=\bigcap_{j\le n}E_j$, so,
because $\phi$ is \imp,

\Centerline{$\mu(\bigcap_{j\le n}E_j)=\hat\mu(\bigcap_{j\le n}E_j)
=\lambda W=\prod_{j=1}^n\mu E_j$.}

\noindent As $E_1,\ldots,E_n$ are arbitrary,
$\langle\tilde\Sigma_s\rangle_{s\in S}$ is independent.
}%end of proof of 272K

\leader{272L}{}\cmmnt{ I give a typical application of this result as
a sample.

\medskip

\noindent}{\bf Corollary} Let $X,X_1,\ldots,X_n$ be independent
real-valued random
variables and $h:\BbbR^n\to\Bbb R$ a Borel 
measurable function.   Then $X$ and $h(X_1,\ldots,X_n)$ are independent.

\proof{ Let $\Sigma_X$, $\Sigma_{X_i}$ be the
$\sigma$-algebras defined by $X$, $X_i$ (272C).   Then
$\Sigma_X,\Sigma_{X_1},\ldots,\Sigma_{X_n}$ are independent (272D).
Let $\Sigma^*$ be the $\sigma$-algebra generated by
$\Sigma_{X_1}\cup\ldots\cup\Sigma_{X_n}$.   Then 272K (perhaps working
in the completion of the original probability space) tells us that
$\Sigma_X$ and $\Sigma^*$ are independent.   But every $X_j$ is
$\Sigma^*$-measurable so $Y=h(X_1,\ldots,X_n)$ is $\Sigma^*$-measurable
(121Kb);  also $\dom Y\in\Sigma^*$, so $\Sigma_Y\subseteq\Sigma^*$ and
$\Sigma_X$, $\Sigma_Y$ are independent.   By 272D again, $X$ and $Y$ are
independent, as claimed.
}%end of proof of 272L

\cmmnt{\medskip

\noindent{\bf Remark} Nearly all of us, when teaching elementary
probability theory, would invite our students to treat this corollary
(with an explicit function $h$, of course) as `obvious'.   In effect,
the proof here is a
confirmation that the formal definition of `independence' offered is
a faithful representation of our intuition of independent events having
independent causes.
}%end of comment

\leader{272M}{}{\bf Products of probability spaces and independent
families of
random \dvrocolon{variables}}\cmmnt{ We have already seen that
the concept of
`independent random variables' is intimately linked with that of
`product measure'.   I now give some further manifestations of the
connexion.

\medskip

\noindent}{\bf Proposition} Let
$\langle(\Omega_i,\Sigma_i,\mu_i)\rangle_{i\in I}$ be a family of
probability spaces, and $(\Omega,\Sigma,\mu)$ their product.

(a) For each $i\in I$ write
$\tilde\Sigma_i=\{\pi_i^{-1}[E]:E\in\Sigma_i\}$, where
$\pi_i:\Omega\to\Omega_i$ is the coordinate map.   Then
$\langle\tilde\Sigma_i\rangle_{i\in I}$ is an independent family of
$\sigma$-subalgebras of $\Sigma$.

(b) For each $i\in I$ let $\langle X_{ij}\rangle_{j\in J(i)}$ be an
independent family of real-valued random variables on
$\Omega_i$, and for $i\in I$, $j\in J(i)$ write $\tilde
X_{ij}(\omega)=X_{ij}(\omega(i))$ for those
$\omega\in\Omega$ such that $\omega(i)\in\dom X_{ij}$.   Then
$\langle \tilde X_{ij}\rangle_{i\in I,j\in J(i)}$ is an independent
family of
random variables, and each $\tilde X_{ij}$ has the same distribution as
the corresponding $X_{ij}$.

\proof{{\bf (a)}  It is easy to check that each $\tilde\Sigma_i$ is a
$\sigma$-algebra of sets.   The rest amounts just to recalling from
254Fb that if $J\subseteq I$ is finite and $E_i\in\Sigma_{i}$ for
$i\in J$, then

\Centerline{$\mu(\bigcap_{i\in J}\pi_i^{-1}[E_i])
=\mu\{\omega:\omega(i)\in E_i\text{ for every }i\in I\}
=\prod_{i\in I}\mu_{i}E_i$}

\noindent if we set $E_i=X_i$ for $i\in I\setminus J$.

\medskip

{\bf (b)} We know also that $(\Omega,\Sigma,\mu)$ is the product of the
completions $(\Omega_i,\hat\Sigma_i,\hat\mu_i)$ (254I).
From this, we see that each $\tilde X_{ij}$ is defined $\mu$-a.e., and
is $\Sigma$-measurable, with the same distribution as $X_{ij}$.    Now
apply condition (iii) of 272D.   Suppose that $\langle
F_{ij}\rangle_{i\in I,j\in J(i)}$ is a family of Borel sets in $\Bbb R$,
and that $\{(i,j):F_{ij}\ne\Bbb R\}$ is finite.   Consider

\Centerline{$E_i
=\bigcap_{j\in J(i)}
(X_{ij}^{-1}[F_{ij}]\cup(\Omega_i\setminus\dom X_{ij}))$,}

\Centerline{$E=\prod_{i\in I}E_i
=\bigcap_{i\in I,j\in J(i)}
(\tilde X_{ij}^{-1}[F_{ij}]\cup(\Omega\setminus\dom \tilde X_{ij}))$.}

\noindent Because each family $\langle X_{ij}\rangle_{j\in J(i)}$ is
independent, and $\{j:F_{ij}\ne\Bbb R\}$ is finite,

\Centerline{$\hat\mu_iE_i=\prod_{j\in J(i)}\Pr(X_{ij}\in E_{ij})$}

\noindent for each $i\in I$.   Because

\Centerline{$\{i:E_i\ne \Omega_i\}\subseteq\{i:\exists\,j\in
J(i),\,F_{ij}\ne\Bbb R\}$}

\noindent is finite,

\Centerline{$\mu E=\prod_{i\in I}\hat\mu_iE_i
=\prod_{i\in I,j\in J(i)}\Pr(\tilde X_{ij}\in F_{ij})$;}

\noindent as $\langle F_{ij}\rangle_{i\in I,j\in J(i)}$ is arbitrary,
$\langle \tilde X_{ij}\rangle_{i\in I,j\in J(i)}$ is independent.
}%end of proof of 272M

\cmmnt{\medskip

\noindent{\bf Remark} The formulation in (b) is more complicated than is
necessary to express the idea, but is what is needed for an application
below.
}

\leader{272N}{}\cmmnt{ A special case of 272J is  of particular
importance in general measure theory, and is most useful in an adapted
form.

\medskip

\noindent}{\bf Proposition} Let $(\Omega,\Sigma,\mu)$ be a complete
probability space, and $\langle E_i\rangle_{i\in I}$ an independent
family in $\Sigma$ such that $\mu E_i=\bover12$ for every $i\in I$.
Define $\phi:\Omega\to\{0,1\}^I$ by setting $\phi(\omega)(i)=1$ if
$\omega\in E_i$, $0$ if $\omega\in\Omega\setminus E_i$.   Then $\phi$ is
inverse-measure-preserving for the usual measure $\lambda$ on
$\{0,1\}^I$\cmmnt{ (254J)}.

\proof{ I use 254G again.   For each $i\in I$ let $\Sigma_i$ be the
algebra $\{\emptyset,E_i,\Omega\setminus E_i,\Omega\}$;  then
$\langle\Sigma_i\rangle_{i\in I}$ is independent (272F).   For $i\in I$
set $\phi_i(\omega)=\phi(\omega)(i)$.   Let $\nu$ be the usual measure
of $\{0,1\}$.   Then it is easy to check that

\Centerline{$\mu\phi_i^{-1}[H]=\Bover12\#(H)=\nu H$}

\noindent for every $H\subseteq\{0,1\}$.
If $\langle H_i\rangle_{i\in I}$ is a family of subsets of $\{0,1\}$,
and $\{i:H_i\ne\{0,1\}\}$ is finite, then

$$\eqalignno{\mu\phi^{-1}[\bigcap_{i\in I}H_i]
&=\mu(\bigcap_{i\in I}\phi_i^{-1}[H_i])
=\prod_{i\in J}\mu\phi_i^{-1}[H_i]\cr
\noalign{\noindent (because $\phi^{-1}[H_i]\in\Sigma_i$ for each $i$,
and $\langle\Sigma_i\rangle_{i\in I}$ is independent)}
&=\prod_{i\in I}\nu H_i
=\lambda(\prod_{i\in I}H_i).\cr}$$

\noindent As $\langle H_i\rangle_{i\in I}$ is arbitrary, 254G gives the
result.
}%end of proof of 272N

\leader{272O}{Tail $\sigma$-algebras and the
zero-one \dvrocolon{law}}\cmmnt{ I have never
been able to make up my mind whether the following result is `deep'
or not.   I think it is one of the many cases in mathematics where a
theorem is surprising and exciting if one comes on it unprepared, but is
natural and straightforward if one approaches it from the appropriate
angle.

\medskip

\noindent}{\bf Proposition} Let $(\Omega,\Sigma,\mu)$ be a probability
space and
$\sequencen{\Sigma_n}$ an independent sequence of $\sigma$-subalgebras
of $\Sigma$.   Let $\Sigma_n^*$ be the $\sigma$-algebra generated by
$\bigcup_{m\ge n}\Sigma_m$ for each $n$, and set
$\Sigma_{\infty}^*=\bigcap_{n\in\Bbb N}\Sigma_n^*$.   Then $\mu E$
is either $0$ or $1$ for every $E\in\Sigma_{\infty}^*$.

\proof{ For each $n$, the family
$(\Sigma_0,\ldots,\Sigma_n,\Sigma^*_{n+1})$ is independent, by 272K.
So $(\Sigma_0,\ldots,\Sigma_n,\Sigma^*_{\infty})$ is independent,
because $\Sigma^*_{\infty}\subseteq\Sigma^*_{n+1}$.   But this means
that every finite subfamily of
$(\Sigma^*_{\infty},\Sigma_0,\Sigma_1,\ldots)$ is independent, and
therefore that the whole family is (272Bb).   Consequently
$(\Sigma^*_{\infty},\Sigma^*_0)$ must be independent, by 272K again.

Now if $E\in\Sigma^*_{\infty}$, then $E$ also belongs to $\Sigma^*_0$,
so we must have

\Centerline{$\mu(E\cap E)=\mu E\cdot\mu E$,}

\noindent that is, $\mu E=(\mu E)^2$;  so that $\mu E\in\{0,1\}$, as
claimed.
}%end of proof of 272O

\leader{272P}{}\cmmnt{ To support the claim that somewhere we have
achieved a
non-trivial insight, I give a corollary, which will be fundamental to
the understanding of the limit theorems in the next section, and does
not seem to be obvious.

\medskip

\noindent}{\bf Corollary} Let $(\Omega,\Sigma,\mu)$ be a probability
space, and $\langle X_n\rangle_{n\in\Bbb N}$ an independent sequence of
real-valued random variables on $\Omega$.   Then

\Centerline{$\limsup_{n\to\infty}\Bover{1}{n+1}(X_0+\ldots+X_n)$}

\noindent is almost everywhere constant\dvro{.}{}\cmmnt{ -- that is,
there is some $u\in[-\infty,\infty]$ such that

\Centerline{$\limsup_{n\to\infty}
\Bover{1}{n+1}(X_0+\ldots+X_n)=u$}

\noindent almost everywhere.
}%end of comment

\proof{ We may suppose that each $X_n$ is $\Sigma$-measurable and
defined everywhere in
$\Omega$,  because (as remarked in 272H) changing the $X_n$ on a
negligible set does not affect their independence, and it affects
$\limsup_{n\to\infty}\Bover1{n+1}(X_0+\ldots+X_n)$ only on a
negligible set.   For each $n$, let $\Sigma_n$ be the
$\sigma$-algebra generated by $X_n$ (272C), and $\Sigma_n^*$ the
$\sigma$-algebra generated by $\bigcup_{m\ge n}\Sigma_m$;  set
$\Sigma_{\infty}^*=\bigcap_{n\in\Bbb N}\Sigma_n^*$.   By 272D,
$\sequencen{\Sigma_n}$ is independent, so $\mu E\in\{0,1\}$ for
every $E\in\Sigma_{\infty}^*$ (272O).

Now take any $a\in\Bbb R$ and set

\Centerline{$E_a=\{\omega:
\limsup_{m\to\infty}\Bover1{m+1}(X_0(\omega)+\ldots+X_m(\omega))
\le a\}$.}

\noindent Then

\Centerline{$\limsup_{m\to\infty}\Bover1{m+1}(X_0+\ldots+X_m)
=\limsup_{m\to\infty}\Bover1{m+1}(X_n+\ldots+X_{m+n})$,}

\noindent so


\Centerline{$E_a
=\{\omega:\limsup_{m\to\infty}
\Bover1{m+1}(X_n(\omega)+\ldots+X_{n+m}(\omega))\le a\}$}

\noindent belongs to $\Sigma^*_n$ for every $n$, because $X_i$ is
$\Sigma_n^*$-measurable for every $i\ge n$.   So $E\in\Sigma_{\infty}^*$
and

\Centerline{$\Pr(\limsup_{m\to\infty}
  \Bover1{m+1}(X_0+\ldots+X_m)\le a)
=\mu E_a$}

\noindent must be either $0$ or $1$.   Setting

\Centerline{$u=\sup\{a:a\in\Bbb R,\,\mu E_a=0\}$}

\noindent (allowing $\sup\emptyset=-\infty$ and $\sup\Bbb R=\infty$, as
usual in such contexts), we see that

\Centerline{$\limsup_{n\to\infty}\Bover1{n+1}(X_0+\ldots+X_n)=u$}

\noindent almost everywhere.
}%end of proof of 272P

\leader{*272Q}{}\cmmnt{ I add here a result which will be useful in
Volume 5 and
which gives further insight into the nature of large independent families.

\medskip

\noindent}{\bf Theorem}\dvAnew{2009.}
Let $(\Omega,\Sigma,\mu)$ be a probability space,
and $\familyiI{\Sigma_i}$ an independent family of $\sigma$-subalgebras of
$\Sigma$.   Let $\Cal E\subseteq\Sigma$ be a family of measurable sets, and
$\Tau$ the $\sigma$-algebra generated by $\Cal E$.   Then there is a set
$J\subseteq I$ such that $\#(I\setminus J)\le\max(\omega,\#(\Cal E))$ and
$\Tau$, $\family{j}{J}{\Sigma_j}$ are independent, in the sense that
$\mu(F\cap\bigcap_{r\le n}E_r)=\mu F\cdot\prod_{r=1}^n\mu E_r$ whenever
$F\in\Tau$, $j_1,\ldots,j_r$ are distinct members of $J$ and
$E_r\in\Sigma_{j_r}$ for each $r\le n$.

\proof{{\bf (a)} As in 272J, give $\Omega^I$ the probability measure
$\lambda$ which is
the product of the measures $\mu\restr\Sigma_i$, and let $\phi:\Omega\to \Omega^I$
be the diagonal map, so that $\phi$ is \imp\ for $\hat\mu$
and $\lambda$, where $\hat\mu$ is the completion of $\mu$.
Write $\Lambda$ for the domain of $\lambda$.   Set
$\kappa=\max(\omega,\#(\Cal E))$, and let $\Cal E^*$ be the set
$\{\bigcap_{r\le n}F_r:n\in\Bbb N$, $F_r\in\Cal E$ for every $r\le n\}$.
Because $\#(\Cal E^n)\le\kappa$ for each $n$ (2A1Lc),
$\#(\Cal E^*)\le\kappa$ (2A1Ld).   For each
$F\in\Cal E^*$, define $\nu_F:\Lambda\to[0,1]$ by setting
$\nu_FW=\hat\mu(F\cap\phi^{-1}[W])$;  then $\nu_F$ is countably additive
and dominated by $\lambda$.
It therefore has a Radon-Nikod\'ym derivative $h_F$ with respect to
$\lambda$, so that $\hat\mu(F\cap\phi^{-1}[W])=\int_Wh_Fd\lambda$ for
every $W\in\Lambda$ (232F).   By 254Qc or 254Rb, we can find a function
$h'_F$ equal $\lambda$-almost everywhere to $h_F$ and determined by
coordinates in a countable set $J_F$, in the sense that $h'_F(w)=h'_F(w')$
whenever $w$, $w'\in \Omega^I$ and $w\restr J_F=w'\restr J_F$.   (I am taking it
for granted that we chose $h'_F$ to be defined everywhere on $\Omega^I$.)

\medskip

{\bf (b)} Set $J=I\setminus\bigcup_{F\in\Cal E^*}J_F$;  by 2A1Ld,
$I\setminus J=\bigcup_{F\in\Cal E^*}J_F$ has cardinal at most $\kappa$.
If $F\in\Cal E^*$, $j_1,\ldots,j_r$ are distinct members of $J$ and
$E_r\in\Sigma_{j_r}$ for each $r\le n$,
$\mu(F\cap\bigcap_{r\le n}E_r)=\mu F\cdot\prod_{r=1}^n\mu E_r$.   \Prf\
Set  $W=\{w:w\in \Omega^I$, $w(j_r)\in E_r$ for each $r\le n\}$.   Then

\Centerline{$\mu(F\cap\bigcap_{r\le n}E_r)
=\hat\mu(F\cap\phi^{-1}[W])
=\int_Wh'_Fd\lambda
=\int h'_F\times\chi W\,d\lambda$.}

\noindent But observe that $W$ is determined by coordinates in $J$, while
$h'_F$ is determined by coordinates in $J_F\subseteq I\setminus J$;
putting 272Ma, 272K and 272R together (or otherwise), we have

\Centerline{$\mu(F\cap\bigcap_{r\le n}E_r)
=\int h'_F\times\chi W\,d\lambda
=\int h'_Fd\lambda\cdot\lambda W
=\mu F\cdot\prod_{r=1}^n\mu E_r$.  \Qed}

\medskip

{\bf (c)} Now
consider the family $\Cal A$ of those sets $F\in\Sigma$ such that
$\mu(F\cap\bigcap_{r\le n}E_r)
=\mu F\cdot\prod_{r=1}^n\mu E_r$ whenever $j_1,\ldots,j_n\in J$ are
distinct and $E_r\in\Sigma_{j_r}$ for every $r\le n$.   It is easy to check
that $\Cal A$ is a Dynkin class, and we have just seen that
$\Cal A$ includes $\Cal E^*$;  as $\Cal E^*$ is closed under $\cap$,
$\Cal A$ includes the $\sigma$-algebra $\Tau$ of sets generated by
$\Cal E^*$ (136B).   And this is just what the theorem asserts.
}%end of proof of 272Q

\leader{272R}{}\dvAformerly{2{}72Q}\cmmnt{ I must
now catch up on some basic facts from elementary probability theory.

\medskip

\noindent}{\bf Proposition} Let $X$, $Y$ be independent
real-valued random variables with finite
expectation\cmmnt{ (271Ab)}.   Then
$\Expn(X\times Y)$ exists and is equal to $\Expn(X)\Expn(Y)$.

\proof{ Let $\nu_{(X,Y)}$ be the joint distribution of the pair $(X,Y)$.
Then $\nu_{(X,Y)}$ is the product of the distributions $\nu_X$ and
$\nu_Y$ (272G).   Also $\int x\nu_X(dx)=\Expn(X)$ and
$\int y\nu_Y(dy)=\Expn(Y)$ exist in $\Bbb R$ (271F).   So

\Centerline{$\int xy\nu_{(X,Y)}d(x,y)$ exists $=\Expn(X)\Expn(Y)$}

\noindent (253D).   But this is just $\Expn(X\times Y)$, by 271E with
$h(x,y)=xy$.
}%end of proof of 272R

\leader{272S}{Bienaym\'e's Equality}\dvAformerly{2{}72R} Let
$X_1,\ldots,X_n$ be independent
real-valued random variables.   Then
$\Var(X_1+\ldots+X_n)=\Var(X_1)+\ldots+\Var(X_n)$.

\proof{{\bf (a)} Suppose first that all the $X_i$ have finite variance.
Set $a_i=\Expn(X_i)$, $Y_i=X_i-a_i$, $X=X_1+\ldots+X_n$,
$Y=Y_1+\ldots+Y_n$;  then $\Expn(X)=a_1+\ldots+a_n$, so $Y=X-\Expn(X)$
and

$$\eqalignno{\Var(X)
&=\Expn(Y^2)
=\Expn(\sum_{i=1}^n Y_i)^2\cr
&=\Expn(\sum_{i=1}^n\sum_{j=1}^n Y_i\times Y_j)
=\sum_{i=1}^n\sum_{j=1}^n\Expn(Y_i\times Y_j).\cr}$$

\noindent Now observe that if $i\ne j$ then
$\Expn(Y_i\times Y_j)=\Expn(Y_i)\Expn(Y_j)=0$, because $Y_i$ and $Y_j$
are independent
(by 272E) and we may use 272R, while if $i=j$ then

\Centerline{$\Expn(Y_i\times Y_j)=\Expn(Y_i^2)=\Expn(X_i-\Expn(X_i))^2
=\Var(X_i)$.}

\noindent So

\Centerline{$\Var(X)=\sum_{i=1}^n\Expn(Y_i^2)=\sum_{i=1}^n\Var(X_i)$.}

\medskip

{\bf (b)(i)} I show next that if $\Var(X_1+X_2)<\infty$ then
$\Var(X_1)<\infty$.   \Prf\ We have

$$\eqalignno{\iint(x+y)^2\nu_{X_1}(dx)\nu_{X_2}(dy)
&=\int (x+y)^2\nu_{(X_1,X_2)}(d(x,y))\cr
\noalign{\noindent (by 272G and Fubini's theorem)}
&=\Expn((X_1+X_2)^2)\cr
\noalign{\noindent (by 271E)}
&<\infty.\cr}$$

\noindent So there must be some $a\in\Bbb R$ such that 
$\int(x+a)^2\mu_{X_1}(dx)$ is finite, that is, $\Expn((X_1+a)^2)<\infty$;
consequently $\Expn(X_1^2)$ and $\Var(X_1)$ are finite.   \Qed

\medskip

\quad{\bf (ii)} Now an easy induction (relying on 272L!) shows that if
$\Var(X_1+\ldots+X_n)$ is finite, so is $\Var X_j$ for every $j$.
Turning this round, if $\sum_{j=1}^n\Var(X_j)=\infty$, then
$\Var(X_1+\ldots+X_n)=\infty$, and again the two are equal.
}%end of proof of 272S

\leader{272T}{The distribution of a sum of independent random variables:
Theorem} Let $X$, $Y$ be independent real-valued random variables on a
probability space $(\Omega,\Sigma,\mu)$, with distributions $\nu_X$,
$\nu_Y$.   Then the distribution of $X+Y$ is the convolution
$\nu_X*\nu_Y$\cmmnt{ (257A)}.

\wheader{272T}{0}{0}{0}{48pt}

\proof{ Set $\nu=\nu_X*\nu_Y$.   Take $a\in\Bbb R$ and set
$h=\chi\ocint{-\infty,a}$.   Then $h$ is $\nu$-integrable, so

$$\eqalignno{\nu\ocint{-\infty,a}
&=\int h\,d\nu
=\int h(x+y)(\nu_X\times\nu_Y)(d(x,y))\cr
\noalign{\noindent (by 257B, writing $\nu_X\times\nu_Y$ for the product
measure on $\BbbR^2$)}
&=\int h(x+y)\nu_{(X,Y)}(d(x,y))\cr
\noalign{\noindent (by 272G, writing $\nu_{(X,Y)}$ for the joint
distribution of $(X,Y)$;  this is where we use the hypothesis that $X$
and $Y$ are independent)}
&=\Expn(h(X+Y))\cr
\noalign{\noindent (applying 271E to the function
$(x,y)\mapsto h(x+y)$)}
&=\Pr(X+Y\le a).\cr}$$

\noindent As $a$ is arbitrary, $\nu_X*\nu_Y$ is the distribution of
$X+Y$.
}%end of proof of 272T

\leader{272U}{Corollary}\dvAformerly{2{}72T} Suppose
that $X$ and $Y$ are independent real-valued random
variables, and that they have densities $f$ and $g$.   Then the convolution
$f*g$ is a density function for $X+Y$.

\proof{ By 257F, $f*g$ is a density function for
$\nu_X*\nu_Y=\nu_{X+Y}$.
}%end of proof of 272U

\leader{272V}{}\dvAformerly{2{}72U}\cmmnt{ The following simple result
will be very useful when we come to stochastic processes in Volume 4,
as well as in the next section.

\medskip

\noindent}{\bf Etemadi's lemma}\cmmnt{ ({\smc Etemadi 96})} Let
$X_0,\ldots,X_n$ be independent real-valued random variables.   For
$m\le n$, set $S_m=\sum_{i=0}^mX_i$.   Then

\Centerline{$\Pr(\sup_{m\le n}|S_m|\ge 3\gamma)
\le 3\max_{m\le n}\Pr(|S_m|\ge\gamma)$}

\noindent for every $\gamma>0$.

\proof{ As in 272P, we may suppose that every $X_i$ is a measurable
function defined everywhere on a measure space $\Omega$.   Set
$\alpha=\max_{m\le n}\Pr(|S_m|\ge\gamma)$.   For each $r\le n$, set

\Centerline{$E_r=\{\omega:|S_m(\omega)|<3\gamma$ for every $m<r$,
$|S_r(\omega)|\ge 3\gamma\}$.}

\noindent Then $E_0,\ldots,E_n$ is a partition of
$\{\omega:\max_{m\le n}|S_m(\omega)|\ge 3\gamma\}$.   Set
$E'_r=\{\omega:\omega\in E_r$, $|S_n(\omega)|<\gamma\}$.   Then
$E'_r\subseteq\{\omega:\omega\in E_r$, $|(S_n-S_r)(\omega)|>2\gamma\}$.
But $E_r$ depends on $X_0,\ldots,X_r$ so is independent of
$\{\omega:|(S_n-S_r)(\omega)|>2\gamma\}$, which can be calculated
from $X_{r+1},\ldots,X_n$ (272K).   So

$$\eqalign{\mu E'_r
&\le\mu\{\omega:\omega\in E_r,\,|(S_n-S_r)(\omega)|>2\gamma\}
=\mu E_r\cdot\Pr(|S_n-S_r|>2\gamma)\cr
&\le\mu E_r(\Pr(|S_n|>\gamma)+\Pr(|S_r|>\gamma))
\le 2\alpha\mu E_r,\cr}$$

\noindent and $\mu(E_r\setminus E'_r)\ge(1-2\alpha)\mu E_r$.   On the
other hand, $\langle E_r\setminus E'_r\rangle_{r\le n}$ is a disjoint
family of sets all included in $\{\omega:|S_n(\omega)|\ge\gamma\}$.
So

\Centerline{$\alpha
\ge\mu\{\omega:|S_n(\omega)|\ge\gamma\}
\ge\sum_{r=0}^n\mu(E_r\setminus E'_r)
\ge(1-2\alpha)\sum_{r=0}^n\mu E_r$,}

\noindent and

\Centerline{$\Pr(\sup_{r\le n}|S_r|\ge 3\gamma)
=\sum_{r=0}^n\mu E_r
\le\min(1,\Bover{\alpha}{1-2\alpha})
\le 3\alpha$,}

\noindent (considering $\alpha\le\bover13$, $\alpha\ge\bover13$
separately), as required.
}%end of proof of 272V

\leader{*272W}{}\dvAnew{2009.}\cmmnt{ The next result is a similarly
direct application
of the ideas of this section.   While it will not be used in this volume,
it is an accessible and useful representative
of a very large number of results on tails of sums of independent random
variables.

\medskip

\noindent}{\bf Theorem}\cmmnt{ ({\smc Hoeffding 63})}
Let $X_0,\ldots,X_n$ be
independent real-valued random variables such that $0\le X_i\le 1$ a.e.\
for every $i$.
Set $S=\bover1{n+1}\sum_{i=0}^nX_i$ and $a=\Expn(S)$.   Then

\Centerline{$\Pr(S-a\ge c)\le\exp(-2(n+1)c^2)$}

\noindent for every $c\ge 0$.

\proof{{\bf (a)} Set $a_i=\Expn(X_i)$ for each $i$.
If  $b\ge 0$ and $i\le n$, then

\Centerline{$\Expn(e^{bX_i})\le\exp(ba_i+\Bover18b^2)$.}

\noindent\Prf\
Set $\phi(x)=e^{bx}$ for $x\in\Bbb R$.   Then $\phi$ is
convex, so

\Centerline{$\phi(x)
\le 1+x(e^b-1)$}

\noindent whenever $x\in[0,1]$,

\Centerline{$\phi(X_i)
\leae 1+(e^b-1)X_i$}

\noindent and

\Centerline{$\Expn(e^{bX_i})
=\Expn(\phi(X_i))
\le 1+(e^b-1)a_i
=e^{h(b)}$}

\noindent where $h(t)=\ln(1-a_i+a_ie^t)$
for $t\in\Bbb R$.   Now $h(0)=0$,

\Centerline{$h'(t)=\Bover{a_ie^t}{1-a_i+a_ie^t}
=1-\Bover{1-a_i}{1-a_i+a_ie^t}$,
\quad$h'(0)=a_i$,}

\Centerline{$h''(t)
=\Bover{1-a_i}{1-a_i+a_ie^t}\cdot\Bover{a_ie^t}{1-a_i+a_ie^t}
\le\Bover14$}

\noindent because $a_ie^t$ and $1-a_i$ are both greater than or equal to
$0$.   By Taylor's theorem with remainder, there is some
$t\in[0,b]$ such that

\Centerline{$h(b)
=h(0)+bh'(0)+\Bover12b^2h_i''(t)
\le ba_i+\Bover18b^2$,}

\noindent and

\Centerline{$\Expn(e^{bX_i})\le\exp(ba_i+\Bover18b^2)$.  \Qed}

\medskip

{\bf (b)} Take any $b\ge 0$.   Then

$$\eqalignno{\Pr(S-a\ge c)
&=\Pr(\sum_{i=0}^n(X_i-a_i-c)\ge 0)
\le\Expn(\exp(b\sum_{i=0}^nX_i-a_i-c))\cr
\displaycause{because $\exp(b\sum_{i=0}^nX_i-a_i-c)\ge 1$ whenever
$\sum_{i=0}^nX_i-a_i-c\ge 0$}
&=e^{-(n+1)bc}\prod_{i=0}^ne^{-ba_i}\Expn(\prod_{i=0}^n\exp(bX_i))\cr
&=e^{-(n+1)bc}\prod_{i=0}^ne^{-ba_i}
  \prod_{i=0}^n\Expn(\exp(bX_i))\cr
\displaycause{because the random variables $\exp(bX_i)$ are independent,
by 272E, so
the expectation of the product is the product of the expectations, by
272R}
&\le e^{-(n+1)bc}\prod_{i=0}^ne^{-ba_i}\exp(ba_i+\Bover18b^2)\cr
\displaycause{(a) above}
&=\exp(-(n+1)bc+\Bover{n+1}8b^2).\cr}$$

\noindent Now the minimum value of the quadratic $\Bover{n+1}8b^2-(n+1)cb$
is $-2(n+1)c^2$ when $b=4c$, so $\Pr(S-a\ge c)\le\exp(-2(n+1)c^2)$, as
claimed.
}%end of proof of 272W

\exercises{
\leader{272X}{Basic exercises (a)}
%\spheader 272Xa
Let $(\Omega,\Sigma,\mu)$ be an atomless probability space, and
$\sequencen{\epsilon_n}$ any sequence in $[0,1]$.   Show that there is
an independent sequence $\sequencen{E_n}$ in $\Sigma$ such that
$\mu E_n=\epsilon_n$ for every $n$.   \Hint{215D.}
%272B

\sqheader 272Xb Let $\langle X_i\rangle_{i\in I}$ be a family of
real-valued random variables.   Show that it is independent iff

\Centerline{$\Expn(h_1(X_{i_1})\times\ldots\times h_n(X_{i_n}))
=\prod_{j=1}^n\Expn(h_j(X_{i_j}))$}

\noindent whenever $i_1,\ldots,i_n$ are distinct members of $I$ and
$h_1,\ldots,h_n$ are Borel measurable functions from $\Bbb R$ to
$\Bbb R$ such that $\Expn(h_j(X_{i_j}))$ are all finite.
%272E

\spheader 272Xc Write out a proof of 272F which does not use the
theory of product measures.
%272F

\spheader 272Xd Let $\pmb{X}=(X_1,\ldots,X_n)$ be a
family of real-valued
random variables all defined on the same probability space,
and suppose that $\pmb{X}$ has a density function $f$ expressible in the
form $f(\xi_1,\ldots,\xi_n)=f_1(\xi_1)f_2(\xi_2)\ldots f_n(\xi_n)$ for
suitable functions $f_1,\ldots,f_n$ of one real variable.   Show that
$X_1,\ldots,X_n$ are independent.
%272G

\spheader 272Xe Let $X_1$, $X_2$ be independent real-valued random
variables both with distribution $\nu$ and distribution function $F$.
Set $Y=\max(X_1,X_2)$.   Show that the distribution of $Y$ is absolutely
continuous with respect to $\nu$, with a Radon-Nikod\'ym derivative
$F+F^{-}$, where $F^{-}(x)=\lim_{t\uparrow x}F(t)$ for every
$x\in\Bbb R$.
%272G

\spheader 272Xf Use 254Sa and the idea of 272J to give another proof of
272O.
%272O

\spheader 272Xg Let $(\Omega,\Sigma,\mu)$ be a probability space and
$\sequencen{\Sigma_n}$ a non-decreasing sequence of $\sigma$-subalgebras
of $\Sigma$.   Let $\Sigma_{\infty}$ be the $\sigma$-algebra generated
by $\bigcup_{n\in\Bbb N}\Sigma_n$.   Let $\Tau$ be another
$\sigma$-subalgebra of $\Sigma$ such that $\Sigma_n$ and $\Tau$ are
independent for each $n$.   Show that $\Sigma_{\infty}$ and $\Tau$ are
independent.   \Hint{apply the Monotone Class Theorem to
$\{E:\mu(E\cap F)=\mu E\cdot\mu F$ for every $F\in\Tau\}$.}   Use this
to prove 272O.
%272O

\spheader 272Xh Let $\sequencen{X_n}$ be a sequence of
real-valued random variables and $Y$ a real-valued
random variable such that $Y$ and $X_n$ are independent for
each $n\in\Bbb N$.   Suppose that $\Pr(Y\in\Bbb N)=1$ and that
$\sum_{n=0}^{\infty}\Pr(Y\ge n)\Expn(|X_n|)$ is finite.   Set
$Z=\sum_{n=0}^YX_n$ (that is,
$Z(\omega)=\sum_{n=0}^{Y(\omega)}X_n(\omega)$ whenever $\omega\in\dom Y$
is such that $Y(\omega)\in\Bbb N$ and $\omega\in\dom X_n$ for every
$n\le Y(\omega)$).   (i) Show that
$\Expn(Z)=\sum_{n=0}^{\infty}\Pr(Y\ge n)\Expn(X_n)$.   \Hint{set
$X'_n(\omega)=X_n(\omega)$ if $Y(\omega)\ge n$, $0$ otherwise.}   (ii)
Show that if $\Expn(X_n)=\gamma$ for every $n\in\Bbb N$ then
$\Expn(Z)=\gamma\Expn(Y)$.   (This is {\bf Wald's equation}.)
%272R

\sqheader 272Xi Let $X_1,\ldots,X_n$ be independent real-valued random
variables.   Show that if $X_1+\ldots+X_n$ has finite expectation so
does every $X_j$.   \Hint{part (b) of the proof of 272S.}
%272S

\sqheader 272Xj Let $X$ and $Y$ be independent real-valued random
variables with densities $f$ and $g$.   Show that $X\times Y$ has
a density function $h$ where
$h(x)=\int_{-\infty}^{\infty}\bover1{|y|}g(y)f(\bover{x}{y})dy$ for
almost every $x$.   \Hint{271K.}
%272U

\spheader 272Xk\dvAnew{2009.} Let
$X_0,\ldots,X_n$ be independent real-valued
random variables such that $d_i\le X_i\le d'_i$ a.e.\
for every $i$.   (i) Show that if $b\ge 0$ then
$\Expn(e^{bX_i})\le\exp(ba_i+\Bover18b^2(d'_i-d_i)^2)$ for each $i$,
where $a_i=\Expn(X_i)$.   (ii)
Set $S=\bover1{n+1}\sum_{i=0}^nX_i$ and $a=\Expn(S)$.   Show that

\Centerline{$\Pr(S-a\ge c)\le\exp(-\Bover{2(n+1)^2c^2}{d})$}

\noindent for every $c\ge 0$, where $d=\sum_{i=0}^n(d'_i-d_i)^2$.
%272W

\spheader 272Xl\dvAnew{2009.} Suppose
that $X_0,\ldots,X_n$ are independent real-valued random
variables, all with expectation $0$, such that $\Pr(|X_i|\le 1)=1$ for
every $i$.   Set $S=\Bover1{\sqrt{n+1}}\sum_{i=0}^nX_i$.   Show that
$\Pr(S\ge c)\le\exp(-c^2/2)$ for every $c\ge 0$.
%272Xk 272W

\leader{272Y}{Further exercises (a)}
%\spheader 272Ya
\dvAformerly{2{}72Yc}
Let $X_0,\ldots,X_n$ be independent real-valued random
variables with distributions $\nu_0,\ldots,\nu_n$ and distribution
functions $F_0,\ldots,F_n$.   Show that, for any Borel set
$E\subseteq\Bbb R$,

\Centerline{$\Pr(\sup_{i\le n}X_i\in E)
=\sum_{i=0}^n\int_E\prod_{j=0}^{i-1}F_j^{-}(x)
  \prod_{j=i+1}^nF_j(x)\nu_i(dx)$,}

\noindent where $F_j^{-}(x)=\lim_{t\uparrow x}F_j(t)$ for each $j$, and
we interpret the empty products $\prod_{j=0}^{-1}F_j^{-}(x)$,
$\prod_{j=n+1}^nF_j(x)$ as $1$.
%272Xe, 272G

\spheader 272Yb\dvAformerly{2{}72Yf}
Let $\pmb{X}=\sequencen{X_n}$ be an independent
sequence of real-valued random variables on a complete probability space
$(\Omega,\Sigma,\mu)$.   Let $\Cal B$ be the Borel $\sigma$-algebra of
$\BbbR^{\Bbb N}$ (271Ya).   Let $\nu_{\pmb{X}}^{(\Cal B)}$ be the
probability measure with domain $\Cal B$ defined by setting
$\nu_{\pmb{X}}^{(\Cal B)}E=\mu\pmb{X}^{-1}[E]$ for every $E\in\Cal B$,
and write $\nu_{\pmb{X}}$ for the completion of $\nu_{\pmb{X}}^{(\Cal
B)}$.   Show that $\nu_{\pmb{X}}$ is just the product of the
distributions $\nu_{X_n}$.
%272G

\spheader 272Yc\dvAformerly{2{}72Ye}
Let $X_1,\ldots,X_n$ be real-valued random variables such
that for each $j<n$ the family

\Centerline{$(X_1,\ldots,X_j,-X_{j+1},\ldots,-X_n)$}

\noindent has the same joint distribution as the original family
$(X_1,\ldots,X_n)$.   Set $S_j=X_1+\ldots+X_j$ for each $j\le n$.   (i)
Show that for any $a\ge 0$

\Centerline{$\Pr(\sup_{1\le j\le n}|S_j|\ge a)
\le 2\Pr(|S_n|\ge a)$.}

\noindent \Hint{show that if
$E_j=\{\omega:\omega\in\bigcap_{i\le n}\dom X_i,\,|S_i(\omega)|<a$ for
$i<j,\,|S_j(\omega)|\ge a\}$ then
$\mu\{\omega:\omega\in E_j,\,
  |S_n(\omega)|\ge|S_j(\omega)|\}\ge\bover12\mu E_j$.}
(ii) Show that $\Expn(\sup_{j\le n}|S_j|)\le 2\Expn(|S_n|)$.  (iii)
Show that $\Expn(\sup_{i\le n}S_i^2)\le 2\Expn(S_n^2)$.
%272V

\spheader 272Yd\dvAnew{2009.}
Let $\sequence{i}{X_i}$ be an independent sequence of
real-valued random variables, and set $S_n=\sum_{i=0}^nX_i$ for each $n$.
Show that if $\sequencen{S_n}$ converges to $S$ in $\eusm L^0$
for the topology of convergence in measure,
then $\sequencen{S_n}$ converges to $S$ a.e.
%272V mt27bits

\spheader 272Ye\dvAformerly{2{}72Yd}
Let $(\Omega,\Sigma,\mu)$ be a probability space.

\quad (i) Let $\sequencen{E_n}$ be an independent sequence in $\Sigma$.
Show that for any real-valued random variable $X$ with finite
expectation,

\Centerline{$\lim_{n\to\infty}\int_{E_n}X\,d\mu-\mu E_n\Expn(X)=0$.}

\noindent \Hint{let $\Tau_0$ be the subalgebra of $\Sigma$
generated by $\{E_n:n\in\Bbb N\}$ and $\Tau$ the $\sigma$-subalgebra of
$\Sigma$ generated by $\{E_n:n\in\Bbb N\}$.   Start by considering
$X=\chi E$ for $E\in\Tau_0$ and then $X=\chi E$ for $E\in\Tau$.   Move
from $\eusm L^1(\mu\restrp\Tau)$ to $\eusm L^1(\mu)$ by using conditional
expectations.}

\quad (ii) Let $\sequencen{X_n}$ be a uniformly integrable independent
sequence of real-valued
random variables on $\Omega$.   Show that for any bounded
real-valued random variable $Y$,

\Centerline{$\lim_{n\to\infty}\Expn(X_n\times Y)-\Expn(X_n)\Expn(Y)=0$.}

\quad(iii) Suppose that $1<p\le\infty$ and set $q=p/(p-1)$ (taking $q=1$
if $p=\infty$).   Let $\sequencen{X_n}$ be an independent sequence of
real-valued
random variables with $\sup_{n\in\Bbb N}\|X_n\|_p<\infty$, and $Y$ a
real-valued random variable with $\|Y\|_q<\infty$.   Show that

\Centerline{$\lim_{n\to\infty}\Expn(X_n\times Y)-\Expn(X_n)\Expn(Y)=0$.}

\spheader 272Yf\dvAformerly{2{}72Yg}
Let $(\Omega,\Sigma,\mu)$ be a probability space and
$\sequencen{Z_n}$  a sequence of
random variables on $\Omega$ such that
$\Pr(Z_n\in\Bbb N)=1$ for each $n$, and $\Pr(Z_m=Z_n)=0$ for all
$m\ne n$.   Let $\sequencen{X_n}$ be a sequence of real-valued random
variables on $\Omega$, all with the same distribution $\nu$, and
independent of each other and the $Z_n$, in the sense that if $\Sigma_n$
is the $\sigma$-algebra defined by $X_n$, and $\Tau_n$ the
$\sigma$-algebra
defined by $Z_n$, and $\Tau$ is the $\sigma$-algebra generated by
$\bigcup_{n\in\Bbb N}\Tau_n$, then $(\Tau,\Sigma_0,\Sigma_1,\ldots)$ is
independent.   Set $Y_n(\omega)=X_{Z_n(\omega)}(\omega)$ whenever this
is defined, that is, $\omega\in\dom Z_n$, $Z_n(\omega)\in\Bbb N$ and
$\omega\in\dom X_{Z_n(\omega)}$.   Show that $\sequencen{Y_n}$ is an
independent sequence of random variables and that every $Y_n$ has the
distribution $\nu$.

\spheader 272Yg\dvAformerly{2{}72Yb}
Show that all the ideas of this section apply
equally to complex-valued random variables, subject to suitable
adjustments (to be devised).

\spheader 272Yh\dvAformerly{2{}72Ya}  Develop a theory of independence
for random variables taking values in $\BbbR^r$, following through as
many as possible of the ideas of this section.
}%end of exercises

\endnotes{
\Notesheader{272} This section is lengthy for two reasons:  I am trying
to pack in the basic results associated with one of the most fertile
concepts of mathematics, and it is hard to know where to stop;  and I am
trying to do this in language appropriate to abstract measure theory,
insisting on a variety of distinctions which are peripheral to the
essential ideas.   For while I am prepared to be flexible on the
question of whether the letter $X$ should denote a space or a function,
some of the applications of these results which are most important to me
are in contexts where we expect to be exactly clear what the domains of
our functions are.   Consequently it is necessary to form an opinion on
such matters as what the $\sigma$-algebra defined by a random variable
really is (272C).

The point of 272Q is that the family $\Cal E$ does not have to be related
in any way to the family $\familyiI{\Sigma_i}$, except, of course, that we
are dealing with measurable sets.   All we need to know is that $I$ should
be large compared with $\Cal E$;  for instance, that $\Cal E$ is countable
and $I$ is uncountable.   The family $\family{j}{J}{\Sigma_j}$ is now a
kind of `tail' of $\familyiI{\Sigma_i}$, safely independent of the `head'
$\sigma$-algebra generated by $\Cal E$.

Of course I should emphasize again that such proofs as those in
272R-272S are to be thought of as confirmations that we have a suitable
model of probability theory, rather than as reasons for believing the
results to be valid in statistical contexts.   Similarly, 272T-272U can
be approached by a variety of intuitions concerning discrete random
variables and random variables with continuous densities, and while the
elegant general results are delightful, they are more important to the
pure mathematician than to the statistician.    But I came to an odd
obstacle in the proof of 272S, when showing that if $X_1+\ldots+X_n$ has
finite variance then so does every $X_j$.   We have done enough measure
theory for this to be readily dealt with, but the connexion with
ordinary probabilistic intuition, both here and in 272Xi, remains
unclear to me.

There are four ideas in 272W worth storing for future use.
The first is the estimate

\Centerline{$\Expn(e^{bX_i})\le 1-a_i+e^ba_i$}

\noindent in part (a), a crude but effective way of using the hypothesis
that $X_i$ is bounded.   The second is the use of Taylor's theorem to
show that $1-a_i+e^ba_i\le\exp(a_i+\bover18b^2)$.
The third is the estimate

\Centerline{$\Pr(Y\ge 0)\le\Expn(e^{bY})$ if $b\ge 0$}

\noindent used in part (b);  and the fourth is 272R.
After this one need only be sufficiently determined to reach 272Xk.   But
even the special case 272W is both striking and useful.
}%end of notes

\discrpage


\frfilename{mt273.tex}
\versiondate{2.12.09}
\copyrightdate{2000}

\def\chaptername{Probability theory}
\def\sectionname{The strong law of large numbers}

\newsection{273}

I come now to the first of the three main theorems of this chapter.
Perhaps I should call it a `principle', rather than a `theorem',
as I shall not attempt to enunciate any fully general form, but will
give three theorems (273D, 273H, 273I), with a variety of
corollaries, each setting out conditions under which the averages of a
sequence of independent random variables will almost surely converge.
At the end of the section (273N) I add a result on norm-convergence of
averages.

\leader{273A}{}\cmmnt{ It will
be helpful to start with an explicit statement of a very simple but very
useful lemma.

\medskip

\noindent}{\bf Lemma} Let $\sequencen{E_n}$ be a sequence of measurable
sets in a measure space $(\Omega,\Sigma,\mu)$, and suppose that
$\sum_{n=0}^{\infty}\mu E_n<\infty$.   Then  $\{n:\omega\in E_n\}$ is
finite for almost every $\omega\in\Omega$.

\proof{ We have

$$\eqalign{\mu\{\omega:\{n:\omega\in E_n\}\text{ is infinite}\}
&=\mu(\bigcap_{n\in\Bbb N}\bigcup_{m\ge n}E_m)
=\inf_{n\in\Bbb N}\mu(\bigcup_{m\ge n}E_m)\cr
&\le\inf_{n\in\Bbb N}\sum_{m=n}^{\infty}\mu E_m
=0.\cr}$$
}%end of proof of 273A

\leader{273B}{Lemma} Let $\sequencen{X_n}$ be an independent sequence of
real-valued random variables, and set $S_n=\sum_{i=0}^nX_i$ for each
$n\in\Bbb N$.

(a) If $\sequencen{S_n}$ is convergent in measure, then it is convergent
almost everywhere.

(b) In particular, if $\Expn(X_n)=0$ for every $n$ and
$\sum_{n=0}^{\infty}\Expn(X_n^2)<\infty$, then
$\sum_{n=0}^{\infty}X_n$ is defined, and finite, almost everywhere.

\proof{{\bf (a)} Let $(\Omega,\Sigma,\mu)$ be the underlying probability
space.   If we change each $X_n$ on a negligible set, we do not change
the independence of $\sequencen{X_n}$ (272H), and the $S_n$ are also
changed only on a negligible set;  so we may suppose from the beginning
that every $X_n$ is a measurable function defined on the whole of
$\Omega$.

Because the functional $X\mapsto\Expn(\min(1,|X|))$ is one of the
pseudometrics defining the topology of convergence in measure (245A),
$\lim_{m,n\to\infty}\Expn(\min(1,|S_m-S_n|))=0$, and we can find for
each $k\in\Bbb N$ an $n_k\in\Bbb N$ such that
$\Expn(\min(1,|S_m-S_{n_k}|))\le 4^{-k}$ for every $m\ge n_k$.   So
$\Pr(|S_m-S_{n_k}|\ge 2^{-k})\le 2^{-k}$ for every $m\ge n_k$.   By
Etemadi's lemma (272V) applied to $\langle X_i\rangle_{i\ge n_k}$,

\Centerline{$\Pr(\sup_{n_k\le m\le n}|S_m-S_{n_k}|\ge 3\cdot 2^{-k})
\le 3\cdot 2^{-k}$}

\noindent for every $n\ge n_k$.   Setting

\Centerline{$H_{kn}
=\{\omega:\sup_{n_k\le m\le n}|S_m(\omega)-S_{n_k}(\omega)|
  \ge 3\cdot 2^{-k}\}$ for $n\ge n_k$,}

\Centerline{$H_k=\bigcup_{n\ge n_k}H_{kn}$,}

\noindent we have

\Centerline{$\mu H_k=\lim_{n\to\infty}\mu H_{kn}\le 3\cdot 2^{-k}$}

\noindent for each $k$, so $\sum_{k=0}^{\infty}\mu H_k$ is finite and
almost every $\omega\in\Omega$ belongs to only finitely many of the
$H_k$ (273A).

Now take any such $\omega$.   Then there is some $r\in\Bbb N$ such that
$\omega\notin H_k$ for any $k\ge r$.   In this case, for every $k\ge r$,
$\omega\notin\bigcup_{n\ge n_k}H_{kn}$, that is,
$|S_n(\omega)-S_{n_k}(\omega)|<3\cdot 2^{-k}$ for every $n\ge n_k$.
But this means that $\sequencen{S_n(\omega)}$ is a Cauchy sequence,
therefore convergent.   Since this is true for almost every $\omega$,
$\sequencen{S_n}$ converges almost everywhere, as claimed.

\medskip

{\bf (b)} Now suppose that $\Expn(X_n)=0$ for every $n$ and that
$\sum_{n=0}^{\infty}\Expn(X_n^2)<\infty$.   In this case, for any
$m<n$,

$$\eqalignno{\|S_n-S_m\|_1^2
&\le\|\chi\Omega\|_2^2\|S_n-S_m\|_2^2\cr
\displaycause{by Cauchy's inequality, 244Eb}
&=\Expn(S_n-S_m)^2
=\Var(S_n-S_m)\cr
\displaycause{because $\Expn(S_n-S_m)=\sum_{i=m+1}^n\Expn(X_i)=0$}
&=\sum_{i=m+1}^n\Var(X_i)\cr
\displaycause{by Bienaym\'e's equality, 272S}
&\to 0\cr}$$

\noindent as $m\to\infty$.   So $\sequencen{S_n^{\ssbullet}}$ is a
Cauchy sequence in $L^1(\mu)$ and converges in $L^1(\mu)$, by 242F;  by
245G, it converges in measure in $L^0(\mu)$, that is, $\sequencen{S_n}$
converges in measure in $\eusm L^0(\mu)$.   By (a), $\sequencen{S_n}$
converges almost everywhere, that is, $\sum_{i=0}^{\infty}X_i$ is
defined and finite almost everywhere.
}%end of proof of 273B

\cmmnt{\medskip

\noindent{\bf Remark} The proof above assumes familiarity with the ideas
of Chapter 24.   However part (b), at least, can be established without
any of these;  see 273Xa.   In 276B there is a generalization of (b)
based on a different approach.
}%end of comment

\leader{273C}{}\cmmnt{ We now need a lemma (part (b) below) from the
theory of
summability.   I take the opportunity to include an elementary fact
which will be useful later in this section and elsewhere.

\medskip

\noindent}
{\bf Lemma} (a) If $\lim_{n\to\infty}x_n=x$, then
$\lim_{n\to\infty}\Bover1{n+1}\sum_{i=0}^nx_i=x$.

(b) Let $\sequencen{x_n}$ be such that $\sum_{i=0}^{\infty}x_i$ is defined
in $\Bbb R$\dvAnew{2012}, and
$\sequencen{b_n}$ a non-decreasing sequence in $\coint{0,\infty}$
diverging to $\infty$.   Then
\ifnum\stylenumber=12

\Centerline{$\lim_{n\to\infty}\Bover1{b_n}\sum_{k=0}^nb_kx_k=0$.}
\else
$\lim_{n\to\infty}\Bover1{b_n}\sum_{k=0}^nb_kx_k=0$.\fi

\proof{{\bf (a)} Let $\epsilon>0$.   Let $m$ be such that
$|x_n-x|\le\epsilon$ whenever $n\ge m$.   Let $m'\ge m$ be such that
$|\sum_{i=0}^{m-1}x-x_i|\le\epsilon m'$.   Then for $n\ge m'$ we have

$$\eqalign{|x-\Bover1{n+1}\sum_{i=0}^nx_i|
&=\Bover1{n+1}|\sum_{i=0}^nx-x_i|\cr
&\le\Bover1{n+1}|\sum_{i=0}^{m-1}x-x_i|
+\Bover1{n+1}\sum_{i=m}^n|x-x_i|\cr
&\le\Bover{\epsilon m'}{n+1}+\Bover{\epsilon (n-m+1)}{n+1}
\le 2\epsilon.\cr}$$

\noindent As $\epsilon$ is arbitrary,
$\lim_{n\to\infty}\Bover1{n+1}\sum_{i=0}^nx_i=x$.

\medskip

{\bf (b)}  Let $\epsilon>0$.   Write $s_n=\sum_{i=0}^nx_i$ for each $n$,
and

\Centerline{$s=\lim_{n\to\infty}s_n=\sum_{i=0}^{\infty}x_i$;}

\noindent set $s^*=\sup_{n\in\Bbb N}|s_n|<\infty$.   Let $m\in\Bbb N$ be
such that $|s_n-s|\le\epsilon$ whenever $n\ge m$;  then $|s_n-s_j|\le
2\epsilon$ whenever $j$, $n\ge m$.   Let $m'\ge m$ be such that
$b_ms^*\le\epsilon b_{m'}$.

Take any $n\ge m'$.   Then

$$\eqalignno{|\sum_{k=0}^nb_kx_k|
&=|b_0s_0+b_1(s_1-s_0)+\ldots+b_n(s_n-s_{n-1})|\cr
&=|(b_0-b_1)s_0+(b_1-b_2)s_1+\ldots+(b_{n-1}-b_n)s_{n-1}+b_ns_n|\cr
&=|b_0s_n+\sum_{i=0}^{n-1}(b_{i+1}-b_i)(s_n-s_i)|\cr
&\le b_0|s_n|+\sum_{i=0}^{m-1}(b_{i+1}-b_{i})|s_n-s_i|
+\sum_{i=m}^{n-1}(b_{i+1}-b_{i})|s_n-s_i|\cr
&\le b_0s^*
+2s^*\sum_{i=0}^{m-1}(b_{i+1}-b_i)
+2\epsilon\sum_{i=m}^{n-1}(b_{i+1}-b_i)\cr
&=b_0s^*+2s^*(b_m-b_0)+2\epsilon(b_n-b_m)
\le 2s^*b_m+2\epsilon b_n.\cr}$$

\noindent Consequently, because $b_n\ge b_{m'}$,

\Centerline{$|\Bover1{b_n}\sum_{k=0}^nb_kx_k|\le
2\Bover{s^*b_m}{b_n}+2\epsilon
\le 4\epsilon$.}

\noindent As $\epsilon$ is arbitrary,

\Centerline{$\lim_{n\to\infty}\Bover1{b_n}\sum_{k=0}^nb_kx_k
=0$,}

\noindent as required.
}%end of proof of 273C

\cmmnt{\medskip

\noindent{\bf Remark} Part (b) above is sometimes called `Kronecker's
lemma'.
}%end of comment

\leader{273D}{The strong law of large numbers:  first form} Let
$\sequencen{X_n}$ be an independent sequence of real-valued random
variables, and suppose that $\sequencen{b_n}$ is a non-decreasing
sequence in $\ooint{0,\infty}$, diverging to $\infty$, such that
$\sum_{n=0}^{\infty}\Bover1{b_n^2}\Var(X_n)<\infty$.   Then

\Centerline{$\lim_{n\to\infty}
\Bover{1}{b_n}\sum_{i=0}^n(X_i-\Expn(X_i))=0$}

\noindent almost everywhere.

\proof{ As usual, write $(\Omega,\Sigma,\mu)$ for the
underlying probability space.   Set

\Centerline{$Y_n=\Bover1{b_n}(X_n-\Expn(X_n))$}

\noindent for each $n$;  then $\sequencen{Y_n}$ is independent (272E),
$\Expn(Y_n)=0$ for each $n$, and

\Centerline{$\sum_{n=0}^{\infty}\Expn(Y_n^2)
=\sum_{n=0}^{\infty}\Bover1{b_n^2}\Var(X_n)<\infty$.}

\noindent By 273B, $\sequencen{Y_n(\omega)}$ is summable for almost
every $\omega\in\Omega$.   But by 273Cb,

\Centerline{$\lim_{n\to\infty}\Bover{1}{b_n}
  \sum_{i=0}^n(X_i(\omega)-\Expn(X_i))
=\lim_{n\to\infty}\Bover{1}{b_n}
\sum_{i=0}^nb_iY_i(\omega)=0$}

\noindent for all such $\omega$.   So we have the result.
}%end of proof of 273D

\leader{273E}{Corollary} Let $\sequencen{X_n}$ be an independent
sequence of random variables such that $\Expn(X_n)=0$ for every $n$ and
$\sup_{n\in\Bbb N}\Expn(X_n^2)<\infty$.   Then

\Centerline{$\lim_{n\to\infty}\Bover{1}{b_n}(X_0+\ldots+X_n)
=0$}

\noindent almost everywhere whenever $\sequencen{b_n}$ is a
non-decreasing sequence of strictly positive numbers and
$\sum_{n=0}^{\infty}\Bover{1}{b_n^2}$ is finite.   In particular,

\Centerline{$\lim_{n\to\infty}\Bover{1}{n+1}(X_0+\ldots+X_n)=0$}

\noindent almost everywhere.

\cmmnt{\medskip

\noindent{\bf Remark} For most of the rest of this section, we shall
take $b_n=n+1$.   The special virtue of 273D is that it allows other
$b_n$, e.g., $b_n=\sqrt{n}\ln n$.   A direct strengthening of this
theorem is in 276C below.
}%end of comment

\leader{273F}{Corollary} Let $\sequencen{E_n}$ be an independent
sequence of measurable sets in a probability space
$(\Omega,\Sigma,\mu)$. and suppose that

\Centerline{$\lim_{n\to\infty}\Bover{1}{n+1}\sum_{i=0}^n\mu
E_i=c$.}

\noindent Then

\Centerline{$\lim_{n\to\infty}\Bover{1}{n+1}\#(\{i:i\le
n,\,\omega\in E_i\})=c$}

\noindent for almost every $\omega\in\Omega$.

\proof{ In 273D, set $X_n=\chi E_n$, $b_n=n+1$.   For
almost every $\omega$, we have

\Centerline{$\lim_{n\to\infty}\Bover1{n+1}
\sum_{i=0}^n(\chi E_i(\omega)-a_i)=0$,}

\noindent writing $a_i=\mu E_i=\Expn(X_i)$ for each $i$.
(I see that I am relying on 272F to support the claim that
$\sequencen{X_n}$ is independent.)   But for any such $\omega$,

$$\eqalign{\lim_{n\to\infty}\bigl(\Bover1{n+1}
&\#(\{i:i\le n,\,\omega\in E_i\})-\Bover1{n+1}\sum_{i=0}^na_i\bigr)\cr
&=\lim_{n\to\infty}\Bover1{n+1}
\sum_{i=0}^n(\chi E_i(\omega)-a_i)
=0;\cr}$$

\noindent because we are supposing that
$\lim_{n\to\infty}\Bover1{n+1}\sum_{i=0}^na_i=c$, we must have

\Centerline{$\lim_{n\to\infty}\Bover{1}{n+1}\#(\{i:i\le
n,\,\omega\in E_i\})=c,$}

\noindent as required.
}%end of proof of 273F

\vleader{48pt}{273G}{Corollary} Let $\mu$ be the usual measure on
$\Cal P\Bbb N$\cmmnt{, as described in 254Jb}.   Then for $\mu$-almost
every set $a\subseteq\Bbb N$,

\Centerline{$\lim_{n\to\infty}\Bover1{n+1}
\#(a\cap\{0,\ldots,n\})
={1\over 2}$.}

\proof{ The sets $E_n=\{a:n\in a\}$ are independent, with
measure ${1\over 2}$.
}%end of proof of 273G

\cmmnt{\medskip

\noindent{\bf Remark} The limit $\lim_{n\to\infty}\Bover1{n+1}
\#(a\cap\{0,\ldots,n\})$, when it is defined, 
is called the {\bf asymptotic density} of $a$.
}%end of comment

\leader{273H}{Strong law of large numbers:  second form} Let
$\sequencen{X_n}$ be an independent sequence of real-valued random
variables, and suppose that
$\sup_{n\in\Bbb N}\Expn(|X_n|^{1+\delta})<\infty$ for
some $\delta>0$.   Then

\Centerline{$\lim_{n\to\infty}\Bover{1}{n+1}\sum_{i=0}^n(X_i-\Expn(X_i))
=0$}

\noindent almost everywhere.

\proof{ As usual, call the underlying probability space
$(\Omega,\Sigma,\mu)$;  as in 273B we can adjust the $X_n$ on negligible
sets so as to make them measurable and defined everywhere on $\Omega$,
without changing $\Expn(X_n)$, $\Expn(|X_n|)$ or the convergence of the
averages except on a negligible set.

\medskip

{\bf (a)} For each $n$, define a random variable $Y_n$ on $\Omega$ by
setting

$$\eqalign{Y_n(\omega)&=X_n(\omega)\text{ if }|X_n(\omega)|\le n,\cr
&=0\text{ if }|X_n(\omega)|>n.\cr}$$

\noindent Then $\sequencen{Y_n}$ is independent (272E).   For each
$n\in\Bbb N$,

\Centerline{$\Var(Y_n)\le
\Expn(Y_n^2)\le\Expn(n^{1-\delta}|X_n|^{1+\delta})\le n^{1-\delta}K$,}

\noindent where $K=\sup_{n\in\Bbb N}\Expn(|X_n|^{1+\delta})$, so

\Centerline{$\sum_{n=0}^{\infty}\Bover1{(n+1)^2}\Var(Y_n)
\le\sum_{n=0}^{\infty}\Bover{n^{1-\delta}}{(n+1)^2}K
<\infty$.}

\noindent By 273D,

\Centerline{$G
=\{\omega:\lim_{n\to\infty}\Bover{1}{n+1}
  \sum_{i=0}^n(Y_i(\omega)-\Expn(Y_i))=0\}$}

\noindent is conegligible.

\medskip

{\bf (b)} On the other hand, setting

\Centerline{$E_n=\{\omega:Y_n(\omega)\ne X_n(\omega)\}
=\{\omega:|X_n(\omega)|>n\}$,}

\noindent we have $K\ge n^{1+\delta}\mu E_n$ for each $n$, so

\Centerline{$\sum_{n=0}^{\infty}\mu E_n
\le 1+K\sum_{n=1}^{\infty}\Bover1{n^{1+\delta}}<\infty$,}

\noindent and the set $H=\{\omega:\{n:\omega\in E_n\}$ is
finite$\}$ is conegligible (273A).   But of course

\Centerline{$\lim_{n\to\infty}\Bover1{n+1}
\sum_{i=0}^n(X_i(\omega)-Y_i(\omega))=0$}

\noindent for every $\omega\in H$.

\medskip

{\bf (c)} Finally,

\Centerline{$|\Expn(Y_n)-\Expn(X_n)|\le\int_{E_n}|X_n|
\le\int_{E_n}n^{-\delta}|X_n|^{1+\delta}\le n^{-\delta}K$}

\noindent whenever $n\ge 1$, so
$\lim_{n\to\infty}\Expn(Y_n)-\Expn(X_n)=0$ and

\Centerline{$\lim_{n\to\infty}\Bover1{n+1}
\sum_{i=0}^n\Expn(Y_i)-\Expn(X_i)=0$}

\noindent (273Ca).   Putting these three together, we get

\Centerline{$\lim_{n\to\infty}\Bover{1}{n+1}
\sum_{i=0}^nX_i(\omega)-\Expn(X_i)=0$}

\noindent whenever $\omega$ belongs to the conegligible set
$G\cap H$.   So

\Centerline{$\lim_{n\to\infty}\Bover{1}{n+1}
\sum_{i=0}^nX_i-\Expn(X_i)=0$}

\noindent almost everywhere, as required.
}%end of proof of 273H

\leader{273I}{Strong law of large numbers:  third form} Let
$\sequencen{X_n}$ be an independent sequence of real-valued random
variables with finite expectation, and suppose that they are
{\bf identically distributed}, that is, all have the same
distribution.   Then

\Centerline{$\lim_{n\to\infty}\Bover{1}{n+1}
\sum_{i=0}^n(X_i-\Expn(X_i))=0$}

\noindent almost everywhere.

\proof{ The proof follows the same line as that of
273H, but some of the inequalities require more delicate arguments.
As usual, call the underlying probability space
$(\Omega,\Sigma,\mu)$ and suppose that the $X_n$ are all measurable and
defined everywhere on $\Omega$.    (We need to remember that changing a
random variable on a negligible set does not change its distribution.)
Let $\nu$ be the common distribution of the $X_n$.

\medskip

{\bf (a)} For each $n$, define a random variable $Y_n$ on $\Omega$ by
setting

$$\eqalign{Y_n(\omega)&=X_n(\omega)\text{ if }|X_n(\omega)|\le n,\cr
&=0\text{ if }|X_n(\omega)|>n.\cr}$$

\noindent Then $\sequencen{Y_n}$ is independent (272E).   For each
$n\in\Bbb N$,

\Centerline{$\Var(Y_n)\le\Expn(Y_n^2)=\int_{[-n,n]}x^2\nu(dx)$}

\noindent (271E).   To estimate
$\sum_{n=0}^{\infty}\Bover1{(n+1)^2}\Expn(Y_n^2)$, set

\Centerline{$f_n(x)=\Bover{x^2}{(n+1)^2}$ if $|x|\le n$,
$0$ if $|x|>n$,}

\noindent so that $\Bover1{(n+1)^2}\Var(Y_n)\le\int f_nd\nu$.   If
$r\ge 1$ and $r<|x|\le r+1$ then

$$\eqalign{\sum_{n=0}^{\infty}f_n(x)
&\le\sum_{n=r+1}^{\infty}\Bover1{(n+1)^2}(r+1)|x|\cr
&\le(r+1)|x|\sum_{n=r+1}^{\infty}(\Bover1n-\Bover1{n+1})
\le |x|,\cr}$$

\noindent while if $|x|\le 1$ then

\Centerline{$\sum_{n=0}^{\infty}f_n(x)
\le\sum_{n=0}^{\infty}\Bover1{(n+1)^2}=\Bover{\pi^2}6\le 2<\infty$.}

\noindent (You do not need to know that the sum is $\bover{\pi^2}6$,
only that it is finite;  but see 282Xo.)   Consequently

\Centerline{$f(x)=\sum_{n=0}^{\infty}f_n(x)\le 2+|x|$}

\noindent for every $x$, and $\int fd\nu<\infty$, because
$\int |x|\nu(dx)$ is the common value of $\Expn(|X_n|)$, and is finite.
By any of the great convergence theorems,

\Centerline{$\sum_{n=0}^{\infty}\Bover1{(n+1)^2}\Var(Y_n)
\le\sum_{n=0}^{\infty}\int f_nd\nu
=\int fd\nu<\infty$.}

\noindent By 273D,

\Centerline{$G
=\{\omega:\lim_{n\to\infty}\Bover{1}{n+1}
   \sum_{i=0}^n(Y_i(\omega)-\Expn(Y_i))=0\}$}

\noindent is conegligible.

\medskip

{\bf (b)} Next, setting

\Centerline{$E_n
=\{\omega:X_n(\omega)\ne Y_n(\omega)\}
=\{\omega:|X_n(\omega)|>n\}$,}

\noindent we have

\Centerline{$E_n=\bigcup_{i\ge n}F_{ni}$,}

\noindent where

\Centerline{$F_{ni}=\{\omega:i<|X_n(\omega)|\le i+1\}$.}

\noindent Now

\Centerline{$\mu F_{ni}=\nu\{x:i<|x|\le i+1\}$}

\noindent for every $n$ and $i$.   So

$$\eqalign{\sum_{n=0}^{\infty}\mu E_n
&=\sum_{n=0}^{\infty}\sum_{i=n}^{\infty}\mu F_{ni}
=\sum_{i=0}^{\infty}\sum_{n=0}^i\mu F_{ni}\cr
&=\sum_{i=0}^{\infty}(i+1)\nu\{x:i<|x|\le i+1\}
\le\int(1+|x|)\nu(dx)
<\infty.\cr}$$

\noindent Consequently the set
$H=\{\omega:\{n:X_n(\omega)\ne Y_n(\omega)\}$ is finite$\}$ is
conegligible (273A).   But of course

\Centerline{$\lim_{n\to\infty}\Bover1{n+1}
  \sum_{i=0}^nX_i(\omega)-Y_i(\omega)
=0$}

\noindent for every $\omega\in H$.

\medskip

{\bf (c)} Finally,

\Centerline{$|\Expn(Y_n)-\Expn(X_n)|\le\int_{E_n}|X_n|
=\int_{\Bbb R\setminus[-n,n]}|x|\nu(dx)$}

\noindent whenever $n\in\Bbb N$, so
$\lim_{n\to\infty}\Expn(Y_n)-\Expn(X_n)=0$ and

\Centerline{$\lim_{n\to\infty}\Bover1{n+1}
\sum_{i=0}^n\Expn(Y_i)-\Expn(X_i)=0$}

\noindent (273Ca).   Putting these three together, we get

\Centerline{$\lim_{n\to\infty}\Bover{1}{n+1}
\sum_{i=0}^nX_i(\omega)-\Expn(X_i)=0$}

\noindent whenever $\omega$ belongs to the conegligible set
$G\cap H$.   So

\Centerline{$\lim_{n\to\infty}\Bover{1}{n+1}
\sum_{i=0}^nX_i-\Expn(X_i)=0$}

\noindent almost everywhere, as required.
}%end of proof of 273I

\cmmnt{\medskip

\noindent{\bf Remarks} In my own experience, this is the most important
form of the strong law from the point of view of `pure' measure
theory.   I note that 273G above can also be regarded as a consequence
of this form.

For a very striking alternative proof, see 275Yq.   Yet another
proof treats this result as a special case of the Ergodic Theorem
(see 372Xg in Volume 3).
}%end of comment

\leader{273J}{Corollary}\discrversionA{\footnote{Extended 2008.}}{}
Let $(\Omega,\Sigma,\mu)$ be a probability space.
If $f$ is a real-valued function such that $\int fd\mu$ is defined in
$[-\infty,\infty]$, then

\Centerline{$\lim_{n\to\infty}\Bover1{n+1}\sum_{i=0}^nf(\omega_i)
=\int fd\mu$}

\noindent for $\lambda$-almost
every $\pmb{\omega}=\sequencen{\omega_n}\in\Omega^{\Bbb N}$,
where $\lambda$ is the product measure on
$\Omega^{\Bbb N}$\cmmnt{ (254A-254C)}.

\proof{{\bf (a)} To begin with, suppose that $f$ is integrable.
Define functions $X_n$ on $\Omega^{\Bbb N}$ by setting

\Centerline{$X_n(\pmb{\omega})=f(\omega_n)$ whenever
$\omega_n\in\dom f$.}

\noindent Then $\sequencen{X_n}$ is an independent sequence of random
variables, all with the same distribution as $f$ (272M).   So

\Centerline{$
\lim_{n\to\infty}\Bover1{n+1}\sum_{i=0}^nf(\omega_i)-\int fd\mu
=\lim_{n\to\infty}\Bover1{n+1}\sum_{i=0}^nX_i(\pmb{\omega})-\Expn(X_i)
=0$}

\noindent for almost every $\pmb{\omega}$, by 273I, and

\Centerline{$\lim_{n\to\infty}\Bover1{n+1}\sum_{i=0}^nf(\omega_i)
=\int fd\mu$}

\noindent for almost every $\pmb{\omega}$.

\medskip

{\bf (b)} Next, suppose that $f\ge 0$ and $\int f=\infty$.   In this case,
for every $m\in\Bbb N$,

$$\eqalign{\liminf_{n\to\infty}\Bover1{n+1}\sum_{i=0}^nf(\omega_i)
&\ge\liminf_{n\to\infty}\Bover1{n+1}\sum_{i=0}^n\min(m,f(\omega_i))\cr
&=\int\min(m,f(\omega))\mu(d\omega)\cr}$$

\noindent for almost every $\pmb{\omega}$, so

\Centerline{$\liminf_{n\to\infty}\Bover1{n+1}\sum_{i=0}^nf(\omega_i)
\ge\sup_{m\in\Bbb N}\int\min(m,f(\omega))\mu(d\omega)
=\infty$}

\noindent and

\Centerline{$\lim_{n\to\infty}\Bover1{n+1}\sum_{i=0}^nf(\omega_i)
=\infty=\int f$}

\noindent for almost every $\pmb{\omega}$.

\medskip

{\bf (c)} In general, if $\int f=\infty$,
this is because $\int f^+=\infty$
and $f^-$ is integrable, so

$$\eqalign{\lim_{n\to\infty}\Bover1{n+1}\sum_{i=0}^nf(\omega_i)
&=\lim_{n\to\infty}\Bover1{n+1}\sum_{i=0}^nf^+(\omega_i)
  -\lim_{n\to\infty}\Bover1{n+1}\sum_{i=0}^nf^-(\omega_i)\cr
&=\infty-\int f^-  %
=\int f\cr}$$

\noindent for almost every $\pmb{\omega}$.   Similarly,

\Centerline{$\lim_{n\to\infty}\Bover1{n+1}\sum_{i=0}^nf(\omega_i)
=-\infty$}

\noindent for almost every $\pmb{\omega}$ if $\int fd\mu=-\infty$.
}%end of proof of 273J

\cmmnt{\medskip

\noindent{\bf Remark} I find myself slipping here into
measure-theorists' terminology;  this corollary is one of the basic
applications of the strong law to measure theory.   Obviously, in view
of 272J and 272M, this corollary covers 273I.   It could also (in
theory) be used as a {\it definition} of integration on a probability
space (see 273Ya);   it is sometimes called the `Monte Carlo'
method of integration.
}%end of comment

\leader{273K}{}\cmmnt{ It is tempting to seek extensions of 273I in
which the $X_n$ are not identically distributed, but are otherwise
well-behaved.   Any such idea should be tested against the following
example.   I find that I need another standard result, complementing
that in 273A.

\medskip

\noindent}{\bf Borel-Cantelli lemma} Let $(\Omega,\Sigma,\mu)$ be a
probability space and $\sequencen{E_n}$ a sequence of
measurable subsets of $\Omega$ such that
$\sum_{n=0}^{\infty}\mu E_n=\infty$ and
$\mu(E_m\cap E_n)\le\mu E_m\cdot\mu E_n$ whenever $m\ne n$.
Then almost every point of
$\Omega$ belongs to infinitely many of the $E_n$.

\proof{ For $n$, $k\in\Bbb N$ set $X_n=\sum_{i=0}^n\chi E_i$,
$\beta_n=\sum_{i=0}^n\mu E_n=\Expn(X_n)$ and
$F_{nk}=\{x:x\in\Omega$, $\#(\{i:i\le n$, $x\in E_i\})\le k\}$.   Then

$$\eqalign{\Expn(X_n^2)
&=\sum_{i=0}^n\sum_{j=0}^n\mu(E_i\cap E_j)
\le\sum_{i=0}^n\mu E_i+\sum_{i=0}^n\sum_{j\ne i}\mu E_i\cdot\mu E_j\cr
&=\beta_n+\beta_n^2-\sum_{i=0}^n(\mu E_i)^2,\cr}$$

\noindent so

\Centerline{$\Var(X_n)=\beta_n-\sum_{i=0}^n(\mu E_i)^2\le\beta_n$.}

\noindent Now if $k<\beta_n$,

\Centerline{$(\beta_n-k)^2\mu F_{nk}
=(\beta_n-k)^2\Pr(X_n\le k)
\le\Expn(X_n-\beta_n)^2=\Var(X_n)\le\beta_n$}

\noindent and $\mu F_{nk}\le\Bover{\beta_n}{(\beta_n-k)^2}$.

Now recall that we are assuming that $\lim_{n\to\infty}\beta_n=\infty$.
So for any $k\in\Bbb N$,

\Centerline{$\mu(\bigcap_{n\in\Bbb N}F_{nk})
=\lim_{n\to\infty}\mu F_{nk}
\le\lim_{n\to\infty}\Bover{\beta_n}{(\beta_n-k)^2}
=0$.}

\noindent Accordingly

\Centerline{$\mu\{x:x$ belongs to only finitely many $E_n\}
=\mu(\bigcup_{k\in\Bbb N}\bigcap_{n\in\Bbb N}F_{nk})
=0$,}

\noindent and almost every point of $\Omega$ belongs to infinitely many
$E_n$.
}%end of proof of 273K

\cmmnt{\medskip

\noindent{\bf Remark} Of course this result is usually applied to an
independent sequence $\sequencen{E_n}$.   But very occasionally it is of
interest to know that it is enough to assume that weaker hypotheses
suffice.   See also 273Yb.}

\vleader{72pt}{273L}{}\cmmnt{ Now for the promised example.

\medskip

\noindent}{\bf Example} There is an independent sequence
$\sequencen{X_n}$ of non-negative random variables such that
$\lim_{n\to\infty}\Expn(X_n)=0$ but

\Centerline{$\limsup_{n\to\infty}\Bover1{n+1}
  \sum_{i=0}^{\infty}X_i-\Expn(X_i)=\infty$,}

\Centerline{$\liminf_{n\to\infty}\Bover1{n+1}
  \sum_{i=0}^{\infty}X_i-\Expn(X_i)=0$}

\noindent almost everywhere.

\proof{ Let $(\Omega,\Sigma,\mu)$ be a probability space
with an independent sequence $\sequencen{E_n}$ of measurable sets such
that $\mu E_n=\Bover{1}{(n+3)\ln(n+3)}$ for each $n$.
(I have nowhere explained
exactly how to build such a sequence.   Two obvious methods are
available to us, and another a trifle less obvious.   (i) Take
$\Omega=\{0,1\}^{\Bbb N}$ and $\mu$ to be
the product of the probabilities $\mu_n$ on $\{0,1\}$, defined by saying
that $\mu_n\{1\}=\Bover1{(n+3)\ln(n+3)}$ for each $n$;  set
$E_n=\{\omega:\omega(n)=1\}$, and appeal to 272M to check that the
$E_n$ are independent.   (ii) Build the $E_n$ inductively as subsets of
$[0,1]$, arranging that each $E_n$ should be a finite union of
intervals, so that when you come to choose $E_{n+1}$ the sets
$E_0,\ldots,E_n$ define a partition $\Cal I_n$ of $[0,1]$ into
intervals, and you
can take $E_{n+1}$ to be the union of (say) the left-hand subintervals
of length a proportion $\Bover1{(n+3)\ln(n+3)}$ of the intervals in
$\Cal I_n$.   (iii) Use 215D to see that the method of (ii) can be used
on any atomless probability space, as in 272Xa.)

Set $X_n=(n+3)\ln\ln(n+3)\chi E_n$ for each $n$;  then $\sequencen{X_n}$
is an independent sequence of real-valued random variables (272F) and
$\Expn(X_n)=\Bover{\ln\ln(n+3)}{\ln(n+3)}$ for each $n$, so that
$\Expn(X_n)\to 0$ as $n\to\infty$.   Thus, for instance,
$\{X_n:n\in\Bbb N\}$ is uniformly integrable and $\sequencen{X_n}\to 0$
in measure (246Jc);  while surely
$\lim_{n\to\infty}\Bover1{n+1}\sum_{i=0}^n\Expn(X_i)=0$.

On the other hand,

$$\eqalign{\sum_{n=0}^{\infty}\mu E_n
&=\sum_{n=0}^{\infty}\Bover1{(n+3)\ln(n+3)}
\ge\int_0^{\infty}\Bover1{(x+3)\ln(x+3)}dx\cr
&=\lim_{a\to\infty}(\ln\ln(a+3)-\ln\ln 3)
=\infty,\cr}$$

\noindent so almost every $\omega$ belongs to infinitely many of the
$E_n$, by the Borel-Cantelli lemma (273K).  Now if we write
$Y_n=\Bover1{n+1}\sum_{i=0}^nX_i$, then if $\omega\in E_n$ we have
$X_n(\omega)=(n+3)\ln\ln(n+3)$ so

\Centerline{$Y_n(\omega)\ge\Bover{n+3}{n+1}\ln\ln(n+3).$}

\noindent   This means that

$$\eqalign{\{\omega:\limsup_{n\to\infty}\Bover1{n+1}
    &\sum_{i=0}^n(X_i(\omega)-\Expn(X_i))=\infty\}
=\{\omega:\limsup_{n\to\infty}
    \Bover{1}{n+1}\sum_{i=0}^nX_i(\omega)=\infty\}\cr
&=\{\omega:\sup_{n\in\Bbb N}Y_n(\omega)=\infty\}
\supseteq\{\omega:\{n:\omega\in E_n\}\text{ is infinite}\}\cr}$$

\noindent is conegligible, and the strong law of large numbers does not
apply to $\sequencen{X_n}$.

Because

\Centerline{$\lim_{n\to\infty}\|Y_n\|_1
=\lim_{n\to\infty}\Expn(Y_n)=\lim_{n\to\infty}\Expn(X_n)=0$}

\noindent (273Ca), $\sequencen{Y_n}\to 0$ for the topology of
convergence in measure, and $\sequencen{Y_n}$ has a
subsequence converging to $0$ almost everywhere (245K).   So

\Centerline{$\liminf_{n\to\infty}\Bover1{n+1}
\sum_{i=0}^n(X_i(\omega)-\Expn(X_i))=\liminf_{n\to\infty}Y_n(\omega)=0$}

\noindent for almost every $\omega$.   The fact that both
$\limsup_{n\to\infty}Y_n$ and $\liminf_{n\to\infty}Y_n$ are constant
almost everywhere is of course a consequence of the zero-one law
(272P).
}%end of example

\leader{*273M}{}\cmmnt{ All the above has been concerned with
pointwise
convergence of the averages of independent random variables, and that is
the important part of the work of this section.   But it is perhaps
worth complementing it with a brief investigation of norm-convergence.
To deal efficiently with convergence in $\eusm L^p$, we need the
following.   (I should perhaps remark that, compared with the general
case treated here, the case $p=2$ is trivial;  see 273Xl.)

\medskip

\noindent}{\bf Lemma} For any $p\in\ooint{1,\infty}$ and $\epsilon>0$,
there is a $\delta>0$ such that $\|S+X\|_p\le 1+\epsilon\|X\|_p$
whenever $S$ and $X$ are independent random variables, $\|S\|_p=1$,
$\|X\|_p\le\delta$ and $\Expn(X)=0$.

\proof{{\bf (a)} Take $\zeta\in\ocint{0,1}$ such that $p\zeta\le 2$ and

\Centerline{$(1+\xi)^p\le 1+p\xi+\Bover{p^2}2\xi^2$}

\noindent whenever $|\xi|\le\zeta$;  such exists because

\Centerline{$\lim_{\xi\to 0}\Bover{(1+\xi)^p-1-p\xi}{\xi^2}
=\Bover{p(p-1)}2<\Bover{p^2}2$.}

\noindent Observe that

\Centerline{$(1+\xi)^p
\le(1+\Bover1{\zeta})^p+\xi^p+2p\xi^{p-1}$}

\noindent for every $\xi\ge 0$.   \Prf\
If $\xi\le\Bover1{\zeta}$, this is trivial.   If $\xi\ge\Bover1{\zeta}$,
then

$$\eqalign{(1+\xi)^p
&=\xi^p(1+\Bover1{\xi})^p
\le\xi^p(1+\Bover{p}{\xi}+\Bover{p^2}{2\xi^2})\cr
&\le\xi^p(1+\Bover{p}{\xi}+\Bover{p^2\zeta}{2\xi})
=\xi^p+p\xi^{p-1}(1+\Bover{p\zeta}{2})
\le\xi^p+2p\xi^{p-1}.  \text{ \Qed}\cr}$$

Define $\eta>0$ by declaring that $3\eta^{p-1}=\Bover{\epsilon}2$ (this
is one of the places where we need to know that $p>1$).
Let $\delta>0$ be such that

\Centerline{$\delta\le\eta\zeta$,
\quad$\Bover{p^2}{2\eta^2}\delta+(1+\Bover1{\zeta})^p\delta^{p-1}
\le\Bover{p\epsilon}2$.}

\medskip

{\bf (b)} Now suppose that $S$ and $X$ are independent random variables
with $\|S\|_p=1$, $\|X\|_p\le\delta$ and $\Expn(X)=0$.   If $\|X\|_p=0$
then of course $\|S+X\|_p\le 1+\epsilon\|X\|_p$,
so suppose that $X$ is non-trivial.   Write
$(\Omega,\Sigma,\mu)$ for the underlying probability space and adjust
$S$ and $X$ on negligible sets so that they are measurable and defined
everywhere on $\Omega$.   Set $\alpha=\|X\|_p$, $\gamma=\alpha/\eta$,

\Centerline{$E=\{\omega:S(\omega)\ne 0\}$,
\quad$F=\{\omega:|X(\omega)|>\gamma|S(\omega)|\}$,
\quad$\beta=\|S\times\chi F\|_p$.}

\noindent Then

$$\eqalignno{\int|S+X|^p
&=\int_F|S+X|^p+\int_{E\setminus F}|S+X|^p\cr
\displaycause{because $S$ and $X$ are both zero on
$\Omega\setminus(E\cup F)$}
&=\|(S\times\chi F)+(X\times\chi F)\|_p^p
  +\int_{E\setminus F}|S|^p|1+\Bover{X}{S}|^p\cr
&\le(\|S\times\chi F\|_p+\|X\times\chi F)\|_p)^p
  +\int_{E\setminus F}|S|^p(1+p\Bover{X}{S}+\Bover{p^2}2\gamma^2)\cr
\displaycause{because
$|\Bover{X}{S}|\le\gamma\le\Bover{\delta}{\eta}\le\zeta\le 1$ everywhere on
$E\setminus F$}
&\le(\beta+\alpha)^p+(1+\Bover{p^2}2\gamma^2)\int_{E\setminus F}|S|^p
  +p\int_{E\setminus F}|S|^{p-1}\times\sgn S\times X\cr
\displaycause{writing $\sgn(\xi)=\xi/|\xi|$ if $\xi\ne 0$, $0$ if
$\xi=0$}
&=(\beta+\alpha)^p+(1+\Bover{p^2}2\gamma^2)\int_{\Omega\setminus F}|S|^p
  +p\int_{\Omega\setminus F}|S|^{p-1}\times\sgn S\times X\cr
\displaycause{because $S=0$ on $\Omega\setminus E$}
&=\alpha^p(1+\Bover{\beta}{\alpha})^p+(1+\Bover{p^2}2\gamma^2)(1-\beta^p)
  -p\int_F|S|^{p-1}\times\sgn S\times X\cr
\displaycause{because $X$ and $|S|^{p-1}\times\sgn S$ are independent,
by 272L, so
$\int|S|^{p-1}\times\sgn S\times X=\Expn(|S|^{p-1}\times\sgn S)\Expn(X)=0$}
&\le\alpha^p\bigl((1+\Bover1{\zeta})^p+2p(\Bover{\beta}{\alpha})^{p-1}
  +(\Bover{\beta}{\alpha})^p\bigr)+(1+\Bover{p^2}2\gamma^2)(1-\beta^p)\cr
 &\qquad\qquad\qquad
  +p\int_F|S|^{p-1}\times|X|\cr
\displaycause{see (a) above}
&\le\alpha^p(1+\Bover1{\zeta})^p+\beta^p+2p\beta^{p-1}\alpha
  +(1+\Bover{p^2}2\gamma^2)(1-\beta^p)\cr
 &\qquad\qquad\qquad
  +p\int_F\Bover{1}{\gamma^{p-1}}|X|^p\cr
&\le\alpha^p(1+\Bover1{\zeta})^p
  +2p\Bover{\alpha^p}{\gamma^{p-1}}
  +1+\Bover{p^2}2\gamma^2
  +p\Bover{\alpha^p}{\gamma^{p-1}}\cr
\displaycause{because $\beta=\|S\times\chi F\|_p
\le\Bover1{\gamma}\|X\times\chi F\|_p\le\Bover{\alpha}{\gamma}$}
&=\alpha^p(1+\Bover1{\zeta})^p
  +3p\eta^{p-1}\alpha
  +1+\Bover{p^2\alpha^2}{2\eta^2}
  \cr
&=1+\bigl(\alpha^{p-1}(1+\Bover1{\zeta})^p
  +3p\eta^{p-1}
  +\Bover{p^2\alpha}{2\eta^2}\bigr)\alpha\cr
&\le 1+\bigl(\delta^{p-1}(1+\Bover1{\zeta})^p
  +3p\eta^{p-1}
  +\Bover{p^2\delta}{2\eta^2}\bigr)\alpha\cr
&\le 1+p\alpha\epsilon\le(1+\epsilon\|X\|_p)^p.\cr}$$

\noindent So $\|S+X\|_p\le 1+\epsilon\|X\|_p$, as required.
}%end of proof of 273M

\cmmnt{\medskip

\noindent*{\bf Remark} What is really happening here is that
$\phi=\|\,\|^p_p:L^p\to\Bbb R$ is differentiable (as a real-valued
function on the normed space $L^p$) and

\Centerline{$\phi'(S^{\ssbullet})(X^{\ssbullet})
=p\int|S|^{p-1}\times\sgn S\times X$,}

\noindent so that in the context here

\Centerline{$\phi((S+X)^{\ssbullet})
=\phi(S^{\ssbullet})+\phi'(S^{\ssbullet})(X^{\ssbullet})
   +o(\|X\|_p)
=1+o(\|X\|_p)$}

\noindent and $\|S+X\|_p=1+o(\|X\|_p)$.    The calculations above are
elaborate partly because they do not appeal to any non-trivial ideas
about normed spaces, and partly because we need the estimates to be
uniform in $S$.
}%end of comment

\leader{273N}{Theorem} Let $\sequencen{X_n}$ be an independent sequence
of real-valued random variables with zero expectation, and set
$Y_n=\bover1{n+1}(X_0+\ldots+X_n)$ for each $n\in\Bbb N$.

(a) If $\sequencen{X_n}$ is uniformly integrable, then
$\lim_{n\to\infty}\|Y_n\|_1=0$.

*(b) If $p\in\ooint{1,\infty}$ and
$\sup_{n\in\Bbb N}\|X_n\|_p<\infty$,
then $\lim_{n\to\infty}\|Y_n\|_p=0$.

\proof{{\bf (a)} Let $\epsilon>0$.   Then there is an $M\ge 0$ such that
$\Expn(|X_n|-M)^+\le\epsilon$ for every $n\in\Bbb N$.   Set

\Centerline{$X_n'=(-M\chi\Omega)\vee(X_n\wedge M\chi\Omega)$,
\quad$\alpha_n=\Expn(X_n')$,
\quad $\tilde X_n=X'_n-\alpha_n$,
\quad $X''_n=X_n-X'_n$}

\noindent for each $n\in\Bbb N$.   Then $\sequencen{X'_n}$ and
$\sequencen{\tilde X_n}$ are independent and uniformly bounded, and
$\|X''_n\|_1\le\epsilon$ for every $n$.   So if we write

\Centerline{$\tilde Y_n=\Bover1{n+1}\sum_{i=0}^n\tilde X_i$,
\quad$Y''_n=\Bover1{n+1}\sum_{i=0}^nX''_i$,}

\noindent $\sequencen{\tilde Y_n}\to 0$ almost everywhere, by 273E (for
instance), while $\|Y''_n\|_1\le\epsilon$ for every $n$.  Moreover,

\Centerline{$|\alpha_n|=|\Expn(X'_n-X_n)|\le\Expn(|X''_n|)\le\epsilon$}

\noindent for every $n$.    As $|\tilde Y_n|\le 2M$ almost everywhere
for each $n$, $\lim_{n\to\infty}\|\tilde Y_n\|_1=0$, by Lebesgue's
Dominated Convergence Theorem.   So

$$\eqalign{\limsup_{n\to\infty}\|Y_n\|_1
&=\limsup_{n\to\infty}\|\tilde Y_n+Y''_n+\alpha_n\|_1\cr
&\le\lim_{n\to\infty}\|\tilde Y_n\|_1+\sup_{n\in\Bbb N}\|Y''_n\|_1
  +\sup_{n\in\Bbb N}|\alpha_n|\cr
&\le 2\epsilon.\cr}$$

\noindent As $\epsilon$ is arbitrary, $\lim_{n\to\infty}\|Y_n\|_1=0$, as
claimed.

\medskip

{\bf *(b)} Set $M=\sup_{n\in\Bbb N}\|X_n\|_p$.   For $n\in\Bbb N$,
set $S_n=\sum_{i=0}^nX_i$.   Let $\epsilon>0$.   Then there is a
$\delta>0$ such that $\|S+X\|_p\le 1+\epsilon\|X\|_p$ whenever $S$ and
$X$ are independent random variables, $\|S\|_p=1$, $\|X\|_p\le\delta$
and $\Expn(X)=0$ (273M).   It follows that
$\|S+X\|_p\le\|S\|_p+\epsilon\|X\|_p$ whenever $S$ and $X$ are
independent random variables, $\|S\|_p$ is finite,
$\|X\|_p\le\delta\|S\|_p$ and $\Expn(X)=0$.   In particular,
$\|S_{n+1}\|_p\le\|S_n\|_p+\epsilon M$ whenever
$\|S_n\|_p\ge M/\delta$.   An easy induction shows that

\Centerline{$\|S_n\|_p\le\Bover{M}{\delta}+M+n\epsilon M$}

\noindent for every $n\in\Bbb N$.   But this means that

\Centerline{$\limsup_{n\to\infty}\|Y_n\|_p
=\limsup_{n\to\infty}\Bover1{n+1}\|S_n\|_p\le\epsilon M$.}

\noindent As $\epsilon$ is arbitrary, $\lim_{n\to\infty}\|Y_n\|_p=0$.
}%end of proof of 273N

\cmmnt{\medskip

\noindent{\bf Remark} There are strengthenings of (a) in
276Xe, and of (b) in 276Ya.
}%end of comment

\exercises{
\leader{273X}{Basic exercises (a)}
%\spheader 273Xa
In part (b) of the proof of 273B, use Bienaym\'e's equality to show that
$\lim_{m\to\infty}\sup_{n\ge m}\discretionary{}
{}{}\Pr(|S_n-S_m|\ge\epsilon)=0$ for
every $\epsilon>0$, so that we can apply the argument of part (a) of the
proof directly, without appealing to 242F or 245G or even 244E.
%273B

\spheader 273Xb Show that
$\sum_{n=0}^{\infty}\Bover{(-1)^{\omega(n)}}{n+1}$ is defined in
$\Bbb R$ for almost every $\pmb{\omega}=\sequencen{\omega(n)}$ in
$\{0,1\}^{\Bbb N}$, where $\{0,1\}^{\Bbb N}$ is given its usual measure
(254J).
%273B

\spheader 273Xc Let $\sequencen{E_n}$ be an independent sequence of
measurable sets in a probability space, all with the same non-zero
measure.   Let $\sequencen{a_n}$ be a sequence of non-negative real
numbers such that $\sum_{n=0}^{\infty}a_n=\infty$.   Show that
$\sum_{n=0}^{\infty}a_n\chi E_n=\infty$ a.e.   \Hint{Take a strictly
increasing sequence $\sequencen{k_n}$ such that
$d_n=\sum_{i=k_n+1}^{k_{n+1}}a_i\ge 1$ for each $n$.   Set
$c_i=\Bover{a_i}{(n+1)d_n}$ for $k_n<i\le k_{n+1}$;  show that
$\sum_{n=0}^{\infty}c_n^2<\infty=\sum_{n=0}^{\infty}c_n$.
Apply 273D with $X_n=c_n\chi E_n$ and $b_n=\sqrt{\sum_{i=0}^nc_i}$.}
%273D  added 2002

\sqheader 273Xd Take any $q\in[0,1]$, and
give $\Cal P\Bbb N$ a measure $\mu$ such that

\Centerline{$\mu\{a:I\subseteq a\}=q^{\#(I)}$}

\noindent for every $I\subseteq\Bbb N$, as in 254Xg.   Show that
for $\mu$-almost every $a\subseteq\Bbb N$,

\Centerline{$\lim_{n\to\infty}\Bover{1}{n+1}
\#(a\cap\{0,\ldots,n\})=q$.}
%273F

\sqheader 273Xe Let $\mu$ be the usual probability measure on
$\Cal P\Bbb N$ (254Jb), and for $r\ge 1$ let $\mu^r$ be the product
probability measure on $(\Cal P\Bbb N)^r$.   Show that

\Centerline{$\lim_{n\to\infty}
  \Bover1{n+1}\#(a_1\cap\ldots\cap a_r\cap\{0,\ldots,n\})=2^{-r}$,}

\Centerline{$\lim_{n\to\infty}
  \Bover1{n+1}\#((a_1\cup\ldots\cup a_r)\cap\{0,\ldots,n\})=1-2^{-r}$}

\noindent for $\mu^r$-almost every
$(a_1,\ldots,a_r)\in(\Cal P\Bbb N)^r$.
%273F

\spheader 273Xf Let $\mu$ be the usual probability measure on
$\Cal P\Bbb N$, and $b$ any infinite subset of $\Bbb N$.   Show that
$\lim_{n\to\infty}
\Bover{\#(a\cap b\cap\{0,\ldots,n\})}{\#(b\cap\{0,\ldots,n\})}
\penalty-100=\Bover12$ for almost every $a\subseteq\Bbb N$.
%273F

\sqheader 273Xg For each $x\in[0,1]$, let $\epsilon_k(x)$ be the
$k$th digit in the decimal expansion of $x$ (choose for yourself what to
do with $0{\cdot}100\ldots=0{\cdot}099\ldots$).   Show that
$\lim_{k\to\infty}\bover1k\#(\{j:j\le k,\,\epsilon_j(x)=7\})
=\bover1{10}$ for almost every $x\in[0,1]$.
%273F

\spheader 273Xh Let $\sequencen{F_n}$ be a sequence of
distribution functions for real-valued random variables, in the sense of
271Ga, and $F$ another
distribution function;  suppose that $\lim_{n\to\infty}F_n(q)=F(q)$ for
every $q\in\Bbb Q$ and $\lim_{n\to\infty}F_n(a^-)=F(a^-)$ whenever
$F(a^-)<F(a)$, where I write $F(a^-)$ for $\lim_{x\uparrow a}F(x)$.
Show that $F_n\to F$ uniformly.
%for 273Xi

\sqheader 273Xi\dvAformerly{2{}73Xh}
Let $(\Omega,\Sigma,\mu)$ be a probability space
and $\sequencen{X_n}$ an independent identically distributed sequence of
real-valued random
variables on $\Omega$ with common distribution function $F$.   For
$a\in\Bbb R$, $n\in\Bbb N$ and $\omega\in\bigcap_{i\le n}\dom X_i$ set

\Centerline{$F_n(\omega,a)
=\Bover1{n+1}\#(\{i:i\le n,\,X_i(\omega)\le a\})$.}

\noindent Show that

\Centerline{$\lim_{n\to\infty}\sup_{a\in\Bbb R}|F_n(\omega,a)-F(a)|
=0$}

\noindent for almost every $\omega\in\Omega$.
%273Xh, 273F

\spheader 273Xj Let $(\Omega,\Sigma,\mu)$ be a probability space, and
$\lambda$ the product measure on $\Omega^{\Bbb N}$.
Let $f:\Omega\to\Bbb R$ be a function, and set $f^*(\pmb{\omega})
=\limsup_{n\to\infty}\Bover1{n+1}\sum_{i=0}^nf(\omega_i)$
for $\pmb{\omega}=\sequencen{\omega_n}\in\Omega^{\Bbb N}$.
Show that  $\overline{\int}f^*d\lambda=\overline{\int}fd\mu$ whenever
the right-hand-side is finite.
\Hint{133J(a-i).}
%273J

\spheader 273Xk Find an independent sequence $\sequencen{X_n}$ of random
variables with zero expectation such that $\|X_n\|_1=1$ and
$\|\bover1{n+1}\sum_{i=0}^nX_i\|_1\ge\bover12$ for every $n\in\Bbb N$.
\Hint{take $\Pr(X_n\ne 0)$ very small.}
%273N

\spheader 273Xl Use 272S to prove 273Nb in the case $p=2$.
%273N

\spheader 273Xm Find an independent sequence $\sequencen{X_n}$ of random
variables with zero expectation such that
$\|X_n\|_{\infty}=\|\bover1{n+1}\sum_{i=0}^nX_i\|_{\infty}=1$ for every
$n\in\Bbb N$.
%273N

\spheader 273Xn Repeat the work of this section for
complex-valued random variables.
%273+

\spheader 273Xo Let $(X,\Sigma,\mu)$ be a probability space and
$\sequencen{E_n}$ an independent sequence in $\Sigma$ such that
$\alpha=\lim_{n\to\infty}\mu E_n$ is defined.   For $x\in X$ set
$I_x=\{n:x\in E_n\}$.   Show that $I_x$ has asymptotic density $\alpha$ for
almost every $x$.
%273G

\leader{273Y}{Further exercises (a)}
%\spheader 273Ya
Let $(\Omega,\Sigma,\mu)$ be a
probability space, and $\lambda$ the product measure on
$\Omega^{\Bbb N}$.   Suppose that $f$ is a real-valued function, defined
on a subset of $\Omega$, such that

\Centerline{$h(\pmb{\omega})
=\lim_{n\to\infty}\Bover1{n+1}\sum_{i=0}^nf(\omega_i)$}

\noindent exists in $\Bbb R$ for $\lambda$-almost every
$\pmb{\omega}=\sequencen{\omega_n}$ in $\Omega^{\Bbb N}$.   Show (i)
that $f$ has conegligible domain (ii) $f$ is $\hat\Sigma$-measurable, where
$\hat\Sigma$ is the domain of the
completion of $\mu$ (iii) there is an $a\in\Bbb R$ such that $h=a$
almost everywhere in $\Omega^{\Bbb N}$ (iv) $f$ is integrable, with
$\int fd\mu=a$.
%273J mt27bits

\spheader 273Yb\dvAnew{2009}
Let $\sequencen{X_n}$ be a sequence of random variables
with finite variance.   Suppose that $\lim_{n\to\infty}\Expn(X_n)=\infty$
and $\liminf_{n\to\infty}
\Bover{\Expn(X_n^2)}{(\Expn(X_n))^2}\le 1$.
Show that $\limsup_{n\to\infty}X_n=\infty$ a.e.	
%273K
}%end of exercises

\endnotes{
\Notesheader{273} I have tried in this section to offer the most useful
of the standard criteria for pointwise convergence of averages of
independent random variables.   In my view the strong law of large
numbers, like Fubini's theorem, is one of the crucial steps in measure
theory, where the subject changes character.   Theorems depending on the
strong law have a kind of depth and subtlety to them which is missing in
other parts of the subject.   I have described only a handful of
applications here, but I hope that 273G, 273J, 273Xd, 273Xg and 273Xi
will give
an idea of what is to be expected.   These do have rather different
weights.   Of the four, only 273J requires the full resources of this
chapter;  the others can be deduced from the essentially simpler version
in 273Xi.

273Xi is the `fundamental theorem of statistics' or `Glivenko-Cantelli
theorem'.   The $F_n(.,a)$
are `statistics', computed from the $X_i$;  they are the `empirical
distributions', and the theorem says that, almost surely, $F_n\to F$
uniformly.   (I say `uniformly' to make the result look more striking,
but of course the real content is that $F_n(.,a)\to F(a)$ almost surely
for each $a$;  the extra step is just 273Xh.)

I have included 273N to show that independence is quite as important in
questions of norm-convergence as it is in questions of pointwise
convergence.   It does not really rely on any form of the strong law;  in
the proof I
quote 273E as a quick way of disposing of the `uniformly bounded parts'
$X'_n$, but of course Bienaym\'e's equality (272S) is already enough
to show that if $\sequencen{X'_n}$ is an independent uniformly bounded
sequence of random variables with zero expectation, then
$\|\bover1{n+1}(X_0+\ldots+X_n)\|_p\to 0$ for $p=2$, and therefore
for every $p<\infty$.

The proofs of 273H, 273I and 273Na all involve
`truncation';  the expression of a random variable $X$ as the sum of a
bounded random variable and a tail.   This is one of the most powerful
techniques in the subject, and will appear again in \S276 and (in a rather
different way) in \S274.
In 273Na I used a slightly different formulation of the method, solely
because it matched the definition of `uniformly integrable' more
closely.
}%end of notes

\discrpage


\frfilename{mt274.tex}
\versiondate{13.4.10}
\copyrightdate{1995}

\def\chaptername{Probability theory}
\def\sectionname{The central limit theorem}

\newsection{274}

The second of the great theorems to which this chapter is devoted is of
a new type.   It is a limit theorem, but the limit involved is a limit
of {\it distributions}, not of functions (as in the strong limit theorem
above or the martingale theorem below), nor of equivalence classes of
functions (as in Chapter 24).   I give three forms of the theorem, in
274I-274K, all drawn as corollaries of Theorem 274G;  the proof is
spread over 274C-274G.   In 274A-274B and 274M I give the most
elementary properties of the normal distribution.

\leader{274A}{The normal distribution }\cmmnt{We need some facts from
basic probability theory.

\medskip

}{\bf (a)}\cmmnt{ Recall that

\Centerline{$\int_{-\infty}^{\infty}e^{-x^2/2}dx=\sqrt{2\pi}$}

\noindent (263G).   Consequently, if we set}\dvro{ If we set}{}

\Centerline{$\mu_G E=\Bover1{\sqrt{2\pi}}\int_Ee^{-x^2/2}dx$}

\noindent for every Lebesgue measurable set $E$, $\mu_G$ is a Radon
probability measure\cmmnt{ (256E)};  we call it the {\bf standard
normal distribution}.   The corresponding distribution function is

\Centerline{$\Phi(a)
\cmmnt{\mskip5mu =\mu_G\ocint{-\infty,a}}
=\Bover1{\sqrt{2\pi}}\int_{-\infty}^ae^{-x^2/2}dx$}

\noindent for $a\in\Bbb R$;  for the rest of this section I will reserve
the symbol $\Phi$ for this function.

\cmmnt{Writing $\Sigma$ for the algebra of Lebesgue
measurable subsets of $\Bbb R$, $(\Bbb R,\Sigma,\mu_G)$ is a probability
space.   Note that it is complete, and has the same
negligible sets as Lebesgue measure, because $e^{-x^2/2}>0$ for every
$x$ (cf.\ 234Lc).}

\header{274Ab}{\bf (b)} A random variable $X$ is {\bf standard normal}
if its distribution is $\mu_G$\cmmnt{;  that is, if the function
$x\mapsto\Bover1{\sqrt{2\pi}}e^{-x^2/2}$ is a density function for $X$.
The point of the remarks in (a) is that there are such random variables;
for instance, take the probability space $(\Bbb R,\Sigma,\mu_G)$ there,
and set $X(x)=x$ for every $x\in\Bbb R$}.

\header{274Ac}{\bf (c)} If $X$ is a standard normal random variable,
then

\Centerline{$\Expn(X)
=\Bover1{\sqrt{2\pi}}\int_{-\infty}^{\infty}xe^{-x^2/2}dx
=0$,}

\Centerline{$\Var(X)
=\Bover1{\sqrt{2\pi}}\int_{-\infty}^{\infty}x^2e^{-x^2/2}dx
=1$\dvro{.}{}}

\cmmnt{\noindent by 263H.}

\header{274Ad}{\bf (d)} More generally, a random variable $X$ is {\bf
normal} if there are $a\in\Bbb R$ and $\sigma>0$ such that
$Z=(X-a)/\sigma$ is standard normal.   In this
case\cmmnt{ $X=\sigma Z+a$ so}
$\Expn(X)\cmmnt{\mskip5mu =\sigma\Expn(Z)+a}=a$,
$\Var(X)\cmmnt{\mskip5mu =\sigma^2\Var(Z)}=\sigma^2$.
\cmmnt{

We have, for any $c\in\Bbb R$,

$$\eqalignno{\Bover1{\sigma\sqrt{2\pi}}
  \int_{\infty}^ce^{-(x-a)^2/2\sigma^2}dx
&=\Bover1{\sqrt{2\pi}}\int_{-\infty}^{(c-a)/\sigma}
  e^{-y^2/2}dy\cr
\noalign{\noindent (substituting $x=a+\sigma y$ for
$-\infty<y\le(c-a)/\sigma$)}
&=\Pr(Z\le\Bover{c-a}{\sigma})
=\Pr(X\le c).\cr}$$

\noindent So}
$x\mapsto\Bover1{\sigma\sqrt{2\pi}}e^{-(x-a)^2/2\sigma^2}$
is a density function for $X$\cmmnt{ (271Ib)}.
Conversely,\cmmnt{ of course,} a random variable
with such a density function is normal,
with expectation $a$ and variance $\sigma^2$.
The {\bf normal distributions} are the distributions with these density
functions.

\header{274Ae}{\bf (e)} If\cmmnt{ $Z$ is standard normal, so is $-Z$,
because

\Centerline{$\Pr(-Z\le a)=\Pr(Z\ge -a)
=\Bover1{\sqrt{2\pi}}\int_{-a}^{\infty}e^{-x^2/2}dx
=\Bover1{\sqrt{2\pi}}\int_{-\infty}^{a}e^{-x^2/2}dx$.}

\noindent The definition in the first sentence of (d) now makes it
obvious that if} $X$ is normal, so is $a+bX$ for any $a\in\Bbb R$ and
$b\in\Bbb R\setminus\{0\}$.

\leader{274B}{Proposition} Let $X_1,\ldots,X_n$ be independent normal
random variables.   Then $Y=X_1+\ldots+X_n$ is normal, with
$\Expn(Y)=\Expn(X_1)+\ldots+\Expn(X_n)$ and
$\Var(Y)=\Var(X_1)+\ldots+\Var(X_n)$.

\proof{ There are innumerable proofs of this fact;  the
following one gives me a chance to show off the power of Chapter 26, but
of course (at the price of some disagreeable algebra) 272U also gives
the result.

\medskip

{\bf (a)} Consider first the case $n=2$.   Setting
$a_i=\Expn(X_i)$, $\sigma_i=\sqrt{\Var(X_i)}$,
$Z_i=(X_i-a_i)/\sigma_i$
we get independent standard normal variables $Z_1$, $Z_2$.   Set
$\rho=\sqrt{\sigma_1^2+\sigma_2^2}$, and express $\sigma_1$, $\sigma_2$
as $\rho\cos\theta$, $\rho\sin\theta$.   Consider $U=\cos\theta
Z_1+\sin\theta Z_2$.   We know that $(Z_1,Z_2)$ has a density function

\Centerline{$(\zeta_1,\zeta_2)\mapsto g(\zeta_1,\zeta_2)=
\Bover1{2\pi\sigma_1\sigma_2}e^{-(\zeta_1^2+\zeta_2^2)/2}$}

\noindent (272I).   Consequently, for any $c\in\Bbb R$,

\Centerline{$\Pr(U\le c)=\int_Fg(z)dz$,}

\noindent where
$F=\{(\zeta_1,\zeta_2):\zeta_1\cos\theta+\zeta_2\sin\theta\le c\}$.
But now let $T$ be the matrix

$$\Matrix{\cos\theta&-\sin\theta\\\sin\theta&\cos\theta}.$$

\noindent Then it is easy to check that

\Centerline{$T^{-1}[F]=\{(\eta_1,\eta_2):\eta_1\le c\}$,}

\Centerline{$\det T=1$,
\quad$g(Ty)=g(y)$ for every $y\in\BbbR^2$,}

\noindent so by 263A

\Centerline{$\Pr(U\le c)=\int_Fg(z)dz=\int_{T^{-1}[F]}g(Ty)dy
=\int_{\ocint{-\infty,c}\times\Bbb R}g(y)dy
=\Pr(Z_1\le c)=\Phi(c)$.}

\noindent As this is true for every $c\in\Bbb R$, $U$ also is standard
normal (I am appealing to 271Ga again).   But

\Centerline{$X_1+X_2=\sigma_1Z_1+\sigma_2Z_2+a_1+a_2
=\rho U+a_1+a_2$,}

\noindent so $X_1+X_2$ is normal.

\medskip

{\bf (b)} Now we can induce on $n$.   If $n=1$ the result is trivial.
For the inductive step to $n+1\ge 2$, we know that $X_1+\ldots+X_n$ is
normal, by the inductive hypothesis, and that $X_{n+1}$ is independent
of $X_1+\ldots+X_n$, by 272L.   So $X_1+\ldots+X_n+X_{n+1}$ is normal,
by (a).

The computation of the expectation and variance of $X_1+\ldots+X_n$ is
immediate from 271Ab and 272S.
}%end of proof of 274B

\leader{274C}{Lemma} Let $U_0,\ldots,U_n,V_0,\ldots,V_n$ be independent
real-valued random variables and $h:\Bbb R\to\Bbb R$ a bounded Borel
measurable function.   Then

\Centerline{$|\Expn\bigl(h(\sum_{i=0}^nU_i)-h(\sum_{i=0}^nV_i)\bigr)|
\le\sum_{i=0}^n\sup_{t\in\Bbb R}|\Expn\bigl(h(t+U_i)-h(t+V_i)\bigr)|$.}

\proof{ For $0\le j\le n+1$, set
$Z_j=\sum_{i=0}^{j-1}U_i+\sum_{i=j}^nV_i$, taking $Z_0=\sum_{i=0}^nV_i$
and $Z_{n+1}=\sum_{i=0}^nU_i$, and for $j\le n$ set
$W_j=\sum_{i=0}^{j-1}U_j+\sum_{i=j+1}^nV_j$, so that $Z_j=W_j+V_j$ and
$Z_{j+1}=W_j+U_j$ and $W_j$, $U_j$ and $V_j$ are independent (I am
appealing to 272K, as in 272L).   Then

$$\eqalign{|\Expn\bigl(h(\sum_{i=0}^nU_i)-h(\sum_{i=0}^nV_i)\bigr)|
&=|\Expn\bigl(\sum_{i=0}^nh(Z_{i+1})-h(Z_i)\bigr)|\cr
&\le\sum_{i=0}^n|\Expn\bigl(h(Z_{i+1})-h(Z_i)\bigr)|\cr
&=\sum_{i=0}^n|\Expn\bigl(h(W_i+U_i)-h(W_i+V_i)\bigr)|.\cr}$$

To estimate this sum I turn it into a sum of integrals, as follows.
For each $i$, let $\nu_{W_i}$ be the distribution of $W_i$, and so on.
Because $(w,u)\mapsto w+u$ is continuous, therefore Borel measurable,
$(w,u)\mapsto h(w,u)$ also is Borel measurable;  accordingly
$(w,u,v)\mapsto h(w+u)-h(w+v)$ is measurable for each of the product
measures $\nu_{W_i}\times\nu_{U_i}\times\nu_{V_i}$ on $\BbbR^3$, and
271E and 272G give us

$$\eqalignno{\bigl|\Expn\bigl(h(W_i+U_i)-&h(W_i+V_i)\bigr)\bigr|\cr
&=\bigl|\int h(w+u)-h(w+v)
(\nu_{W_i}\times\nu_{U_i}\times\nu_{V_i})d(w,u,v)\bigr|\cr
&=\bigl|\int\bigl(\int h(w+u)-h(w+v)(\nu_{U_i}\times\nu_{V_i})d(u,v)
\bigr)\nu_{W_i}(dw)\bigr|\cr
&\le\int\bigl|\int h(w+u)-h(w+v)(\nu_{U_i}\times\nu_{V_i})d(u,v)\bigr|
\nu_{W_i}(dw)\cr
&=\int\bigl|\Expn\bigl(h(w+U_i)-h(w+V_i)\bigr)\bigr|\nu_{W_i}(dw)\cr
&\le\sup_{t\in\Bbb R}
    \bigl|\Expn\bigl(h(t+U_i)-h(t+V_i)\bigr)\bigr|.\cr}$$

\noindent So we get

$$\eqalign{|\Expn\bigl(h(\sum_{i=0}^nU_i)-h(\sum_{i=0}^nV_i)\bigr)|
&\le\sum_{i=0}^n|\Expn\bigl(h(W_i+U_i)-h(W_i+V_i)\bigr)|\cr
&\le\sum_{i=0}^n
\sup_{t\in\Bbb R}|\Expn\bigl(h(t+U_i)-h(t+V_i)\bigr)|,\cr}$$

\noindent as required.
}%end of proof of 274C

\leader{274D}{Lemma} Let $h:\Bbb R\to\Bbb R$ be a bounded
three-times-differentiable function such that
$M_2=\sup_{x\in\Bbb R}|h''(x)|$, $M_3=\sup_{x\in\Bbb R}|h'''(x)|$ are
both finite.   Let $\epsilon>0$.

(a) Let $U$ be a real-valued random variable with zero expectation and
finite variance $\sigma^2$.   Then for any $t\in\Bbb R$ we have

\Centerline{$|\Expn(h(t+U))-h(t)-\Bover{\sigma^2}2h''(t)|
\le\Bover16\epsilon M_3\sigma^2+M_2\Expn(\psi_{\epsilon}(U))$}

\noindent where $\psi_{\epsilon}(x)=0$ if $|x|\le \epsilon$, $x^2$ if
$|x|>\epsilon$.

(b) Let $U_0,\ldots,U_n,V_0,\ldots,V_n$ be independent random variables
with finite variances, and suppose that $\Expn(U_i)=\Expn(V_i)=0$ and
$\Var(U_i)=\Var(V_i)=\sigma_i^2$ for every $i\le n$.   Then

$$\eqalign{|\Expn\bigl(h(\sum_{i=0}^n&U_i)-h(\sum_{i=0}^nV_i)\bigr)|\cr
&\le\Bover13\epsilon M_3\sum_{i=0}^n\sigma_i^2
+M_2\sum_{i=0}^n\Expn\bigl(\psi_{\epsilon}(U_i)\bigr)
+M_2\sum_{i=0}^n\Expn\bigl(\psi_{\epsilon}(V_i)\bigr).\cr}$$

\proof{{\bf (a)}   The point is that, by Taylor's theorem with
remainder,

\Centerline{$|h(t+x)-h(t)-xh'(t)|\le\Bover12M_2x^2$,}

\Centerline{$|h(t+x)-h(t)-xh'(t)-\Bover12x^2h''(t)|
\le\Bover16M_3|x|^3$}

\noindent for every $x\in\Bbb R$.   So

\Centerline{$|h(t+x)-h(t)-xh'(t)-\Bover12x^2h''(t)|
\le\min(\Bover16M_3|x|^3,M_2x^2)
\le \Bover16\epsilon M_3x^2+M_2\psi_{\epsilon}(x)$.}

\noindent Integrating with respect to the distribution of $U$, we get

$$\eqalign{|\Expn\bigl(h(t+U))-h(t)-\Bover12h''(t)\sigma^2\bigr)|
&=|\Expn(h(t+U))-h(t)-h'(t)\Expn(U)
-\Bover12h''(t)\Expn(U^2)|\cr
&=|\Expn\bigl(h(t+U)-h(t)-h'(t)U-\Bover12h''(t)U^2\bigr)|\cr
&\le\Expn\bigl(|h(t+U)-h(t)-h'(t)U-\Bover12h''(t)U^2|\bigr)\cr
&\le\Expn\bigl(\Bover16\epsilon M_3U^2+M_2\psi_{\epsilon}(U)\bigr)\cr
&=\Bover16\epsilon M_3\sigma^2+M_2\Expn(\psi_{\epsilon}(U)),\cr}$$

\noindent as claimed.

\medskip

{\bf (b)} By 274C,

$$\eqalign{|\Expn\bigl(h(\sum_{i=0}^nU_i)-h(\sum_{i=0}^nV_i)\bigr)|
&\le\sum_{i=0}^n\sup_{t\in\Bbb R}|\Expn\bigl(h(t+U_i)-h(t+V_i)\bigr)|\cr
&\le\sum_{i=0}^{n}\sup_{t\in\Bbb R}\bigl(
|\Expn(h(t+U_i))-h(t)-\Bover12h''(t)\sigma_i^2|\cr
&\qquad\qquad\qquad
+|\Expn(h(t+V_i))-h(t)-\Bover12h''(t)\sigma_i^2|\bigr),\cr}$$

\noindent which by (a) above is at most

\Centerline{$\sum_{i=0}^n\Bover13\epsilon  M_3\sigma_i^2
+M_2\Expn(\psi_{\epsilon}(U_i))+M_2\Expn(\psi_{\epsilon}(V_i))$,}

\noindent as claimed.
}%end of proof of 274D

\leader{274E}{Lemma} For any $\epsilon>0$, there is a
three-times-differentiable function $h:\Bbb R\to[0,1]$, with continuous
third derivative, such that $h(x)=1$ for $x\le -\epsilon$ and $h(x)=0$
for $x\ge\epsilon$.

\proof{ Let $f:\ooint{-\epsilon,\epsilon}
\to\ooint{0,\infty}$
be any twice-differentiable function such that

\Centerline{
$\lim_{x\downarrow-\epsilon}f^{(n)}(x)
=\lim_{x\uparrow\epsilon}f^{(n)}(x)=
0$}

\noindent for $n=0$, $1$ and $2$, writing $f^{(n)}$ for the $n$th
derivative
of $f$;  for instance, you could take $f(x)=(\epsilon^2-x^2)^3$, or
$f(x)=\exp(-\Bover1{\epsilon^2-x^2})$.   Now set

\Centerline{$h(x)=1-\int_{-\epsilon}^xf/\int_{-\epsilon}^{\epsilon}f$}

\noindent for $|x|\le\epsilon$.
}%end of proof of 274E

\leader{274F}{Lindeberg's theorem} Let $\epsilon>0$.   Then there is a
$\delta>0$ such that whenever $X_0,\ldots,X_n$ are independent
real-valued random variables such that

\Centerline{$\Expn(X_i)=0$ for every $i\le n$,}

\Centerline{$\sum_{i=0}^n\Var(X_i)=1$,}

\Centerline{$\sum_{i=0}^n\Expn(\psi_{\delta}(X_i))\le\delta$}

\noindent (writing $\psi_{\delta}(x)=0$ if $|x|\le\delta$, $x^2$ if
$|x|>\delta$), then

\Centerline{$\bigl|\Pr(\sum_{i=0}^nX_i\le a)
-\Phi(a)\bigr|
\le\epsilon$}

\noindent for every $a\in\Bbb R$.

\proof{{\bf (a)} Let $h:\Bbb R\to[0,1]$ be a
three-times-differentiable function, with continuous third derivative,
such that
$\chi\ocint{-\infty,-\epsilon}\le h\le\chi\ocint{-\infty,\epsilon}$, as
in 274E.   Set

\Centerline{$M_2=\sup_{x\in\Bbb
R}|h''(x)|=\sup_{|x|\le\epsilon}|h''(x)|$,}

\Centerline{$M_3=\sup_{x\in\Bbb
R}|h'''(x)|=\sup_{|x|\le\epsilon}|h'''(x)|$;}

\noindent because \hbox{$h'''$} is continuous, both are finite.   Write
$\epsilon'=\epsilon(1-\Bover2{\sqrt{2\pi}})>0$,   and let $\eta>0$ be
such that

\Centerline{$(\Bover13M_3+2M_2)\eta
\le \epsilon'$.}

\noindent Note that $\lim_{m\to\infty}\psi_m(x)=0$ for every $x$, so if
$X$ is a random variable with finite variance we must have
$\lim_{m\to\infty}\Expn(\psi_m(X))\discretionary{}{}{}=0$,
by Lebesgue's Dominated
Convergence Theorem;  let $m\ge 1$ be such that
$\Expn(\psi_m(Z))\le\eta$, where $Z$ is some (or any) standard normal
random variable.   Finally, take $\delta>0$ such that
$\delta+\delta^2\le(\eta/m)^2$;  note that $\delta\le\eta$.

(I hope that you have seen enough $\epsilon$-$\delta$ arguments not to
be troubled by any expectation of understanding the reasons for each
particular formula here before reading the rest of the argument.   But
the formula
$\bover13M_3+2M_2$, in association with $\psi_{\delta}$, should recall
274D.)

\medskip

{\bf (b)} Let $X_0,\ldots,X_n$ be independent random
variables with zero expectation such that $\sum_{i=0}^n\Var(X_i)=1$ and
$\sum_{i=0}^n\Expn(\psi_{\delta}(X_i))\le\delta$.   We need an auxiliary
sequence $Z_0,\ldots,Z_n$ of standard
normal random variables to match against the $X_i$.   To create this, I
use the following device.   Suppose that the probability space
underlying $X_0,\ldots,X_n$ is $(\Omega,\Sigma,\mu)$.
Set $\Omega'=\Omega\times\BbbR^{n+1}$, and let $\mu'$ be the product
measure on $\Omega'$, where $\Omega$ is given the measure $\mu$ and each
factor $\Bbb R$ of $\BbbR^{n+1}$ is given the measure $\mu_G$.   Set
$X'_i(\omega,z)=X_i(\omega)$ and $Z_i(\omega,z)=\zeta_i$ for
$\omega\in\dom
X_i$, $z=(\zeta_0,\ldots,\zeta_n)\in\BbbR^{n+1}$, $i\le n$.   Then
$X'_0,\ldots,X'_n,Z_0,\ldots,Z_n$ are independent, and each $X'_i$ has
the same distribution as $X_i$ (272Mb).   Consequently
$S'=X'_0+\ldots+X'_n$ has the same distribution
as $S=X_0+\ldots+X_n$ (using 272T, or otherwise);  so that
$\Expn(g(S'))=\Expn(g(S))$ for any
bounded Borel measurable function $g$ (using 271E).   Also each $Z_i$
has distribution $\mu_G$, so is standard normal.

\medskip

{\bf (c)} Write $\sigma_i=\sqrt{\Var(X_i)}$ for each $i$, and set
$K=\{i:i\le n,\,\sigma_i>0\}$.   Observe that $\eta/\sigma_i\ge m$ for
each $i\in K$.   \Prf\ We know that

\Centerline{$\sigma_i^2=\Var(X_i)=\Expn(X_i^2)
\le\Expn(\delta^2+\psi_{\delta}(X_i))=\delta^2+\Expn(\psi_{\delta}(X_i))
\le\delta^2+\delta$,}

\noindent so

\Centerline{$\Bover{\eta}{\sigma_i}\ge\Bover{\eta}{\sqrt{\delta+\delta^2}}
\ge m$}

\noindent by the choice of $\delta$.   \Qed

\medskip

{\bf (d)} Consider the independent normal random variables
$\sigma_iZ_i$.   We have $\Expn(\sigma_iZ_i)=\Expn(X'_i)=0$ and
$\Var(\sigma_iZ_i)=\Var(X'_i)=\sigma_i^2$ for each $i$, so that
$Z=\sigma_0Z_0+\ldots+\sigma_nZ_n$ has expectation $0$ and variance
$\sum_{i=0}^n\sigma_i^2=1$;
moreover, by 274B, $Z$ is normal, so in fact it is standard normal.
Now we have

$$\eqalignno{\sum_{i=0}^n\Expn(\psi_{\eta}(\sigma_iZ_i))
&=\sum_{i\in K}\Expn(\psi_{\eta}(\sigma_iZ_i))
=\sum_{i\in K}\sigma_i^2\Expn(\psi_{\eta/\sigma_i}(Z_i))\cr
\noalign{\noindent (because
$\sigma^2\psi_{\eta/\sigma}(x)=\psi_{\eta}(\sigma x)$ whenever
$x\in\Bbb R$, $\sigma>0$)}
&=\sum_{i\in K}\sigma_i^2\Expn(\psi_{\eta/\sigma_i}(Z))
\le\sum_{i\in K}\sigma_i^2\Expn(\psi_{m}(Z))\cr
\noalign{\noindent (because, by (c),
$\eta/\sigma_i\ge m$ for every $i\in K$, so
$\psi_{\eta/\sigma_i}(t)\le\psi_m(t)$ for every $t$)}
&=\Expn(\psi_m(Z))
\le\eta\cr}$$

\noindent (by the choice of $m$).   On the other hand, we surely have

\Centerline{$\sum_{i=0}^n\Expn(\psi_{\eta}(X'_i))
=\sum_{i=0}^n\Expn(\psi_{\eta}(X_i))
\le\sum_{i=0}^n\Expn(\psi_{\delta}(X_i))
\le\delta\le\eta$.}

\medskip

{\bf (e)} For any real number $t$, set

\Centerline{$h_{t}(x)=h(x-t)$}

\noindent for each $x\in\Bbb R$.   Then $h_t$ is
three-times-differentiable, with $\sup_{x\in\Bbb R}|h_t''(x)|=M_2$ and
$\sup_{x\in\Bbb R}|h'''(x)|=M_3$.   Consequently

\Centerline{$|\Expn(h_t(S))-\Expn(h_t(Z))|
\le\epsilon'$.}

\noindent\vthsp\Prf\ By 274Db,

$$\eqalign{|\Expn(h_t(S))-\Expn(h_t(Z))|
&=|\Expn(h_t(S'))-\Expn(h_t(Z))|\cr
&=|\Expn(h_t(\sum_{i=0}^nX'_i))-\Expn(h_t(\sum_{i=0}^n\sigma_iZ_i))|\cr
&\le\Bover13\eta M_3\sum_{i=0}^n\sigma_i^2
+M_2\sum_{i=0}^n\Expn(\psi_{\eta}(X_i))
+M_2\sum_{i=0}^n\Expn(\psi_{\eta}(\sigma_iZ_i))\cr
&\le\Bover13\eta M_3+M_2\eta+M_2\eta
\le\epsilon',\cr}$$

\noindent by the choice of $\eta$.  \Qed


\medskip

{\bf (f)} Now take any $a\in\Bbb R$.   We have

\Centerline{$\chi\ocint{-\infty,a-2\epsilon}
\le h_{a-\epsilon}\le\chi\ocint{\infty,a}
\le h_{a+\epsilon}\le\chi\ocint{-\infty,a+\epsilon}$.}

\noindent Note also that, for any $b$,

\Centerline{$\Phi(b+2\epsilon)
=\Phi(b)+\Bover1{\sqrt{2\pi}}\int_b^{b+2\epsilon}e^{-x^2/2}dx
\le\Phi(b)+\Bover{2\epsilon}{\sqrt{2\pi}}
=\Phi(b)+\epsilon-\epsilon'$.}


\noindent Consequently

$$\eqalign{\Phi(a)-\epsilon
&\le\Phi(a-2\epsilon)-\epsilon'
=\Pr(Z\le a-2\epsilon)-\epsilon'
\le\Expn(h_{a-\epsilon}(Z))-\epsilon'
\le\Expn(h_{a-\epsilon}(S))\cr
&\le\Pr(S\le a)\cr
&\le\Expn(h_{a+\epsilon}(S))
\le\Expn(h_{a+\epsilon}(Z))+\epsilon'
\le\Pr(Z\le a+2\epsilon)+\epsilon'
=\Phi(a+2\epsilon)+\epsilon'\cr
&\le\Phi(a)+\epsilon.\cr}$$

\noindent But this means just that

\Centerline{$\bigl|\Pr(\sum_{i=0}^nX_i\le a)
-\Phi(a)\bigr|
\le\epsilon$,}

\noindent as claimed.
}%end of proof of 274F

\leader{274G}{Central Limit Theorem} Let $\sequencen{X_n}$ be
an independent sequence of random variables,
all with zero expectation and finite
variance;  write $s_n=\sqrt{\sum_{i=0}^n\Var(X_i)}$ for each $n$.
Suppose that

\Centerline{$\lim_{n\to\infty}\Bover1{s_n^2}\sum_{i=0}^n
\Expn(\psi_{\delta s_n}(X_i))=0$ for every $\delta>0$,}

\noindent writing $\psi_{\delta}(x)=0$ if $|x|\le\delta$, $x^2$ if
$|x|>\delta$.   Set

\Centerline{$S_n=\Bover1{s_n}(X_0+\ldots+X_n)$}

\noindent for each $n\in\Bbb N$ such that $s_n>0$.   Then

\Centerline{$\lim_{n\to\infty}\Pr(S_n\le a)
=\Phi(a)$}

\noindent uniformly for $a\in\Bbb R$.

\proof{  Given $\epsilon>0$, take $\delta>0$ as in
Lindeberg's theorem (274F).   Then for all $n$ large enough,

\Centerline{$\Bover1{s_n^2}\sum_{i=0}^n
\Expn(\psi_{\delta s_n}(X_i))\le\delta$.}

\noindent Fix on any such $n$.   Of course we have $s_n>0$.   Set

\Centerline{$X'_i=\Bover1{s_n}X_i$ for $i\le n$;}

\noindent then $X'_0,\ldots,X'_n$ are independent, with zero expectation,

\Centerline{$\sum_{i=0}^n\Var(X'_i)=\sum_{i=0}^n\Bover1{s_n^2}\Var(X_i)
=1$,}

\Centerline{$\sum_{i=0}^n\Expn(\psi_{\delta}(X'_i))
=\sum_{i=0}^n\Bover1{s_n^2}\Expn(\psi_{\delta s_n}(X_i))
\le\delta$.}

\noindent By 274F,

\Centerline{$\bigl|\Pr(S_n\le a)
-\Phi(a)\bigr|
=\bigl|\Pr(\sum_{i=0}^nX'_i\le a)
-\Phi(a)\bigr|
\le\epsilon$}

\noindent for every $a\in\Bbb R$.   Since this is true for all $n$ large
enough, we have the result.
}%end of proof of 274G

\leader{274H}{Remarks (a)} The condition

\Centerline{$\lim_{n\to\infty}\Bover1{s_n^2}\sum_{i=0}^n
\Expn(\psi_{\epsilon s_n}(X_i))=0$ for every $\epsilon>0$}

\noindent is called {\bf Lindeberg's condition}\cmmnt{, following
{\smc Lindeberg 1922}}.

\header{274Hb}{\bf (b)} Lindeberg's condition is necessary as well as
sufficient, in the following sense.   Suppose that $\sequencen{X_n}$ is
an independent sequence of real-valued random variables with zero expectation and
finite variance;  write $\sigma_n=\sqrt{\Var(X_n)}$,
$s_n=\sqrt{\sum_{i=0}^n\Var(X_i)}$ for each $n$.   Suppose that
$\lim_{n\to\infty}s_n=\infty$,
$\lim_{n\to\infty}\Bover{\sigma_n}{s_n}=0$ and that
$\lim_{n\to\infty}\Pr(S_n\le a)=\Phi(a)$ for each $a\in\Bbb R$, where
$S_n=\Bover1{s_n}(X_0+\ldots+X_n)$.   Then

\Centerline{$\lim_{n\to\infty}\Bover1{s_n^2}\sum_{i=0}^n
\Expn(\psi_{\epsilon s_n}(X_i))=0$}

\noindent for every $\epsilon>0$.   \cmmnt{({\smc Feller 66}, \S XV.6,
Theorem 3;  {\smc Lo\`eve 77}, \S21.2.)
}%end of comment

\cmmnt{\header{274Hc}{\bf (c)}
The proof of 274F-274G
here is adapted from {\smc Feller 66}, \S VIII.4.   It has the virtue
of being `elementary', in that it does not involve characteristic
functions.   Of course this has to be paid for by a number of
detailed estimations;  and -- what is much more serious -- it leaves us
without one of the most powerful techniques for describing
distributions.   The proof does offer a method of bounding

\Centerline{$|\Pr(S_n\le a)-\Phi(a)|$;}

\noindent but it should be said that the bounds obtained are not useful
ones, being grossly over-pessimistic, at least in the readily analysable
cases.   (For instance, a better bound, in many cases, is given by the
Berry-Ess\'een theorem:  if $\sequencen{X_n}$ is independent and
identically distributed, with zero expectation, and the common values of
$\sqrt{\Expn(X_n^2)}$,  $\Expn(|X_n|^3)$ are $\sigma$, $\rho<\infty$,
then

\Centerline{$|\Pr(S_n\le a)-\Phi(a)|
\le\Bover{33\rho}{4\sigma^3\sqrt{n+1}}$;}

\noindent see {\smc Feller 66}, \S XVI.5, {\smc Lo\`eve 77},
\S21.3, or {\smc Hall 82}.)   Furthermore, when $|a|$ is large,
$\Phi(a)$ is exceedingly
close to either $0$ or $1$, so that any uniform bound for
$|\Pr(S\le a)-\Phi(a)|$ gives very little information;  a great deal of
work has been done on estimating the tails of such distributions more
precisely, subject to special conditions.   For instance, if
$X_0,\ldots,X_n$ are independent random variables with zero expectation,
uniformly bounded with $|X_i|\le K$ almost everywhere for each $i$,
$Y=X_0+\ldots+X_n$, $s=\sqrt{\Var(Y)}>0$, $S=\bover1sY$, then for any
$\alpha\in[0,s/K]$

$$\Pr(|S|\ge\alpha)
\le 2\exp\bigl(\bover{-\alpha^2}{2(1+\bover{\alpha K}{2s})^2}\bigr)
\bumpeq 2e^{-\alpha^2/2}$$

\noindent if $s\gg\alpha K$ ({\smc R\'enyi 70}, \S VII.4, Theorem 1).
A less precise result of the same kind is in 272Xl.

I now list some of the standard cases in which Lindeberg's condition
is satisfied, so that we may apply the theorem.
}%end of comment

\leader{274I}{Corollary} Let $\sequencen{X_n}$ be an independent
sequence of real-valued random variables, all with the same
distribution, and suppose that their common expectation is $0$ and their
common
variance is finite and not zero.   Write $\sigma$ for the common value
of $\sqrt{\Var(X_n)}$, and set

\Centerline{$S_n=\Bover1{\sigma\sqrt{n+1}}(X_0+\ldots+X_n)$}

\noindent for each $n\in\Bbb N$.   Then

\Centerline{$\lim_{n\to\infty}\Pr(S_n\le a)=\Phi(a)$}

\noindent uniformly for $a\in\Bbb R$.

\proof{ In the language of 274G-274H, we have $\sigma_n=\sigma$,
$s_n=\sigma\sqrt{n+1}$ and $S_n=\Bover1{s_n}\sum_{i=0}^nX_i$.
Moreover, if $\nu$ is the common distribution of the $X_n$, then

\Centerline{$\Expn(\psi_{\epsilon s_n}(X_n))
=\int_{\{x:|x|>\epsilon\sigma\sqrt{n}\}}x^2\nu(dx)\to 0$}

\noindent by Lebesgue's Dominated Convergence Theorem;  so that

\Centerline{$\Bover1{s_n^2}\sum_{i=0}^n\Expn(\psi_{\epsilon s_n}(X_n))
\to 0$}

\noindent by 273Ca.   Thus Lindeberg's condition is satisfied and 274G
gives the result.
}%end of proof of 274I

\leader{274J}{Corollary} Let $\sequencen{X_n}$ be an independent
sequence of real-valued random variables with zero expectation, and
suppose that $\{X_n^2:n\in\Bbb N\}$ is uniformly
integrable and that

\Centerline{$\liminf_{n\to\infty}\Bover1{n+1}\sum_{i=0}^n\Var(X_i)>0$.}

\noindent Set

\Centerline{$s_n=\sqrt{\sumop_{i=0}^n\Var(X_i)}$,
\quad$S_n=\Bover1{s_n}(X_0+\ldots+X_n)$}
%\sumop needed because inside \sqrt{}

\noindent for large $n\in\Bbb N$.   Then

\Centerline{$\lim_{n\to\infty}\Pr(S_n\le a)=\Phi(a)$}

\noindent uniformly for $a\in\Bbb R$.

\proof{ The condition

\Centerline{$\liminf_{n\to\infty}\Bover1{n+1}\sum_{i=0}^n\Var(X_i)>0$}

\noindent means that there are $c>0$, $n_0\in\Bbb N$ such that
$s_n\ge c\sqrt{n+1}$ for every $n\ge n_0$.   Let the
underlying space be $(\Omega,\Sigma,\mu)$, and take $\epsilon$, $\eta>0$.
Writing
$\psi_{\delta}(x)=0$ for $|x|\le\delta$, $x^2$ for $|x|>\delta$, as in
274F-274G, we have

\Centerline{$\Expn(\psi_{\epsilon s_n}(X_i))
\le\Expn(\psi_{c\epsilon\sqrt{n+1}}(X_i))
=\int_{F(i,c\epsilon\sqrt{n+1})}X_i^2d\mu$}

\noindent for $n\ge n_0$, $i\le n$, where
$F(i,\gamma)=\{\omega:\omega\in\dom X_i,\,|X_i(\omega)|>\gamma\}$.
Because $\{X_i^2:i\in\Bbb N\}$ is uniformly integrable, there is a
$\gamma\ge 0$ such that $\int_{F(i,\gamma)}X_i^2d\mu\le\eta c^2$ for
every $i\in\Bbb N$ (246I).   Let $n_1\ge n_0$ be such that
$c\epsilon\sqrt{n_1+1}\ge\gamma$;  then for any $n\ge n_1$

\Centerline{$\Bover1{s_n^2}\sum_{i=0}^{n}\Expn(\psi_{\epsilon s_n}(X_i))
\le\Bover1{c^2(n+1)}\sum_{i=0}^n\eta c^2=\eta$.}

\noindent As $\epsilon$ and $\eta$ are arbitrary, the conditions of 274G
are satisfied and the result follows.
}%end of proof of 274J

\leader{274K}{Corollary} Let $\sequencen{X_n}$ be an independent
sequence of real-valued random variables with zero expectation,
and suppose that

(i) there is some $\delta>0$ such that
$\sup_{n\in\Bbb N}\Expn(|X_n|^{2+\delta})<\infty$,

(ii) $\liminf_{n\to\infty}\Bover1{n+1}\sum_{i=0}^n\Var(X_i)>0$.

\noindent Set $s_n=\sqrt{\sum_{i=0}^n\Var(X_i)}$ and

\Centerline{$S_n=\Bover1{s_n}(X_0+\ldots+X_n)$}

\noindent for large $n\in\Bbb N$.   Then

\Centerline{$\lim_{n\to\infty}\Pr(S_n\le a)
=\Phi(a)$}

\noindent uniformly for $a\in\Bbb R$.

\proof{ The point is that $\{X_n^2:n\in\Bbb N\}$ is
uniformly integrable.   \Prf\ Set
$K=1+\sup_{n\in\Bbb N}\Expn(|X_n|^{2+\delta})$.   Given $\epsilon>0$,
set $M=(K/\epsilon)^{1/\delta}$.   Then
$(X_n^2-M)^+\le M^{-\delta}|X_n|^{2+\delta}$, so

\Centerline{$\Expn(X_n^2-M)^+\le KM^{-\delta}=\epsilon$}

\noindent for every $n\in\Bbb N$.   As $\epsilon$ is arbitrary,
$\{X_n^2:n\in\Bbb N\}$ is uniformly integrable.   \Qed

Accordingly the conditions of 274J are satisfied and we have the result.
}

\cmmnt{
\leader{274L}{Remarks (a)} All the theorems of this section are devoted
to finding conditions under which a random variable $S$ is `nearly'
standard normal, in the sense that $\Pr(S\le a)\bumpeq\Pr(Z\le a)$
uniformly for $a\in\Bbb R$, where $Z$ is some (or any) standard normal
random variable.   In all cases the random variable $S$ is normalized to
have expectation $0$ and variance $1$, and is a sum of a large number of
independent random variables.   (In 274G and
274I-274K it is explicit that there must be many $X_i$, since they refer
to
a limit as $n\to\infty$.   This is not said in so many words in the
formulation I give of Lindeberg's theorem, but the proof makes it
evident that $n(\delta+\delta^2)\ge 1$, so surely $n$ will have to
be large there also.)

\header{274Lb}{\bf (b)} I cannot leave this section without remarking
that the form of the definition of `nearly standard normal' may lead
your intuition astray if you try to apply it to other distributions.
If we take $F$ to be the distribution function of $S$, so that
$F(a)=\Pr(S\le a)$, I am saying that $S$ is `nearly standard normal'
if $\sup_{a\in\Bbb R}|F(a)-\Phi(a)|$ is small.   It is
natural to think of this as approximation in a metric, writing

\Centerline{$\tilde\rho(\nu,\nuprime)
=\sup_{a\in\Bbb R}|F_{\nu}(a)-F_{\nuprime}(a)|$}

\noindent for distributions $\nu$, $\nuprime$ on $\Bbb R$, where
$F_{\nu}(a)=\nu\ocint{-\infty,a}$.   In this form, the theorems above
can be read as finding conditions under which
$\lim_{n\to\infty}\tilde\rho(\nu_{S_n},\mu_G)=0$.   But the point is
that $\tilde\rho$ is not really the right metric to use.   It works here
because $\mu_G$ is atomless.   But suppose, for instance, that $\nu$ is
the Dirac measure on $\Bbb R$ concentrated at $0$,
and that $\nu_n$ is the distribution of a normal random variable with
expectation $0$ and variance ${1\over n}$, for each $n\ge 1$.   Then
$F_{\nu}(0)=1$ and $F_{\nu_n}(0)={1\over 2}$, so
$\tilde\rho(\nu_n,\nu)={1\over 2}$ for each $n\ge 1$.   However, for
most
purposes one would regard the difference between $\nu_n$ and $\nu$ as
small, and surely $\nu$ is the only distribution which one could
reasonably call a limit of the $\nu_n$.

\header{274Lc}{\bf (c)} The difficulties here present themselves in more
than one form.   A statistician would be unhappy with the idea that the
$\nu_n$ of (b) above were far from $\nu$ (and from each other),
on the grounds that any measurement involving random variables with
these distributions must be subject to error, and small errors of
measurement will render them indistinguishable.   A pure mathematician,
looking forward to the possibility of generalizing these results, will
be unhappy with the emphasis given to the values of
$\nu\ocint{-\infty,a}$, for which it may be difficult to find suitable
equivalents in more abstract spaces.

\header{274Ld}{\bf (d)} These considerations join together to lead us to
a rather different definition for a topology on the space $P$ of
probability distributions on $\Bbb R$.   For any bounded continuous function $h:\Bbb R\to\Bbb R$ we have a pseudometric
$\rho_h:P\times P\to\coint{0,\infty}$
defined by writing

\Centerline{$\rho_h(\nu,\nuprime)=|\int h\,d\nu-\int h\,d\nuprime|$}

\noindent for all $\nu$, $\nuprime\in P$.   The {\bf vague topology} on $P$
is that generated by the pseudometrics $\rho_h$ (2A3F).   I will
not go into its properties in detail here (some are sketched in
274Yc-274Yf %274Yc 274Yd 274Ye 274Yf
below;  see also 285K-285L, 285S and 437J-437T
%437J 437K 437L 437M 437N (4{}37Q) 437P (4{}37T) 437R (4{}37O, 4{}37R)
%437S (4{}37P}) 437T (4{}37S)
and 454T-454V
in Volume 4).  But I maintain that the right way to look at the
results of this chapter is to say that (i) the distributions $\nu_S$ are
close to $\mu_G$ for the vague topology (ii) the sets
$\{\nu:\tilde\rho(\nu,\mu_G)<\epsilon\}$ are open for that topology, and
that is why $\tilde\rho(\nu_S,\mu_G)$ is small.
}%end of comment

\leader{*274M}{}\cmmnt{ I conclude with a simple pair of inequalities
which are frequently useful when studying normal random variables.

\medskip

\noindent}{\bf Lemma} (a)
$\int_x^{\infty}e^{-t^2/2}dt\le\Bover1xe^{-x^2/2}$ for every $x>0$.

(b) $\int_x^{\infty}e^{-t^2/2}dt\ge\Bover1{2x}e^{-x^2/2}$ for every
$x\ge 1$.

\proof{{\bf (a)}

$$\eqalign{\int_x^{\infty}e^{-t^2/2}dt
=\int_0^{\infty}e^{-(x+s)^2/2}ds
\le e^{-x^2/2}\int_0^{\infty}e^{-xs}ds
=\Bover1xe^{-x^2/2}.\cr}$$

\medskip

{\bf (b)} Set

\Centerline{$f(t)=e^{-t^2/2}-(1-x(t-x))e^{-x^2/2}$.}

\noindent Then $f(x)=f'(x)=0$ and $f''(t)=(t^2-1)e^{-t^2/2}$ is positive
for $t\ge x$ (because $x\ge 1$).   Accordingly $f(t)\ge 0$ for every
$t\ge x$, and $\int_x^{x+1/x}f(t)dt\ge 0$.   But this means just that

$$\eqalign{\int_x^{\infty}e^{-t^2/2}dt
\ge\int_x^{x+\bover1x}e^{-t^2/2}dt
\ge\int_x^{x+\bover1x}(1-x(t-x))e^{-x^2/2}dt
=\Bover1{2x}e^{-x^2/2},\cr}$$

\noindent as required.
}%end of proof of 274M

\exercises{
\leader{274X}{Basic exercises $\pmb{>}$(a)}
%\spheader 274Xa
Use 272U to give an alternative proof of 274B.
%274B

\spheader 274Xb Suppose that $f:\Bbb R\to\Bbb R$ is absolutely continuous
on every closed bounded interval, and that
$\int_{-\infty}^{\infty}|f'(x)|e^{-ax^2}dx\penalty-100<\infty$ for every $a>0$.
Let $X$ be a
normal random variable with zero expectation.   Show that $\Expn(Xf(X))$
and $\Expn(X^2)\Expn(f'(X))$ are defined and equal.
%274B

\spheader 274Xc Prove 274D when $h''$ is $M_3$-Lipschitz but not
necessarily differentiable.
%274D

\spheader 274Xd Let $\sequence{k}{m_k}$ be a strictly increasing
sequence in $\Bbb N$ such that $m_0=0$ and
$\lim_{k\to\infty}m_k/m_{k+1}=0$.   Let $\sequencen{X_n}$ be an
independent sequence of random variables such that
$\Pr(X_n=\sqrt{m_{k}})=\Pr(X_n=-\sqrt{m_{k}})=1/2m_{k}$,
$\Pr(X_n=0)=1-1/m_{k}$ whenever $m_{k-1}\le n<m_{k}$.    Show that the
Central Limit Theorem is not valid for $\sequencen{X_n}$.
\Hint{setting $W_k=(X_0+\ldots+X_{m_k-1})/\sqrt{m_k}$, show that
$\Pr(W_k\in[\epsilon,1-\epsilon])\to 0$ for every $\epsilon>0$.}
%274G mt27bits

\spheader 274Xe Let $\sequencen{X_n}$ be any independent
sequence of random variables all with the same distribution;  suppose
that they all have finite variance $\sigma^2>0$, and that their common
expectation is $c$.   Set $S_n=\Bover1{\sqrt{n+1}}(X_0+\ldots+X_n)$ for each $n$, and let $Y$ be a normal random variable with expectation $c$ and variance $\sigma^2$.   Show that
$\lim_{n\to\infty}\Pr(S_n\le a)=\Pr(Y\le a)$
uniformly for $a\in\Bbb R$.
%274I

%\sqheader 274Xf
\wheader{274Xf}{10}{4}{4}{48pt}$\pmb{>}${\bf (f)}
Show that for any $a\in\Bbb R$,

$$\lim_{n\to\infty}\bover{1}{2^n}
\sum_{r=0}^{\lfloor\bover{n}2+a\bover{\sqrt{n}}2\rfloor}
\bover{n!}{r!(n-r)!}
=\lim_{n\to\infty}\bover1{2^n}\#(\{I:I\subseteq
n,\,\#(I)\le\Bover{n}2+a\Bover{\sqrt{n}}2\})
=\Phi(a).$$
%274I

\spheader 274Xg Show that 274I is a special case of 274J.
%274J

\spheader 274Xh Let $\sequencen{X_n}$ be an independent sequence
of real-valued random variables with zero expectation.   Set
$s_n=\sqrt{\sum_{i=0}^n\Var(X_i)}$ and

\Centerline{$S_n=\Bover1{s_n}(X_0+\ldots+X_n)$}

\noindent for each $n\in\Bbb N$.   Suppose that there is some $\delta>0$
such that

\Centerline{$\lim_{n\to\infty}\Bover1{s_n^{2+\delta}}
\sum_{i=0}^n\Expn(|X_i|^{2+\delta})=0$.}

\noindent  Show that $\lim_{n\to\infty}\Pr(S_n\le a)=\Phi(a)$ uniformly
for $a\in\Bbb R$.   (This is a form of {\bf Liapounoff's central limit
theorem};  see {\smc Liapounoff 1901}.)
%274K

\spheader 274Xi Let $P$ be the set of Radon probability measures
on $\Bbb R$.   Let $\nu_0\in P$, $a\in\Bbb R$.   Show that the map
$\nu\mapsto\nu\ocint{-\infty,a}:P\to[0,1]$ is continuous at $\nu_0$ for
the vague topology on $P$ iff $\nu_0\{a\}=0$.
%274L

\spheader 274Xj Let $\sequencen{X_n}$ be an independent identically
distributed sequence of random variables with non-zero
finite variance.   Let
$\sequencen{t_n}$ be a sequence in $\Bbb R$ such that
$\sum_{n=0}^{\infty}t_n^2=\infty$.   Show that $\sum_{n=0}^{\infty}t_nX_n$
is undefined or infinite a.e.   \Hint{First deal with the case in which
$\sequencen{t_n}$ does not converge to $0$.
Otherwise, use 274G to show that, for any $n\in\Bbb N$,
$\lim_{m\to\infty}\Pr(|\sum_{i=n}^mt_iX_i|\ge 1)\ge\bover12)$.  See also
276Xd.}
%274G
%query:  do we really need `finite variance'?

\spheader 274Xk Let $\sequencen{X_n}$ be an independent
sequence of real-valued random variables with zero expectation.
Suppose that $M\ge 0$ is such that $|X_n|\le M$ a.e.\ for every $n$, and
that $\sum_{n=0}^{\infty}\Var(X_n)=\infty$.   Set
$s_n=\sqrt{\sum_{i=0}^n\Var(X_i)}$ for each $n$, and
$S_n=\Bover1{s_n}\sum_{i=0}^nX_i$ when $s_n>0$.   Show that
$\lim_{n\to\infty}\Pr(S_n\le a)=\Phi(a)$ for every $a\in\Bbb R$.
%274G
%out of order query

\leader{274Y}{Further exercises (a)}
%\spheader 274Ya
({\smc Steele 86})
Suppose that $X_0,\ldots,X_n,Y_0,\ldots,Y_n$ are independent random
variables such that, for each $i\le n$,
$X_i$ and $Y_i$ have the same distribution.
Let $h:\BbbR^{n+1}\to\Bbb R$ be a Borel measurable function, and set
$Z=h(X_0,\ldots,X_n)$, $Z_i=h(X_0,\ldots,X_{i-1},Y_i,X_{i+1},\ldots,X_n)$
for each $i$
(with $Z_0=h(Y_0,X_1,\ldots,X_n)$ and $Z_n=h(X_0,\ldots,X_{n-1},Y_n)$,
of course).   Suppose that $Z$ has finite expectation.   Show that
$\Var(Z)\le\bover12\sum_{i=0}^n\Expn(Z_i-Z)^2$.
%274C mt27bits

\spheader 274Yb\dvAnew{2010} Show that for any $\epsilon>0$ there is a
smooth function $h:\Bbb R\to[0,1]$ such that
$\chi\ocint{-\infty,-\epsilon}\le h\le\chi\coint{\epsilon,\infty}$.
%274E

\spheader 274Yc\dvAformerly{2{}74Ya}
Write $P$ for the set of Radon
probability measures on $\Bbb R$.   For $\nu$, $\nuprime\in P$ set

$$\eqalign{\rho(\nu,\nuprime)
=\inf\{\epsilon:\epsilon\ge 0,\,\nu\ocint{-\infty,a-\epsilon}-\epsilon
\le\nuprime\ocint{-\infty,a}
&\le\nu\ocint{-\infty,a+\epsilon}+\epsilon\cr
&\quad\quad\text{ for every }a\in\Bbb R\}.\cr}$$

\noindent Show that $\rho$ is a metric on $P$ and that it defines the
vague topology on $P$.   ($\rho$ is called {\bf L\'evy's metric}.)
%274L

\spheader 274Yd Write $P$ for the set of Radon probability
measures on $\Bbb R$, and let $\tilde\rho$ be the metric on $P$ defined
in 274Lb.   Show that if $\nu\in P$ is atomless and $\epsilon>0$, then
$\{\nuprime:\nuprime\in P,\,\tilde\rho(\nuprime,\nu)<\epsilon\}$ is open
for the vague topology on $P$.
%274L

\spheader 274Ye Let $\sequencen{S_n}$ be a sequence of
real-valued random variables, and $Z$ a standard normal random variable.
Show that the following are equiveridical:

\quad(i) $\mu_G=\lim_{n\to\infty}\nu_{S_n}$ for the vague topology,
writing $\nu_{S_n}$ for the distribution of $S_n$;

\quad(ii) $\Expn(h(Z))=\lim_{n\to\infty}\Expn(h(S_n))$ for every bounded
continuous function $h:\Bbb R\to\Bbb R$;

\quad(iii) $\Expn(h(Z))=\lim_{n\to\infty}\Expn(h(S_n))$ for every
bounded function $h:\Bbb R\to\Bbb R$ such that ($\alpha$) $h$ has
continuous derivatives of all orders ($\beta$) $\{x:h(x)\ne 0\}$ is
bounded;

\quad(iv) $\lim_{n\to\infty}\Pr(S_n\le a)=\Phi(a)$ for every
$a\in\Bbb R$;

\quad(v) $\lim_{n\to\infty}\Pr(S_n\le a)=\Phi(a)$ uniformly for
$a\in\Bbb R$;

\quad(vi) $\{a:\lim_{n\to\infty}\Pr(S_n\le a)=\Phi(a)\}$ is dense in
$\Bbb R$.

\noindent (See also 285L.)
%274L

\spheader 274Yf Let $(\Omega,\Sigma,\mu)$ be a probability space
and $P$ the set of Radon probability measures on
$\Bbb R$.   Show that $X\mapsto\nu_X:\eusm L^0(\mu)\to P$ is continuous
for the topology of convergence in measure on $\eusm L^0(\mu)$ and the
vague topology on $P$.
%274L 454U

\spheader 274Yg Let $\sequencen{X_n}$ be an independent
sequence of real-valued random variables.   Suppose that
there is an $M\ge 0$ such that $|X_n|\le M$ a.e.\ for every $n\in\Bbb N$,
and that $\sum_{n=0}^{\infty}X_n$ is defined, as a real number, almost
everywhere.   Show that $\sum_{n=0}^{\infty}\Var(X_n)<\infty$.
%274Xk 274G 274Yf

}%end of exercises

\endnotes{
\Notesheader{274} For more than two hundred years the Central Limit
Theorem has been one of the glories of mathematics, and no branch of
mathematics or science would be the same without it.   I suppose it is
the most important single theorem of probability theory;  and I observe
that the proof hardly uses measure theory.   To be sure, I have clothed
the arguments above in the language of measure and integration.   But if
you look at their essence, the vital elements of the proof are

\inset{(i) a linear combination of independent normal random variables
is normal (274Ae, 274B);}

\inset{(ii) if $U$, $V$, $W$ are independent random variables, and $h$
is a bounded continuous function, then
$|\Expn(h(U,V,W))|\le\sup_{t\in\Bbb R}|\Expn(h(U,V,t))|$ (274C);}

\inset{(iii) if $(X_0,\ldots,X_n)$ are independent random variables,
then we can find independent random variables
$(X_0',\ldots,X_n',Z_0,\ldots,Z_n)$ such that $Z_j$ is standard normal
and $X_j'$ has the same distribution as $X_j$, for each $j$ (274F).}

\noindent The rest of the argument consists of elementary calculus,
careful estimations and a few of the most fundamental properties of
expectations and independence.   Now (ii) and (iii) are justified above
by appeals to Fubini's theorem, but surely they belong to the list of
probabilistic intuitions which take priority over the identification of
probabilities with countably additive functionals.   If they had given
any insuperable difficulty it would have been a telling argument against
the model of probability we were using, but would not have affected the
Central Limit Theorem.   In fact (i) seems to be the place where we
really need a mathematical  model of the concept of `distribution', and
all the relevant calculations can be done in terms of the Riemann
integral on the plane, with no mention of countable additivity.   So
while I am happy and proud to have written out a version of these
beautiful ideas, I have to admit that they are in no essential way
dependent on the rest of this treatise.

In \S285 I will describe a quite different approach to the theorem,
using much more sophisticated machinery;  but it will again be the case,
perhaps more thoroughly hidden, that the relevance of measure theory
will not be to the theorem itself, but to our imagination of what an
arbitrary distribution is.   For here I do have a claim to make for my
subject.   The characterization of distribution functions as arbitrary
monotonic functions, continuous on the right, and with the correct limits
at $\pm\infty$ (271Xb), together with the analysis of monotonic
functions in \S226, gives us a chance of forming a mental picture of
the proper class of objects to which such results as the Central Limit
Theorem can be applied.

Theorem 274F is a trifling modification of Theorem 3 of
{\smc Lindeberg 1922}.   Like the original, it emphasizes what I
believe to be vital to all the limit theorems of this chapter:  they are
best founded on a proper understanding of finite sequences of random
variables.   Lindeberg's condition was the culmination of a long search
for the most general conditions under which the Central Limit Theorem
would be valid.  I offer a version of Laplace's theorem (274Xf)
as the starting place, and Liapounoff's condition (274Xh) as an example
of one of the intermediate stages.   Naturally the corollaries 274I,
274J, 274K and 274Xe are those one seeks to apply by choice.   There is
an intriguing, but as far as I know purely coincidental, parallel between
273H/274K and 273I/274Xe.   As an example of an independent sequence
$\sequencen{X_n}$ of random variables, all with expectation zero and
variance $1$, to which the Central Limit Theorem does {\it not} apply, I
offer 274Xd.
}%end of notes

\discrpage

\frfilename{mt275.tex}
\versiondate{3.12.12}

\def\chaptername{Probability theory}
\def\sectionname{Martingales}
\copyrightdate{2001}

\newsection{275}

This chapter so far has been dominated by independent sequences of
random variables.   I now turn to another of the remarkable concepts to
which probabilistic intuitions have led us.   Here we study evolving
systems, in which we gain progressively more information as time
progresses.   I give the basic theorems on pointwise convergence of
martingales (275F-275H, %275F 275G 275H
275K) and a very brief account of `stopping times'
(275L-275P). %275L 275M 275N 275O 275P

\leader{275A}{Definition} Let $(\Omega,\Sigma,\mu)$ be a probability
space with completion $(\Omega,\hat\Sigma,\hat\mu)$, and
$\sequencen{\Sigma_n}$ a non-decreasing sequence of
$\sigma$-subalgebras of $\hat\Sigma$.   \cmmnt{(Such sequences
$\sequencen{\Sigma_n}$ are called {\bf filtrations}.)}
A {\bf martingale adapted to
$\sequencen{\Sigma_n}$} is a sequence $\sequencen{X_n}$ of integrable
real-valued random variables on $\Omega$ such that (i)
$\dom X_n\in\Sigma_n$ and $X_n$ is
$\Sigma_n$-measurable for each $n\in\Bbb N$
(ii) whenever $m\le n\in\Bbb N$
and $E\in\Sigma_m$ then $\int_EX_{n}=\int_EX_m$.

\cmmnt{Note that for (ii)
it is enough if $\int_EX_{n+1}=\int_EX_n$ whenever $n\in\Bbb N$
and $E\in\Sigma_n$.}

\leader{275B}{Examples }\cmmnt{We have seen many contexts in which
such sequences appear naturally;  here are a few.

\medskip

}{\bf (a)} Let $(\Omega,\Sigma,\mu)$ be a probability space
and $\sequencen{\Sigma_n}$ a non-decreasing sequence of
$\sigma$-subalgebras of $\Sigma$.   Let $X$ be any
real-valued random variable on $\Omega$ with finite expectation, and for
each $n\in\Bbb N$ let
$X_n$ be a conditional expectation of $X$ on $\Sigma_n$\cmmnt{, as in
\S233}.   Subject to the conditions that $\dom X_n\in\Sigma_n$ and $X_n$
is actually $\Sigma_n$-measurable for each $n$\cmmnt{ (a purely
technical
point -- see 232He)}, $\sequencen{X_n}$ will be a martingale adapted to
$\sequencen{\Sigma_n}$\cmmnt{, because
$\int_EX_{n+1}=\int_EX=\int_EX_n$ whenever $E\in\Sigma_n$}.

\spheader 275Bb Let $(\Omega,\Sigma,\mu)$ be a probability space
and $\sequencen{X_n}$ an independent sequence of random
variables all with zero expectation.   For each $n\in\Bbb N$ let
$\tilde\Sigma_n$ be the
$\sigma$-algebra generated by $\bigcup_{i\le n}\Sigma_{X_i}$, writing
$\Sigma_{X_i}$ for the
$\sigma$-algebra defined by $X_i$\cmmnt{ (272C)}, and set
$S_n=X_0+\ldots+X_n$.   Then $\sequencen{S_n}$ is a martingale adapted
to $\tilde\Sigma_n$.   \cmmnt{(Use 272K to see that $\Sigma_{X_{n+1}}$
is independent of $\tilde\Sigma_n$, so that
$\int_EX_{n+1}=\int X_{n+1}\times\chi E=0$ for every
$E\in\tilde\Sigma_n$, by 272R.)}

\spheader 275Bc Let $(\Omega,\Sigma,\mu)$ be a probability space
and $\sequencen{X_n}$ an independent sequence of
random variables all with expectation $1$.   For each $n\in\Bbb N$ let
$\tilde\Sigma_n$ be the
$\sigma$-algebra generated by $\bigcup_{i\le n}\Sigma_{X_i}$, writing
$\Sigma_{X_i}$ for the
$\sigma$-algebra defined by $X_i$, and set
$W_n=X_0\times\ldots\times X_n$.   Then $\sequencen{W_n}$ is a
martingale adapted to $\sequencen{\tilde\Sigma_n}$.
%\sequencen{\tilde\Sigma_n} is the {\bf natural filtration} for
%\sequencen{X_n}

\leader{275C}{Remarks }\cmmnt{{\bf (a)} It seems appropriate to the
concept of a random variable $X$ being `adapted' to a
$\sigma$-algebra $\Sigma$ to require that $\dom X\in\Sigma$ and that $X$
should be $\Sigma$-measurable, even though this may mean that other
random variables, equal almost everywhere to $X$, may fail to be
`adapted' to $\Sigma$.

\spheader 275Cb Technical problems of this kind evaporate, of
course, if all
$\mu$-negligible subsets of $X$ belong to $\Sigma_0$.   But examples
such as 275Bb make it seem unreasonable to insist on such a
simplification as a general rule.

\header{275Cc}}%end of comment
{\bf (c)} The concept of `martingale' can readily be
extended to other index sets than $\Bbb N$;  indeed, if $I$ is any
partially ordered set, we can say that $\langle X_i\rangle_{i\in I}$ is
a martingale on $(\Omega,\Sigma,\mu)$ adapted to
$\langle\Sigma_i\rangle_{i\in I}$ if (i) each $\Sigma_i$ is a
$\sigma$-subalgebra of $\hat\Sigma$ (ii) each $X_i$ is an integrable
real-valued $\Sigma_i$-measurable random variable such that
$\dom X_i\in\Sigma_i$ (iii) whenever $i\le j$ in $I$, then
$\Sigma_i\subseteq\Sigma_j$ and $\int_EX_i=\int_EX_j$ for every
$E\in\Sigma_i$.   \cmmnt{The principal case, after $I=\Bbb N$, is
$I=\coint{0,\infty}$;  $I=\Bbb Z$ also is interesting, and I think it is
fair to say that the most important ideas can already be expressed in
theorems about martingales indexed by finite sets $I$.   But in this
volume I will generally take martingales to be indexed by $\Bbb N$.
}%end of comment

\spheader 275Cd Given just a sequence $\sequencen{X_n}$ of
integrable
real-valued random variables on a probability space
$(\Omega,\Sigma,\mu)$, we can say simply that $\sequencen{X_n}$ is a
{\bf martingale} on $(\Omega,\Sigma,\mu)$ if there is some
non-decreasing
sequence $\sequencen{\Sigma_n}$ of $\sigma$-subalgebras of
$\hat\Sigma$\cmmnt{ (the completion of $\Sigma$)} such that
$\sequencen{X_n}$ is a martingale
adapted to $\sequencen{\Sigma_n}$.   If we write $\tilde\Sigma_n$ for
the $\sigma$-algebra generated by $\bigcup_{i\le n}\Sigma_{X_i}$, where
$\Sigma_{X_i}$ is the
$\sigma$-algebra defined by $X_i$\cmmnt{, as in 275Bb},
then\cmmnt{ it is easy to see that} $\sequencen{X_n}$ is a martingale
iff it is a martingale adapted to
$\sequencen{\tilde\Sigma_n}$.

\spheader 275Ce Continuing from (d),\cmmnt{ it is also easy to
see that} if $\sequencen{X_n}$
is a martingale on $(\Omega,\Sigma,\mu)$, and
$X'_n\eae X_n$ for every $n$, then $\sequencen{X'_n}$ is a martingale
on $(\Omega,\Sigma,\mu)$.   \cmmnt{(The point is that if
$\sequencen{X_n}$ is
adapted to $\sequencen{\Sigma_n}$, then both $\sequencen{X_n}$ and
$\sequencen{X'_n}$ are adapted to $\sequencen{\hat\Sigma_n}$, where

\Centerline{$\hat\Sigma_n
=\{E\symmdiff F:E\in\Sigma_n,\,F\text{ is negligible}\}$.)}

\noindent}Consequently
we have a concept of `martingale' as a sequence in $L^1(\mu)$, saying
that a sequence $\sequencen{X_n^{\ssbullet}}$ in $L^1(\mu)$ is a
martingale iff $\sequencen{X_n}$ is a martingale.

\cmmnt{Nevertheless, I think that the concept of `martingale
adapted
to a sequence of $\sigma$-algebras' is the primary one, since in all the
principal applications the $\sigma$-algebras reflect some essential
aspect of the problem, which may not be fully encompassed by the random
variables alone.}

\cmmnt{
\spheader 275Cf The word `martingale' originally (in English;
the history in French is more complex) referred to
a strap used to prevent a horse from throwing its head back.   Later it
was used as the name of a gambling system in which the gambler doubles
his stake each time he loses, and (in French) as a general term for
gambling systems.   These may be regarded as a class of
`stopped-time martingales', as described in
275L-275P %275L 275M 275N 275O 275P
below.
}%end of comment

\leader{275D}{}\cmmnt{ A large part of the theory of martingales
consists of inequalities of various kinds.   I give two of the most
important, both due to J.L.Doob.   (See also 276Xa-276Xb.)

\medskip

\noindent}{\bf Lemma} Let $(\Omega,\Sigma,\mu)$ be a probability space,
and $\sequencen{X_n}$ a
martingale on $\Omega$.   Fix $n\in\Bbb N$ and set
$X^*=\max(X_0,\ldots,X_n)$.   Then for any $\epsilon>0$,

\Centerline{$\Pr(X^*\ge\epsilon)
\le\Bover1{\epsilon}\Expn(X_n^+)$,}

\noindent writing $X_n^+=\max({0},X_n)$.

\proof{ Write $\hat\mu$ for the completion of $\mu$, and $\hat\Sigma$
for its domain.   Let $\sequencen{\Sigma_n}$ be a
non-decreasing sequence of
$\sigma$-subalgebras of $\hat\Sigma$ to which $\sequencen{X_n}$ is
adapted.   For each $i\le n$ set

\Centerline{$E_i=\{\omega:\omega\in\dom
X_i,\,X_i(\omega)\ge\epsilon\}$,}

\Centerline{$F_i=E_i\setminus\bigcup_{j<i}E_j$.}

\noindent Then $F_0,\ldots,F_n$ are disjoint and
$F=\bigcup_{i\le n}F_i=\bigcup_{i\le n}E_i$;  moreover, writing $H$ for the conegligible set $\bigcap_{i\le n}\dom X_i$,

\Centerline{$\{\omega:X^*(\omega)\ge\epsilon\}=F\cap H$,}

\noindent so that

\Centerline{$\Pr(X^*\ge\epsilon)=
\hat\mu\{\omega:X^*(\omega)\ge\epsilon\}
=\hat\mu F=\sum_{i=0}^n\hat\mu F_i$.}

\noindent On the other hand, $E_i$ and $F_i$ belong to $\Sigma_i$ for
each $i\le n$, so

\Centerline{$\int_{F_i}X_n=\int_{F_i}X_i\ge \epsilon\hat\mu F_i$}

\noindent for every $i$, and

\Centerline{$\epsilon\hat\mu F=\epsilon\sum_{i=0}^n\hat\mu F_i
\le\sum_{i=0}^n\int_{F_i}X_n
=\int_FX_n
\le\int_FX_n^+
\le\Expn(X_n^+)$,}

\noindent as required.
}%end of proof of 275D

\cmmnt{\medskip

\noindent{\bf Remark} Note that in fact we have $\epsilon\hat\mu
F\le\int_FX_n$, where $F=\{\omega:X^*(\omega)\ge\epsilon\}$;  this is of
great importance in many applications.
}%end of comment

\leader{275E}{Up-crossings}\cmmnt{ The next lemma depends on the
notion of
`up-crossing'.}   Let $x_0,\ldots,x_n$ be any list of real numbers,
and $a<b$ in $\Bbb R$.   The {\bf number of up-crossings from $a$ to
$b$} in the list $x_0,\ldots,x_n$ is the number of pairs $(j,k)$ such
that $0\le j<k\le n$, $x_j\le a$, $x_k\ge b$ and $a<x_i<b$ for $j<i<k$.
\cmmnt{Note that this is also the largest $m$ such that $s_m<\infty$,
if we
write

\Centerline{$r_1=\inf\{i:i\le n,\,x_i\le a\}$,}

\Centerline{$s_1=\inf\{i:r_1<i\le n,\,x_i\ge b\}$,}

\Centerline{$r_2=\inf\{i:s_1<i\le n,\,x_i\le a\}$,}

\Centerline{$s_2=\inf\{i:r_2<i\le n,\,x_i\ge b\}$}

\noindent and so on, taking $\inf\emptyset=\infty$.
}%end of comment

\leader{275F}{Lemma} Let $(\Omega,\Sigma,\mu)$ be a probability space
and $\sequencen{X_n}$ a martingale on $\Omega$.   Suppose that $n\in\Bbb
N$ and that
$a<b$ in $\Bbb R$.   For each $\omega\in\bigcap_{i\le n}\dom X_i$, let
$U(\omega)$ be the number of up-crossings from $a$ to $b$ in the list
$X_0(\omega),\ldots,X_n(\omega)$.   Then

\Centerline{$\Expn(U)\le\Bover1{b-a}\Expn((X_n-X_0)^+)$,}

\noindent writing $(X_n-X_0)^+(\omega)=\max(0,X_n(\omega)-X_0(\omega))$
for $\omega\in\dom X_n\cap\dom X_0$.

\proof{ Each individual step in the proof is `elementary', but the
structure as a whole is non-trivial.

\medskip

{\bf (a)}  The following fact will be useful.   Suppose that
$x_0,\ldots,x_n$ are real numbers;  let $u$ be the number of
up-crossings from $a$ to $b$ in the list $x_0,\ldots,x_n$.   Set
$y_i=\max(x_i,a)$ for each $i$;  then $u$ is also the number of
up-crossings from $a$ to $b$ in the list $y_0,\ldots,y_n$.   For each
$k\le n$, set $c_k=1$ if there is a $j\le k$ such that $x_j\le a$ and
$x_i<b$ for $j\le i\le k$, $0$ otherwise.   Then

\Centerline{$(b-a)u\le\sum_{k=0}^{n-1}c_k(y_{k+1}-y_{k})$.}

\noindent\vthsp\Prf\ I induce on $m$ to show that (defining $r_m$, $s_m$
as in 275E)

\Centerline{$(b-a)m\le\sum_{k=0}^{s_m-1}c_k(y_{k+1}-y_{k})$}

\noindent whenever $m\le u$.   For $m=0$ (taking $s_0=-1$) we have
$0=0$.   For the inductive step to $m\ge 1$, we have
$s_{m-1}<r_{m}<s_m\le n$ (because I am supposing that $m\le u$), and
$c_k=0$ if $s_{m-1}\le k<r_{m}$, $c_k=1$ if $r_{m}\le k<s_m$.   So

$$\eqalignno{\sum_{k=0}^{s_{m}-1}c_k(y_{k+1}-y_{k})
&=\sum_{k=0}^{s_{m-1}-1}c_k(y_{k+1}-y_{k})
+\sum_{k=r_m}^{s_m-1}(y_{k+1}-y_k)\cr
&\ge(b-a)(m-1)+y_{s_m}-y_{r_m}\cr
\noalign{\noindent (by the inductive hypothesis)}
&\ge(b-a)m\cr}$$

\noindent (because $y_{s_m}\ge b$, $y_{r_m}=a$), and the induction
proceeds.


Accordingly

\Centerline{$\sum_{k=0}^{s_u-1}c_k(y_{k+1}-y_{k})\ge(b-a)u$.}

\noindent As for the sum $\sum_{k=s_u}^{n-1}c_k(y_{k+1}-y_k)$, we have
$c_k=0$ for $s_u\le k<r_{u+1}$, $c_k=1$ for $r_{u+1}\le k<s_{u+1}$, while
$s_{u+1}>n$, so if $n\le r_{u+1}$ we have

\Centerline{$\sum_{k=0}^{n-1}c_k(y_{k+1}-y_{k})
=\sum_{k=0}^{s_u-1}c_k(y_{k+1}-y_{k})\ge(b-a)u$,}

\noindent while if $n>r_{u+1}$ we have

$$\eqalign{\sum_{k=0}^{n-1}c_k(y_{k+1}-y_{k})
&=\sum_{k=0}^{s_u-1}c_k(y_{k+1}-y_{k})
+\sum_{k=r_{u+1}}^{n-1}y_{k+1}-y_k\cr
&\ge(b-a)u+y_n-y_{r_{u+1}}\cr
&\ge(b-a)u\cr}$$


\noindent because $y_n\ge a=y_{r_{u+1}}$.   Thus in both cases we have
the required result.\ \Qed

\medskip

{\bf (b)(i)} Now define

\Centerline{$Y_k(\omega)=\max(a,X_k(\omega))$ for $\omega\in\dom X_k$,}

\Centerline{$F_k=\{\omega:\omega\in\bigcap_{i\le k}\dom X_i,\,
\exists\,j\le k,\,X_j(\omega)\le a,\,X_i(\omega)<b$ if $j\le i\le k\}$}

\noindent for each $k\in\Bbb N$.   If $\sequencen{\Sigma_n}$ is a
non-decreasing sequence of $\sigma$-algebras to which $\sequencen{X_n}$
is adapted, then $F_k\in\Sigma_k$ (because
if $j\le k$ all the sets $\dom X_j$, $\{\omega:X_j(\omega)\le a\}$,
$\{\omega:X_j(\omega)<b\}$ belong to $\Sigma_j\subseteq\Sigma_k$).

\medskip

\quad{\bf (ii)} We find that $\int_FY_k\le\int_FY_{k+1}$ if
$F\in\Sigma_k$.   \Prf\ Set $G=\{\omega:X_k(\omega)>a\}\in\Sigma_k$.
Then

$$\eqalign{\int_FY_k
&=\int_{F\cap G}X_k+a\hat\mu(F\setminus G)\cr
&=\int_{F\cap G}X_{k+1}+a\hat\mu(F\setminus G)\cr
&\le\int_{F\cap G}Y_{k+1}+\int_{F\setminus G}Y_{k+1}
=\int_FY_{k+1}.\,\,\Qed\cr}$$

\medskip

\quad{\bf (iii)}
Consequently $\int_FY_{k+1}-Y_k\le\int Y_{k+1}-Y_k$ for every
$F\in\Sigma_k$.

\Centerline{\Prf\enskip $\int (Y_{k+1}-Y_k)-\int_F(Y_{k+1}-Y_k)
=\int_{\Omega\setminus F}Y_{k+1}-\int_{\Omega\setminus F}Y_k\ge 0$.
\Qed}

\medskip

{\bf (c)} Let $H$ be the conegligible set
$\dom U=\bigcap_{i\le n}\dom X_i\in\Sigma_n$.   We ought to check at
some point that $U$ is
$\Sigma_n$-measurable;  but this is clearly true, because all the
relevant sets $\{\omega:X_i(\omega)\le a\}$, $\{\omega:X_i(\omega)\ge
b\}$ belong to $\Sigma_n$.   For each $\omega\in H$, apply (a) to the
list $X_0(\omega),\ldots,X_n(\omega)$ to see that

\Centerline{$(b-a)U(\omega)\le\sum_{k=0}^{n-1}\chi F_k(\omega)
(Y_{k+1}(\omega)-Y_k(\omega))$.}

\noindent Because $H$ is conegligible, it follows that

$$\eqalignno{(b-a)\Expn(U)
&\le\sum_{k=0}^{n-1}\int_{F_k}Y_{k+1}-Y_k
\le\sum_{k=0}^{n-1}\int Y_{k+1}-Y_k\cr
\noalign{\noindent (using (b-iii))}
&=\Expn(Y_n-Y_0)
\le\Expn((X_n-X_0)^+)\cr}$$

\noindent because $Y_n-Y_0\le(X_n-X_0)^+$ everywhere  on $\dom
X_n\cap\dom X_0$.   This completes the proof.
}%end of proof of 275F


\leader{275G}{}\cmmnt{ We are now ready for the principal theorems of
this section.

\medskip

\noindent}{\bf Doob's Martingale Convergence Theorem} Let
$\sequencen{X_n}$ be a
martingale on a probability space $(\Omega,\Sigma,\mu)$, and suppose
that
$\sup_{n\in\Bbb N}\Expn(|X_n|)<\infty$.   Then
$\lim_{n\to\infty}X_n(\omega)$ is defined in $\Bbb R$ for almost every
$\omega$ in $\Omega$.

\proof{{\bf (a)} Set $H=\bigcap_{n\in\Bbb N}\dom X_n$, and for
$\omega\in H$ set

\Centerline{$Y(\omega)=\liminf_{n\to\infty}X_n(\omega)$,
\quad$Z(\omega)=\limsup_{n\to\infty}X_n(\omega)$,}

\noindent allowing $\pm\infty$ in
both cases.   But note that $Y\le\liminf_{n\to\infty}|X_n|$, so by
Fatou's Lemma $Y(\omega)<\infty$ for almost every $\omega$;  similarly
$Z(\omega)>-\infty$ for almost every $\omega$.   It will therefore be
enough if I can show that $Y\eae Z$, for then
$Y(\omega)=Z(\omega)\in\Bbb R$ for almost every $\omega$, and
$\sequencen{X_n(\omega)}$ will be convergent for almost every $\omega$.

\medskip

{\bf (b)} \Quer\ So suppose, if possible, that $Y$ and $Z$ are not equal
almost everywhere.   Of course both are $\hat\Sigma$-measurable, where
$(\Omega,\hat\Sigma,\hat\mu)$ is the completion of
$(\Omega,\Sigma,\mu)$, so we
must have

\Centerline{$\hat\mu\{\omega:\omega\in H,\,Y(\omega)<Z(\omega)\}>0$.}

\noindent    Accordingly there are rational numbers
$q$, $q'$ such that $q<q'$ and $\hat\mu G>0$, where

\Centerline{$G=\{\omega:\omega\in H,\,Y(\omega)<q<q'<Z(\omega)\}$.}

\noindent Now, for each $\omega\in H$ and $n\in\Bbb N$, let $U_n(\omega)$
be the number of up-crossings from $q$ to $q'$ in the list
$X_0(\omega),\ldots,
\ifdim\pagewidth>467pt\penalty-50\fi
X_n(\omega)$.   Then 275F tells us that

\Centerline{$\Expn(U_n)\le\Bover1{q'-q}\Expn((X_n-X_0)^+)
\le\Bover1{q'-q}\Expn(|X_n|+|X_0|)
\le\Bover{2M}{q'-q}$,}

\noindent if we write $M=\sup_{i\in\Bbb N}\Expn(|X_i|)$.   By B.Levi's
theorem, $U(\omega)=\sup_{n\in\Bbb N}U_n(\omega)<\infty$ for almost
every $\omega$.   On the other hand, if $\omega\in G$, then there are
arbitrarily large $j$, $k$ such that $X_j(\omega)<q$ and
$X_k(\omega)>q'$, so $U(\omega)=\infty$.   This means that $\hat\mu G$
must be $0$, contrary to the choice of $q$ and $q'$.   \Bang

\medskip

{\bf (c)} Thus we must in fact have $Y\eae Z$, and
$\sequencen{X_n(\omega)}$ is convergent for almost every $\omega$, as
claimed.
}%end of proof of 275G

\leader{275H}{Theorem}
Let $(\Omega,\Sigma,\mu)$ be a probability space, and
$\sequencen{\Sigma_n}$ a non-decreasing sequence of
$\sigma$-subalgebras of $\Sigma$.   Let $\sequencen{X_n}$ be a
martingale adapted to $\sequencen{\Sigma_n}$.   Then the following are
equiveridical:

\quad(i) there is a random variable $X$, of finite expectation, such
that $X_n$ is a conditional expectation of $X$ on $\Sigma_n$ for every
$n$;

\quad (ii) $\{X_n:n\in\Bbb N\}$ is uniformly integrable;

\quad (iii) $X_{\infty}(\omega)=\lim_{n\to\infty}X_n(\omega)$ is defined
in $\Bbb R$ for almost every $\omega$, and
$\Expn(|X_{\infty}|)=\lim_{n\to\infty}\Expn(|X_n|)<\infty$.

\proof{{\bf(i)$\Rightarrow$(ii)} By 246D, the set of all
conditional expectations of $X$ is uniformly integrable, so
$\{X_n:n\in\Bbb N\}$ is surely uniformly integrable.

\medskip

{\bf (ii)$\Rightarrow$(iii)} If $\{X_n:n\in\Bbb N\}$ is uniformly
integrable,  we surely have $\sup_{n\in\Bbb N}\Expn(|X_n|)<\infty$, so
275G tells us that $X_{\infty}$ is
defined almost everywhere.   By 246Ja, $X_{\infty}$ is integrable and
$\lim_{n\to\infty}\Expn(|X_n-X_{\infty}|)=0$.    Consequently
$\Expn(|X_{\infty}|)=\lim_{n\to\infty}\Expn(|X_n|)<\infty$.

\medskip

{\bf (iii)$\Rightarrow$(i)} Because
$\Expn(|X_{\infty}|)=\lim_{n\to\infty}\Expn(|X_n|)$,
$\lim_{n\to\infty}\Expn(|X_n-X_{\infty}|)=0$ (245H(a-ii)).     Now take
$n\in\Bbb N$ and $E\in\Sigma_n$.   Then

\Centerline{$\int_EX_n=\lim_{m\to\infty}\int_EX_m=\int_EX_{\infty}$.}

\noindent As $E$ is arbitrary, $X_n$ is a conditional expectation of
$X_{\infty}$ on $\Sigma_n$.
}%end of proof of 275H

\leader{275I}{Theorem} Let $(\Omega,\Sigma,\mu)$ be a probability space,
and $\sequencen{\Sigma_n}$ a non-decreasing sequence of
$\sigma$-subalgebras of $\Sigma$;  write $\Sigma_{\infty}$ for the
$\sigma$-algebra generated by $\bigcup_{n\in\Bbb N}\Sigma_n$.   Let $X$
be any real-valued random variable on $\Omega$ with finite expectation,
and for each $n\in\Bbb N$ let $X_n$ be a conditional expectation of $X$
on $\Sigma_n$.   Then $X_{\infty}(\omega)=\lim_{n\to\infty}X_n(\omega)$
is defined almost everywhere;
$\lim_{n\to\infty}\Expn(|X_{\infty}-X_n|)=0$, and $X_{\infty}$ is a
conditional expectation of $X$ on $\Sigma_{\infty}$.

\proof{ By 275G-275H, we know that $X_{\infty}$ is defined
almost everywhere, and, as remarked in the proof of 275H,
$\lim_{n\to\infty}\Expn(|X_{\infty}-X_n|)=0$.   To see that $X_{\infty}$
is a conditional expectation of $X$ on $\Sigma_{\infty}$, set

\Centerline{$\Cal A
=\{E:E\in\Sigma_{\infty},\,\int_EX_{\infty}=\int_EX\}$,
\quad$\Cal I=\bigcup_{n\in\Bbb N}\Sigma_n$.}

\noindent Now $\Cal I$ and $\Cal A$ satisfy the
conditions of the Monotone Class Theorem (136B).  \Prf\ {\bf
($\pmb{\alpha}$)} Of course
$\Omega\in\Cal I$ and $\Cal I$ is
closed under finite intersections, because $\sequencen{\Sigma_n}$ is a
non-decreasing sequence of $\sigma$-algebras;  in fact $\Cal I$ is a
subalgebra of $\Cal P\Omega$, and is closed under finite unions and
complements.   {\bf ($\pmb{\beta}$)} If $E\in\Cal I$, say
$E\in\Sigma_n$;  then

\Centerline{$\int_EX_{\infty}=\lim_{m\to\infty}\int_EX_m=\int_EX$,}

\noindent as in (iii)$\Rightarrow$(i) of 275H, so $E\in\Cal A$.   Thus
$\Cal I\subseteq\Cal A$.   {\bf ($\pmb{\gamma}$)} If $E$, $F\in\Cal A$
and $E\subseteq F$, then

\Centerline{$\int_{F\setminus E}X_{\infty}
=\int_FX_{\infty}-\int_EX_{\infty}
=\int_FX-\int_EX
=\int_{F\setminus E}X$,}

\noindent so $F\setminus E\in\Cal A$.   {\bf($\pmb{\delta}$)}
If $\sequencen{E_k}$ is a non-decreasing sequence in
$\Cal A$ with union $E$, then

\Centerline{$\int_{E}X_{\infty}
=\lim_{k\to\infty}\int_{E_k}X_{\infty}
=\lim_{k\to\infty}\int_{E_k}X
=\int_{E}X$,}

\noindent so $E\in\Cal A$.   Thus $\Cal A$ is a Dynkin class.\ \Qed

Consequently, by 136B, $\Cal A$ includes $\Sigma_{\infty}$;  that is,
$X_{\infty}$ is a conditional expectation of $X$ on $\Sigma_{\infty}$.
}%end of proof of 275I

\cmmnt{\medskip

\noindent{\bf Remark} I have written
`$\lim_{n\to\infty}\Expn(|X_n-X_{\infty}|)=0$';  but you may prefer to
say `$X^{\ssbullet}_{\infty}=\lim_{n\to\infty}X_n^{\ssbullet}$ in
$L^1(\mu)$', as in Chapter 24.

The importance of this theorem is such that you may be interested in a
proof based on 275D rather than 275E-275G;  see 275Xd.
}

\leader{*275J}{}\cmmnt{ As a corollary of this theorem I give an
important result, a kind of density theorem for product measures.

\medskip

\noindent}{\bf Proposition} Let $\sequencen{(\Omega_n,\Sigma_n,\mu_n)}$
be a sequence of probability spaces with product $(\Omega,\Sigma,\mu)$.
Let $X$ be a real-valued random variable on $\Omega$ with finite
expectation.   For each $n\in\Bbb N$ define $X_n$ by setting

\Centerline{$X_n(\pmb{\omega})
=\int X(\omega_0,\ldots,\omega_n,\xi_{n+1},\ldots)d(\xi_{n+1},\ldots)$}

\noindent wherever this is defined, where I write
`$\int\ldots d(\xi_{n+1},\ldots)$' to mean integration with respect to
the product measure $\lambda'_n$ on $\prod_{i\ge n+1}\Omega_i$.   Then
$X(\pmb{\omega})=\lim_{n\to\infty}X_n(\pmb{\omega})$ for almost every
$\pmb{\omega}=(\omega_0,\omega_1,\ldots)$ in $\Omega$, and
$\lim_{n\to\infty}\Expn(|X-X_n|)=0$.

\proof{ For each $n$, we can identify $\mu$ with the product of
$\lambda_n$ and $\lambda'_n$, where $\lambda_n$ is the product measure
on $\Omega_0\times\ldots\times\Omega_n$ (254N).   So 253H tells us that
$X_n$ is a conditional expectation of $X$ on the $\sigma$-algebra
$\Lambda_n=\{E\times\prod_{i>n}\Omega_i:E\in\dom\lambda_n\}$.   Since
(by 254N again) we can think of $\lambda_{n+1}$ as the product of
$\lambda_n$ and $\mu_{n+1}$, $\Lambda_n\subseteq\Lambda_{n+1}$ for each
$n$.   So 275I tells us that $\sequencen{X_n}$ converges almost
everywhere to a conditional expectation $X_{\infty}$ of $X$ on the
$\sigma$-algebra $\Lambda_{\infty}$ generated by
$\bigcup_{n\in\Bbb N}\Lambda_n$.   Now $\Lambda_{\infty}\subseteq\Sigma$
and also $\Tensorhat_{n\in\Bbb N}\Sigma_n\subseteq\Lambda_{\infty}$, so
every member of $\Sigma$ is sandwiched between two members of
$\Lambda_{\infty}$ of the same measure (254Ff), and $X_{\infty}$ must be
equal to $X$ almost everywhere.   Moreover, 275I also tells us that

\Centerline{$\lim_{n\to\infty}\Expn(|X-X_n|)
=\lim_{n\to\infty}\Expn(|X_{\infty}-X_n|)=0$,}

\noindent as required.
}%end of proof of 275J

\leader{275K}{Reverse \dvrocolon{martingales}}\cmmnt{ We have a result
corresponding to 275I for {\it decreasing} sequences of
$\sigma$-algebras.   While this is used less often than
275G-275I, %275G 275H 275I
it does have very important applications.

\medskip

\noindent}{\bf Theorem} Let $(\Omega,\Sigma,\mu)$ be a probability
space, and $\sequencen{\Sigma_n}$ a non-increasing sequence of
$\sigma$-subalgebras of $\Sigma$, with intersection $\Sigma_{\infty}$.
Let $X$ be any real-valued random variable with finite expectation, and
for each $n\in\Bbb N$ let $X_n$ be a conditional expectation of $X$ on
$\Sigma_n$.   Then $X_{\infty}=\lim_{n\to\infty}X_n$ is defined almost
everywhere and is a conditional expectation of $X$ on $\Sigma_{\infty}$.

\proof{{\bf (a)} Set $H=\bigcap_{n\in\Bbb N}\dom X_n$, so that $H$ is
conegligible.   For $n\in\Bbb N$, $a<b$ in $\Bbb R$, and $\omega\in H$,
write $U_{abn}(\omega)$ for the number of up-crossings from $a$ to $b$
in the list $X_n(\omega),X_{n-1}(\omega),\ldots,X_0(\omega)$ (275E).
Then

$$\eqalignno{\Expn(U_{abn})
&\le\Bover1{b-a}\Expn((X_0-X_n)^+)\cr
\displaycause{275F}
&\le\Bover1{b-a}\Expn(|X_0|+|X_n|)
\le\Bover2{b-a}\Expn(|X_0|)<\infty.\cr}$$

\noindent So $\lim_{n\to\infty}U_{abn}(\omega)$ is finite for almost
every $\omega$.   But this means that

\Centerline{$\{\omega:\liminf_{n\to\infty}X_n(\omega)<a,\,
  \limsup_{n\to\infty}X_n(\omega)>b\}$}

\noindent is negligible.   As $a$ and $b$ are arbitrary,
$\sequencen{X_n}$ is convergent a.e., just as in 275G.   Set
$X_{\infty}(\omega)=\lim_{n\to\infty}X_n(\omega)$ whenever this is
defined in $\Bbb R$.

\medskip

{\bf (b)} By 246D, $\{X_n:n\in\Bbb N\}$ is uniformly integrable, so
$\Expn(|X_n-X_{\infty}|)\to 0$ as $n\to\infty$ (246Ja), and

\Centerline{$\int_EX_{\infty}=\lim_{n\to\infty}\int_EX_n=\int_EX_0$}

\noindent for every $E\in\Sigma_{\infty}$.

\medskip

{\bf (c)} Now there is a conegligible set $G\in\Sigma_{\infty}$ such
that $G\subseteq\dom X_{\infty}$ and $X_{\infty}\restr G$ is
$\Sigma_{\infty}$-measurable.   \Prf\ For each $n\in\Bbb N$, there is a
conegligible set $G_n\in\Sigma_n$ such that $G_n\subseteq\dom X_n$ and
$X_n\restr G_n$ is $\Sigma_n$-measurable.   Set
$G'=\bigcup_{n\in\Bbb N}\bigcap_{m\ge n}G_m$;  then, for any
$r\in\Bbb N$, $G'=\bigcup_{n\ge r}\bigcap_{m\ge n}G_m$ belongs to
$\Sigma_r$, so $G'\in\Sigma_{\infty}$, while of course $G'$ is
conegligible.   For $n\in\Bbb N$, set $X'_n(\omega)=X_n(\omega)$ for
$\omega\in G_n$, $0$ for $\omega\in\Omega\setminus G_n$;  then for
$\omega\in G'$,
$\lim_{n\to\infty}X'_n(\omega)=\lim_{n\to\infty}X_n(\omega)$ if either
is defined in $\Bbb R$.   Writing $X'_{\infty}=\lim_{n\to\infty}X'_n$
whenever this is defined in $\Bbb R$, 121F and 121H tell us that
$X'_{\infty}$ is $\Sigma_r$-measurable and $\dom X'_{\infty}\in\Sigma_r$
for every $r\in\Bbb N$, so that $G''=\dom X'_{\infty}$ belongs to
$\Sigma_{\infty}$ and $X'_{\infty}$ is $\Sigma_{\infty}$-measurable.
We also know, from (a), that $G''$ is conegligible.   So setting
$G=G'\cap G''$ we have the result.\ \Qed

Thus $X_{\infty}$ is a conditional expectation of $X$ on
$\Sigma_{\infty}$.
}%end of proof of 275K

\leader{275L}{Stopping\dvro{ times: }{ times}}\cmmnt{ In a sense, the
main work of this section
is over;  I have no room for any more theorems of importance comparable
to 275G-275I.   However, it would be wrong to leave this chapter without
briefly describing one of the most fruitful ideas of the subject.

\medskip

\noindent}{\bf Definition} Let $(\Omega,\Sigma,\mu)$ be a probability
space, with completion $(\Omega,\hat\Sigma,\hat\mu)$, and
$\sequencen{\Sigma_n}$ a non-decreasing sequence of
$\sigma$-subalgebras of $\hat\Sigma$.   A {\bf stopping time adapted to
$\sequencen{\Sigma_n}$}\cmmnt{ (also called `{\bf optional time}',
`{\bf Markov time}')} is a function $\tau$ from $\Omega$ to
$\Bbb N\cup\{\infty\}$ such that
$\{\omega:\tau(\omega)\le n\}\in\Sigma_n$ for every $n\in\Bbb N$.

\cmmnt{\medskip

\noindent{\bf Remark} Of course the condition

\Centerline{$\{\omega:\tau(\omega)\le n\}\in\Sigma_n$ for every
$n\in\Bbb N$}

\noindent can be replaced by the equivalent condition

\Centerline{$\{\omega:\tau(\omega)=n\}\in\Sigma_n$
for every $n\in\Bbb N$.}

\noindent I give priority to the former expression because it is more
appropriate to other index sets (see 275Cc).
}%end of comment

\leader{275M}{Examples (a)} If $\sequencen{X_n}$ is a martingale adapted
to a sequence $\sequencen{\Sigma_n}$ of $\sigma$-algebras, and ${H_n}$
is a Borel subset of $\BbbR^{n+1}$ for each $n$, then we have a
stopping time $\tau$ adapted to $\sequencen{\Sigma_n}$ defined by the
formula

\Centerline{$\tau(\omega)
=\inf\{n:\omega\in\bigcap_{i\le n}\dom X_i,\,
  (X_0(\omega),\ldots,X_n(\omega))\in H_n\}$,}

\noindent setting $\inf\emptyset=\infty$\cmmnt{ as usual}.
\cmmnt{(For by 121Ka the
set $E_n=\{\omega:(X_0(\omega),\ldots,X_n(\omega))\in H_n\}$ belongs to
$\Sigma_n$ for each $n$, and
$\{\omega:\tau(\omega)\le n\}=\bigcup_{i\le n}E_i$.)}   In particular,
\cmmnt{for instance,}  the formulae

\Centerline{$\inf\{n:X_n(\omega)\ge a\}$, \quad
$\inf\{n:|X_n(\omega)|> a\}$}

\noindent define stopping times.

\header{275Mb}{\bf (b)} Any constant function
$\tau:\Omega\to\Bbb N\cup\{\infty\}$ is a stopping time.   If $\tau$,
$\tau'$ are two
stopping times adapted to the same sequence $\sequencen{\Sigma_n}$ of
$\sigma$-algebras, then $\tau\wedge\tau'$ is a stopping time adapted to
$\sequencen{\Sigma_n}$, setting
$(\tau\wedge\tau')(\omega)=\min(\tau(\omega),\tau'(\omega))$ for
$\omega\in\Omega$\prooflet{, because

\Centerline{$\{\omega:(\tau\wedge\tau')(\omega)\le n\}
=\{\omega:\tau(\omega)\le n\}\cup\{\omega:\tau'(\omega)\le n\}
\in\Sigma_n$}

\noindent for every $n\in\Bbb N$}.

\leader{275N}{Lemma} Let $(\Omega,\Sigma,\mu)$ be a complete probability
space, and $\sequencen{\Sigma_n}$ a non-decreasing sequence of
$\sigma$-subalgebras of $\Sigma$.   Suppose that $\tau$ and $\tau'$ are
stopping times on $\Omega$, and $\sequencen{X_n}$ a martingale, all
adapted to $\sequencen{\Sigma_n}$.

(a) The family

\Centerline{$\tilde\Sigma_{\tau}
=\{E:E\in\Sigma,\,E\cap\{\omega:\tau(\omega)\le n\}\in\Sigma_n$ for
every $n\in\Bbb N\}$}

\noindent is a $\sigma$-subalgebra of $\Sigma$.

(b) If $\tau(\omega)\le\tau'(\omega)$ for every
$\omega$, then $\tilde\Sigma_{\tau}\subseteq\tilde\Sigma_{\tau'}$.

(c) Now suppose that $\tau$ is finite almost everywhere.   Set

\Centerline{$\tilde X_{\tau}(\omega)=X_{\tau(\omega)}(\omega)$}

\noindent whenever $\tau(\omega)<\infty$ and
$\omega\in\dom X_{\tau(\omega)}$.   Then
$\dom\tilde X_{\tau}\in\tilde\Sigma_{\tau}$ and $\tilde X_{\tau}$ is
$\tilde\Sigma_{\tau}$-measurable.

(d) If $\tau$ is essentially bounded, that is, there is some
$m\in\Bbb N$ such that $\tau\le m$ almost everywhere, then
$\Expn(\tilde X_{\tau})$ exists and is equal to $\Expn(X_0)$.

(e) If $\tau\le\tau'$ almost everywhere, and $\tau'$ is essentially
bounded, then $\tilde X_{\tau}$ is a conditional expectation of
$\tilde X_{\tau'}$ on $\tilde\Sigma_{\tau}$.

\proof{{\bf (a)}  This is elementary.   Write
$H_n=\{\omega:\tau(\omega)\le n\}$ for each $n\in\Bbb N$.   The empty
set belongs to $\tilde\Sigma_{\tau}$ because it belongs to $\Sigma_n$
for every $n$.
If $E\in\tilde\Sigma_{\tau}$, then

\Centerline{$(\Omega\setminus E)\cap H_n=H_n\setminus(E\cap
H_n)\in\Sigma_n$}

\noindent because $H_n\in\Sigma_n$;  this is true for for every $n$, so
$X\setminus E\in\tilde\Sigma_{\tau}$.   If $\sequence{k}{E_k}$ is any
sequence in $\tilde\Sigma_{\tau}$ then

\Centerline{$(\bigcup_{k\in\Bbb N}E_k)\cap H_n=\bigcup_{k\in\Bbb
N}E_k\cap H_n\in\Sigma_n$}

\noindent for every $n$, so $\bigcup_{k\in\Bbb
N}E_k\in\tilde\Sigma_{\tau}$.

\medskip

{\bf (b)} If $E\in\tilde\Sigma_{\tau}$ then of course $E\in\Sigma$,
and if $n\in\Bbb N$ then $\{\omega:\tau'(\omega)\le
n\}\subseteq\{\omega:\tau(\omega)\le n\}$, so that

\Centerline{$E\cap\{\omega:\tau'(\omega)\le n\}
=E\cap\{\omega:\tau(\omega)\le n\}\cap\{\omega:\tau'(\omega)\le
n\}$}

\noindent belongs to $\Sigma_n$;  as $n$ is arbitrary,
$E\in\tilde\Sigma_{\tau'}$.

\medskip

{\bf (c)} Set $H_n=\{\omega:\tau(\omega)\le n\}$ for each $n\in\Bbb N$.
For any $a\in\Bbb R$,

$$\eqalign{H_n\cap\{\omega:\omega\in
&\dom\tilde X_{\tau},\,\tilde X_{\tau}(\omega)\le a\}\cr
&=\bigcup_{k\le n}\{\omega:\tau(\omega)=k,\,\omega\in\dom X_k,
  \,X_k(\omega)\le a\}
\in\Sigma_n.\cr}$$

\noindent As $n$ is arbitrary,

\Centerline{$G_a=\{\omega:\omega\in\dom\tilde X_{\tau},\,
\tilde X_{\tau}(\omega)\le a\}\in\tilde\Sigma_{\tau}$.}

\noindent As $a$ is arbitrary, $\dom\tilde X_{\tau}=\bigcup_{m\in\Bbb
N}G_m\in\tilde\Sigma_{\tau}$ and $\tilde X_{\tau}$ is
$\tilde\Sigma_{\tau}$-measurable.

\medskip

{\bf (d)} Set $H_k=\{\omega:\tau(\omega)=k\}$ for $k\le m$.   Then
$\bigcup_{k\le m}H_k$ is conegligible, so

\Centerline{$\Expn(\tilde X_{\tau})
=\sum_{k=0}^m\int_{H_k}X_k
=\sum_{k=0}^m\int_{H_k}X_m
=\int_{\Omega}X_m
=\int_{\Omega}X_0$.}

\medskip

{\bf (e)} Suppose $\tau'\le n$ almost everywhere.   Set
$H_k=\{\omega:\tau(\omega)=k\}$,
$H'_k=\{\omega:\tau'(\omega)=k\}$ for each $k$;  then both
$\langle H_k\rangle_{k\le n}$ and $\langle H'_k\rangle_{k\le n}$ are
partitions of conegligible subsets of $X$.
Now suppose that $E\in\tilde\Sigma_{\tau}$.   Then

\Centerline{$\int_E\tilde X_{\tau}
=\sum_{k=0}^n\int_{E\cap H_k}\tilde X_{\tau}
=\sum_{k=0}^n\int_{E\cap H_k}X_k
=\sum_{k=0}^n\int_{E\cap H_k}X_n
=\int_EX_n$}

\noindent because $E\cap H_k\in\Sigma_k$ for every $k$.   By (b),
$E\in\tilde\Sigma_{\tau'}$, so we also have
$\int_E\tilde X_{\tau'}=\int_EX_n$.   Thus
$\int_E\tilde X_{\tau}=\int_E\tilde X_{\tau'}$ for every
$E\in\tilde\Sigma_{\tau}$, as claimed.
}%end of proof of 275N

\leader{275O}{Proposition} Let $\sequencen{X_n}$ be a martingale and
$\tau$ a stopping time, both adapted to the same sequence
$\sequencen{\Sigma_n}$ of
$\sigma$-algebras.   For each $n$, set $(\tau\wedge
n)(\omega)=\min(\tau(\omega),n)$ for $\omega\in\Omega$;  then
$\tau\wedge n$ is a stopping time, and $\sequencen{\tilde X_{\tau\wedge
n}}$ is a martingale
adapted to $\sequencen{\tilde\Sigma_{\tau\wedge n}}$, defining
$\tilde X_{\tau\wedge n}$ and $\tilde\Sigma_{\tau\wedge n}$ as in 275N.

\proof{ As remarked in 275Mb, each $\tau\wedge n$ is a
stopping time.   If $m\le n$, then $\tilde\Sigma_{\tau\wedge
m}\subseteq\tilde\Sigma_{\tau\wedge n}$ by 275Nb.   Each
$\tilde X_{\tau\wedge m}$ is
$\tilde\Sigma_{\tau\wedge m}$-measurable, with domain belonging to
$\tilde\Sigma_{\tau\wedge m}$,  by 275Nc, and has finite expectation, by
275Nd;  finally, if $m\le n$, then $\tilde X_{\tau\wedge m}$ is a
conditional expectation of
$\tilde X_{\tau\wedge n}$ on $\tilde\Sigma_{\tau\wedge m}$, by 275Ne.
}%end of proof of 275O

\leader{275P}{Corollary} Suppose that $(\Omega,\Sigma,\mu)$ is a
probability space and $\sequencen{X_n}$ is a martingale on $\Omega$ such
that $W=\sup_{n\in\Bbb N}|X_{n+1}-X_n|$ is finite almost everywhere and
has finite expectation.   Then for almost every $\omega\in\Omega$,
either $\lim_{n\to\infty}X_n(\omega)$ exists in $\Bbb R$ or
$\sup_{n\in\Bbb N}X_n(\omega)=\infty$ and $\inf_{n\in\Bbb
N}X_n(\omega)=-\infty$.

\proof{ Let $\sequencen{\Sigma_n}$ be a non-decreasing sequence of
$\sigma$-algebras to which $\sequencen{X_n}$ is adapted.    Let $H$ be
the conegligible set $\bigcap_{n\in\Bbb N}\dom
X_n\cap\{\omega:W(\omega)<\infty\}$.   For each $m\in\Bbb N$, set

\Centerline{$\tau_m(\omega)=\inf\{n:\omega\in\dom
X_n,\,X_n(\omega)>m\}$.}

\noindent As in 275Ma, $\tau_m$ is a stopping time adapted to
$\sequencen{\Sigma_n}$.   Set

\Centerline{$Y_{mn}=\tilde X_{\tau_m\wedge n}$,}

\noindent defined as in 275O, so that $\sequencen{Y_{mn}}$ is a
martingale.   If
$\omega\in H$, then either $\tau_m(\omega)>n$ and

\Centerline{$Y_{mn}(\omega)=X_n(\omega)\le m$,}

\noindent or $0<\tau_m(\omega)\le n$ and

\Centerline{$Y_{mn}(\omega)=X_{\tau_m(\omega)}(\omega)
\le W(\omega)+X_{\tau_m(\omega)-1}(\omega)\le W(\omega)+m$,}

\noindent or $\tau_m(\omega)=0$ and

\Centerline{$Y_{mn}(\omega)=X_0(\omega)$.}

\noindent Thus

\Centerline{$Y_{mn}(\omega)\le|X_0(\omega)|+W(\omega)+m$}

\noindent for every $\omega\in H$, and

\Centerline{$|Y_{mn}(\omega)|=2\max(0,Y_{mn}(\omega))-Y_{mn}(\omega)
\le 2(|X_0(\omega)|+W(\omega)+m)-Y_{mn}(\omega)$,}

\Centerline{$\Expn(|Y_{mn}|)\le 2\Expn(|X_0|)+2\Expn(W)+2m-\Expn(Y_{mn})
= 2\Expn(|X_0|)+2\Expn(W)+2m-\Expn(X_0)$}

\noindent by 275Nd.   As this is true for every $n\in\Bbb N$,
$\sup_{n\in\Bbb N}\Expn(|Y_{mn}|)<\infty$, and $\lim_{n\to\infty}Y_{mn}$
is defined in $\Bbb R$ almost everywhere, by Doob's Martingale
Convergence Theorem (275G).   Let $F_m$ be the conegligible set on which
$\sequencen{Y_{mn}}$ converges.
Set $H^*=H\cap\bigcap_{m\in\Bbb N}F_m$, so that $H^*$ is conegligible.

Now consider

\Centerline{$E=\{\omega:\omega\in H^*,\,\sup_{n\in\Bbb
N}X_n(\omega)<\infty\}$.}

\noindent For any $\omega\in E$, there must be an $m\in\Bbb N$ such that
$\sup_{n\in\Bbb N}X_n(\omega)\le m$.   Now this means that
$Y_{mn}(\omega)=X_n(\omega)$ for every $n$, and as $\omega\in F_m$ we
have

\Centerline{$\lim_{n\to\infty}X_n(\omega)
=\lim_{n\to\infty}Y_{mn}(\omega)\in\Bbb R$.}

\noindent This means that $\sequencen{X_n(\omega)}$ is convergent for
almost every $\omega$ such that $\{X_n(\omega):n\in\Bbb N\}$ is bounded
above.

Similarly, $\sequencen{X_n(\omega)}$ is convergent for almost every
$\omega$ such that $\{X_n(\omega):n\in\Bbb N\}$ is bounded below, which
completes the proof.
}%end of proof of 275P


\exercises{
\leader{275X}{Basic exercises $\pmb{>}$(a)}
%\sqheader 275Xa
Let $\sequencen{X_n}$ be an
independent sequence of random variables with zero expectation and finite
variance.   Set $s_n=(\sum_{i=0}^n\Var(X_i))^{1/2}$,
$Y_n=(X_0+\ldots+X_n)^2-s_n^2$ for each $n$.   Show that
$\sequencen{Y_n}$ is a martingale.
%275B

\sqheader 275Xb Let $\sequencen{X_n}$ be a martingale.   Show that for
any $\epsilon>0$, $\Pr(\sup_{n\in\Bbb N}|X_n|)\ge\epsilon)
\le\bover1\epsilon\sup_{n\in\Bbb N}\Expn(|X_n|)$.
%275D

\spheader 275Xc {\bf P\'olya's urn scheme} Imagine a box containing
red and white balls.   At each move, a ball is drawn at random from the
box and replaced together with another of the same colour.  (i) Writing
$R_n$, $W_n$ for the numbers of red and white balls after the $n$th move
and $X_n=R_n/(R_n+W_n)$,
show that $\sequencen{X_n}$ is a martingale.   (ii) Starting from
$R_0=W_0=1$, find the distribution of $(R_n,W_n)$ for each $n$.
(iii) Show that $X=\lim_{n\to\infty}X_n$ is defined almost everywhere,
and find its distribution when $R_0=W_0=1$.   (See {\smc Feller 66} for
a discussion of other starting values.)
%275G

\sqheader 275Xd Let $(\Omega,\Sigma,\mu)$ be a probability space, and
$\sequencen{\Sigma_n}$ a non-decreasing sequence of $\sigma$-subalgebras
of $\Sigma$;  for each $n\in\Bbb N$ let $P_n:L^1\to L^1$ be the
conditional expectation operator corresponding to $\Sigma_n$, where
$L^1=L^1(\mu)$ (242J).   (i) Show that
$V=\{u:u\in L^1,\,\lim_{n\to\infty}\|P_nu-u\|_1=0\}$ is a
$\|\,\|_1$-closed linear subspace of $L^1$.   (ii) Show that
$\{E:E\in\Sigma,\,\chi E^{\ssbullet}\in V\}$ is a Dynkin class including
$\bigcup_{n\in\Bbb N}\Sigma_n$, so includes the $\sigma$-algebra
$\Sigma_{\infty}$ generated by $\bigcup_{n\in\Bbb N}\Sigma_n$.   (iii)
Show that if $u\in L^1$ then
$v=\sup_{n\in\Bbb N}P_n|u|$ is defined in $L^1$ and is of the form
$W^{\ssbullet}$ where $\Pr(W\ge\epsilon)\le\bover1{\epsilon}\|u\|_1$ for
every $\epsilon>0$.   \Hint{275D.}   (iv) Show that if $X$ is a
$\Sigma_{\infty}$-measurable random variable with finite expectation,
and for each $n\in\Bbb N\,\,X_n$ is a conditional expectation of $X$ on
$\Sigma_n$, then $X^{\ssbullet}\in V$ and $X\eae\lim_{n\to\infty}X_n$.
\Hint{apply (iii) to $u=(X-X_m)^{\ssbullet}$ for large $m$.}
%275I

\spheader 275Xe Let $(\Omega,\Sigma,\mu)$ be a probability space,
$\sequencen{\Sigma_n}$ a non-decreasing sequence of $\sigma$-subalgebras
of $\Sigma$, and $\Sigma_{\infty}$ the $\sigma$-algebra generated by
$\bigcup_{n\in\Bbb N}\Sigma_n$.   For each $n\in\Bbb N\cup\{\infty\}$
let $P_n:L^1\to L^1$ be the conditional expectation operator
corresponding to $\Sigma_n$, where $L^1=L^1(\mu)$.   Show that
$\lim_{n\to\infty}\|P_nu-P_{\infty}u\|_p=0$ whenever $p\in\coint{1,\infty}$ and
$u\in L^p(\mu)$.
\Hint{275Xd, 233J/242K, 246Xg.}
%275Xd, 275I

\spheader 275Xf Let $\sequencen{X_n}$ be a martingale, and
suppose that $p\in\ooint{1,\infty}$ is such that
$\sup_{n\in\Bbb N}\|X_n\|_p<\infty$.   Show that
$X=\lim_{n\to\infty}X_n$ is defined almost everywhere and that
$\lim_{n\to\infty}\|X_n-X\|_p=0$.
%275Xe, 275H, 275I

\sqheader 275Xg Let $(\Omega,\Sigma,\mu)$ be $[0,1]$ with
Lebesgue measure.   For each $n\in\Bbb N$ let $\Sigma_n$ be the finite
subalgebra of $\Sigma$ generated by intervals of the type $[0,2^{-n}r]$
for $r\le 2^{-n}$.   Use 275I to show that for any integrable
$X:[0,1]\to\Bbb R$ we must have
$X(t)=\lim_{n\to\infty}2^n\int_{I_n(t)}X$ for almost every
$t\in\coint{0,1}$, where $I_n(t)$ is
the interval of the form $\coint{2^{-n}r,2^{-n}(r+1)}$ containing $t$.
Compare this result with 223A and 261Yd.
%275I

\spheader 275Xh In 275K, show that
$\lim_{n\to\infty}\|X_n-X_{\infty}\|_p=0$ for any $p\in\coint{1,\infty}$
such that $\|X_0\|_p$ is finite.   (Compare 275Xe.)
%275K

\spheader 275Xi\dvAnew{2012} Let $(\Omega,\Sigma,\mu)$ be a probability
space, with completion $(\Omega,\hat\Sigma,\hat\mu)$, and
$\sequencen{\Sigma_n}$ a non-decreasing sequence of
$\sigma$-subalgebras of $\hat\Sigma$.   Show that if $\sequence{i}{\tau_i}$
is a sequence of stopping times adapted to $\sequencen{\Sigma_n}$, and we
set $\tau(\omega)=\sup_{i\in\Bbb N}\tau_i(\omega)$ for $\omega\in\Omega$,
then $\tau$ is a stopping time adapted to $\sequencen{\Sigma_n}$.
%275M

\spheader 275Xj Let $(\Omega,\Sigma,\mu)$ be a probability
space, with completion $(\Omega,\hat\Sigma,\hat\mu)$, and
$\sequencen{\Sigma_n}$ a non-decreasing sequence of
$\sigma$-subalgebras of $\hat\Sigma$.   Let $\sequencen{X_n}$ be a
uniformly integrable martingale adapted to $\Sigma_n$, and set
$X_{\infty}=\lim_{n\to\infty}X_n$.   Let $\tau$ be a stopping time
adapted to $\sequencen{\Sigma_n}$, and set
$\tilde X_{\tau}(\omega)=X_{\tau(\omega)}(\omega)$ whenever
$\omega\in\dom X_{\tau(\omega)}$,
allowing $\infty$ as a value of $\tau(\omega)$.   Show that
$\tilde X_{\tau}$
is a conditional expectation of $X_{\infty}$ on $\tilde\Sigma_{\tau}$,
as defined in 275N.
%275N

\spheader 275Xk Let $(\Omega,\Sigma,\mu)$ be a probability
space, with completion $(\Omega,\hat\Sigma,\hat\mu)$, and
$\sequencen{\Sigma_n}$ a non-decreasing sequence of
$\sigma$-subalgebras of $\hat\Sigma$.   Let $\sequencen{X_n}$ be a
martingale and $\tau$ a stopping time, both adapted to
$\sequencen{\Sigma_n}$.   Suppose that
$\sup_{n\in\Bbb N}\Expn(|X_n|)<\infty$ and that $\tau$ is finite almost
everywhere.
Show that $\tilde X_{\tau}$, as defined in 275Nc, has finite
expectation, but
that $\Expn(\tilde X_{\tau})$ need not be equal to $\Expn(X_0)$.
%275N

\spheader 275Xl(i)\dvAnew{2014} Show that if $\sequencen{X_n}$ is a martingale
such that $\sup_{n\in\Bbb N}\Expn(|X_n|)$ is finite and $\sequencen{X_n}$ is
convergent in measure, then $\sequencen{X_n}$ is convergent a.e.
(ii) Find a martingale $\sequencen{X_n}$ such that
$\sequencen{X_{2n}}\to 0$ a.e.\ but $|X_{2n+1}|\ge 1$ a.e.\ for every
$n\in\Bbb N$.   (iii) Find a martingale which converges in measure but is
not convergent a.e.   
%275D+  

\leader{275Y}{Further exercises (a)}%
%\spheader 275Ya
\dvAnew{2012} Let $(\Omega,\Sigma,\mu)$ be a probability space,
$\sequencen{\Sigma_n}$ an independent sequence of $\sigma$-subalgebras
of $\Sigma$, and $X$ a random variable on $\Omega$ with finite variance.
Let $X_n$ be a conditional expectation of $X$ on $\Sigma_n$ for each
$n$.   Show that $\lim_{n\to\infty}X_n=\Expn(X)$ almost everywhere.
\Hint{consider $\sum_{n=0}^{\infty}\Var(X_n)$.}
%-

\spheader 275Yb
Let $(\Omega,\Sigma,\mu)$ be a complete probability space,
$\sequencen{\Sigma_n}$ a non-decreasing sequence of
$\sigma$-subalgebras of $\Sigma$ all containing every negligible set,
and $\sequencen{X_n}$ a martingale adapted to $\sequencen{\Sigma_n}$.
Let $\nu$ be another probability measure with domain $\Sigma$ which is
absolutely continuous with respect to $\mu$, with Radon-Nikod\'ym
derivative $Z$.   For each $n\in\Bbb N$ let $Z_n$ be a conditional
expectation of $Z$ on $\Sigma_n$ (with respect to the measure $\mu$).
(i) Show that $Z_n$ is a Radon-Nikod\'ym derivative of
$\nu\restr\Sigma_n$ with respect to $\mu\restr\Sigma_n$, for each
$n\in\Bbb N$.   (ii) Defining $X_n/Z_n$ as in 121E, so that its domain is
$\{\omega:\omega\in\dom X_n\cap\dom Z_n$, $Z_n(\omega)\ne 0\}$,
show that $\sequencen{X_n/Z_n}$ is a martingale with respect to
the measure $\nu$.
%275B

\spheader 275Yc Combine the ideas of 275Cc with those of
275Cd-275Ce to describe a notion of `martingale indexed by $I$', where
$I$ is an arbitrary partially ordered set.
%275C

\spheader 275Yd\dvAformerly{2{}75Yc} 
Let $\sequence{k}{X_k}$ be a martingale on a complete
probability space $(\Omega,\Sigma,\mu)$, and fix
$n\in\Bbb N$.   Set $X^*=\max(|X_0|,\ldots,\discretionary{}{}{}|X_n|)$.
Let $p\in\ooint{1,\infty}$.
Show that $\|X^*\|_p\le\Bover{p}{p-1}\|X_n\|_p$.   ({\it Hint\/}:  set
$F_t=\{\omega:X^*(\omega)\ge t\}$.   Show that
$t\mu F_t\le\int_{F_t}|X_n|$.   Using Fubini's
theorem on $\Omega\times\coint{0,\infty}$ and on
$\Omega\times\coint{0,\infty}\times\coint{0,\infty}$, show that

\Centerline{$\Expn((X^*)^p)=p\int_0^{\infty}t^{p-1}\mu F_tdt$,}

\Centerline{$\int_0^{\infty}t^{p-2}\int_{F_t}|X_n|dt
=\Bover1{p-1}\Expn(|X_n|\times(X^*)^{p-1})$,}

\Centerline{$\Expn(|X_n|\times(X^*)^{p-1})
\le\|X_n\|_p\|X^*\|_p^{p-1}$.}

\noindent Compare 286A below.)
%275D

\spheader 275Ye\dvArevised{2012}(i)
Show that if $a$, $b\ge 0$ then
$a\ln^+b\le a\ln^+a+\Bover{b}e$, where $\ln^+t=0$ if $t\le 1$,
$\ln t$ if $t\ge 1$.
(ii) Let $(\Omega,\Sigma,\mu)$ be a
complete probability space and $X$, $Y$
non-negative random variables on $\Omega$ such that
$t\mu F_t\le\int_{F_t}X$ for every $t\ge 0$, where
$F_t=\{\omega:Y(\omega)\ge t\}$.   Show that
$\int_{F_1}Y\le\int_{F_1}X\times\ln^+Y$, and hence that
$\Expn(Y)\le\Bover{e}{e-1}(1+\Expn(X\times\ln^+X))$.   (iii) Show that
if $\sequencen{X_n}$ is a martingale on $\Omega$, $n\in\Bbb N$ and
$X^*=\sup_{i\le n}|X_i|$, then
$\Expn(X^*)\le\Bover{e}{e-1}(1+\Expn(|X_n|\times\ln^+|X_n|))$.
%275Yd, 275D {\smc Doob 53}

\spheader 275Yf Let $(\Omega,\Sigma,\mu)$ be a probability space
and $\langle\Sigma_i\rangle_{i\in I}$ a countable family  of
$\sigma$-subalgebras of $\Sigma$ such that for any $i$, $j\in I$ either
$\Sigma_i\subseteq\Sigma_j$ or $\Sigma_j\subseteq\Sigma_i$.   Let $X$ be
a real-valued random variable on $\Omega$ such that $\|X\|_p<\infty$,
where $1<p<\infty$, and suppose that $X_i$ is a conditional expectation
of $X$ on $\Sigma_i$ for each $i\in I$.   Show that
$\|\sup_{i\in I}|X_i|\|_p\le\Bover{p}{p-1}\|X\|_p$.
%275Yd, 275D

\spheader 275Yg Let $(\Omega,\Sigma,\mu)$ be a probability
space, with completion $(\Omega,\hat\Sigma,\hat\mu)$, and let
$\sequencen{\Sigma_n}$ be a non-decreasing sequence of
$\sigma$-subalgebras of $\hat\Sigma$.   Let $\sequencen{X_n}$ be a
sequence of $\mu$-integrable real-valued functions such that
$\dom X_n\in\Sigma_n$ and $X_n$ is
$\Sigma_n$-measurable for each $n\in\Bbb N$.   We say that
$\sequencen{X_n}$ is a {\bf submartingale adapted to
$\sequencen{\Sigma_n}$} if
$\int_EX_{n+1}\ge\int_EX_n$ for every
$n\in\Bbb N$ and every $E\in\Sigma_n$.
Prove versions of 275D, 275F, 275G, 275Xf for submartingales.
%275G

\spheader 275Yh Let $\sequencen{X_n}$ be a martingale, and
$\phi:\Bbb R\to\Bbb R$ a convex function.   Show that
$\sequencen{\phi(X_n)}$ is a submartingale.   \Hint{233J.}
Re-examine part (b-ii) of the proof of 275F in the light of
this fact.
%275F, 275Yg, 275G

\spheader 275Yi Let $\sequencen{X_n}$ be an independent sequence of
non-negative random variables all with expectation $1$.   Set
$W_n=X_0\times\ldots\times X_n$ for every $n$.   (i) Show that
$W=\lim_{n\to\infty}W_n$ is defined a.e.\   (ii) Show that $\Expn(W)$ is
either $0$ or $1$.   \Hint{suppose $\Expn(W)>0$.   Set
$Z_n=\lim_{m\to\infty}X_n\times\ldots\times X_m$.   Show that
$\lim_{n\to\infty}Z_n=1$ when $0<W<\infty$, therefore a.e., by the
zero-one law, while $\Expn(Z_n)\le 1$, by Fatou's lemma, so
$\lim_{n\to\infty}\Expn(Z_n)=1$, while
$\Expn(W)=\Expn(W_n)\Expn(Z_{n+1})$ for every $n$.}   (iii) Set
$\gamma=\prod_{n=0}^{\infty}\Expn(\sqrt{X_n})$.   Show that $\gamma>0$
iff $\Expn(W)=1$.
\Hint{$\Pr(W_n\ge\bover14\gamma^2)\ge\bover14\gamma^2$ for every $n$, so
if $\gamma>0$ then $W$ cannot be zero a.e.;  while
$\Expn(\sqrt{W})\le\gamma$.}
%275G

\spheader 275Yj Let $\sequencen{(\Omega_n,\Sigma_n,\mu_n)}$ be a
sequence of probability spaces with product $(\Omega,\Sigma,\mu)$.
Suppose that
for each $n\in\Bbb N$ we have a probability measure $\nu_n$, with domain
$\Sigma_n$, which is absolutely continuous with respect to $\mu_n$, with
Radon-Nikod\'ym derivative $f_n$, and suppose that
$\prod_{n=0}^{\infty}\int\sqrt{f_n}d\mu_n>0$.   Let $\nu$ be the product of $\sequencen{\nu_n}$.   Show that $\nu$ is an
indefinite-integral measure over $\mu$, with Radon-Nikod\'ym derivative
$f$, where $f(\pmb{\omega})=\prod_{n=0}^{\infty}f_n(\omega_n)$ for
$\mu$-almost every $\pmb{\omega}=\sequencen{\omega_n}$ in $\Omega$.
\Hint{use 275Yi to show that $\int fd\mu=1$.}
%275G, 275Yi

\spheader 275Yk Let $\sequencen{p_n}$ be a sequence in $[0,1]$.  Let
$\mu$ be the usual measure on $\{0,1\}^{\Bbb N}$ (254J) and $\nu$ the
product of $\sequencen{\nu_n}$, where $\nu_n$ is the probability measure
on $\{0,1\}$ defined by setting $\nu_n\{1\}=p_n$.   Show that $\nu$ is
an indefinite-integral measure over $\mu$ iff
$\sum_{n=0}^{\infty}|p_n-\bover12|^2<\infty$.
%275Yj, 275G

\spheader 275Yl
Find a martingale $\sequencen{X_n}$ such that the
sequence $\nu_{X_n}$ of distributions (271C) is convergent for the vague
topology (274Ld), but $\sequencen{X_n}$ is not convergent in measure.
%275G added 2002

\spheader 275Ym Let $\sequencen{X_n}$ be an independent sequence of
real-valued random variables such that $\sum_{n=0}^{\infty}X_n$ is defined
in $\Bbb R$ almost everywhere.   Suppose that there is an $M\ge 0$ such
that $|X_n|\le M$ a.e.\ for every $n$.   Show that
$\sum_{n=0}^{\infty}\Expn(X_n)$ is defined in $\Bbb R$.
\Hint{274Yg, 275G.}
%275G

\spheader 275Yn Let $(\Omega,\Sigma,\mu)$ be a probability space and
$\sequencen{X_n}$ an independent sequence of
real-valued random variables on $\Omega$;  set
$E_n=\{\omega:\omega\in\dom X_n$, $|X_n(\omega)|>1\}$,
$Y_n=X_n\times\chi(\Omega\setminus E_n)$ for each $n$, and
$Z_n(\omega)=\med(-1,X_n(\omega),1)$ for $n\in\Bbb N$ and
$\omega\in\dom X_n$.   Show that the following are equiveridical:
(i) $\sum_{n=0}^{\infty}X_n(\omega)$ is defined in $\Bbb R$ for almost
every $\omega$;  (ii) $\sum_{n=0}^{\infty}\hat\mu E_n<\infty$,
$\sum_{n=0}^{\infty}\Expn(Y_n)$ is defined in $\Bbb R$, and
$\sum_{n=0}^{\infty}\Var(Y_n)<\infty$, where $\hat\mu$ is the completion of
$\mu$;
(iii) $\sum_{n=0}^{\infty}\hat\mu E_n<\infty$,
$\sum_{n=0}^{\infty}\Expn(Z_n)$ is defined in $\Bbb R$, and
$\sum_{n=0}^{\infty}\Var(Z_n)<\infty$.   \Hint{273K, 275Ym.}
(This is a version of the {\bf Three Series Theorem}.)
%275Ym 275G 273K

\spheader 275Yo Let $(\Omega,\Sigma,\mu)$ be a probability
space, $\sequencen{\Sigma_n}$ a non-decreasing sequence of
$\sigma$-subalgebras of $\Sigma$ and $\sequencen{X_n}$ a sequence of
random variables on $\Omega$ such that
$\Expn(\sup_{n\in\Bbb N}|X_n|)$ is finite
and $X=\lim_{n\to\infty}X_n$ is defined almost everywhere.
For each $n$, let $Y_n$ be a conditional expectation of $X_n$ on
$\Sigma_n$.   Show that $\sequencen{Y_n}$ converges almost everywhere to
a conditional expectation of $X$ on the $\sigma$-algebra generated by
$\bigcup_{n\in\Bbb N}\Sigma_n$.
%275I %mt27bits

\spheader 275Yp Show that 275Yo can fail if $\sequencen{X_n}$ is
merely uniformly integrable, rather than dominated by an integrable
function.
%275Yo, 275I  %mt27bits

\spheader 275Yq Let $(\Omega,\Sigma,\mu)$ be a complete probability
space, and $\sequencen{X_n}$ an independent sequence of random variables
on $\Omega$, all with the same distribution, and of finite expectation.
For each $n$, set $S_n=\bover1{n+1}(X_0+\ldots+X_n)$;  let $\Sigma_n$ be
the $\sigma$-algebra defined by $S_n$ and $\Sigma_n^*$ the
$\sigma$-algebra generated by $\bigcup_{m\ge n}\Sigma_m$.   Show that
$S_n$ is a conditional expectation of $X_0$ on $\Sigma_n^*$.
\Hint{assume every $X_i$ defined everywhere on $\Omega$.   Set
$\phi(\omega)=\sequence{i}{X_i(\omega)}$.   Show that
$\phi:\Omega\to\BbbR^{\Bbb N}$ is \imp\ for a suitable product measure
on $\BbbR^{\Bbb N}$, and that every set in $\Sigma_n^*$ is of the form
$\phi^{-1}[H]$ where $H\subseteq\BbbR^{\Bbb N}$ is a Borel set
invariant under permutations of coordinates in the set $\{0,\ldots,n\}$,
so that $\int_EX_i=\int_EX_j$ whenever $i\le j\le n$ and
$E\in\Sigma_n^*$.}   Hence show that $\sequencen{S_n}$ converges almost
everywhere.   (Compare 273I.)
%275K

\spheader 275Yr Formulate and prove versions of the
results of this section for martingales consisting of functions taking
values in $\Bbb C$ or $\BbbR^r$ rather than $\Bbb R$.
}%end of exercises

\cmmnt{
\Notesheader{275} I hope that the sketch above, though distressingly
abbreviated, has suggested some of the richness of the concepts
involved, and will provide a foundation for further study.   All the
theorems of this section have far-reaching implications, but the one
which is simply indispensable in advanced measure theory is 275I,
`L\'evy's martingale convergence theorem', which I will use in the proof
of the Lifting Theorem in Chapter 34 of the next volume.

As for stopping times, I mention them partly in an attempt to cast
further light on what martingales are for (see 276Ed below), and partly
because the ideas of 275N-275O are so important in modern probability
theory that, just as a matter of general knowledge, you should be aware
that there is something there.   I add 275P as one of the most
accessible of the standard results which may be obtained by this method.

}%end of notes

\discrpage

\frfilename{mt276.tex}
\versiondate{16.4.13}
\copyrightdate{2000}

\def\chaptername{Probability theory}
\def\sectionname{Martingale difference sequences}

\newsection{276}

Hand in hand with the concept of `martingale' is that of `martingale
difference sequence' (276A), a direct generalization of the notion of
`independent sequence'.   In this section I collect results which can be
naturally expressed in terms of difference sequences, including yet
another strong law of large numbers (276C).   I end the section with a
proof of Koml\'os's theorem (276H).

\leader{276A}{Martingale difference sequences (a)}\cmmnt{ If
$\sequencen{X_n}$
is a martingale adapted to a sequence $\sequencen{\Sigma_n}$ of
$\sigma$-algebras, then we have

\Centerline{$\int_EX_{n+1}-X_n=0$}

\noindent whenever $E\in\Sigma_n$.}   Let us say that if
$(\Omega,\Sigma,\mu)$ is a probability space, with completion
$(\Omega,\hat\Sigma,\hat\mu)$, and $\sequencen{\Sigma_n}$ is a
non-decreasing sequence of $\sigma$-subalgebras of $\hat\Sigma$, then a
{\bf martingale difference sequence adapted to $\sequencen{\Sigma_n}$}
is a sequence $\sequencen{X_n}$ of real-valued random variables on
$\Omega$, all with finite expectation, such that (i)
$\dom X_n\in\Sigma_n$ and
$X_n$ is $\Sigma_n$-measurable, for each $n\in\Bbb N$ (ii)
$\int_EX_{n+1}=0$ whenever $n\in\Bbb N$ and $E\in\Sigma_n$.

\header{276Ab}{\bf (b)}
\cmmnt{ Evidently} $\sequencen{X_n}$ is a martingale difference
sequence adapted
to $\sequencen{\Sigma_n}$ iff $\sequencen{\sum_{i=0}^nX_i}$ is a
martingale adapted to $\sequencen{\Sigma_n}$.

\header{276Ac}{\bf (c)}
Just as in 275Cd, we can say that a sequence $\sequencen{X_n}$ is in
itself a {\bf martingale difference sequence} if
$\sequencen{\sum_{i=0}^nX_i}$ is a martingale\cmmnt{, that is, if
$\sequencen{X_n}$ is a martingale difference sequence adapted to
$\sequencen{\tilde\Sigma_n}$, where $\tilde\Sigma_n$ is the
$\sigma$-algebra generated by $\bigcup_{i\le n}\Sigma_{X_i}$}.

\header{276Ad}{\bf (d)} If $\sequencen{X_n}$ is a martingale difference
sequence then $\sequencen{a_nX_n}$ is a martingale difference sequence
for any real $a_n$.

\header{276Ae}{\bf (e)}  If $\sequencen{X_n}$ is a martingale difference
sequence and $X'_n\eae X_n$ for every $n$, then $\sequencen{X'_n}$ is
a martingale difference sequence.   \cmmnt{(Compare 275Ce.)}

\cmmnt{\header{276Af}{\bf (f)} Of course the most important example of
`martingale difference sequence' is that of 275Bb:  any independent
sequence of random variables with zero expectation is a martingale difference
sequence.   It turns out that some of the theorems of \S273 concerning
such independent sequences may be generalized to martingale difference
sequences.}

\leader{276B}{Proposition} Let $\sequencen{X_n}$ be a martingale
difference sequence such that $\sum_{n=0}^{\infty}\Expn(X_n^2)<\infty$.
Then $\sum_{n=0}^{\infty}X_n$ is defined, and finite, almost everywhere.

\proof{{\bf (a)} Let $(\Omega,\Sigma,\mu)$ be the
underlying probability space, $(\Omega,\hat\Sigma,\hat\mu)$ its
completion,  and $\sequencen{\Sigma_n}$ a non-decreasing sequence of
$\sigma$-subalgebras of $\hat\Sigma$ such that $\sequencen{X_n}$ is
adapted to $\sequencen{\Sigma_n}$.  Set
$Y_n=\sum_{i=0}^nX_i$ for each $n\in\Bbb N$.   Then $\sequencen{Y_n}$ is
a martingale adapted to $\sequencen{\Sigma_n}$.

\medskip

{\bf (b)} $\Expn(Y_{n}\times X_{n+1})=0$ for each $n$.   \Prf\ $Y_n$ is
a sum of random variables with finite variance, so
$\Expn(Y_n^2)<\infty$, by 244Ba;  it follows that
$Y_n\times X_{n+1}$ has finite expectation, by 244Eb.   Because
the constant function $\pmb{0}$ is a conditional expectation of
$X_{n+1}$ on $\Sigma_n$,

\Centerline{$\Expn(Y_n\times X_{n+1})=\Expn(Y_n\times\pmb{0})=0$,}

\noindent by 242L.\ \Qed

\medskip

{\bf (c)} It follows that $\Expn(Y_n^2)=\sum_{i=0}^n\Expn(X_i^2)$ for
every $n$.   \Prf\ Induce on $n$.   For the inductive step, we have

\Centerline{$\Expn(Y_{n+1}^2)=\Expn(Y_n^2+2Y_n\times X_{n+1}+X_{n+1}^2)
=\Expn(Y_n^2)+\Expn(X_{n+1}^2)$}

\noindent because, by (b), $\Expn(Y_n\times X_{n+1})=0$.\ \Qed\

\medskip

{\bf (d)} Of course

\Centerline{$\Expn(|Y_n|)
=\int|Y_n|\times\chi\Omega\le\|Y_n\|_2\|\chi\Omega\|_2
=\sqrt{\Expn(Y_n^2)}$,}

\noindent so

\Centerline{$\sup_{n\in\Bbb N}\Expn(|Y_n|)
\le\sup_{n\in\Bbb N}\sqrt{\Expn(Y_n^2)}
=\sqrt{\sumop_{i=0}^{\infty}\Expn(X_i^2)}<\infty$.}

\noindent By 275G, $\lim_{n\to\infty}Y_n$ is defined and finite
almost everywhere, that is, $\sum_{i=0}^{\infty}X_i$ is defined and
finite almost everywhere.
}%end of proof of 276B

\leader{276C}{The strong law of large numbers:  fourth form}  Let
$\sequencen{X_n}$ be a martingale difference sequence, and suppose that
$\sequencen{b_n}$ is a non-decreasing
sequence in $\ooint{0,\infty}$, diverging to $\infty$, such that
$\sum_{n=0}^{\infty}\Bover1{b_n^2}\Var(X_n)<\infty$.   Then

\Centerline{$\lim_{n\to\infty}\Bover{1}{b_n}
\sum_{i=0}^n X_i=0$}

\noindent almost everywhere.

\proof{ (Compare 273D.) As usual, write
$(\Omega,\Sigma,\mu)$ for the
underlying probability space.   Set

\Centerline{$\tilde X_n=\Bover1{b_n}X_n$}

\noindent for each $n$;  then $\sequencen{\tilde X_n}$ also is a
martingale difference sequence, and

\Centerline{$\sum_{n=1}^{\infty}\Expn(\tilde X_n^2)
=\sum_{n=1}^{\infty}\Bover1{b_n^2}\Var(X_n)<\infty$.}

\noindent By 276B, $\sequencen{\tilde X_n(\omega)}$ is summable for
almost every $\omega\in\Omega$.   But by 273Cb,

\Centerline{$\lim_{n\to\infty}\Bover{1}{b_n}\sum_{i=0}^n X_i(\omega)
=\lim_{n\to\infty}\Bover{1}{b_n}\sum_{i=0}^nb_i\tilde X_i(\omega)
=0$}

\noindent for all such $\omega$.   So we have the result.
}%end of proof of 276C

\leader{276D}{Corollary} Let $\sequencen{X_n}$ be a martingale such that
$b_n=\Expn(X_n^2)$ is finite for each $n$.

(a) If $\sup_{n\in\Bbb N}b_n$ is infinite, then
$\lim_{n\to\infty}\bover1{b_n}X_n=0$ a.e.

(b) If $\sup_{n\ge 1}\bover1nb_n<\infty$, then
$\lim_{n\to\infty}\bover1nX_n=0$ a.e.

\proof{ Consider the martingale difference sequence
$\sequencen{Y_n}=\sequencen{X_{n+1}-X_n}$.   Then
$\Expn(Y_n\times X_n)=0$, so
$\Expn(Y_n^2)+\Expn(X_{n}^2)=\Expn(X_{n+1}^2)$ for each $n$.
In particular, $\sequencen{b_n}$ must be non-decreasing.

\medskip

{\bf (a)} If $\lim_{n\to\infty}b_n=\infty$, take $m$ such that $b_m>0$;
then

\Centerline{$\sum_{n=m}^{\infty}\Bover1{b_{n+1}^2}\Var(Y_n)
=\sum_{n=m}^{\infty}\Bover1{b_{n+1}^2}(b_{n+1}-b_n)
\le\int_{b_m}^{\infty}\Bover1{t^2}dt<\infty$.}

\noindent By 276C (modifying $b_i$ for $i<m$, if necessary),

\Centerline{$\lim_{n\to\infty}\Bover1{b_n}X_n
=\lim_{n\to\infty}\Bover1{b_{n+1}}(X_0+\sum_{i=0}^nY_i)
=\lim_{n\to\infty}\Bover1{b_{n+1}}\sum_{i=0}^nY_i=0$}

\noindent almost everywhere.

\medskip

{\bf (b)} If $\gamma=\sup_{n\ge 1}\bover1nb_n<\infty$, then
$\bover1{(n+1)^2}\le\min(1,\gamma^2/t^2)$ for $b_n<t\le b_{n+1}$, so

\Centerline{$\sum_{n=0}^{\infty}\Bover1{(n+1)^2}(b_{n+1}-b_n)
\le \gamma+\gamma^2\int_{\gamma}^{\infty}\Bover1{t^2}dt<\infty$,}

\noindent and, by the same argument as before,
$\lim_{n\to\infty}\bover1nX_n=0$ a.e.

}%end of proof of 276D

\leader{276E}{`Impossibility of systems'}\cmmnt{ {\bf (a)} I return
to the word
`martingale' and the idea of a gambling system.   Consider a gambler
who takes a sequence of `fair' bets, that is, bets which have payoff
expectations of zero, but who chooses which bets to take on the basis of
past experience.   The appropriate model for such a sequence of random
events is a martingale in the sense of 275A, taking $\Sigma_n$ to be the
algebra of all events which are observable up to and including the
outcome of the $n$th bet, and $X_n$ to be the gambler's net gain at that
time.   (In this model it is natural to take
$\Sigma_0=\{\emptyset,\Omega\}$ and $X_0=\tbf{0}$.)   Certain paradoxes can
arise if we try to imagine this model with atomless $\Sigma_n$;  to
begin with it is perhaps easier to work with the discrete case, in which
each $\Sigma_n$ is finite, or is the set of unions of some countable
family of atomic events.   Now suppose that the bets involved are just
two-way bets, with two equally likely outcomes, but that the gambler
chooses his stake each time.   In this case we can think of the outcomes
as corresponding to an independent sequence $\sequencen{W_n}$ of random
variables, each taking the values $\pm 1$ with equal probability.
The gambler's system must be of the form

\Centerline{$X_{n+1}=X_n + Z_{n+1}\times W_{n+1}$,}

\noindent where $Z_{n+1}$ is his stake on the $(n+1)$-st bet, and must
be constant on each atom of the $\sigma$-algebra $\Sigma_n$ generated by
$W_1,\ldots,W_n$.   The point is that because $\int_EW_{n+1}=0$ for each
$E\in\Sigma_n$, $\Expn(Z_{n+1}\times W_{n+1})=0$, so
$\Expn(X_{n+1})=\Expn(X_n)$.

\header{276Eb}} %end of comment
{\bf (b)}\cmmnt{ The general result, of which this is a special
case, is the following.}   If $\sequencen{W_n}$ is a martingale
difference sequence adapted to $\sequencen{\Sigma_n}$, and
$\langle{Z_n}\rangle_{n\ge 1}$ is a sequence of random variables such
that (i) $Z_n$ is $\Sigma_{n-1}$-measurable (ii) $Z_n\times W_{n}$ has
finite expectation for each $n\ge 1$, then
$W_0,Z_1\times W_1,Z_2\times W_2,\ldots$
is a martingale difference sequence adapted to
$\sequencen{\Sigma_n}$\cmmnt{;  the proof that
$\int_EZ_{n+1}\times W_{n+1}=0$
for every $E\in\Sigma_n$ is exactly the argument of (b) of the proof of
276B}.

\cmmnt{
\header{276Ec}{\bf (c)} I invited you to restrict your ideas to the
discrete case for a moment;  but if you feel that you understand what it
means to say that a `system' or {\bf predictable sequence}
$\langle{Z_n}\rangle_{n\ge 1}$ must be adapted to
$\sequencen{\Sigma_n}$, in the sense that every $Z_n$ is
$\Sigma_{n-1}$-measurable, then any further difficulty lies in the
measure theory needed to show that the integrals
$\int_EZ_{n+1}\times W_{n+1}$ are zero, which is what this book is
about.

\header{276Ed}{\bf (d)} Consider the gambling system mentioned in 275Cf.
Here the idea is that $W_n=\pm 1$, as in (a), and $Z_{n+1}=2^na$ if
$X_n\le 0$, $0$ if $X_n>0$;  that is, the gambler doubles his stake each
time until he wins, and then quits.   Of course he is almost sure to win
eventually, so we have $\lim_{n\to\infty}X_n=a$ almost everywhere,
even though
$\Expn(X_n)=0$ for every $n$.   We can compute the distribution of
$X_n$:  for $n\ge 1$ we have $\Pr(X_n=a)=1-2^{-n}$,
$\Pr(X_n=-(2^n-1)a)=2^{-n}$.   Thus $\Expn(|X_n|)=(2-2^{-n+1})a$ and the
almost-everywhere convergence of the $X_n$ is an example of Doob's
Martingale Convergence Theorem.

In the language of stopping times (275N),
$X_n=\tilde Y_{\tau\wedge n}$,
where $Y_n=\sum_{k=0}^n2^kaW_k$ and $\tau=\min\{n:Y_n>0\}$.
}

\leader{*276F}{}\cmmnt{ I come now to Koml\'os's theorem.   The first
step is a trifling refinement of 276C.

\medskip

\noindent}{\bf Lemma} Let $(\Omega,\Sigma,\mu)$ be a probability space,
and $\sequencen{\Sigma_n}$ a non-decreasing sequence of
$\sigma$-subalgebras of $\Sigma$.   Suppose that $\sequencen{X_n}$ is a
sequence of random variables on $\Omega$ such that (i) $X_n$ is
$\Sigma_n$-measurable for each $n$ (ii)
$\sum_{n=0}^{\infty}\Bover1{(n+1)^2}\Expn(X_n^2)$ is finite (iii)
$\lim_{n\to\infty}X'_n=0$ a.e., where $X_n'$ is a conditional
expectation of $X_n$ on $\Sigma_{n-1}$ for each $n\ge 1$.   Then
$\lim_{n\to\infty}\Bover1{n+1}\sum_{k=0}^nX_k=0$ a.e.

\proof{ Making suitable adjustments on a negligible set if necessary, we
may suppose that $X'_n$ is $\Sigma_{n-1}$-measurable for $n\ge 1$ and
that every $X_n$ and $X'_n$ is defined on the whole of $\Omega$.   Set
$X'_0=X_0$ and $Y_n=X_n-X'_n$ for $n\in\Bbb N$.   Then $\sequencen{Y_n}$
is a martingale difference sequence adapted to $\sequencen{\Sigma_n}$.
Also $\Expn(Y_n^2)\le\Expn(X_n^2)$ for every $n$.   \Prf\ If $n\ge 1$,
$X'_n$ is square-integrable (244M), and $\Expn(Y_n\times X'_n)=0$, as in
part (b) of the proof of 276B.   Now

\Centerline{$\Expn(X_n^2)
=\Expn(Y_n+X'_n)^2
=\Expn(Y_n^2)+2\Expn(Y_n\times X'_n)+\Expn(X'_n)^2
\ge\Expn(Y_n^2)$.  \Qed}

This means that $\sum_{n=0}^{\infty}\Bover1{(n+1)^2}\Expn(Y_n^2)$ must
be finite.   By 276C,
$\lim_{n\to\infty}\Bover1{n+1}\sum_{i=0}^nY_i=0$ a.e.
But by 273Ca we also have
$\lim_{n\to\infty}\Bover1{n+1}\sum_{i=0}^nX'_i=0$ whenever
$\lim_{n\to\infty}X'_n=0$, which is almost everywhere.   So
$\lim_{n\to\infty}\Bover1{n+1}\sum_{i=0}^nX_i=0$ a.e.
}%end of proof of 276F

\leader{*276G}{Lemma} Let $(\Omega,\Sigma,\mu)$ be a probability space,
and $\sequencen{X_n}$ a sequence of random
variables on $\Omega$ such that $\sup_{n\in\Bbb N}\Expn(|X_n|)$ is
finite.   For
$k\in\Bbb N$ and $x\in\Bbb R$ set $F_k(x)=x$ if $|x|\le k$, $0$
otherwise.   Let $\Cal F$ be an ultrafilter on $\Bbb N$.

(a) For each $k\in\Bbb N$ there is a measurable function
$Y_k:\Omega\to[-k,k]$ such that
$\lim_{n\to\Cal F}\int_EF_k(X_n)=\int_EY_k$ for every $E\in\Sigma$.

(b) $\lim_{n\to\Cal F}\Expn((F_k(X_n)-Y_k)^2)
\le\lim_{n\to\Cal F}\Expn(F_k(X_n)^2)$ for each $k$.

(c) $Y=\lim_{k\to\infty}Y_k$ is defined a.e.\ and
$\lim_{k\to\infty}\Expn(|Y-Y_k|)=0$.

\proof{{\bf (a)} For each $k$, $|F_k(X_n)|\leae k\chi\Omega$ for
every $n$, so that $\{F_k(X_n):n\in\Bbb N\}$ is uniformly integrable,
and $\{F_k(X_n)^{\ssbullet}:n\in\Bbb N\}$ is relatively weakly compact
in $L^1=L^1(\mu)$ (247C).   Accordingly
$v_k=\lim_{n\to\Cal F}F_k(X_n)^{\ssbullet}$ is defined in $L^1$ for the
weak topology
(2A3Se);  take $Y_k:\Omega\to\Bbb R$ to be a measurable function
such that $Y_k^{\ssbullet}=v_k$.   For any $E\in\Sigma$,

\Centerline{$\int_EY_k
=\int v_k\times(\chi E)^{\ssbullet}
=\lim_{n\to\Cal F}\int_EF_k(X_n)$.}

\noindent In particular,

\Centerline{$|\int_EY_k|\le\sup_{n\in\Bbb N}|\int_EF_k(X_n)|\le k\mu E$}

\noindent for every $E$, so that $\{\omega:Y_k(\omega)>k\}$ and
$\{\omega:Y_k(\omega)<-k\}$ are both negligible;  changing $Y_k$ on a
negligible set if necessary, we may suppose that $|Y_k(\omega)|\le k$
for every $\omega\in\Omega$.

\medskip

{\bf (b)} Because $Y_k$ is bounded,
$Y_k^{\ssbullet}\in L^{\infty}(\mu)$, and

\Centerline{$\lim_{n\to\Cal F}\int F_k(X_n)\times Y_k
=\lim_{n\to\Cal F}\int F_k(X_n)^{\ssbullet}\times Y_k^{\ssbullet}
=\int Y_k^{\ssbullet}\times Y_k^{\ssbullet}
=\int Y_k^2$.}

\noindent Accordingly

$$\eqalignno{\lim_{n\to\Cal F}\int(F_k(X_n)-Y_k)^2
&=\lim_{n\to\Cal F}\int F_k(X_n)^2
  -2\lim_{n\to\Cal F}\int F_k(X_n)\times Y_k
  +\int Y_k^2\cr
&=\lim_{n\to\Cal F}\int F_k(X_n)^2-\int Y_k^2
\le\lim_{n\to\Cal F}\int F_k(X_n)^2.\cr}$$

\medskip

{\bf (c)} Set $W_0=Y_0=\tbf{0}$, $W_k=Y_k-Y_{k-1}$ for $k\ge 1$.   Then
$\Expn(|W_k|)\le\lim_{n\to\Cal F}\Expn(|F_k(X_n)-F_{k-1}(X_n)|)$ for
every $k\ge 1$.   \Prf\ Set $E=\{\omega:W_k(\omega)\ge 0\}$.   Then

$$\eqalign{\int_EW_k
&=\int_EY_k-\int_EY_{k-1}\cr
&=\lim_{n\to\Cal F}\int_EF_k(X_n)-\lim_{n\to\Cal F}\int_EF_{k-1}(X_n)\cr
&=\lim_{n\to\Cal F}\int_EF_k(X_n)-F_{k-1}(X_n)
\le\lim_{n\to\Cal F}\int_E|F_k(X_n)-F_{k-1}(X_n)|.\cr}$$

\noindent Similarly,

\Centerline{$|\int_{X\setminus E}W_k|
\le\lim_{n\to\Cal F}\int_{X\setminus E}|F_k(X_n)-F_{k-1}(X_n)|$.}

\noindent So

\Centerline{$\Expn(|W_k|)
=\int_EW_k-\int_{X\setminus E}W_k
\le\lim_{n\to\Cal F}\int|F_k(X_n)-F_{k-1}(X_n)|$.  \Qed}

It follows that $\sum_{k=0}^{\infty}\Expn(|W_k|)$ is finite.   \Prf\ For
any $m\ge 1$,

$$\eqalign{\sum_{k=0}^m\Expn(|W_k|)
&\le\sum_{k=1}^m\lim_{n\to\Cal F}\Expn(|F_k(X_n)-F_{k-1}(X_n)|)\cr
&=\lim_{n\to\Cal F}\Expn(\sum_{k=1}^m|F_k(X_n)-F_{k-1}(X_n)|)\cr
&=\lim_{n\to\Cal F}\Expn(|F_m(X_n)|)
\le\sup_{n\in\Bbb N}\Expn(|X_n|).\cr}$$

\noindent So
$\sum_{k=0}^{\infty}\Expn(|W_k|)\le\sup_{n\in\Bbb N}\Expn(|X_n|)$ is
finite.\ \Qed

By B.Levi's theorem (123A), $\lim_{m\to\infty}\sum_{k=0}^m|W_k|$ is
finite a.e., so that

\Centerline{$Y=\lim_{m\to\infty}Y_m=\sum_{k=0}^{\infty}W_k$}

\noindent is defined a.e.;  and moreover

\Centerline{$\Expn(|Y-Y_k|)
\le\lim_{m\to\infty}\Expn(\sum_{j=k+1}^m|W_j|)
\to 0$}

\noindent as $k\to\infty$.
}%end of proof of 276G

\leader{*276H}{Koml\'os's theorem}\cmmnt{ ({\smc Koml\'os 67})}
Let $(\Omega,\Sigma,\mu)$ be any
measure space, and $\sequencen{X_n}$ a sequence of integrable
real-valued functions on $\Omega$ such that $\sup_{n\in\Bbb N}\int|X_n|$
is finite.   Then there are a subsequence $\sequencen{X'_n}$ of
$\sequencen{X_n}$ and an integrable function $Y$ such that
$Y\eae\lim_{n\to\infty}\Bover1{n+1}\sum_{i=0}^nX''_i$
whenever $\sequencen{X''_n}$ is a subsequence of $\sequencen{X'_n}$.

\proof{ Since neither the hypothesis nor the conclusion is affected by
changing the $X_n$ on a negligible set, we may suppose throughout that
every $X_n$ is measurable and defined on the whole of $\Omega$.   In
addition, to begin with (down to the end of (e) below), let us suppose
that $\mu X=1$.   As in 276G, set $F_k(x)=x$ for $|x|\le k$, $0$ for $|x|>k$.

\medskip

{\bf (a)} Let $\Cal F$ be any non-principal ultrafilter on $\Bbb N$
(2A1O).   For $j\in\Bbb N$ set $p_j=\lim_{n\to\Cal F}\Pr(|X_n|>j)$.
Then $\sum_{j=0}^{\infty}p_j$ is finite.   \Prf\ For any $k\in\Bbb N$,

$$\eqalignno{\sum_{j=0}^kp_j
&=\sum_{j=0}^k\lim_{n\to\Cal F}\Pr(|X_n|>j)
=\lim_{n\to\Cal F}\sum_{j=0}^k\Pr(|X_n|>j)\cr
&\le\lim_{n\to\Cal F}(1+\int|X_n|)
\le 1+\sup_{n\in\Bbb N}\int|X_n|.\cr}$$

\noindent So $\sum_{j=0}^{\infty}p_j\le 1+\sup_{n\in\Bbb N}\int|X_n|$ is
finite.\ \Qed

Setting

\Centerline{$p'_j=p_j-p_{j+1}=\lim_{n\to\Cal F}\Pr(j<|X_n|\le j+1)$}

\noindent for each $j$, we have

$$\eqalign{\sum_{j=0}^{\infty}(j+1)p'_j
&=\lim_{m\to\infty}
  \bigl(\sum_{j=0}^m(j+1)p_j-\sum_{j=0}^m(j+1)p_{j+1}\bigr)\cr
&=\lim_{m\to\infty}\sum_{j=0}^mp_j-(m+1)p_{m+1}
\le\sum_{j=0}^{\infty}p_j
<\infty.\cr}$$

Next,

\Centerline{$\lim_{n\to\Cal F}\int F_k(X_n)^2
\le\sum_{j=0}^k(j+1)^2p'_j$}

\noindent for each $k$.   \Prf\ Setting
$E_{jn}=\{\omega:j<|X_n(\omega)|\le j+1\}$ for
$j$, $n\in\Bbb N$, $F_k(X_n)^2\le\sum_{j=0}^k(j+1)^2\chi E_{jn}$, so

\Centerline{$\lim_{n\to\Cal F}\int F_k(X_n)^2
\le\lim_{n\to\Cal F}\sum_{j=0}^k(j+1)^2\mu E_{jn}
=\sum_{j=0}^k(j+1)^2p'_j$.  \Qed}

\medskip

{\bf (b)} Define $\sequence{k}{Y_k}$ and $Y\eae\lim_{k\to\infty}Y_k$
from $\sequencen{X_n}$ and $\Cal F$ as in Lemma 276G.   Then

\Centerline{$J_k
=\{n:n\in\Bbb N,\,\int(F_k(X_n)-Y_k)^2\le 1+\sum_{j=0}^k(j+1)^2p'_j\}$}

\noindent belongs to $\Cal F$ for every $k\in\Bbb N$.   \Prf\ By (a)
above and 276Gb,

\Centerline{$\lim_{n\to\Cal F}\int(F_k(X_n)-Y_k)^2
\le\lim_{n\to\Cal F}\int F_k(X_n)^2
\le\sum_{j=0}^k(j+1)^2p'_j$.  \Qed}

\noindent Also, of course,

\Centerline{$K_k=\{n:n\in\Bbb N,\,\Pr(F_j(X_n)\ne X_n)\le p_j+2^{-j}$
for every $j\le k\}$}

\noindent belongs to $\Cal F$ for every $k$.

\medskip

{\bf (c)} For $n$, $k\in\Bbb N$ let $Z_{kn}$ be a simple function such
that $|Z_{kn}|\le|F_k(X_n)-Y_k|$ and
$\int|F_k(X_n)-Y_k-Z_{kn}|\le 2^{-k}$.   For $m\in\Bbb N$ let $\Sigma_m$
be the algebra of subsets of $\Omega$ generated by sets of the form
$\{\omega:Z_{kn}(\omega)=\alpha\}$ for $k$, $n\le m$ and
$\alpha\in\Bbb R$.   Because each $Z_{kn}$ takes only finitely many
values, $\Sigma_m$ is finite (and is therefore a $\sigma$-subalgebra of
$\Sigma$);  and of course $\Sigma_m\subseteq\Sigma_{m+1}$ for every $m$.

We need to look at conditional expectations on the $\Sigma_m$, and
because $\Sigma_m$ is always finite these have a particularly
straightforward
expression.   Let $\Cal A_m$ be the set of `atoms', or minimal non-empty
sets, in $\Sigma_m$;  that is, the set of equivalence classes in
$\Omega$ under the relation $\omega\sim\omega'$ if
$Z_{kn}(\omega)=Z_{kn}(\omega')$ for all $k$, $n\le m$.   For any
integrable random variable $X$ on $\Omega$, define $\Expn_m(X)$ by
setting

$$\eqalign{\Expn_m(X)(\omega)
&=\Bover1{\mu A}\int_AX\text{ if }x\in A\in\Cal A_m
  \text{ and }\mu A>0,\cr
&=0\text{ if }x\in A\in\Cal A_m\text{ and }\mu A=0.\cr}$$

\noindent Then $\Expn_m(X)$ is a conditional expectation of $X$ on
$\Sigma_m$.
%It will be useful later to note that $\Expn_l(\Expn_m(X))=\Expn_l(X)$
%whenever $l\le m$.

Now

$$\eqalignno{\lim_{n\to\Cal F}\int|\Expn_m(F_k(X_n)-Y_k)|
&=\lim_{n\to\Cal F}\sum_{A\in\Cal A_m}\int_A|\Expn_m(F_k(X_n)-Y_k)|\cr
&=\lim_{n\to\Cal F}\sum_{A\in\Cal A_m}|\int_A\Expn_m(F_k(X_n)-Y_k)|\cr
\displaycause{because $\Expn_m(F_k(X_n)-Y_k)$ is constant on each
$A\in\Cal A_m$}
&=\lim_{n\to\Cal F}\sum_{A\in\Cal A_m}|\int_AF_k(X_n)-Y_k|\cr
&=\sum_{A\in\Cal A_m}\lim_{n\to\Cal F}|\int_AF_k(X_n)-Y_k|
=0\cr}$$

\noindent by the choice of $Y_k$.   So if we set

\Centerline{$I_m
=\{n:n\in\Bbb N,\,\int|\Expn_m(F_k(X_n)-Y_k)|\le 2^{-k}$ for every
$k\le m\}$,}

\noindent then $I_m\in\Cal F$ for every $m$.

\medskip

{\bf (d)} Suppose that $\sequencen{r(n)}$ is any strictly increasing
sequence in $\Bbb N$ such that $r(0)>0$, $r(n)\in J_n\cap K_n$ for every
$n$ and
$r(n)\in I_{r(n-1)}$ for $n\ge 1$.   Then
$\Bover1{n+1}\sum_{i=0}^nX_{r(i)}\to Y$ a.e.\ as $n\to\infty$.
\Prf\   Express $X_{r(n)}$ as

\Centerline{$(X_{r(n)}-F_n(X_{r(n)}))
+(F_n(X_{r(n)})-Y_n-Z_{n,r(n)})+Y_n+Z_{n,r(n)}$}

\noindent for each $n$.   Taking these pieces in turn:

\medskip

\quad{\bf (i)}

$$\eqalignno{\sum_{n=0}^{\infty}\Pr(X_{r(n)}\ne F_n(X_{r(n)}))
&\le\sum_{n=0}^{\infty}p_n+2^{-n}\cr
\displaycause{because $r(n)\in K_n$ for every $n$}
&<\infty\cr}$$

\noindent by (a).   But this means that
$X_{r(n)}-F_n(X_{r(n)})\to 0$ a.e.,
since the sequence is eventually zero at almost every point, and
$\Bover1{n+1}\sum_{i=0}^nX_{r(i)}-F_i(X_{r(i)})\to 0$ a.e.\ by 273Ca again.

\medskip

\quad{\bf (ii)} By the choice of the $Z_{n,r(n)}$,

\Centerline{$\sum_{n=0}^{\infty}\int|F_n(X_{r(n)})-Y_n-Z_{n,r(n)}|
\le\sum_{n=0}^{\infty}2^{-n}$}

\noindent is finite, so $F_n(X_{r(n)})-Y_n-Z_{n,r(n)}\to 0$ a.e.\ and
$\Bover1{n+1}\sum_{i=0}^nF_i(X_{r(i)})-Y_i-Z_{i,r(i)}\to 0$ a.e.

\medskip

\quad{\bf (iii)} By 276G, $Y_n\to Y$ a.e.\ and
$\Bover1{n+1}\sum_{i=0}^nY_i\to Y$ a.e.

\medskip

\quad{\bf (iv)} We know that, for each $n\ge 1$,
$r(n)\in I_{r(n-1)}$.   So (because $r(n-1)\ge n$)
$\int|\Expn_{r(n-1)}(F_n(X_{r(n)})-Y_n)|\le 2^{-n}$.   But as also

\Centerline{$\int\bigl|\Expn_{r(n-1)}
  \bigl(F_n(X_{r(n)})-Y_n-Z_{n,r(n)}\bigr)\bigr|
\le\int|F_n(X_{r(n)})-Y_n-Z_{n,r(n)}|\le 2^{-n}$}

\noindent by 244M and the choice of $Z_{n,r(n)}$,

$$\eqalign{\int|\Expn_{r(n-1)}Z_{n,r(n)}|
&=\int\bigl|\Expn_{r(n-1)}(F_n(X_{r(n)})-Y_n)
  -\Expn_{r(n-1)}(F_n(X_{r(n)})-Y_n-Z_{n,r(n)})\bigr|\cr
&\le 2^{-n+1}\cr}$$

\noindent for every $n$.   Accordingly $\Expn_{r(n-1)}Z_{n,r(n)}\to 0$
a.e.

On the other hand,

$$\eqalignno{\sum_{n=0}^{\infty}\Bover1{(n+1)^2}\int Z_{n,r(n)}^2
&\le\sum_{n=0}^{\infty}\Bover1{(n+1)^2}\int F_n(X_{r(n)}-Y_n)^2\cr
&\le\sum_{n=0}^{\infty}\Bover1{(n+1)^2}(1+\sum_{j=0}^n(j+1)^2p'_j)\cr
\displaycause{because $r(n)\in J_n$}
&\le\sum_{n=0}^{\infty}\Bover1{(n+1)^2}
  +\sum_{j=0}^{\infty}(j+1)^2p'_j\sum_{n=j}^{\infty}\Bover1{(n+1)^2}\cr
&\le\Bover{\pi^2}6
  +2\sum_{j=0}^{\infty}(j+1)p'_j\cr}$$

\noindent is finite.   (I am using the estimate

\Centerline{$\sum_{n=j}^{\infty}\Bover1{(n+1)^2}
\le\sum_{n=j}^{\infty}\Bover2{n+1}-\Bover2{n+2}=\Bover2{j+1}$.)}

\noindent By 276F, applied to $\sequencen{\Sigma_{r(n)}}$
and $\sequencen{Z_{n,r(n)}}$, $\Bover1{n+1}\sum_{i=0}^nZ_{i,r(i)}\to 0$
a.e.

\medskip

\quad{\bf (v)} Adding these four components, we see that
$\Bover1{n+1}\sum_{i=0}^nX_{r(i)}\to 0$, as claimed.\ \Qed

\medskip

{\bf (e)} Now fix any strictly increasing sequence $\sequencen{s(n)}$ in
$\Bbb N$ such that $s(0)>0$, $s(n)\in\bigcap_{m\le n}J_m\cap K_n$
for every $n$ and
$s(n)\in I_{s(n-1)}$ for $n\ge 1$;  such a sequence exists because
$J_m\cap K_m\cap I_{s(n-1)}$ belongs to $\Cal F$, so is infinite, for
every $n\ge 1$ and $m\in\Bbb N$.   Set $X'_n=X_{s(n)}$ for every $n$.   If
$\sequencen{X''_n}$ is a subsequence of $\sequencen{X'_n}$, then it is
of the form $\sequencen{X_{s(r(n))}}$ for some strictly increasing
sequence $\sequencen{r(n)}$.   In this case

\Centerline{$s(r(0))\ge s(0)>0$,}

\Centerline{$s(r(n))\in J_{r(n)}\cap K_{r(n)}\subseteq J_n\cap K_n$ for
every $n$,}

\Centerline{$s(r(n))\in I_{s(r(n)-1)}\subseteq I_{s(r(n-1))}$ for every
$n\ge 1$.}

\noindent So (d) tells us that $\Bover1{n+1}\sum_{i=0}^nX''_i\to Y$ a.e.

\medskip

{\bf (f)} Thus the theorem is proved in the case in which
$(\Omega,\Sigma,\mu)$ is a probability space.   Now suppose that $\mu$
is $\sigma$-finite and $\mu\Omega>0$.   In this case there is a strictly
positive measurable function $f:\Omega\to\Bbb R$ such that
$\int fd\mu=1$ (215B(ix)).   Let $\nu$ be the corresponding
indefinite-integral measure (234J), so that $\nu$ is a probability
measure on $\Omega$, and $\sequencen{\Bover1f\times X_n}$ is a sequence
of $\nu$-integrable functions such that
$\sup_{n\in\Bbb N}\int\Bover1f\times X_nd\nu$ is finite (235K).   From
(a)-(e) we see that there must be a $\nu$-integrable function $Y$ and a
subsequence $\sequencen{X'_n}$ of $\sequencen{X_n}$ such that
$\Bover1{n+1}\sum_{i=0}^n\Bover1f\times X''_i\to Y\,\,\nu$-a.e.\ for
every subsequence $\sequencen{X''_n}$ of $\sequencen{X'_n}$.   But $\mu$
and $\nu$ have the same negligible sets (234Lc), so
$\Bover1{n+1}\sum_{i=0}^nX''_i\to f\times Y\,\,\mu$-a.e.\ for every
subsequence $\sequencen{X''_n}$ of $\sequencen{X'_n}$.

\medskip

{\bf (g)} Since the result is trivial if $\mu\Omega=0$, the theorem is
true whenever $\mu$ is $\sigma$-finite.   For the general case, set

\Centerline{$\tilde\Omega
=\bigcup_{n\in\Bbb N}\{\omega:X_n(\omega)\ne 0\}
=\bigcup_{m,n\in\Bbb N}\{\omega:|X_n(\omega)|\ge 2^{-m}\}$,}

\noindent so that the subspace measure $\mu_{\tilde\Omega}$ is
$\sigma$-finite.   Then there are a $\mu_{\tilde\Omega}$-integrable
function $\tilde Y$ and a subsequence $\sequencen{X'_n}$ of
$\sequencen{X_n}$ such that
$\Bover1{n+1}\sum_{i=0}^nX''_i\restrp\tilde\Omega\to\tilde Y\,\,
\mu_{\tilde\Omega}$-a.e.\ for every subsequence $\sequencen{X''_n}$ of
$\sequencen{X'_n}$.   Setting $Y(\omega)=\tilde Y(\omega)$ if
$\omega\in\tilde\Omega$, $0$ for $\omega\in\Omega\setminus\tilde\Omega$,
we see that $Y$ is $\mu$-integrable and that
$\Bover1{n+1}\sum_{i=0}^nX''_i\to Y\,\,\mu$-a.e. whenever
$\sequencen{X''_n}$ is a subsequence of $\sequencen{X'_n}$.   This
completes the proof.
}%end of proof of 276H

\exercises{\leader{276X}{Basic exercises $\pmb{>}$(a)}
%\sqheader 276Xa
Let $\sequencen{X_n}$ be a martingale adapted to
a sequence $\sequencen{\Sigma_n}$ of $\sigma$-algebras.   Show that
$\int_EX_n^2\le\int_EX_{n+1}^2$ whenever $n\in\Bbb N$ and $E\in\Sigma_n$
(allowing $\infty$ as a value of an integral).   \Hint{see the proof of
276B.}
%276B

\sqheader 276Xb Let $\sequencen{X_n}$ be a martingale.   Show
that for any $\epsilon>0$,

\Centerline{$\Pr(\sup_{n\in\Bbb N}|X_n|\ge\epsilon)
\le\Bover1{\epsilon^2}\sup_{n\in\Bbb N}\Expn(X_n^2)$.}

\noindent\Hint{put 276Xa together with the argument for 275D.}
%276Xa, 276B

\spheader 276Xc When does 276Xb give a sharper result than 275Xb?
%276Xb,  276B

\spheader 276Xd Let $\sequencen{X_n}$ be an independent identically
distributed sequence of random variables with zero expectation and non-zero
finite variance, and $\sequencen{t_n}$ a sequence in $\Bbb R$.   Show that
(i) if $\sum_{n=0}^{\infty}t_n^2<\infty$, then $\sum_{n=0}^{\infty}t_nX_n$
is defined in $\Bbb R$ a.e.  (ii) if $\sum_{n=0}^{\infty}t_n^2=\infty$ then
$\sum_{n=0}^{\infty}t_nX_n$ is undefined a.e.   \Hint{276B, 274Xj.}
%276B

\spheader 276Xe Let $\sequencen{X_n}$ be a martingale difference
sequence and set $Y_n=\bover1{n+1}(X_0+\ldots+X_n)$ for each
$n\in\Bbb N$.   Show that if $\sequencen{X_n}$ is uniformly integrable
then $\lim_{n\to\infty}\|Y_n\|_1=0$.   \Hint{use the argument of 273Na,
with 276C in place of 273D, and setting $\tilde X_n=X'_n-Z_n$, where
$Z_n$ is an appropriate conditional expectation of $X'_n$.}
%276C

\spheader 276Xf Suppose that
$\sequencen{X_n}$ is a uniformly bounded martingale
difference sequence and $\sequencen{a_n}\in\ell^2$.
Show that $\lim_{n\to\infty}\prod_{i=0}^n(1+a_iX_i)$ is defined and finite
almost everywhere.   \Hint{$\sequencen{a_nX_n}$ is summable and
square-summable a.e.}
%276C

\sqheader 276Xg\dvAformerly{2{}76Xe} {\bf Strong law of large numbers:  
fifth form} A sequence $\sequencen{X_n}$ of random variables is
{\bf exchangeable} if $(X_{n_0},\ldots,X_{n_k})$ has the same
joint distribution as $(X_0,\ldots,X_k)$ whenever $n_0,\ldots,n_k$ are
distinct.
Show that if $\sequencen{X_n}$ is an exchangeable sequence of random
variables with finite expectation, then
$\sequencen{\Bover1{n+1}\sum_{i=0}^{\infty}X_i}$ converges a.e.
\Hint{276H.}
%276H

\spheader 276Xh\dvAnew{2017} In 276B, show that 
$\Expn((\sum_{n=0}^{\infty}X_n)^2)\le\sum_{n=0}^{\infty}\Expn(X_n^2)$.
%276B out of order query

\leader{276Y}{Further exercises (a)}
%\spheader 276Ya
Let $\sequencen{X_n}$ be a martingale difference
sequence such that $\sup_{n\in\Bbb N}
\ifdim\pagewidth>467pt\penalty-100\fi
\|X_n\|_p$ is finite, where
$p\in\ooint{1,\infty}$.   Show that
$\lim_{n\to\infty}\|\bover1{n+1}\sum_{i=0}^nX_i\|_p=0$.   \Hint{273Nb.}
%276A

\spheader 276Yb Let $\sequencen{X_n}$ be a uniformly integrable
martingale difference sequence and $Y$ a bounded random variable.   Show
that $\lim_{n\to\infty}\Expn(X_n\times Y)=0$.   (Compare 272Ye.)
%276A

\spheader 276Yc Use 275Yh to prove 276Xa.
%276Xa, 276B

\spheader 276Yd Let $\sequencen{X_n}$ be a sequence of random
variables such that, for some $\delta>0$,
$\sup_{n\in\Bbb N}n^{\delta}\Expn(|X_n|)$ is finite.   Set
$S_n=\bover1{n+1}(X_0+\ldots+X_n)$ for each $n$.   Show that
$\lim_{n\to\infty}S_n=0$ a.e.   \Hint{set
$Z_k=2^{-k}(|X_0|+\ldots+|X_{2^k-1}|)$.    Show that
$\sum_{k=0}^{\infty}\Expn(Z_k)<\infty$.   Show that $|S_n|\le 2Z_{k+1}$
if $2^k<n\le 2^{k+1}$.}
%for 276Ye

\spheader 276Ye {\bf Strong law of large numbers:  sixth form} Let
$\sequencen{X_n}$ be a martingale difference sequence such that, for
some $\delta>0$, $\sup_{n\in\Bbb N}\Expn(|X_n|^{1+\delta})$ is finite.
Set $S_n=\bover1{n+1}(X_0+\ldots+X_n)$ for each $n$.   Show that
$\lim_{n\to\infty}S_n=0$ a.e.   \Hint{take a non-decreasing
sequence $\sequencen{\Sigma_n}$ to which $\sequencen{X_n}$ is adapted.
Set $Y_n=X_n$ when $|X_n|\le n$, $0$ otherwise.   Let $U_n$ be a
conditional expectation of $Y_n$ on $\Sigma_{n-1}$ and set
$Z_n=Y_n-U_n$.   Use ideas from 273H, 276C and 276Yd above to show that
$\bover{1}{n+1}\sum_{i=0}^nV_i\to 0$ a.e.\ for $V_i=Z_i$, $V_i=U_i$,
$V_i=X_i-Y_i$.}
%276Yd, 276C mt27bits

\spheader 276Yf Show that there is a martingale $\sequencen{X_n}$ which
converges in measure but is not convergent a.e.   (Compare 273Ba.)
\Hint{arrange that
$\{\omega:X_{n+1}(\omega)\ne 0\}=E_n
\subseteq\{\omega:|X_{n+1}(\omega)-X_n(\omega)|\ge 1\}$, where
$\sequencen{E_n}$ is an independent sequence of sets and
$\mu E_n=\Bover1{n+1}$ for each $n$.}
%with 276Yg

\spheader 276Yg Give an example of an identically distributed
martingale difference sequence $\sequencen{X_n}$ such that
$\sequencen{\bover1{n+1}(X_0+\ldots+X_n)}$ does not
converge to $0$ almost everywhere.   \Hint{start by devising a
uniformly bounded sequence $\sequencen{U_n}$ such that
$\lim_{n\to\infty}\Expn(|U_n|)=0$ but
$\sequencen{\bover1{n+1}(U_0+\ldots+U_n)}$ does not converge to $0$
almost everywhere.   Now repeat your construction in such a context that
the $U_n$ can be derived from an identically distributed martingale
difference sequence by the formulae of 276Ye.}
%276Ye, 276C

\spheader 276Yh Construct a proof of Koml\'os's theorem which does not
involve ultrafilters, or any other use of the full axiom of choice, but
proceeds throughout by selecting appropriate sub-subsequences.
Remember to check that you can prove any fact you use about weakly
convergent sequences in $L^1$ on the same rules.
}%end of exercises

\endnotes{
\Notesheader{276}
I include two more versions of the strong law of large numbers (276C,
276Ye) not because I have any applications in mind but because I think
that if you know the strong law for $\|\,\|_{1+\delta}$-bounded
independent sequences, and what a martingale difference
sequence is, then there is something missing if you do not know the
strong law for $\|\,\|_{1+\delta}$-bounded martingale difference
sequences.   And then, of course, I have to add 276Yf and 276Yg (which
seems to be difficult), lest
you be tempted to think that the strong law is `really' about martingale
difference sequences rather than about independent sequences.
(Compare 272Yd and 275Xl.)

Koml\'os's theorem is rather outside the scope of this volume;  it is
quite hard work and surely much less important, to most probabilists,
than many results I have omitted.   It does provide a quick proof of
276Xg.   However it is relevant to questions arising in some topics
treated in Volumes 3 and 4, and the proof fits naturally into this
section.
}%end of notes

\discrpage

%\ifresultsonly\else
\frfilename{mt27conc.tex}
\versiondate{8.4.09}
\copyrightdate{2009}

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

\def\chaptername{Probability theory}
\def\sectionname{Concordance}

\newsection{}

I list here the section and paragraph numbers which have (to my
knowledge) appeared in print in references to this chapter, and which
have since been changed.

\medskip

\leader{2{}72S}{Distribution of a sum of independent random variables}
This result, referred to in the 2002 and 2004 editions of Volume 3, 
and the 2003 and 2006 editions of Volume 4, is now 272T.

\leader{2{}72U}{Etemadi's lemma} This result, referred to in the 2003 and
2006 editions of Volume 4, is now 272V.

\leader{2{}72Yd}{} This exercise, referred to in the 2002 and 2004 editions
of Volume 3, is now 272Ye.

\leader{2{}73Xh}{} This exercise, referred to in the 2006 edition of Volume
4, is now 273Xi.

\leaveitout{2{}73M, inequalities in $L^p$, {\smc Bogachev 07}, 273M.}

\leader{2{}76Xe}{} This exercise, referred to in the 2003 and 2006 editions
of Volume 4, is now 276Xg.

\frnewpage
%\fi

\references{
\frfilename{mt2r.tex}
\versiondate{17.12.12}
\copyrightdate{2000}

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

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\gdef\headlinesectionname{References}

\centerline{\bf References for Volume 2}

\medskip

%A

Alexits G.\ [78] (ed.) {\it Fourier Analysis and Approximation Theory.}
North-Holland, 1978 (Colloq.\ Math.\ Soc.\ Janos Bolyai 19).

Antonov N.Yu.\ [96] `Convergence of Fourier series', East J.\ Approx.\ 7 (1996) 187-196.
\cmmnt{[\S286 {\it notes\/}.]}

Arias de Reyna J.\ [02] {\it Pointwise Convergence of Fourier Series.} Springer, 2002 (Lecture Notes in Mathematics 1785).
\cmmnt{[\S286 {\it notes\/}.]}

\medskip%B

Baker R.\ [04] `{}``Lebesgue measure'' on $\Bbb R^{\infty}$, II',
Proc.\ Amer.\ Math.\ Soc.\ 132 (2004) 2577-2591.
\cmmnt{[254Yb.]}

Bergelson V., March P.\ \& Rosenblatt J.\ [96] (eds.) {\it Convergence
in Ergodic Theory and Probability.}  de Gruyter, 1996.

\indexheader{Bogachev}
Bogachev V.I.\ [07] {\it Measure theory.}  Springer, 2007.

du Bois-Reymond P.\ [1876] `Untersuchungen \"uber die Convergenz und
Divergenz der Fouriersche Darstellungformeln', Abh.\ Akad.\ M\"unchen 12
(1876) 1-103.   \cmmnt{[\S282 {\it notes}.]}

Bourbaki N.\ [66] {\it General Topology.}  Hermann/Addison-Wesley, 1968.
\cmmnt{[2A5F.]}

Bourbaki N.\ [87] {\it Topological Vector Spaces.}  Springer, 1987.
\cmmnt{[2A5E.]}

\medskip%C

Carleson L.\ [66] `On convergence and growth of partial sums of
Fourier series', Acta Math.\ 116 (1966) 135-157.  \cmmnt{[\S282 {\it
notes}, \S286 {\it intro.}, \S286 {\it notes}.]}

Clarkson J.A.\ [1936] `Uniformly convex spaces', Trans.\ Amer.\ Math.\ Soc.\
40 (1936) 396-414.
\cmmnt{[244O.]}

\medskip%D

Defant A.\ \& Floret K.\ [93] {\it Tensor Norms and Operator Ideals},
North-Holland, 1993.   \cmmnt{[\S253 {\it notes}.]}

Doob J.L.\ [53] {\it Stochastic Processes.}  Wiley, 1953.
%used for 275Ye

Dudley R.M.\ [89] {\it Real Analysis and Probability.} Wadsworth \&
Brooks/Cole, 1989.   \cmmnt{[\S282 {\it notes}.]}

Dunford N.\ \& Schwartz J.T.\ [57] {\it Linear Operators I.}  Wiley,
1957 (reprinted 1988).  \cmmnt{[\S244 {\it notes}, 2A5J.]}

\medskip%E

Enderton H.B.\ [77] {\it Elements of Set Theory.}   Academic, 1977.
\cmmnt{[\S2A1.]}

Engelking R.\ [89] {\it General Topology.} Heldermann, 1989 (Sigma
Series in Pure Mathematics 6).   \cmmnt{[2A5F.]}

Etemadi N.\ [96] `On convergence of partial sums of independent random
variables', pp.\ 137-144 in {\smc Bergelson March \& Rosenblatt 96}.
\cmmnt{[272V.]}

Evans L.C.\ \& Gariepy R.F.\ [92] {\it Measure Theory and Fine
Properties of Functions.}   CRC Press, 1992.
\cmmnt{[263Ec, \S265 {\it notes}.]}
%Kirszbraun's theorem mentioned but not proved

\medskip%F

Federer H.\ [69] {\it Geometric Measure Theory.}  Springer, 1969
(reprinted 1996).
\cmmnt{[262C, 263Ec, \S264 {\it notes}, \S265 {\it notes},
\S266 {\it notes}.]}

Feller W.\ [66] {\it An Introduction to Probability Theory and its
Applications,} vol.\ II.   Wiley, 1966.  \cmmnt{[Chap.\ 27 {\it
intro.}, 274H, 275Xc, 285N.]}

Fremlin D.H.\ [74] {\it Topological Riesz Spaces and
Measure Theory.}   Cambridge U.P., 1974.
\cmmnt{[\S232 {\it notes}, 241F, \S244 {\it notes}, \S245 {\it notes},
\S247 {\it notes}.]}

Fremlin D.H.\ [93] `Real-valued-measurable cardinals', pp.\ 151-304 in
{\smc Judah 93}.  \cmmnt{[232H.]}

\medskip%G\medskip%H

Haimo D.T.\ [67] (ed.) {\it Orthogonal Expansions and their Continuous
Analogues.}   Southern Illinois University Press, 1967.

Hall P.\ [82] {\it Rates of Convergence in the Central Limit Theorem.}
Pitman, 1982.  \cmmnt{[274H.]}

Halmos P.R.\ [50] {\it Measure Theory.}  Van Nostrand, 1950.
\cmmnt{[\S251 {\it notes}, \S252 {\it notes}, 255Yn.]}

Halmos P.R.\ [60] {\it Naive Set Theory.} Van Nostrand, 1960.
\cmmnt{[\S2A1.]}

Hanner O.\ [56] `On the uniform convexity of $L^p$ and $l^p$',
Arkiv f\"or Matematik 3 (1956) 239-244.
\cmmnt{[244O.]}

Henle J.M.\ [86] {\it An Outline of Set Theory.}  Springer, 1986.
\cmmnt{[\S2A1.]}

Hoeffding W.\ [63]
`Probability inequalities for sums of bounded random variables',
J.\ Amer.\ Statistical Association 58 (1963) 13-30.
\cmmnt{[272W.]}

Hunt R.A.\ [67] `On the convergence of Fourier series', pp.\ 235-255
in {\smc Haimo 67}.  \cmmnt{[\S286 {\it notes}.]}

\medskip%I\medskip%J

Jorsb{\o}e O.G.\ \& Mejlbro L.\ [82] {\it The Carleson-Hunt Theorem on
Fourier Series.}  Springer, 1982 (Lecture Notes in Mathematics 911).
\cmmnt{[\S286 {\it notes}.]}

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

\medskip%K

\indexheader{Kelley}
Kelley J.L.\ [55] {\it General Topology.} Van Nostrand, 1955.
\cmmnt{[2A5F.]}

Kelley J.L.\ \& Namioka I.\ [76] {\it Linear Topological Spaces.}
Springer, 1976.   \cmmnt{[2A5C.]}

\indexheader{Kirszbraun}
Kirszbraun M.D.\ [1934] `\"Uber die zusammenziehenden und Lipschitzian
Transformationen', Fund.\ Math.\ 22 (1934) 77-108.  \cmmnt{[262C.]}

Kolmogorov A.N.\ [1926] `Une s\'erie de Fourier-Lebesgue divergente
partout', C.\ R.\ Acad.\ Sci.\ Paris 183 (1926) 1327-1328.
\cmmnt{[\S282 {\it notes}.]}

Koml\'os J.\ [67] `A generalization of a problem of Steinhaus', Acta
Math.\ Acad.\ Sci.\ Hung.\ 18 (1967) 217-229.  \cmmnt{[276H.]}

K\"orner T.W.\ [88] {\it Fourier Analysis.}  Cambridge U.P., 1988.
\cmmnt{[\S282 {\it notes}.]}

K\"othe G.\ [69] {\it Topological Vector Spaces I.}   Springer, 1969.
\cmmnt{[2A5C, 2A5E, 2A5J.]}

Krivine J.-L.\ [71] {\it Introduction to Axiomatic Set Theory.}
D.~Reidel, 1971.  \cmmnt{[\S2A1.]}

\medskip%L

Lacey M.T.\ [05] {\it Carleson's Theorem: Proof, Complements, Variations.}
{\tt http://arxiv.org/pdf/math{\bsp}0307008v4{\penalty-100}.pdf}.
%expanded from 
% Publ.\ Mat.\ 48 (2004) 251-307
\cmmnt{[\S286 {\it notes}.]}

Lacey M.T.\ \& Thiele C.M.\ [00] `A proof of boundedness of the Carleson
operator', Math.\ Research Letters 7 (2000) 1-10.
\cmmnt{[\S286 {\it intro.}, 286H.]}

Lebesgue H.\ [72] {\it Oeuvres Scientifiques.}   L'Enseignement
Math\'ematique, Institut de Math\'ematiques, Univ.\ de Gen\`eve, 1972.
\cmmnt{[Chap.\ 27 {\it intro.}]}

Liapounoff A.\ [1901] `Nouvelle forme du th\'eor\`eme sur la limite
de probabilit\'e', M\'em.\ Acad.\ Imp.\ Sci.\ St-P\'etersbourg 12(5)
(1901) 1-24. \cmmnt{[274Xh.]}

Lighthill M.J.\ [59] {\it Introduction to Fourier Analysis and
Generalised Functions.}  Cambridge U.P., 1959.
\cmmnt{[\S284 {\it notes}.]}

Lindeberg J.W.\ [1922] `Eine neue Herleitung des Exponentialgesetzes in
der Wahrscheinlichkeitsrechnung', Math.\ Zeitschrift 15 (1922) 211-225.
\cmmnt{[274H, \S274 {\it notes}.]}

Lipschutz S.\ [64] {\it Set Theory and Related Topics.}  McGraw-Hill,
1964 (Schaum's Outline Series).  \cmmnt{[\S2A1.]}

Lo\`eve M.\ [77] {\it Probability Theory I.} Springer, 1977.
\cmmnt{[Chap.\ 27 {\it intro.}, 274H.]}

Luxemburg W.A.J.\ \& Zaanen A.C.\ [71] {\it Riesz Spaces
I.} North-Holland, 1971.
\cmmnt{[241F.]}


\medskip%M

Mozzochi C.J.\ [71] {\it On the Pointwise Convergence of Fourier
Series.}   Springer, 1971 (Lecture Notes in Mathematics 199).
\cmmnt{[\S286 {\it notes}.]}

\medskip%N

Naor A.\ [04] `Proof of the uniform convexity lemma',
{\tt http://www.cims.nyu.edu/\penalty-100$^{\sim}$naor/homepage files/\ifnum\stylenumber=12\break\fi{\vthsp}inequality\discretionary{}{}{}.pdf},
26.2.04.
\cmmnt{[244O.]}
%anon04.PDF

\medskip%O\medskip%P\medskip%Q\medskip%R

R\'enyi A.\ [70] {\it Probability Theory.}  North-Holland, 1970.
\cmmnt{[274H.]}

Roitman J.\ [90] {\it  An Introduction to Set Theory.}  Wiley, 1990.
\cmmnt{[\S2A1.]}

Roselli P.\ \& Willem M.\ [02] `A convexity inequality',
Amer.\ Math.\ Monthly 109 (2002) 64-70.
\cmmnt{[244Ym.]}

\medskip%S

%Saks S.\ [1924] 'Sur les nombres d\'eriv\'es des fonctions', 
%  Fundamenta Math.\ 5 (1924) 98-104.
% D-Y-S theorem
% books/saks.24.pdf   http://matwbn.icm.edu.pl/ksiazki/fm/fm5/fm5112.pdf

Saks S.\ [1924] 'Sur les nombres d\'eriv\'es des fonctions', Fund

Schipp F.\ [78] `On Carleson's method', pp.\ 679-695 in
{\smc Alexits 78}.   \cmmnt{[\S286 {\it notes}.]}

Semadeni Z.\ [71] {\it Banach spaces of continuous functions I.}
Polish Scientific Publishers, 1971.  \cmmnt{[\S253 {\it notes}.]}

Shiryayev A.N.\ [84] {\it Probability.} Springer, 1984. \cmmnt{[285N.]}

Steele J.M.\ [86] `An Efron-Stein inequality of nonsymmetric statistics',
Annals of Statistics 14 (1986) 753-758.
\cmmnt{[274Ya.]}

\medskip%T\medskip%U\medskip%V\medskip%W\medskip%Y\medskip%Z

Zaanen A.C.\ [83] {\it Riesz Spaces II.} North-Holland, 1983.
\cmmnt{[241F.]}

Zygmund A. [59] {\it Trigonometric Series.}   Cambridge U.P, 1959.
\cmmnt{[\S244 {\it notes}, \S282 {\it notes}, 284Xk.]}

\frnewpage

}%end of references

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