% The following TeXfile comprises part of

%     MEASURE THEORY, by D.H.Fremlin
%          (University of Essex, Colchester, England;
%                   e-mail  fremdh@essex.ac.uk)

% Please contact me with any difficulties not covered by the remarks
% below.

% The file is set up for processing by plain TeX in the presence of
% files  amstex.tex ,  amssym.tex  and  amsppt.sty  suitably supported by the
% appropriate fonts.   If these are unavailable, a readable version
% can be obtained by removing the comment symbol  %  from the line

%        %   \amstexsupportedfalse

% below.   Some systems do not allow access to files  amstex.tex  and
% amsppt.sty , but instead employ a command "amstex <filename.tex>";
% for such systems it is sometimes right to delete or disable the
% commands

%        \input amstex
%        \documentstyle{amsppt}

% (I have little experience with this and should welcome comments.)

% Various formats of output are obtainable.   The principal variations
% are

%    1. "Standard" output uses the command
%               \smallprint
% and gives output adapted to A4 paper (21 cm x 29.7 cm).

%    2.  Using the command
%               \USsmallprint
% gives output adapted to 8.5" x 11" .

%    3.  Using the command
%               \noproofs
% produces an abridged version, in which proofs are omitted.

%    4.   Using the command
%               \resultsonly
% produces a super-abridged version, omitting comments and exercises
% as well as proofs.

% These commands are grouped together at the end of the introductory
% definitions;  they may be activated by deleting the initial comment
% symbols  % .

% Any diagrams included will be called for by inputting postscript files.
% (You can look for these by hunting for   \picture  in the code below.)
% To get these you will need the relevant  .ps  files with the expected
% names  ( \picture{mt133g}{...}  demands the file  mt133g.ps , for instance).


%(If you can read TeX fluently, you may wish to fast-forward to a line
%beginning   %file starts here   or   \frfilename
%if one of these is present.)

%TeXfiles with this heading are designed for processing with AmSTeX,
%accessed through TeXfiles  amstex.tex (with amssym.tex  and
% amsppt.sty  and possibly  amsppt.sti)  during compilation.   
%If these are unavailable, a readable
%version can (I hope) be produced by deleting the  %  before
% \amstexsupportedfalse  below.

%Some special difficulties arise with the importation of  .ps  files
%in diagrams.   Look for  \picture  below for notes on these.

%The following \count registers are used for special purposes:
%   \count1=\stylenumber  to specify style
%   \count2=\footnotenumber  to count footnotes

%AmSTeX declaration
\newif\ifamstexsupported
     \amstexsupportedtrue
%     \amstexsupportedfalse               %use when AMSTeX unavailable
\newif\ifDHFamstexloaded        %for detection of a non-standard AMSTeX
     \DHFamstexloadedfalse    
\newif\ifDHFamspptloaded        
     \DHFamspptloadedfalse    
\newif\iffourteensavailable
     \fourteensavailabletrue
    \fourteensavailablefalse
\newif\ifathome
     \athometrue     %to cope with a quirk     of my son's Ghostscript
     \athomefalse
\newif\iffirstpage
     \firstpagetrue

%Style assignments
\countdef\stylenumber=1
\countdef\footnotenumber=2
\def\fullsize{\magnification=\magstep1
     \def\pagewidth{390pt}
     \def\pageheight{9truein}
     \hsize\pagewidth
     \vsize\pageheight
     \stylenumber=1}
\def\smallprint{\magnification=\magstep0
     \overfullrule0pt
     \def\pagewidth{468pt}
     \def\pageheight{9truein}
     \hsize\pagewidth
     \vsize\pageheight
     \stylenumber=2}
\def\printersize{\magnification=\magstep1
     \baselineskip24pt
     \overfullrule0pt
     \def\pagewidth{330pt}
     \def\pageheight{8.5truein}
     \hsize\pagewidth
     \vsize\pageheight
     \stylenumber=3}
\def\refsize{\magnification=\magstep1
     \overfullrule0pt
     \def\pagewidth{390pt}
     \def\pageheight{9truein}
     \hsize\pagewidth
     \vsize\pageheight
     \stylenumber=4}
\def\rrversion{\magnification=\magstep1
     \overfullrule0pt
     \def\pagewidth{330pt}
     \def\pageheight{9truein}
     \hsize\pagewidth
     \vsize\pageheight
     \stylenumber=5}
\def\bigsize{\magnification=\magstep2
     \overfullrule0pt
     \def\pagewidth{325pt}
     \def\pageheight{9truein}
     \hsize\pagewidth
     \vsize\pageheight
     \stylenumber=6}
\def\bbigsize{\magnification=\magstep3
     \overfullrule0pt
     \def\pagewidth{271pt}
     \def\pageheight{9truein}
     \hsize\pagewidth
     \vsize\pageheight
     \stylenumber=7}
\def\USsmallprint{\magnification=\magstep0
     \overfullrule0pt
     \def\pagewidth{482pt}
     \def\pageheight{8.7truein}
     \hsize\pagewidth
     \vsize\pageheight
     \stylenumber=8}
\def\halfmagsize{\magnification=\magstephalf
     \def\pagewidth{427pt}
     \def\pageheight{9truein}
     \hsize\pagewidth
     \vsize\pageheight
     \stylenumber=9}
           % register  stylenumber  is used to control the output format

%addresses, letterheads
\def\deptaddress{\hskip 3truein Department of Mathematical Sciences\par
   \noindent e-mail: {\tt fremdh}\hskip 2.9truein\today\par
   \medskip}
\long\def\univaddress#1{\hskip 3truein Department of Mathematical Sciences\par
   \hskip 3truein University of Essex\par
   \hskip 3truein Colchester CO4 3SQ\par
   \hskip 3truein England\par
   %\hskip 3truein van Vleck Hall\par
   %\hskip 3truein 480 Lincoln Drive\par
   %\hskip 3truein Madison\par
   %\hskip 3truein WI 53706-1388, U.S.A.\par
   \hskip 3.2truein#1\par
   \noindent e-mail: {\tt david\@fremlin.org}
       %{\tt fremlin\@math.wisc.edu}
   \bigskip}
\long\def\homeaddress#1{\hskip 3truein 25 Ireton Road\par
   \hskip 3truein Colchester CO3 3AT\par
   \hskip 3truein England\par
   \hskip 3.2truein#1\par
   \noindent e-mail: {\tt david\@fremlin.org}
   \bigskip}
\long\def\letterhead#1#2#3{\univaddress{#1}
   {\parindent=0pt
    #2
   }
   \medskip

   \noindent Dear #3,
     }
\def\shortaddress{\noindent Departmentof Mathematical Sciences, University of Essex,
   Colchester CO4 3SQ, England\par
   \noindent e-mail: david\@fremlin.org}
\def\valediction#1#2{\bigskip
   \hskip 3truein #1,
   \vskip 1truein
   \hskip 3.3truein #2
   \bigskip}

%picture importation
   %It is clear that there is no general standard.   The syntax I
   %use when calling for a  .ps  file is
   %    \def\Caption{<...>}
   %    \picture{<filename>}{<dim>}
   %which is supposed to import  <filename>.ps  at height  <dim> ,
   %centered, and adding a one-line centered caption  <...> .
   %The basic macro listed in  \def\picture  is that which
   %used to work at Essex.   I now generally use the macro in
   % \def\picturewithpsfig , which requires a previous input
   % psfig.tex  or  psfig.sty ;
   %if you interpolate  \atUEssex  early in your file it will
   %give you the latter.
   %   I find that  psfig  is erratic in its centering, especially
   %under magnification, so I have written a routine
   % \centerpicturewithpsfig  (which may be called from
   % \centerpicture ) with the syntax
   %    \centerpicture{<filename>}{<dim>}{<dim>}
   %the first dimension being width and the second height.

   % A second package is called for by  \usegraphicx .   This requires
   % miniltx.tex  and  graphicx.sty ;  miniltx.tex  is currently (April 2016)
   % available from
   % http://mirror.ox.ac.uk/sites/ctan.org/macros/plain/graphics/miniltx.tex
   % The advantage of this is that it seems to allow manipulation of the
   % dvi>pdf stage using  dvipdfmx .

\def\picture#1#2{\bigskip
  \bigskip
  \immediate\write0{importing diagram #1.ps at height #2
         with dvitops}
  \vbox{\vskip #2
    \special{dvitops: import #1.ps \pagewidth #2}
    \Centerline{\Caption}
    }
   \medskip
   \def\Caption{}}
%an alternative version, without caption
\def\centerpicture#1#2#3{\bigskip
  \bigskip
  \immediate\write0{importing diagram #1.ps at width #2
         and height #3 with dvitops}
  \vbox{\vskip #3
    \special{dvitops: import #1.ps \pagewidth #3}
  }}
%to put pictures on same line
\def\sideshiftedpicture#1#2#3#4{\bigskip
   \bigskip
   \immediate\write0{importing diagram #1.ps at height #4
        and width #3 with dvitops}
   \vbox{\special{dvitops:  import #1.ps #3 #4}}
   \hskip #2 plus 0pt minus 0pt
   }
\def\startsideshiftedpicture{}
\newcount\scaleup
\newcount\scaledown
\scaleup=1
\scaledown=1
\def\scalesideshiftedpicture#1#2#3#4{
  \dimen4=#4
  \dimen3=#3
  \multiply\dimen4 by\scaleup
  \divide\dimen4 by \scaledown
  \multiply\dimen3 by\scaleup
  \divide\dimen3 by \scaledown
  \sideshiftedpicture{#1}{#2}{\dimen3}{\dimen4}}

\def\picturewithpsfig#1#2{\bigskip
   \immediate\write0{importing diagram #1.ps at height #2 with psfig}
   \vbox{\hsize\pagewidth
   \centerline{\psfig{figure=#1.ps,height=#2}}
   \medskip
   \centerline{\Caption}}
   \medskip
   \def\Caption{}}
\def\centerpicturewithpsfig#1#2#3{
  \immediate\write0{importing diagram #1.ps at width #2
         and height #3 with psfig}
  \dimen 3 = #2
  \dimen 4 = #3
  \dimen 2 = \pagewidth
  \advance\dimen2 by -\dimen3
  \divide\dimen2 by 2
  \discrversionB{\shrinkonestep}{}
  \discrversionH{\shrinkonestep}{}
  \discrversionF{\enlargeonestep}{}
  \discrversionG{\enlargeonestep\enlargeonestep}{}
  \dimen 6 = \dimen4
  \divide\dimen6 by 3
  \vskip\dimen6
  \ifdim\dimen2>0\hskip\dimen2\fi
     \vbox{\psfig{file=#1.ps,width=\dimen3,height=\dimen4}}
  \discrversionG{\vskip-\dimen6}{}
  }
\def\enlargeonestep{\dimen5 = 5truein
   \dimen6 = 7truein
   \ifdim\dimen3<\dimen5
      \ifdim\dimen4<\dimen6
         \divide\dimen3 by 5
         \multiply\dimen3 by 6
         \divide\dimen4 by 5
         \multiply\dimen4 by 6
      \fi\fi}
\def\shrinkonestep{\divide\dimen3 by 6
         \multiply\dimen3 by 5
         \divide\dimen4 by 6
         \multiply\dimen4 by 5}
\def\sideshiftedpicturewithpsfig#1#2#3#4{
   \immediate\write0{importing diagram #1.ps at height #4
      and width #3 with psfig}
   \dimen 4 = #4
   \dimen 3 = #3
   \discrversionB{\shrinkonestep}{}
   \discrversionH{\shrinkonestep}{}
   \discrversionF{\enlargeonestep}{}
   \discrversionG{\enlargeonestep\enlargeonestep}{}
%   \vbox{\psfig{figure=#1.ps,height=\dimen4,width=\dimen3}}
   \psfig{figure=#1.ps,height=\dimen4,width=\dimen3}
   \hskip #2 plus 0pt minus 0pt}

\def\picturewithgraphicx#1#2{\bigskip
   \immediate\write0{importing diagram #1.ps at height #2 with graphicx}
   \vbox{\hsize\pagewidth
   \centerline{\includegraphics[height=#2]{#1.ps}}
   \medskip
   \centerline{\Caption}}
   \medskip
   \def\Caption{}}
\def\centerpicturewithgraphicx#1#2#3{
  \immediate\write0{importing diagram #1.ps at width #2
         and height #3 with graphicx}
  \dimen 3 = #2
  \dimen 4 = #3
  \dimen 2 = \pagewidth
  \advance\dimen2 by -\dimen3
  \divide\dimen2 by 2
  \discrversionB{\shrinkonestep}{}
  \discrversionH{\shrinkonestep}{}
  \discrversionF{\enlargeonestep}{}
  \discrversionG{\enlargeonestep\enlargeonestep}{}
  \dimen 6 = \dimen4
  \divide\dimen6 by 3
  \vskip\dimen6
  \ifdim\dimen2>0\hskip\dimen2\fi
     \vbox{\includegraphics[width=\dimen3,height=\dimen4]{#1.ps}}
  \discrversionG{\vskip-\dimen6}{}
  }
\def\enlargeonestep{\dimen5 = 5truein
   \dimen6 = 7truein
   \ifdim\dimen3<\dimen5
      \ifdim\dimen4<\dimen6
         \divide\dimen3 by 5
         \multiply\dimen3 by 6
         \divide\dimen4 by 5
         \multiply\dimen4 by 6
      \fi\fi}
\def\shrinkonestep{\divide\dimen3 by 6
         \multiply\dimen3 by 5
         \divide\dimen4 by 6
         \multiply\dimen4 by 5}
\def\sideshiftedpicturewithgraphicx#1#2#3#4{
   \immediate\write0{importing diagram #1.ps at height #4
      and width #3 with graphicx}
   \dimen 4 = #4
   \dimen 3 = #3
   \discrversionB{\shrinkonestep}{}
   \discrversionH{\shrinkonestep}{}
   \discrversionF{\enlargeonestep}{}
   \discrversionG{\enlargeonestep\enlargeonestep}{}
%   \vbox{\includegraphics[height=\dimen4,width=\dimen3]{#1.ps}}
   \includegraphics[height=\dimen4,width=\dimen3]{#1.ps}
   \hskip #2 plus 0pt minus 0pt}

%alternative picture importation code;  I've been told about this
  %but have not used it successfully myself
%\def\Caption{}
%\input epsf
%\epsfverbosetrue
%\def\picture#1#2{\bigskip
%    \bigskip
%    \immediate\write0{importing diagram #1.ps at height #2 with epsf}
%    \epsfysize=#2
%    \centerline{\epsffile{#1.ps}}
%    \medskip
%    \Centerline{\Caption}
%    \medskip
%    }

%in addition it may be necessary to amend the imported  .ps  file
%by replacing the   %%BoundingBox   header by  %%BoundingBox:

\def\atUEssex{
   \input psfig.sty
   \def\picture##1##2{\picturewithpsfig{##1}{##2}}
   \def\sideshiftedpicture##1##2##3##4{
     \sideshiftedpicturewithpsfig{##1}{##2}{##3}{##4}}
   \def\centerpicture##1##2##3{
       \centerpicturewithpsfig{##1}{##2}{##3}}
   \def\startsideshiftedpicture{
       \sideshiftedpicture{empty}{1pt}{1pt}{1pt}}
   }
\def\usegraphicx{
   \input miniltx
   \def\Gin@driver{dvips.def}
   \input graphicx.sty
   \def\picture##1##2{\picturewithgraphicx{##1}{##2}}
   \def\sideshiftedpicture##1##2##3##4{
     \sideshiftedpicturewithgraphicx{##1}{##2}{##3}{##4}}
   \def\centerpicture##1##2##3{
       \centerpicturewithgraphicx{##1}{##2}{##3}}
   \def\startsideshiftedpicture{
       \sideshiftedpicture{empty}{1pt}{1pt}{1pt}}
}


%special fonts
\def\biggerscriptfonts{\scriptfont0=\eightrm
     \scriptfont1=\eighti
     \scriptfont2=\eightsy
     \scriptfont3=\eightex
     \scriptfont\bffam=\eightbf
     \scriptfont\msbfam=\eightmsb
     \scriptfont\eufmfam=\eighteufm
     }
\def\Loadcyrillic{\ifamstexsupported
      \input cyracc.def
      \font\tencyr=wncyr10
      \def\cyr{\tencyr\cyracc} \def\itcyr{\tenitcyr\cyracc}
    \else\def\cyr{\smc}\def\cdprime{u\i}\def\Cdprime{UI}\fi}
%http://wkweb3.cableinet.co.uk/linux/LDP/HOWTO/Cyrillic-HOWTO-7.html
%cyrillic.tex

\def\Loadfourteens{\iffourteensavailable
       \font\fourteenrm=cmr14
       \font\fourteenbf=cmbx14
       \font\fourteenit=cmsl14
   \else\font\fourteenrm=cmr10 scaled \magstep2
      \font\fourteenbf=cmbx10 scaled \magstep2
      \font\fourteenit=cmsl10 scaled \magstep2
   \fi}
\def\smallerfonts{\eightpoint
   \textfont\msbfam=\eightmsb
   \textfont\eufmfam=\eighteufm
   \textfont\bffam=\eightbf
   \textfont\eurmfam=\eighteurm
   \textfont\eusmfam=\eighteusm
   }
\font\twelveex=cmex10 scaled \magstep1


%Copyright declarations
\newcount\oldcopyrightdate
\newcount\newcopyrightdate
\oldcopyrightdate=0
\def\copyrightdate#1{\newcopyrightdate=#1
       \ifnum\newcopyrightdate=\oldcopyrightdate\else
       \oldcopyrightdate=#1
       \oldfootnote{}{\eightrm \copyright\ #1 D.~H.~Fremlin}
       \immediate\write0{Copyright #1}
       \fi}

%hyphenation
\hyphenation{equi-ver-id-ical}

%miscellaneous definitions
\def\Aut{\mathop{\text{Aut}}}
\def\backstep#1{\hskip -0.#1em plus 0 pt minus 0 pt}
\def\Bang{\vthsp\hbox{\bf X\hskip0.07em\llap X\hskip0.07em\llap X\rm}}
\def\BanG{\Bang{\hskip 1.111pt plus 3.333pt minus 0pt}}
%\def\BbbN{\mathchoice{\hbox{$\Bbb N$\hskip0.02em}}
%  {\hbox{$\Bbb N$\hskip0.02em}}
%  {\hbox{$\scriptstyle\Bbb N$\hskip0.04em}}
%  {\hbox{$\scriptscriptstyle\Bbb N$\hskip0.04em}}}
%\def\BbbQ{\mathchoice{\hbox{$\Bbb Q$\hskip0.02em}}
%  {\hbox{$\Bbb Q$\hskip0.02em}}
%  {\hbox{$\scriptstyle\Bbb Q$\hskip0.04em}}
%  {\hbox{$\scriptscriptstyle\Bbb Q$\hskip0.04em}}}
%\def\BbbR{\mathchoice{\hbox{$\Bbb R$\hskip0.02em}}
%  {\hbox{$\Bbb R$\hskip0.02em}}
%  {\hbox{$\scriptstyle\Bbb R$\hskip0.04em}}
%  {\hbox{$\scriptscriptstyle\Bbb R$\hskip0.04em}}}
%  %do not use with $\Bbb R^X$
%\def\BbbZ{\mathchoice{\hbox{$\Bbb Z$\hskip0.02em}}
%  {\hbox{$\Bbb Z$\hskip0.02em}}
%  {\hbox{$\scriptstyle\Bbb Z$\hskip0.04em}}
%  {\hbox{$\scriptscriptstyle\Bbb Z$\hskip0.04em}}}
\def\BbbN{\openBbb{N}}
\def\BbbQ{\openBbb{Q}}
\def\BbbR{\openBbb{R}}
  %do not use with $\Bbb R^X$  do not use in $_{\Bbb R^r}$
\def\BbbZ{\openBbb{Z}}
\def\Bbbone{\hbox{1\hskip0.13em\llap{1}}}
\def\biggerint{\textfont3=\twelveex\intop\nolimits}
  % you can get this size into a displayed formula with
  %  $$\hbox{$\biggerint_{\Cal S}\pmb{u}\,d\pmb{v}$}$$
  % but it may then seem cramped
  % or try \hbox{$\biggerint_{\Cal S}\,$}
\def\biggeriint{\textfont3=\twelveex\iint}
  % remarkable fact:  this seems to infect all \int within a given
  % $..$ ; but  $\biggerint$$\int$ seems to leave the $\int$ normal
  % really  \textfont3=\twelveex  is more reliable, since  \biggerint
  % as written gets turned off rather easily (e.g. by  \comment,
  % or by an extra  {...})
\def\bover#1#2{{{#1}\over{#2}}}
\def\Bover#1#2{\backstep4\hbox{
     \biggerscriptfonts${\shortstrut#1}\over{\shortstrut#2}$}}
  % when there are superscripts in the denominator, as in \Bover1{2^m}
  % or  \Bover1{\sqrt{2\pi}^r} , these can sometimes come out with better
  % horizontal alignment if you use  \Bover1{{\sshortstrut 2}^m}  or
  % \Bover1{{\sqrt{2\pi}}^r}
\def\bsover#1#2{\textstyle\bover{#1}{#2}}
\def\bsp{/\penalty-100}
  % \penalty seems to behave oddly in a \tt string in \fullsize mode
\def\Caption{}
\def\Center#1{

   \Centerline{#1}

   \noindent}
\def\Centerline#1{\vskip 5pt
     {\def\int{\text{}$$\textfont3=\twelveex\intop\nolimits}
      \def\dashint{\text{}$$\textfont3=\twelveex
         \mskip2mu\vrule height2.65pt width4.5pt depth-2.35pt\mskip-11mu
	 \intop\nolimits\mskip-2mu}
      \def\idashint{\text{}$$\int\mskip-7mu\dashint}	 
      \def\dashiint{\text{}$$\dashint\mskip-6mu\int}	 
      \def\dashidashint{\text{}$$\dashint\mskip-6mu\dashint}	 
      \def\underlineint{
           \text{}$$\textfont3=\twelveex\underline{\intop\nolimits}}
      \def\overlineint{
           \text{}$$\textfont3=\twelveex\overline{\intop\nolimits}}
      \def\iint{\text{}$$\textfont3=\twelveex
           \intop\nolimits\hskip-0.5em\intop\nolimits}
      \def\sum{\text{}$$\sumop}
      \def\prod{\text{}$$\prodop}
      \def\bigcap{\text{}$$\bigcapop}
      \def\bigcup{\text{}$$\bigcupop}
   \centerline{#1}}\vskip 5pt}
   \mathchardef\bigcapop="1354
   \mathchardef\bigcupop="1353
   \mathchardef\prodop="1351
   \mathchardef\sumop="1350
   %in  \Centerline{$\overline{}$, $\cmmnt{}$} must use \sumop etc
\def\cf{\mathop{\text{cf}}}
\def\circumflex#1{{\accent"5E #1}}
\def\closeplus{\hbox{+}}
\long\def\cmmnt#1{\ifresultsonly\else{#1}\fi}
\def\coint#1{\left[#1\right[}
\def\curl{\mathop{\text{curl}}}
\def\dashint{\mathchoice{-\mskip-20mu\intop\nolimits}
   {{\vrule height2.75pt width4.5pt depth-2.45pt\mskip-10mu\intop\nolimits}}
      {-\mskip-16.5mu\intop\nolimits} %not yet used
      {-\mskip-16.5mu\intop\nolimits}} %not yet used
\def\idashint{\mathchoice{\int\mskip-6mu\dashint}
      {\int\mskip-5mu\dashint}
      {\int\mskip-5mu\dashint} %not yet used
      {\int\mskip-5mu\dashint}} %not yet used
\def\dashiint{\mathchoice{\dashint\mskip-10mu\int}
      {\dashint\mskip-6mu\int}
      {\dashint\mskip-6mu\int} %not yet used
      {\dashint\mskip-6mu\int}} %not yet used
\def\dashidashint{\mathchoice{\dashint\mskip-10mu\dashint}
      {\dashint\mskip-6mu\dashint}
      {\dashint\mskip-6mu\dashint} %not yet used
      {\dashint\mskip-6mu\dashint}} %not yet used
\def\dashidashiint{\mathchoice{\dashint\mskip-8mu\dashint\mskip-10mu\int}
      {\dashint\mskip-4mu\dashint\mskip-8mu\int} 
      {\dashint\mskip-6mu\int} %not yet used
      {\dashint\mskip-6mu\int}} %not yet used
\def\dashiidashint{\mathchoice{\dashint\mskip-10mu\int\mskip-8mu\dashint}
      {\dashint\mskip-4mu\dashint\mskip-8mu\int} 
      {\dashint\mskip-6mu\int} %not yet used
      {\dashint\mskip-6mu\int}} %not yet used
\def\idashidashint{\mathchoice{\int\mskip-8mu\dashint\mskip-8mu\dashint}
      {\dashint\mskip-6mu\int} %not yet used
      {\dashint\mskip-6mu\int} %not yet used
      {\dashint\mskip-6mu\int}} %not yet used
\def\diam{\mathop{\text{diam}}}
\def\discrpage{\ifnum\stylenumber=1{\frnewpage}  %redefined in mt.tex
           \else\ifnum\stylenumber=3{\frnewpage}
           \else\ifnum\stylenumber=5{\frnewpage}
           \else\bigskip\fi\fi\fi}
\long\def\discrversionA#1#2{\ifnum\stylenumber=1#1\else#2\fi}
\long\def\discrversionB#1#2{\ifnum\stylenumber=2#1\else#2\fi}
\long\def\discrversionC#1#2{\ifnum\stylenumber=3#1\else#2\fi}
\long\def\discrversionD#1#2{\ifnum\stylenumber=4#1\else#2\fi}
\long\def\discrversionE#1#2{\ifnum\stylenumber=5#1\else#2\fi}
\long\def\discrversionF#1#2{\ifnum\stylenumber=6#1\else#2\fi}
\long\def\discrversionG#1#2{\ifnum\stylenumber=7#1\else#2\fi}
\long\def\discrversionH#1#2{\ifnum\stylenumber=8#1\else#2\fi}
           % facility for special interpolations in particular output
           % versions
\def\displaycause#1{\noalign{\noindent (#1)}}
\def\diverg{\mathop{\text{div}}}
\def\dom{\mathop{\text{dom}}}
\def\eae{=_{\text{a.e.}}}

\def\esssup{\mathop{\text{ess sup}}}
\mathchardef\Exists="3239

\def\family#1#2#3{\langle#3\rangle_{#1\in#2}}
\def\familyi#1#2{\family{i}{#1}{#2}}
\def\familyiI#1{\familyi{I}{#1}}

\def\frfilename#1{\ifnum\stylenumber=1{\noindent filename #1}
               \else\ifnum\stylenumber=6{\noindent{\fiverm filename #1}}
               \else\ifnum\stylenumber=7{\noindent{\fiverm filename #1}}
               \else\fi\fi\fi}
\def\Footnote#1#2{\footnote"#1"{#2}}
\mathchardef\Forall="3238
\def\frakc{\frak c\phantom{.}}
\def\frnewpage{\vskip \pageheight plus 0pt minus\pageheight
           \eject
           \gdef\topparagraph{}
           \gdef\bottomparagraph{}
           }
\def\frsmallcirc{{\hskip 1pt plus 0pt minus 0pt}
     {\raise 1pt\hbox{$\smallcirc$}}{\hskip 1pt plus 0pt minus 0pt}}

\def\geae{\ge_{\text{a.e.}}}
\def\grad{\mathop{\text{grad}}}
\def\grheada{{\bf ($\pmb{\alpha}$)}}
\def\grheadb{{\bf ($\pmb{\beta}$)}}
\def\grheadc{{\bf ($\pmb{\gamma}$)}}
\def\grheadd{{\bf ($\pmb{\delta}$)}}
\def\grheade{{\bf ($\pmb{\epsilon}$)}}
\def\grheadz{{\bf ($\pmb{\zeta}$)}}
\def\grv#1{{\accent"12 #1}}

\def\hfillall{\hskip\pagewidth plus0pt minus\pagewidth}

\def\Imag{\mathop{\Cal I\text{m}}}
\long\def\inset#1{{\narrower\narrower{\vskip 1pt plus 1pt minus 0pt}
     #1
     {\vskip 1pt plus 1pt minus 0pt}}}
\def\Int#1#2{\displaystyle{\int_{#1}^{#2}}}
\def\interior{\mathop{\text{int}}}

\def\leae{\le_{\text{a.e.}}}
\long\def\leaveitout#1{}
\def\LLcorner{\mathchoice{\,\hbox{\vrule height7pt width0.5pt depth0pt
       \vrule height0.5pt width5pt depth0pt}\,}
   {\,\hbox{\vrule height7pt width0.5pt depth0pt
       \vrule height0.5pt width5pt depth0pt}\,}
   {\,\hbox{\vrule height5pt width0.3pt depth0pt
       \vrule height0.3pt width4pt depth0pt}\,}
   {\,\hbox{\vrule height4pt width0.25pt depth0pt
       \vrule height0.25pt width3pt depth0pt}\,}
  }
\def\Ln{\mathop{\text{Ln}}}
\def\Loadfont#1{\font#1}
\def\loibr{\hbox{\hskip0.1em]\hskip0.05em}}
\def\Matrix#1{\pmatrix#1\endpmatrix}
\def\Mu{\text{M}}
\def\noamslogo{\def\amslogo{}}
\def\normalsubgroup{\vartriangleleft}
\def\Nu{\text{N}}
\def\ocint#1{\left]#1\right]}
\def\ofamily#1#2#3{\langle#3\rangle_{#1<#2}}
\let\oldcases=\cases
\let\oldfootnote=\footnote
\def\On{\mathop{\text{On}}}
\def\ooint#1{\left]#1\right[}
\def\openBbb#1{\Bbb#1\mskip1mu}
\def\pd#1#2{\bover{\partial#1}{\partial#2}}
\def\Pd#1#2{\Bover{\partial#1}{\partial#2}}
\def\pdd#1#2{{{\partial^2#1}\over{\partial#2^2}}}
\def\Pdd#1#2{\Bover{\partial^2#1}{\partial#2^2}}
\def\pde#1#2#3{{{\partial^2#1}\over{\partial#2\partial#3}}}
\def\Pde#1#2#3{\Bover{\partial^2#1}{\partial#2\partial#3}}
\def\Prf{\vthsp\hbox{\bf P\hskip0.07em\llap P\hskip0.07em\llap P\rm}}
\def\pushbottom#1{\hbox{\vrule height0pt depth#1 width0pt}}
\def\Qed{\vthsp\hbox{\bf Q\hskip0.07em\llap Q\hskip0.07em\llap Q\rm}}
\def\QeD{\Qed{\hskip 1.111pt plus 3.333pt minus 0pt}}
\def\Quer{\vthsp\hbox{\bf ?\hskip0.05em\llap ?\hskip0.05em\llap
   ?\thinspace\rm}}
\def\query{\discrversionA{\immediate\write0{query}
    \global\advance\footnotenumber by 1
    \oldfootnote{$^{\the\footnotenumber}$}{query}}{}}
\def\Real{\mathop{\Cal R\text{e}}}
\def\restr{\mathchoice
  {\hbox{$\restriction$}\mskip 1mu}
  {\hbox{$\restriction$}\mskip 1mu}
  {\hbox{$\scriptstyle\restriction$}\mskip 1mu}
  {\hbox{$\scriptscriptstyle\restriction$}\mskip 1mu}}
\def\restrp{\mathchoice
  {\hbox{$\restriction$}\mskip 2mu}
  {\hbox{$\restriction$}\mskip 2mu}
  {\mskip-1mu\hbox{$\scriptstyle\restriction$}\mskip 2mu}
  {\mskip-1mu\hbox{$\scriptscriptstyle\restriction$}\mskip 1mu}}
    %use in \restrp V \restrp W \restrp Y
    % \restrp\Cal P \restrp\Bbb R \restrp\frak
    % \restrp\Omega \restrp\Tau \restrp\Upsilon
    % \restrp[ \restrp\widehat{}
\def\Rho{\text{P}}
\def\roibr{\hbox{\hskip0.05em[\hskip0.1em}}
\def\RoverC{\hbox{\biggerscriptfonts${{\Bbb R}\atop{\Bbb C}}$}}
\def\sequence#1#2{\family{#1}{\Bbb N}{#2}}
\def\sequencen#1{\sequence{n}{#1}}
\def\setover#1#2{\overset{#1}\to{#2}}
\def\shortstrut{\hbox{\vrule height7.5pt depth2.5pt width0pt}}
\def\sshortstrut{\hbox{\vrule height5.8pt depth2.5pt width0pt}}
\def\smallcirc{{\scriptstyle\circ}}
\def\ssbullet{{\scriptscriptstyle\bullet}}
\def\symmdiff{\triangle}
\def\thicken#1{#1\hskip0.03em\llap #1\hskip0.03em\llap
   #1\hskip0.03em\llap #1}
\def\Tau{{\text{T}}}
\def\tbf#1{\text{\bf #1}}
\def\terror#1#2#3{#1:  `#2' for `#3'}
\def\today{\number\day.\number\month.\number\year}
\def\UEMDRR{University of Essex Mathematics Department Research Report}
\def\UEMDWP{University of Essex Mathematics Department Working Paper}
\def\versiondate#1{
    \immediate\write0{Version of #1}
    \def\vsdate{#1}
               \ifnum\stylenumber=1\hfill Version of #1
                              \medskip
               \else\ifnum\stylenumber=2\hfill Version of #1
                              \medskip
               \else\ifnum\stylenumber=6\hfill{\fiverm Version of #1}
                              \medskip
               \else\ifnum\stylenumber=7\hfill{\fiverm Version of #1}
                              \medskip
               \else\ifnum\stylenumber=8\hfill Version of #1
                              \medskip
               \else\fi\fi\fi\fi\fi}
\def\versionhead{\ifnum\stylenumber=3 \hfill PRINTER'S VERSION

                        \bigskip
                   \else\ifnum\stylenumber=4 \hfill REFEREE'S VERSION

                        \bigskip
                   \else \fi\fi}
\def\versionref{\ifnum\stylenumber=1\workingref
           \else\ifnum\stylenumber=2\workingref
           \else\ifnum\stylenumber=8\workingref
           \else\printerref\fi\fi\fi}
\def\vthsp{\hskip0.001em}

%Missing AmSTeX can be coped with by using the following lines:

\ifamstexsupported\else
   \immediate\write0{AmSTeX-unsupported version of  fremtex.tex }
   \fourteensavailablefalse
   \def\@{@}
   \def\\{\cr}
   \def\AmSTeX{AmS\TeX}
   \def\Bbb#1{\underline{\hbox{\bf #1}}}
   \def\biggerscriptfonts{}
   \def\bumpeq{=_{\text{approx}}}
   \def\Cal#1{{\cal#1}}
   \def\curlyeqprec{\preceq}
   \def\eightit{\it}
   \def\eightrm{\rm}
   \def\eightsmc{\smc}
   \def\eurm#1{\underline{\cal#1}}
   \def\Footnote#1#2{\footnote#1#2}
   \def\frac#1#2{{#1}\over{#2}}
   \def\frak#1{{\hbox{\bf #1\hskip0.1em\llap #1}}}
   \def\frnewpage{\vfill\eject}
   \def\iint{\int\!\!\!\int}
   \def\iiint{\int\!\!\!\int\!\!\!\int}
   \def\iiiint{\int\!\!\!\int\!\!\!\int\!\!\!\int}
   \def\llcorner{\lfloor}
   \def\loadeurm{}
   \def\loadeusm{}
%   \def\loadeusmx#1{\def\eusm#1{\underline{\cal#1}}}
   \def\ltimes{|\hskip-0.5em\times}
   \def\eusm#1{\underline{\cal#1}}
   \def\ltimes{|\times}
   \def\Loadfont#1{}
   \def\Matrix#1{\pmatrix{#1}}
   \def\noamslogo{}
   \def\nrightarrow{\not\rightarrow}
   \def\preccurlyeq{\preceq}
   \def\query{\discrversionA{\immediate\write0{query}
                             \footnote{$^*$}{query}}{}}
   \def\restriction{|}
   \def\rtimes{\times|}
   \def\setover#1#2{#2^{#1}}
   \def\shortstrut{\strut}
   \def\smallerfonts{\eightrm}
   \def\smallfrown{\wedge}
   \def\smc{}
   \def\spcheck{\check{\phantom{x}}}
   \def\sphat{\hat{\phantom{x}}}
   \def\square{\mathchoice\sqr64\sqr64\sqr43\sqr33}
   \def\sqr#1#2{{\vcenter{\vbox{\hrule height.#2pt
          \hbox{\vrule width.#2pt height#1pt \kern#1pt
                  \vrule width.#2pt}
          \hrule height.#2pt}}}}
   \def\pmb#1{\underline{#1}}
   \def\text#1{\hbox{#1}}
   \def\therefore{\ssbullet\raise.9ex\hbox{$\ssbullet$}\ssbullet}
   \def\twelvebf{\bf}
   \def\vartriangleleft{\triangleleft}
   \def\Vdash{|\vdash}
   \def\vDash{|=}
   \fi

%End of lines for missing AmSTeX

\ifamstexsupported\input amstex
%              \documentstyle{amsppt}
               \input amsppt.sty
               \PSAMSFonts
	       \tenpoint
               %\font\eightmsb=msym8
               \font\eightmsb=msbm10
               \font\eighteufm=eufm10 at 8 pt
               \font\eighteurm=eurm8
               \font\eighteusm=eusm8
               \font\varseveneufm=eufm10 at 7 pt
               \scriptfont\eufmfam=\varseveneufm
   \def\.#1{{\accent"5F #1}}
   \def\=#1{{\accent"16 #1}}
   \gdef\nopagenumbers{\NoPageNumbers}
   \else\fi

\newcount\footmarkcount
\ifDHFamstexloaded
    \loadmsBm
    \def\eightBbb#1{{\fam\msBfam#1}}
    \loadmsFm
    \def\eightfrak#1{{\fam\msFfam#1}}
  \else
    \def\eightBbb#1{{\scriptstyle\Bbb#1}}
    \def\eightfrak#1{{\scriptstyle\frak#1}}
  \fi
\ifathome\athome\fi

%interword glue (\tenrm standard):
%  (space) 3.333 + 1.667 - 1.111
%  (,)     3.333 + 2.083 - 0.889
%  (.)     4.444 + 5.    - 0.370

%very odd fact:  \discrversionA{2.}{}  doesn't work,
%   \discrversionA{ 2.}{} does

%The false line feed introduced by my Fortran programs, leading to
%a surplus  ae  at the top of the next page, seems to be associated
%with  Fk(\032)3081  or something similar in the PostScript file.


\immediate\write0{Extract from MEASURE THEORY,
   by D.H.Fremlin, University of Essex, Colchester.
   This material is copyright.   It is issued under the terms of the
   Design Science License as published in
   http://dsl.org/copyleft/dsl.txt.   This is a development version and the
   source files are not permanently archived.
   For further information contact david\@fremlin.org.}

%hyphenations
\hyphenation{Archi-me-dean}

%new style
\def\smallprintersize{\magnification=\magstep0
     \overfullrule0pt
     \def\pagewidth{468pt}
     \def\pageheight{9truein}
     \hsize\pagewidth
     \vsize\pageheight
     \nologo
     \def\copyrightdate##1{}
     \def\folio{\ifnum\pageno<0{}\else\number\pageno\fi}
     \def\mtfootline{}
     \stylenumber=10
     \pageno=-1000}

%With-proofs, results-only versions
\newif\ifwithproofs
\withproofstrue
\newif\ifresultsonly
\resultsonlyfalse
\def\noproofs{\withproofsfalse}
\def\resultsonly{\resultsonlytrue\withproofsfalse}
\long\def\proof#1{\ifwithproofs{\medskip

     \noindent{\bf proof} #1}\else\fi}
\long\def\prooflet#1{\ifwithproofs{#1}\else\fi}
%\long\def\Prooflet#1{\prooflet{\Prf\ #1 \Qed\ }}
\long\def\endnotes#1{\ifresultsonly\else{#1}\fi}
\long\def\exercises#1{\ifresultsonly\else{#1}\fi}
\long\def\references#1{\ifresultsonly\else{#1}\fi}
\long\def\dvro#1#2{\ifresultsonly{#1}\else{#2}\fi}

%Headline assignments

\def\topparagraph{}
\def\bottomparagraph{}
\def\newparagraph{}
\newif\ifpageturned

\newdimen\rhoffset
\newdimen\lhoffset
\lhoffset=0cm
\rhoffset=0cm

\newcount\volumeno
\volumeno=5
\newcount\luluvolumeno
\luluvolumeno=0

\let\plainpagebody=\pagebody
\def\pagebody{\ifodd\pageno\global\hoffset=\rhoffset
         \else\global\hoffset=\lhoffset\fi
     \plainpagebody
     \turnpage}
\def\turnpage{
     \iffirstpage\else{\baselineskip=24pt\line{\mtfootline}}\fi
%     \immediate\write0{turning page \folio,
%           top paragraph \topparagraph,   bottom \bottomparagraph,
%           new \newparagraph, pagetotal \the\pagetotal}
     \global\pageturnedtrue
     \global\firstpagefalse
     \global\let\topparagraph=\bottomparagraph
     \global\let\bottomparagraph=\newparagraph
     }

\newdimen\pagetotalplus
\newdimen\pagetotalminus

%wheader parameters:  #1 new paragraph name
 % #2#3#4  \vskip parameters if page break not forced (no. of points)
 % #5  spare height demanded if page break not to be forced
\long\def\wheader#1#2#3#4#5{\gdef\newparagraph{#1}
  \pagetotalplus=\the\pagetotal
  \advance\pagetotalplus by #5
  \pagetotalminus=\the\pagetotal
  \advance\pagetotalminus by -\pageshrink
  %\showthe\pagetotal \showthe\pageshrink \showthe\pagegoal
  \global\pageturnedfalse
  \ifdim\pagetotalminus<\pagegoal\ifdim\pagetotalplus>\pagegoal
%     \immediate\write0{forcing page break}
     \eject
     \else\vskip #2pt plus #3pt minus #4pt\fi
     \else\vskip #2pt plus #3pt minus #4pt\fi
  \ifpageturned\global\let\topparagraph=\newparagraph
%     \immediate\write0{paragraph change on turning page}
     \fi
%  \immediate\write0{top paragraph \topparagraph, bottom
%       \bottomparagraph, pagetotal \the\pagetotal}
  \ifdim\pagetotal>\pagegoal\else\gdef\bottomparagraph{#1}\fi
%   \immediate\write0{top paragraph \topparagraph, bottom
%       \bottomparagraph, pagetotal \the\pagetotal}
  }
\def\header#1{
     \wheader{#1}{10}{4}{4}{24pt}}
\def\leader#1#2{\immediate\write0{#1 #2}
     \wheader{#1}{14}{7}{5}{24pt}
     {\bf #1 #2}}
\def\vleader#1#2#3{\immediate\write0{#2 #3}
     \wheader{#2}{14}{7}{5}{#1}
     {\bf #2 #3}}
\def\Notesheader#1{\immediate\write0{#1 Notes and comments}
     \wheader{#1 Notes}{16}{8}{5}{48pt}
     \noindent{\bf #1 Notes and comments}}
\def\vNotesheader#1#2{\immediate\write0{#2 Notes and comments}
     \wheader{#2 Notes}{16}{8}{5}{#1}
     \noindent{\bf #2 Notes and comments}}
\def\spheader#1#2#3#4#5{\header{#1#2#3#4#5}{\bf (#5)}}
\def\spstheader#1#2#3#4#5{\header{*#1#2#3#4#5}{\bf (#5)}}
\def\vspheader#1#2#3#4#5#6{
     \wheader{#2#3#4#5#6}{10}{4}{4}{#1}{\bf (#6)}}
\def\sqheader#1#2#3#4#5{\header{#1#2#3#4#5}{$\pmb{>}${\bf (#5)}}}
\def\errheader#1#2#3{
     \wheader{#3}{6}{4}{2}{24pt}
     {\bf p #1 l #2}}
\long\def\woddheader#1#2#3#4#5{
   \ifodd\pageno\wheader{#1}{#2}{#3}{#4}{#5}
   \else\vskip #2pt plus #3pt minus #4pt\fi}

\def\indexmedskip{
  \pagetotalplus=\the\pagetotal
  \advance\pagetotalplus by 18 pt
  \pagetotalminus=\the\pagetotal
  \advance\pagetotalminus by -\pageshrink
  \ifdim\pagetotalminus<\pagegoal\ifdim\pagetotalplus>\pagegoal
      \else\medskip\fi\else\medskip\fi}

\def\indexheader#1{\immediate\write0{#1}\vindexheader{#1}{24}}
\def\vindexheader#1#2{
  \gdef\newparagraph{#1}
  \pagetotalplus=\the\pagetotal
  \advance\pagetotalplus by #2 pt
  \pagetotalminus=\the\pagetotal
  \advance\pagetotalminus by -\pageshrink
  %\showthe\pagetotal \showthe\pageshrink \showthe\pagegoal
  \global\pageturnedfalse
%  \ifdim\pagetotal>\pagegoal
%      \immediate\write0{warning: pagetotal greater than pagegoal}
%      \immediate\write0{...   last paragraph reached is #1}
%      \immediate\write0{...   check for underfull vbox here}
%      \else\fi
  \ifdim\pagetotalminus<\pagegoal\ifdim\pagetotalplus>\pagegoal{\eject}
      \fi\fi
  \ifdim\pagetotal>\pagegoal\else\gdef\bottomparagraph{#1}\fi
  \ifpageturned{\global\let\topparagraph=\newparagraph
%     \immediate\write0{paragraph change on turning page}
      }\else\fi
%  \immediate\write0{top paragraph \topparagraph, bottom
%    \bottomparagraph}
  }

\def\leftheadline{\rlap{\folio}\hfill{\eightit\chaptername}\hfill
       \llap{\eightbf\topparagraph}}
\def\rightheadline{\rlap{\eightbf\bottomparagraph}\hfill
       {\eightit\headlinesectionname}\hfill\llap{\folio}}

%Footline assignments
\def\mtfootline{\ifodd\pageno\rightfootline\else\leftfootline\fi}
\def\rightfootline{\hfill\eightsmc D.H.Fremlin}
\def\leftfootline{\ifwithproofs{\eightsmc Measure Theory\hfill}
    \else{\eightsmc Measure Theory \eightit (abridged version)\hfill}
    \fi}

%Style assignments
\def\newchapter#1{\gdef\topparagraph{}
   \gdef\bottomparagraph{Chap.\ #1 {\it intro.}}
   \gdef\newparagraph{Chap.\ #1 {\it intro.}}
   \footmarkcount=0
   \gdef\sectionname{Introduction}
   \gdef\headlinesectionname{Introduction}
   \centerline{\bf Chapter #1}
   \medskip
   \centerline{\bf \chaptername}
   \medskip
   }
\def\newsection#1{\gdef\bottomparagraph{\S#1 {\it intro.}}
   \noindent{\bf #1 \sectionname}
   \smallskip
   \let\headlinesectionname=\sectionname}
\def\newvolume#1{\gdef\topparagraph{}
   \gdef\bottomparagraph{Vol.\ #1 {\it intro.}}
   \Loadfont{\twelvebf=cmbx12}
   \def\chaptername{Volume #1}
   \def\sectionname{\volumename}
   \def\headlinesectionname{\volumename}
   \centerline{\twelvebf Volume #1:  \volumename}
   \bigskip
   }

%font variations
\newif\ifvolwp
\volwpfalse
\def\tenrmstretch#1{\fontdimen3\tenrm=#1}
\font\elevenex=cmex10 at 11pt

%contents and introductions
\newif\iflargelogo
\def\vtmpb#1{}
\def\newvolume#1{}
\def\pagereference#1#2{\hfill\hbox{\ifresultsonly#2\qquad \hskip 1truein
                                                 \else#1\qquad \fi}}
\long\def\section#1#2#3#4#5#6{\ifresultsonly
    \qquad #1 #2\pagereference{#4}{#5}
  \else\vskip 2pt plus 1pt minus 0pt
    \vbox{\qquad #1 #2\pagereference{#4}{#5}\par
    \abstr{#6}
    }\vskip -1pt plus 1pt minus 0pt
  \fi}
\long\def\chapintrosection#1#2#3{\ifresultsonly
    \qquad Introduction\pagereference{#2}{#3}
  \else\vskip 1pt plus 1pt minus 0pt
    \qquad Introduction\pagereference{#2}{#3}
    \vskip 1pt plus 1pt minus 0pt
  \fi}
\def\abstr#1{\ifresultsonly{}
   \else{\advance\leftskip by\parindent
     \advance\rightskip by\parindent
     \advance\leftskip by\parindent
     \advance\rightskip by\parindent
     \advance\leftskip by\parindent
     \advance\rightskip by\parindent
     \advance\leftskip by\parindent
     \vskip1pt plus 0pt minus0pt
     \parindent=0pt
     \smallerfonts #1%
     \hbox{\vrule height0pt depth2pt width0pt}
     \vskip1pt plus 0pt minus0pt}\fi}
     
%language variations
\newif\ifUSEnglish
\USEnglishfalse
%\USEnglishtrue

%"Later editions only"
\newcount\latereditionpage
\newcount\latereditionmark
\latereditionpage=-1
\latereditionmark=0
\newcount\newpageno

%removing the \write0 instruction altogether disorganizes it!
\def\Latereditions{\newpageno=\pageno%
  \ifdim\pagetotal>\pagegoal\advance\newpageno by 1\fi%
     \divide\newpageno by 2%
  \immediate\write0{%
%     "Later editions only";
%      pagetotal \the\pagetotal, pagegoal \the\pagegoal,
%      pageno \number\pageno,\par newpageno \number\newpageno,
%      latereditionpage \number\latereditionpage%
  }\ifnum\latereditionpage=\newpageno%
     \ifDHFamspptloaded$^{\number\latereditionmark}$%
     \else\footnote{Later editions only.}\fi%
  \else\footnote{Later editions only.}%
  \global\latereditionpage=\newpageno%
  \global\latereditionmark=\footmarkcount\fi}%
%  \immediate\write0{newpageno \number\newpageno,
%      latereditionpage \number\latereditionpage}

   
%general definitions
\def\action{\mathchoice{\lower-.3ex\hbox{$\ssbullet$}}
                       {\lower-.3ex\hbox{$\ssbullet$}}
                       {\ssbullet}
                       {\ssbullet}}  %see also \varaction
\def\add{\mathop{\text{add}}\nolimits}
\long\def\allowmorestretch#1#2{
  \ifdim\pagewidth<#1pt\else
    \fontdimen3\tenrm=2pt\fontdimen4\tenrm=1.67pt\fi
  #2
    \fontdimen3\tenrm=1.67pt\fontdimen4\tenrm=1.11pt}
\def\am{atom\discretionary{-}{}{}lessly-measurable}
\def\AmuA{\Aut_{\bar\mu}\mskip-2mu\frak A}
\def\ASint{\mathchoice{\Bbb A\mskip-5mu}
                  {\Bbb A\mskip-3mu}
                  {\Bbb A\mskip-2mu}
                  {\Bbb A\mskip-2mu}\Sint}
\def\Asisfour{$(\frak A,\bar\mu,T,\family{t}{T}{\frak A_t})$}
\def\Asissix{$(\frak A,\bar\mu,T,
   \penalty-100\family{t}{T}{\frak A_t},
   \penalty-100\Cal T,\family{\tau}{\Cal T}{\frak A_{\tau}})$}
\def\Asisfourt{$(\frak A,\bar\mu,\coint{0,\infty},
   \langle\frak A_t\rangle_{t\ge 0})$}
\def\Asissixt{$(\frak A,\bar\mu,\coint{0,\infty},
   \penalty-100\langle\frak A_t\rangle_{t\ge 0},
   \penalty-100\Cal T,\family{\tau}{\Cal T}{\frak A_{\tau}})$}
\def\Atop#1#2{{\scriptstyle #1}\atop{\scriptstyle #2}}
\def\Aut{\mathop{\text{Aut}}}

\def\Bcap{\Booleanop{\cap}}
\def\Bcup{\Booleanop{\cup}}
\def\Booleanop#1{
   \ifamstexsupported{\mathchoice
    {\hbox{\hskip.2em\scriptfont2=\eightsy$\scriptstyle#1$\hskip.2em}}
    {\hbox{\hskip.2em\scriptfont2=\eightsy$\scriptstyle#1$\hskip.2em}}
 %   {{\mskip2mu\scriptstyle#1\mskip2mu}}  %until 15.10.17
    {{\scriptscriptstyle#1}}  %from 15.10.17
    {{\scriptscriptstyle#1}}}
   \else{\mathchoice{{\scriptstyle#1}}{{\scriptstyle#1}}
    {{\scriptstyle#1}}{{\scriptscriptstyle#1}}}
    \fi}
\def\Bsetminus{\Booleanop{\setminus}}
\def\Bsubseteq{\hskip.1em\Booleanop{\subseteq}\hskip.1em}
\def\Bsubset{\Booleanop{\subset}}
\def\Bsupseteq{\Booleanop{\supseteq}}
\def\Bsupset{\Booleanop{\supset}}
\def\Bsymmdiff{\Booleanop{\symmdiff}}
\def\Bcapshort{\Booleanopshort{\cap}}
\def\Bcupshort{\Booleanopshort{\cup}}
\def\Booleanopshort#1{
   \ifamstexsupported{\mathchoice
    {\hbox{\scriptfont2=\eightsy$\scriptstyle#1$}}
    {\hskip-1pt\hbox{\scriptfont2=\eightsy$\scriptstyle#1$}}
    {{\scriptstyle#1}}
    {{\scriptscriptstyle#1}}}
   \else{\mathchoice{{\scriptstyle#1}}{{\scriptstyle#1}}
    {{\scriptstyle#1}}{{\scriptscriptstyle#1}}}
    \fi}
\def\Bsetminusshort{\Booleanopshort{\setminus}}
\def\BSint{\mathchoice{\Bbb B\mskip-5mu}
                  {\Bbb B\mskip-3mu}
                  {\Bbb B\mskip-2mu}
                  {\Bbb B\mskip-2mu}\Sint}
%\def\Bsubseteqshort{\hskip.1em\Booleanopshort{\subseteq}\hskip.1em}
\def\Bsubseteqshort{\hskip.1em\Booleanopshort{\subseteq}}
\def\Bsubsetshort{\Booleanopshort{\subset}}
\def\Bsupseteqshort{\Booleanopshort{\supseteq}}
\def\Bsymmdiffshort{\Booleanopshort{\symmdiff}}

\def\closeBooleanop#1{
   \ifamstexsupported{\mathchoice
    {\hbox{\scriptfont2=\eightsy$\scriptstyle#1$}}
    {\hbox{\scriptfont2=\eightsy$\scriptstyle#1$}}
    {{\scriptstyle#1}}
    {{\scriptscriptstyle#1}}}
   \else{\mathchoice{{\scriptstyle#1}}{{\scriptstyle#1}}
    {{\scriptstyle#1}}{{\scriptscriptstyle#1}}}
    \fi}
\def\closeBcup{\closeBooleanop{\cup}}
\def\closeBcap{\closeBooleanop{\cap}}
\def\closeBsetminus{\closeBooleanop{\setminus}}
\def\closeBsymmdiff{\closeBooleanop{\symmdiff}}

\def\biggerHeint{\textfont3=\twelveex
   \raise 0.6pt\hbox to 0pt{\hskip 2.6pt{\sevenrm H}\hss}
   \intop\nolimits}
\def\biggerPfint{\textfont3=\twelveex
   \raise 0.6pt\hbox to 0pt{\hskip 1.3pt{\sevenrm Pf}\hss}
   \intop\nolimits\hskip 1.2pt}
\def\biggerSint{\textfont3=\twelveex
   \raise 0.6pt\hbox to 0pt{\hskip 3.6pt{\sevenrm S}\hss}
   \intop\nolimits}
\def\bu{\mathop{\text{bu}}}
\def\Bvalue#1{\mathchoice
   {\hbox{$[\![#1]\!]$}}
   {\hbox{$[\![#1]\!]$}}
   {\hbox{$\scriptstyle[\![#1]\!]$}}
   {\hbox{$\scriptscriptstyle[\![#1]\!]$}}}
\def\cadlag{c\`adl\`ag}
\def\CalBa{\Cal{B}\eurm{a}}
   %note: neither  \cal  nor  \eusm  has lower case
\def\CalFr{\Cal F_{\text{Fr}}}
\def\callal{c\`all\`al}
\def\CalNwd{\Cal{N}\hskip-1.2pt\eurm{w}\hskip-.8pt\eurm{d}}

\def\CalRbg{\Cal{R}\mskip-1mu\eurm{b}\mskip-1mu\eurm{g}}
\def\CalSmz{\Cal{S}\mskip-2mu\eurm{mz}}
\def\CalUn{\Cal{U}\eurm{n}}
\def\capacity{\mathop{\text{cap}}\nolimits}
\def\Caratheodory{Cara\-th\'eo\-dory}
\def\Cdlg{C_{\text{dlg}}}
\def\cff{\cf\mskip1mu}
\def\checkpmb#1{\check{\pmb #1}}
\def\ci{\mathop{\text{ci}}}
\def\cl{\mathop{\text{cl}}}
\def\Clll{\hbox{$C$
  \hskip-4.5pt\raise4pt\hbox{\char'23
  \hskip-1.6pt\vrule height6.5pt width.5pt depth0pt
  \hskip.7pt\vrule height6.5pt width.5pt depth0pt
  \hskip-2.5pt\lower5.4pt\hbox{\char'23}}}}
\def\clMsimp{\overline{M}_{\text{simp}}}
\def\closeuparrow{\mathchar"0222}
\def\closedownarrow{\mathchar"0223}
\def\clstar{\text{cl*}}
\def\cov{\mathop{\text{cov}}}
\def\covar#1#2{\mathchoice
  {[#1{\lower4pt\hbox{${\scriptstyle *}\atop{\vrule height 18pt depth -10pt}$}}#2]}
  {[#1{*\atop{\vrule height 8pt depth 0pt}}#2]}
  {[#1{*\atop{\vrule height 6pt depth 0pt}}#2]}
  {[#1{*\atop{\vrule height 6pt depth 0pt}}#2]}}
\def\covSh{\mathop{\text{cov}}\nolimits_{\text{Sh}}}
\def\cdownarrow{c\mskip2mu^{\downarrow}}
\def\cycle#1{(\overleftarrow{#1})}
\def\ddaggerloc{$(\ddagger)_{\text{loc}}$}
\def\ddownarrow{d\mskip2mu^{\downarrow}}
%\def\DiniD{\setover{
%  \hbox{\hskip.1em\vrule height-1pt width.5em depth1.4pt\hfil}}D}
%\def\Dinid{\hskip.1em
%  \hbox{\hskip.05em\vrule height-1.3pt width.5em depth1.7pt}\hskip-.65em D}
\def\Dinid{\hbox{$D$\hskip-8.3pt\vrule height-1pt width7pt depth1.85pt}
   \hskip1pt}
\def\DiniD{\hbox{$D$\hskip-8.3pt\vrule height9.2pt width7ptdepth-8.35pt}
   \hskip1pt}

\def\discrcenter#1#2{\ifdim\pagewidth>#1{#2}\else{

  \Centerline{#2}

  \noindent}\fi}
    %note:  in \discrcenter, make sure that #2 ends in a space
    % (if you want one to appear) and that the continuation
    % follows without
    % a space after the final } of #2, so that it will be
    % truly unindented
\def\discrpage{\ifnum\stylenumber=1{\frnewpage}  %redefined in mt.tex
        \else\ifnum\stylenumber=3{\frnewpage}
        \else\ifnum\stylenumber=5{\frnewpage}
        \else\wheader{}{18}{14}{2}{72pt}\fi\fi\fi}
\def\displaycause#1{\noalign{\noindent (#1)}}
\def\Displaycause#1{\cr\displaycause{#1}&}
\def\dotproduct{\hskip.1em\tbf{.}\hskip.1em}
\def\doubleheadrightarrow{\rightarrow\mskip-12mu\rightarrow}
\long\def\doubleinset#1{\inset{\inset{\parindent=-20pt #1}}}
\def\duparrow{d\mskip2mu^{\uparrow}}
\def\dvAformerly#1{\discrversionA{\footnote{Formerly #1.}}{}}
\def\dvAnew#1{\discrversionA{\footnote{New #1.}}{}}
\def\dvArevised#1{\discrversionA{\footnote{Revised #1.}}{}}
\def\dvrocolon#1{\dvro{#1: }{#1}}
\def\eightVVdash{\mskip5mu\vrule height 6pt depth 2pt width 0.4pt
  \mskip2mu\vrule height 6pt depth 2pt width 0.4pt
  \vrule height 2.25pt depth -1.8pt width 3.2pt\mskip2mu}
\def\embedsinto{\mskip5mu\raise1.5pt\hbox{$\subset$}\mskip-15mu
   \lower3.5pt\hbox{$\rightarrow$}\mskip5mu}
\def\energy{\mathop{\text{energy}}\nolimits}
%\eqalign:  see \S478 for breaks during \eqalign
\def\equivGT{\equiv_{\text{GT}}}
\def\equivT{\equiv_{\text{T}}}
\def\Eta{\text{H}}
\def\Expn{\Bbb E}
\long\def\Extract{\oldfootnote{}{
   \noindent{\eightit
       \ifresultsonly{Extract from {\eightsmc Measure Theory,}
          results-only version,
          by {\eightsmc D.H.Fremlin,}  University of Essex,
Colchester.}
       \else\ifwithproofs{Extract from {\eightsmc Measure Theory,}
          by {\eightsmc D.H.Fremlin,}  University of Essex,
Colchester.}
       \else{Extract from {\eightsmc Measure Theory,} abridged version,
          by {\eightsmc D.H.Fremlin,}  University of Essex,
Colchester.}
       \fi\fi
   This material is copyright.   It is issued under the terms of the
   Design Science License as published in
   {\eighttt http://dsl.org/copyleft/dsl.txt}.
   This is a development version and the
   source files are not permanently archived, but current versions are
normally
   accessible through
   {\eighttt https://www1.essex.ac.uk/maths/people/fremlin/mt.htm}.
   For further information contact {\eighttt david\@fremlin.org}.}
   }}
\def\Fn{\mathop{\text{Fn}}\nolimits}
\def\FN{\mathop{\text{FN}}\nolimits}
\def\formerly#1{\footnote{Formerly #1.}}
  %use only once in a chapter, never in same volume as reference
\def\fraction#1{\mathchoice{\hbox{$<$}#1\hbox{$>$}}
     {\hbox{$<$}#1\hbox{$>$}}
     {\hbox{$\scriptstyle<$}#1\hbox{$\scriptstyle>$}}
     {\hbox{$\scriptscriptstyle<$}#1\hbox{$\scriptscriptstyle>$}}}
\def\frakmctbl{\frak m_{\text{countable}}}
\def\frmedcirc{{\raise 2.5pt\hbox{$_{\bigcirc}$}}}
\def\grad{\mathop{\text{grad}}\nolimits}
\def\hatpmb#1{\hat{\pmb{#1}}}
\def\hattilde#1{\skew2\hat{\tilde#1}}
\def\hattildepmb#1{\skew1\hat{\tildepmb#1}}
\def\hcov{\mathop{\text{hcov}}\nolimits}
\def\Heint{\mathchoice{\hbox to 0pt{\hskip 3.1pt H\hss}}
     {\raise 0.6pt\hbox to 0pt{\hskip 1.9pt{\sevenrm H}\hss}}
     {\raise 0.4pt\hbox to 0pt{\hskip 1.5pt{\fiverm H}\hss}}
     {\raise 0.4pt\hbox to 0pt{\hskip 1.5pt{\fiverm H}\hss}}
     \intop\nolimits}
\def\high#1{\lceil#1\rceil}
\def\highoverline#1{\overline{\vrule height6pt width0pt depth 0pt#1}}
\def\Hint#1{({\it Hint\/}: #1)}
\def\hL{\mathop{\text{hL}}}
\def\Hmp{\text{Hom}_{\bar\mu,\pi}}
\def\hp{\mathop{\text{hp}}\nolimits}
\def\hwtr{hereditarily weakly $\theta$-{\vthsp}refinable}
\def\iiv{ii_{\pmb{v}}}
\def\iivu{\iiv(\pmb{u})}
\def\imp{inverse-{\vthsp}measure-{\vthsp}preserving}
\def\innerprod#1#2{(#1|#2)}
\def\intstar{\text{int*}}
\def\CalIS{\Cal I(\Cal S)}
\def\jump{\mathop{\text{jump}}\nolimits}
\def\Jump{\mathop{\text{Jump}}\nolimits}
\def\Krein{Kre\v{\i}n}
\def\Lambdapv{\Lambda_{\text{pv}}}
\def\leRB{\le_{\text{RB}}}
\def\leRK{\le_{\text{RK}}}
\def\link{\mathord{\text{link}}}
\def\locmafp{\widehat{\otimes}_{\text{loc}}}
\def\Locmafp{\widehat{\bigotimes}^{\text{loc}}}
\def\low#1{\lfloor#1\rfloor}
\def\lti{left-\vthsp translation-\vthsp invariant}
\def\magnitude{\mathop{\text{mag}}}
\def\Mah{\mathop{\text{Mah}}}
\def\Mahcr{\Mah_{\text{cr}}}
\def\MahcrR{\Mah_{\text{crR}}}
\def\MahqR{\Mah_{\text{qR}}}
\def\MahR{\Mah_{\text{R}}}
\def\MahspcrR{\Mah_{\text{spcrR}}}
\def\Malm{M_{\text{alm}}}
\def\Marik{Ma\v r\'\i k} 
\def\Mbv{M_{\text{bv}}}
\def\Mcdlg{M_{\text{cdlg}}}
\def\Mdm{M_{\text{D-M}}}
\def\med{\mathop{\text{med}}\nolimits}
\def\Mfa{M_{\text{fa}}}
\def\Mhsr{\mathop{\text{Mhsr}}\nolimits}
\def\Mjf{\mathop{M_{\text{j-f}}}\nolimits}
\def\Mlbv{M_{\text{lbv}}}
\def\Mlob{M_{\text{lo-b}}}
\def\Mlns{M_{\text{ln-s}}}
\def\Mmo{M_{\text{mo}}}
\def\Mlmo{M_{\text{lmo}}}
\def\Mns{M_{\text{n-s}}}
\def\Mnsu{M_{\text{n-s}}^{\uparrow}}
%\def\Mnsu{M_{\text{n-s}\uparrow}}
\def\MMah{M_{\text{Mah}}}
\def\Mob{M_{\text{o-b}}}
\def\Mpob{M_{\text{po-b}}}
\def\Mpv{M_{\text{pv}}}
\def\MSi{M_{\text{S-i}}}
\def\Msimp{M_{\text{simp}}}
\def\Mth{Ma\-ha\-ram-{\vthsp}type-{\vthsp}homogeneous}
\def\Mvlm{M_{\text{vlm}}}
\def\negquad{\hskip-1em\relax}
\def\NN{\BbbN^{\Bbb N}}
\def\non{\mathop{\text{non}}}
\def\notBooleanop#1{
   \ifamstexsupported{\mathchoice
{\hbox{\hskip.2em$\not\scriptfont2=\eightsy\scriptstyle#1$\hskip.2em}}
{\hbox{\hskip.2em$\not\scriptfont2=\eightsy\scriptstyle#1$\hskip.2em}}
    {{\not\scriptstyle#1}}
    {{\not\scriptscriptstyle#1}}}
   \else{\mathchoice{{\not\scriptstyle#1}}{{\not\scriptstyle#1}}
    {{\not\scriptstyle#1}}{{\not\scriptscriptstyle#1}}}
    \fi}
\def\notBsubseteq{\notBooleanop{\subseteq}}
\def\notBsupseteq{\notBooleanop{\supseteq}}
\def\notVVdash{\not\mskip4mu\vrule height 7.5pt depth 2.5pt width 0.5pt
  \mskip2.5mu\vrule height 7.5pt depth 2.5pt width 0.5pt
  \vrule height 2.75pt depth -2.25pt width 4pt\mskip-2mu}
\def\nuprime{\nu^{\mskip0.7mu\prime}} %better  \nu'  in superscripts
\def\nw{\mathop{\text{nw}}}
\def\Osclln{\mathop{\text{Osclln}}\nolimits}
\def\otp{\mathop{\text{otp}}}
\def\partstar{\mathchoice{\partial\hbox{*}}
     {\partial\hbox{*}}
     {\partial\hbox{\sevenrm *}}
     {\partial\hbox{\fiverm *}}}
\def\per{\mathop{\text{per}}}
\def\Pfint{\mathchoice{\hbox to 0pt{\hskip 3.1pt Pf\hss}}
     {\raise 0.6pt\hbox to 0pt{\hskip 1.3pt{\sevenrm Pf}\hss}}
     {\raise 0.4pt\hbox to 0pt{\hskip 1.0pt{\fiverm Pf}\hss}}
     {\raise 0.4pt\hbox to 0pt{\hskip 1.0pt{\fiverm Pf}\hss}}
     \intop\nolimits
     \mathchoice{\hskip 2pt}{\hskip 1.2pt}{\hskip 1pt}{\hskip 1pt}}
\def\Pou{\mathop{\text{Pou}}\nolimits}
\def\prforc{\preccurlyeq_{\text{forcing}}}
\def\prGT{\preccurlyeq_{\text{GT}}}
\def\prT{\preccurlyeq_{\text{T}}}
\def\pssqa{power set $\sigma$-quotient algebra}
\def\qm{quasi-{\vthsp}measurable}
\def\Reverse#1{\setover
   {\lower0.4ex\hbox{$\scriptscriptstyle\leftrightarrow$}}{#1}}
\def\rhoKR{\rho_{\text{KR}}} 
\def\rhotv{\rho_{\text{tv}}} 
\def\Rint{\mathchoice{\hbox to 0pt{\hskip 3.1pt R\hss}}
     {\raise 0.6pt\hbox to 0pt{\hskip 1.9pt{\sevenrm R}\hss}}
     {\raise 0.4pt\hbox to 0pt{\hskip 1.5pt{\fiverm R}\hss}}
     {\raise 0.4pt\hbox to 0pt{\hskip 1.5pt{\fiverm R}\hss}}
     \intop\nolimits}
\def\RO{\text{RO}}
\def\rti{right-{\vthsp}translation-{\vthsp}invariant}
\def\rvm{real-{\vthsp}valued-{\vthsp}measurable}
\def\sat{\mathop{\text{sat}}}
\def\sfamilysS#1{\family{\sigma}{\Cal S}{#1_{\sigma}}}
\def\sfcc{$\sigma$-finite-cc}
\def\sgn{\mathop{\text{sgn}}}
\def\shr{\mathop{\text{shr}}\nolimits}
\def\Sii{\text{S}ii}
\def\Siiv{\Sii_{\pmb v}}
\def\Sint{\mathchoice{\hbox to 0pt{\hskip 4.1pt\ninerm S\hss}} 
     {\raise 0.6pt\hbox to 0pt{\hskip 3.0pt{\sevenrm S}\hss}}
     {\raise 0.4pt\hbox to 0pt{\hskip 1.5pt{\fiverm S}\hss}} %unchecked
     {\raise 0.4pt\hbox to 0pt{\hskip 1.5pt{\fiverm S}\hss}} %unchecked
     \intop\nolimits}
\def\Sinteightpoint{\raise 0.6pt\hbox to 0pt{\hskip 2.3pt{\fiverm S}\hss}
     \intop\nolimits}
\def\SL{\text{SL}}
\def\sm{semi-martingale} %hyphen to remind me that it's the "virtual" kind
\def\spread{\mathop{\text{spr}}}
\def\ssplrarrow{^{\scriptscriptstyle\leftrightarrow}}
\def\ssptilde{{\raise1.2ex\hbox{$\scriptscriptstyle\sim$}}}
%\def\sspvec{{\raise1.2ex\hbox{$\scriptscriptstyle\rightarrow$}}}
\def\sspvec{\mskip3mu\vec{\null}\mskip5mu}
\def\Sti{\mathop{\text{Sti}}\nolimits}
\def\strbar{^{\scriptscriptstyle\pmb{-}}}
\def\strplus{^{\scriptscriptstyle\pmb{+}}}
\def\strprime{^{\mathchoice{\mskip1.5mu\pmb{\prime}}
                  {\mskip1.5mu\pmb{\prime}}
                  {\mskip1.5mu\pmb{\prime}}
                  {\mskip1mu\pmb{\prime}}}}
%\def\strprime{\,\pmb{\prime}}
\def\supp{\mathop{\text{supp}}}
\def\tausm{\tau_{\sigma\text{-m}}}
\def\tbfoneminus#1{\rlap{$\tbf 1^{(#1)}$}\rlap{\hskip7pt$_<$}
   \phantom{\tbf 1^{(#1)}}}
%\def\tbfoneminus#1{\rlap{$\tbf 1^{(#1)}$}\rlap{\hskip7pt$_-$}
%   \phantom{\tbf 1^{(#1)}}}
\def\tensorhat{\widehat{\otimes}}
\def\Tensorhat{\widehat{\bigotimes}}
\def\tensorhatsigma{\widehat{\otimes}_{\sigma}}
\def\tildeClll{\hbox{$\tilde C$
  \hskip-4.5pt\raise4pt\hbox{\char'23
  \hskip-1.6pt\vrule height6.5pt width.5pt depth0pt
  \hskip.7pt\vrule height6.5pt width.5pt depth0pt
  \hskip-2.5pt\lower5.4pt\hbox{\char'23}}}}
\def\tildehat#1{\skew2\tilde{\hat#1}}
\def\tildehatpmb#1{\tilde{\hatpmb#1}}
\def\tildepmb#1{\tilde{\pmb{#1}}}
\def\Tr{\mathop{\text{Tr}}}
\def\trs{^{\top}\hskip-1pt}
\def\triplepc#1#2#3{(#1,#2,\hbox{$<$}#3)}
\def\TSi{\frak T_{\text{S-i}}}
\def\undphi{\underline{\phi}{}}
\def\undpsi{\underline{\psi}{}}
\def\undtheta{\underline{\theta}{}}
\def\updown{\mathchoice
  {\hbox{\hskip-1.5pt\char'23
    \hskip-1.7pt\vrule height6.5pt width.7pt depth1.3pt
    \hskip.9pt\vrule height6.5pt width.7pt depth1.3pt
    \hskip-3.3pt\lower6.7pt\hbox{\char'23}\hskip-1.4pt}}
  {\hbox{\hskip-2pt\char'23
    \hskip-1.7pt\vrule height6.5pt width.7pt depth1.3pt
    \hskip.9pt\vrule height6.5pt width.7pt depth1.3pt
    \hskip-2.5pt\lower6.7pt\hbox{\char'23}\hskip-1.4pt}}
  {\hbox{\hskip-2pt\char'23  %OK for subscript, fails for superscript
    \hskip-1.7pt\vrule height6.5pt width.7pt depth0pt
    \hskip.9pt\vrule height6.5pt width.7pt depth0pt
    \hskip-2.5pt\lower5.4pt\hbox{\char'23}\hskip-1.4pt}}
  {\hbox{\hskip-2pt\char'23  %not tested
    \hskip-1.7pt\vrule height6.5pt width.7pt depth1.3pt
    \hskip.9pt\vrule height6.5pt width.7pt depth1.3pt
    \hskip-2.5pt\lower6.7pt\hbox{\char'23}\hskip-1.4pt}}}
\def\updownarrows{\uparrow\mskip-4mu\downarrow}
\def\updowncgt{$\updown$-convergent}
\def\upr{\mathop{\text{upr}}\nolimits}
\def\ureg{\mathop{\text{ureg}}}

\def\Var{\mathop{\text{Var}}\nolimits}

\def\varaction{\mathchoice{\lower-.2ex\hbox{$\smallcirc$}}
                       {\lower-.2ex\hbox{$\smallcirc$}}
                       {\ssbullet}
                       {\ssbullet}}
\def\varcheck#1{{\setover
       {\lower 0.4ex\hbox{$\scriptscriptstyle\vee$}}{#1}}}
\def\varcheckf{{\setover{\,\lower 0.4ex
              \hbox{$\scriptscriptstyle\vee$}}{f}}}
\def\varchecki{\mskip-3mu\text{\it{\accent"14{\hbox{\hskip-3.5pt\i}}}}\mskip2mu}
\def\varspcheck{^{\scriptscriptstyle\vee}}
%\def\varspcheck{{\setover{\scriptscriptstyle\vee}{\phantom{g}}}}
%\def\varspcheckf{{\setover{\lower 0.4ex
%              \hbox{$\scriptscriptstyle\vee$}}{\phantom{f}}}}
\def\varhat#1{{\setover{\lower 0.4ex
              \hbox{$\scriptscriptstyle\wedge$}}{#1}}}
\def\varhatf{{\setover{\,\lower 0.4ex
              \hbox{$\scriptscriptstyle\wedge$}}{f}}}
\def\varinnerprod#1#2{#1\dotproduct#2}
\def\varsphat{^{\scriptscriptstyle\wedge}}
% or use  \hbox{\^{\null}}
%\def\varsphat{{\setover{\scriptscriptstyle\wedge}{\phantom{g}}}}
%\def\varsphatf{{\setover{\lower 0.4ex
%              \hbox{$\scriptscriptstyle\wedge$}}{\phantom{f}}}}
\def\varBspcheck{\mathchoice
  {\hskip-2pt
  {\fourteenrm\hbox{\lower2pt\hbox{\v{\null}}}}\hskip-1pt}
  {\hskip-2pt
  {\fourteenrm\hbox{\lower2pt\hbox{\v{\null}}}}\hskip-1pt}
  {\hskip-2.5pt
  {\fourteenrm\hbox{\lower4pt\hbox{\v{\null}}}}\hskip-1pt}
  {\hskip-2.5pt
  {\fourteenrm\hbox{\lower4pt\hbox{\v{\null}}}}\hskip-1pt}}
\def\varCspcheck{\mathchoice
  {\hskip-1pt\check{\phantom x}}   %not tested
  {\hskip-1pt\check{\phantom x}}
  {\hskip-1.2pt\check{\phantom x}}
  {\hskip-1.5pt\check{\phantom x}}}  %not tested
\def\varCsphat{\mathchoice
  {\hskip-1pt\hat{\phantom x}}   %not tested
  {\hskip-1pt\hat{\phantom x}}
  {\hskip-1.2pt\hat{\phantom x}}  %not tested
  {\hskip-1.5pt\hat{\phantom x}}}  %not tested
\def\VeqL{$V\hbox{$=$}L$}
\def\VVdash{\mskip5mu\vrule height 7.5pt depth 2.5pt width 0.5pt
  \mskip2.5mu\vrule height 7.5pt depth 2.5pt width 0.5pt
  \vrule height 2.75pt depth -2.25pt width 4pt\mskip2mu}
\def\VVdP{\VVdash_{\Bbb P}}
\def\VVdPk{\VVdash_{\Bbb P_{\kappa}}}
\def\wdistr{\mathop{\text{wdistr}}}
\def\weight{\mathop{\text{wt}}}
\def\w1lim{\mathop{\text{w1lim}}}
\def\wsid{weakly $(\sigma,\infty)$-{\vthsp}distributive}
\def\1lim{\mathop{1\mskip-1.5mu\text{lim}}}
\def\2vm{two-{\vthsp}valued-{\vthsp}measurable}

%Cara\-th\'e\-o\-dory

\def\chaptername{}
\def\sectionname{}
\def\headlinesectionname{}

\loadeusm\loadeurm
\Loadfourteens
\Loadfont{\ninerm=cmr9}

%note:  $..\cmmnt{=..}..$  is better as  \cmmnt{\mskip5mu
 
  
%----------------------------------------------------------%  
  
\smallprint  
%\USsmallprint  
%\noproofs  
%\resultsonly  
%\atUEssex  
\usegraphicx
  
\Extract  
  
%----------------------------------------------------------%  

\frfilename{mt48.tex} 
\versiondate{9.5.11} 
\copyrightdate{2001} 
 
\def\chaptername{Gauge integrals} 
\def\sectionname{Introduction} 
 
\newchapter{48} 
 
For the penultimate chapter of this volume I turn to a completely different 
approach to integration which has been developed in the last fifty 
years, following {\smc Kurzweil 57} and {\smc Henstock 63}.   This 
depends for its inspiration on a formulation of the Riemann 
integral (see 481Xe), and leads in particular to some remarkable 
extensions of the Lebesgue integral (\S\S483-484).   While (in my view) 
it remains peripheral to the most important parts of measure theory, it 
has deservedly attracted a good deal of interest in recent years, and is 
entitled to a place here. 
 
From the very beginning, in the definitions of \S122, I have presented 
the Lebesgue integral in terms of almost-everywhere approximations by 
simple functions.   Because the integral $\int\lim_{n\to\infty}f_n$ of a 
limit is {\it not} always the limit $\lim_{n\to\infty}\int f_n$ of the 
integrals, we are forced, from the start, to constrain ourselves by 
the ordering, and to work with monotone or dominated sequences.   This 
almost automatically leads us to an `absolute' integral, in which $|f|$ 
is integrable whenever $f$ is, whether we start from measures (as in 
Chapter 11) or from linear functionals (as in \S436).   For four volumes 
now I have been happily developing the concepts and intuitions 
appropriate to such integrals.   But if we return to one of the 
foundation stones of Lebesgue's theory, the Fundamental Theorem of 
Calculus, we find that it is easy to construct a differentiable 
function $f$ such that the absolute value $|f'|$ of its derivative is 
not integrable (483Xd).   It was observed very early ({\smc Perron 1914}) 
that the Lebesgue integral can be extended to integrate the derivative 
of any function which is differentiable everywhere.   The achievement of 
{\smc Henstock 63} was to find a formulation of this extension which was 
conceptually transparent enough to lend itself to a general theory, 
fragments of which I will present here. 
 
The first step is to set out the essential structures on which the 
theory depends (\S481), with a first attempt at a classification scheme. 
(One of the most interesting features of the Kurzweil-Henstock approach 
is that we have an extraordinary degree of freedom in describing our 
integrals, and apart from the Henstock integral itself it is not clear 
that we have yet found the right canonical forms to use.)   In \S482 I 
give a handful of general theorems showing what kinds of result can be 
expected and what difficulties arise.   In \S483, I work through the 
principal properties of the Henstock integral on the real line, showing, 
in particular, that it coincides with the Perron and special Denjoy 
integrals.   Finally, in \S484, I look at a very striking 
integral on $\BbbR^r$, due to W.F.Pfeffer. 
 
\discrpage 
 
\frfilename{mt481.tex}
\versiondate{4.9.09/30.11.12}
\copyrightdate{2000}

\def\Epsilon{\text{E}}
\def\Eta{\text{H}}

\def\chaptername{Gauge integrals}
\def\sectionname{Tagged partitions}

\newsection{481}

I devote this section to establishing some terminology (481A-481B,
481E-481G) %481E 481F 481G
and describing a variety of examples
(481I-481Q), %481I 481J 481K 481L 481M 481N 481O 481P 481Q
some of which will be elaborated later.   The clearest, simplest and
most important example is surely Henstock's integral on a closed bounded
interval (481J), so I recommend turning immediately to that paragraph
and keeping it in mind while studying the notation here.   It may also
help you to make sense of the definitions here if you glance at the
statements of some of the results in \S482;   in this section I give
only the formula defining gauge integrals (481C)\cmmnt{, with some
elementary examples of its use
(481Xb-481Xh)}.  %481Xb 481Xc 481Xd 481Xe 481Xf 481Xg 481Xh

\leader{481A}{Tagged partitions and Riemann sums} The common idea
underlying\cmmnt{ all the constructions of} this chapter is the following.   We
have a set $X$ and a functional $\nu$ defined on some family $\Cal C$ of
subsets of $X$.   We seek to define an integral
$\int fd\nu$, for functions $f$ with domain $X$, as a limit of
{\it finite} {\bf Riemann sums} $\sum_{i=0}^nf(x_i)\nu C_i$, where
$x_i\in X$ and $C_i\in\Cal C$ for $i\le n$.
\dvro{ A}{ There is no strict reason, at this stage, to forbid
repetitions in the string $(x_0,C_0),\ldots,(x_n,C_n)$, but also little
to be gained from allowing them, and it will simplify some of the
formulae below if I say from the outset
that a} {\bf tagged partition} on $X$ will be a finite subset $\pmb{t}$
of $X\times\Cal PX$.

\cmmnt{So one necessary element of the definition will be a
declaration of which tagged partitions
$\{(x_0,C_0),\ldots,\discretionary{}{}{}(x_n,C_n)\}$ will be
employed, in terms, for instance, of which sets $C_i$ are
permitted, whether they are allowed to overlap at
their boundaries, whether they are required to cover the space, and
whether each {\bf tag} $x_i$ is required to belong to the corresponding
$C_i$.}   The next element of the definition will be a description of a
filter $\Cal F$ on the set $T$ of tagged partitions, so that the
integral will be the limit (when it exists) of the sums along the
filter\cmmnt{, as in 481C below}.

\cmmnt{In the formulations studied in this chapter, the $C_i$ will
generally be disjoint, but this is not absolutely essential, and it is
occasionally
convenient to allow them to overlap in `small' sets, as in 481Ya.   In
some cases, we can restrict attention to families for which the $C_i$
are non-empty and have union $X$, so that $\{C_0,\ldots,C_n\}$ is a
partition of $X$ in the strict sense.}%end of comment

\woddheader{481B}{0}{0}{0}{90pt}

\leader{481B}{Notation}\cmmnt{ Let me immediately introduce notations
which will be in general use throughout the chapter.

\medskip

} {\bf (a)}\cmmnt{ First, a shorthand to describe a particular class of
sets of tagged partitions.}   If $X$ is a set, a {\bf straightforward
set of tagged partitions} on $X$ is a set of the form

\Centerline{$T=\{\pmb{t}:\pmb{t}\in[Q]^{<\omega},\,C\cap C'=\emptyset$
whenever $(x,C)$, $(x',C')$ are distinct members of $\pmb{t}\}$}

\noindent where $Q\subseteq X\times\Cal PX$;  I will say that $T$ is
{\bf generated} by $Q$.   \cmmnt{In this case, of
course, $Q$ can be recovered from $T$, since $Q=\bigcup T$.   Note that
no control is imposed on the tags at this point.   It remains
theoretically possible that a pair $(x,\emptyset)$ should belong to $Q$,
though in many applications this will be excluded in one way
or another.}

\spheader 481Bb If $X$ is a set and $\pmb{t}\subseteq X\times\Cal PX$ is
a tagged partition, \cmmnt{I write}

\Centerline{$W_{\pmb{t}}=\bigcup\{C:(x,C)\in\pmb{t}\}$.}

\spheader 481Bc If $X$ is a set, $\Cal C$ is a family of subsets of $X$,
$f$ and $\nu$ are real-valued functions,
and $\pmb{t}\in[X\times\Cal C]^{<\omega}$ is a tagged
partition, then

\Centerline{$S_{\pmb{t}}(f,\nu)=\sum_{(x,C)\in\pmb{t}}f(x)\nu C$}

\noindent whenever $\pmb{t}\subseteq\dom f\times\dom\nu$.

\leader{481C}{Proposition} Let $X$ be a set, $\Cal C$ a family of
subsets of $X$, $T\subseteq[X\times\Cal C]^{<\omega}$ a
non-empty set of tagged partitions and
$\Cal F$ a filter on $T$.   For real-valued functions $f$ and
$\nu$, set

\Centerline{$I_{\nu}(f)=\lim_{\pmb{t}\to\Cal F}S_{\pmb{t}}(f,\nu)$}

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

(a) $I_{\nu}$ is a linear functional defined on a linear subspace of
$\Bbb R^X$.

(b) Now suppose that $\nu C\ge 0$ for every $C\in\Cal C$.   Then

\quad(i) $I_{\nu}$ is a positive
linear functional\cmmnt{ (definition:  351F)};

\quad(ii) if $f$, $g:X\to\Bbb R$ are such that $|f|\le g$ and
$I_{\nu}(g)$ is defined and equal to $0$, then $I_{\nu}(f)$ is defined
and equal to $0$.

\proof{{\bf (a)} We have only to observe that if $f$, $g$ are real-valued
functions and
$\alpha\in\Bbb R$, then

\Centerline{$S_{\pmb{t}}(f+g,\nu)
=S_{\pmb{t}}(f,\nu)+S_{\pmb{t}}(g,\nu)$,
\quad$S_{\pmb{t}}(\alpha f,\nu)=\alpha S_{\pmb{t}}(f,\nu)$}

\noindent whenever the right-hand sides are defined,
and apply 2A3Sf.

\medskip

{\bf (b)} If $g\ge 0$ in $\Bbb R^X$, that is, $g(x)\ge 0$ for every
$x\in X$, then $S_{\pmb{t}}(g,\nu)\ge 0$ for every $\pmb{t}\in T$, so
the limit $I_{\nu}(g)$, if defined, will also be non-negative.   Next,
if $|f|\le g$, then $|S_{\pmb{t}}(f,\nu)|\le S_{\pmb{t}}(g,\nu)$ for
every $\pmb{t}$, so if $I_{\nu}(g)=0$ then $I_{\nu}(f)$ also is zero.
}%end of proof of 481C

\leader{481D}{Remarks (a)} Functionals
$I_{\nu}=\lim_{\pmb{t}\to\Cal F}S_{\pmb{t}}(.,\nu)$, as described above,
are called {\bf gauge integrals}.

\cmmnt{\spheader 481Db In fact even greater generality is
possible at this point.   There is no reason why $f$ and $\nu$ should
take real values.   All we actually need is an interpretation of sums of
products $f(x)\times\nu C$ in a space in which we can define limits.
So for any linear spaces $U$, $V$ and $W$ with a bilinear functional
$\phi:U\times V\to W$ (253A) and a Hausdorff linear space topology on
$W$, we can set out to construct an integral of a function $f:X\to U$
with respect to a functional $\nu:\Cal C\to V$ as a limit of sums
$S_{\pmb{t}}(f,\nu)=\sum_{(x,C)\in\pmb{t}}\phi(f(x),\nu C)$ in $W$.   I
will not
go farther along this path here.   But it is worth noting that the
constructions of this chapter lead the way to interesting vector
integrals of many types.
}%end of comment

\spheader 481Dc An extension which is\cmmnt{, however,} sometimes useful
is to allow $\nu$ to be undefined\cmmnt{ (or take values outside
$\Bbb R$, such as $\pm\infty$)} on part of $\Cal C$.   In this case, set
$\Cal C_0=\nu^{-1}[\Bbb R]$.   Provided that
$T\cap[X\times\Cal C_0]^{<\omega}$ belongs to $\Cal F$, we can still
define $I_{\nu}$, and 481C will still be true.

\leader{481E}{Gauges}\cmmnt{ The most useful method (so far) of
defining filters on sets of
tagged partitions is the following.

\medskip

}{\bf (a)} If $X$ is a set, a {\bf gauge} on
$X$ is a subset $\delta$ of $X\times\Cal PX$.   For a gauge $\delta$, a
tagged partition $\pmb{t}$ is
{\bf $\delta$-fine} if $\pmb{t}\subseteq\delta$.   Now,
for a set $\Delta$ of gauges and a non-empty
set $T$ of tagged partitions, we can
seek to define a filter $\Cal F$ on $T$ as the filter generated by sets
of the form
$T_{\delta}=\{\pmb{t}:\pmb{t}\in T$ is $\delta$-fine$\}$ as $\delta$
runs over $\Delta$.   \cmmnt{Of course we shall need to establish that
$T$ and $\Delta$ are compatible in the sense that
$\{T_{\delta}:\delta\in\Delta\}$ has the finite intersection
property;  this will ensure that there is indeed a filter containing
every $T_{\delta}$ (4A1Ia).}

\cmmnt{In nearly all cases, $\Delta$ will be non-empty and
downwards-directed (that is, for any $\delta_1$, $\delta_2\in\Delta$
there will be a $\delta\in\Delta$ such that
$\delta\subseteq\delta_1\cap\delta_2$);  in this case, we shall
need only to establish that
$T_{\delta}$ is non-empty for every $\delta\in\Delta$.   Note
that the filter on $T$ generated by $\{T_{\delta}:\delta\in\Delta\}$
depends only on $T$ and the filter on $X\times\Cal PX$ generated by
$\Delta$.}

\spheader 481Eb\cmmnt{ The most important gauges (so far) are
`neighbourhood gauges'.}
If $(X,\frak T)$ is a topological space, a {\bf neighbourhood gauge} on
$X$ is a set expressible in the form
$\delta=\{(x,C):x\in X,\,C\subseteq G_x\}$
where $\family{x}{X}{G_x}$ is a family of open sets such that $x\in G_x$
for every $x\in X$.   It is useful to note (i) that the family
$\family{x}{X}{G_x}$ can be recovered from $\delta$\cmmnt{, since
$G_x=\bigcup\{A:(x,A)\in\delta\}$} (ii) that $\delta_1\cap\delta_2$ is a
neighbourhood gauge whenever $\delta_1$ and $\delta_2$ are.
When $(X,\rho)$ is a metric space, \dvro{we have}{we can define a
neighbourhood gauge $\delta_h$ from any function
$h:X\to\ooint{0,\infty}$, setting

\Centerline{$\delta_h
=\{(x,C):x\in X,\,C\subseteq X,\,\rho(y,x)<h(x)$ for every $y\in C\}$.}

\noindent The set of gauges expressible in this form is coinitial with
the set of all neighbourhood gauges and therefore defines the same
filter on any compatible set $T$ of tagged partitions.   Specializing
yet further, we can restrict attention to constant functions $h$,
obtaining} the {\bf uniform metric gauges}

\Centerline{$\delta_{\eta}
=\{(x,C):x\in X,\,C\subseteq X,\,\rho(x,y)<\eta$ for every $y\in C\}$}

\noindent for $\eta>0$, used in the Riemann integral\cmmnt{ (481I)}.
\cmmnt{(The use of the letter `$\delta$' to represent a gauge has
descended from its
traditional appearance in the definition of the Riemann integral.)}

\spheader 481Ec\dvAformerly{4{}82Jb} If $X$ is a set and
$\Delta\subseteq\Cal P(X\times\Cal PX)$ is a family of gauges on $X$, I
will say that $\Delta$ is {\bf countably full} if whenever
$\sequencen{\delta_n}$ is a sequence in $\Delta$, and $\phi:X\to\Bbb N$
is a function, then there is a $\delta\in\Delta$ such that
$(x,C)\in\delta_{\phi(x)}$ whenever $(x,C)\in\delta$.   I will say that
$\Delta$ is {\bf full} if whenever $\family{x}{X}{\delta_x}$ is a family
in $\Delta$, then there is a $\delta\in\Delta$ such that
$(x,C)\in\delta_x$ whenever $(x,C)\in\delta$.

Of course a full set of gauges is countably full.   Observe that if
$(X,\frak T)$ is any topological space, the set of all
neighbourhood gauges on $X$ is full.

\leader{481F}{Residual sets}\dvro{ Let $\frak R$ be a collection }{ The
versatility and power of the methods being introduced here derives from
the insistence on taking {\it finite} sums
$\sum_{(x,C)\in\pmb{t}}f(x)\nu C$, so
that all questions about convergence are concentrated in the final limit
$\lim_{{\pmb{t}}\to\Cal F}S_{\pmb{t}}(f,\nu)$.   Since in a given
Riemann sum we can look at only finitely many sets $C$ of finite
measure, we cannot insist, even when $\nu$ is Lebesgue measure on
$\Bbb R$, that $W_{\pmb{t}}$ should always be $X$.   There are many
other cases in which it is impossible or inappropriate to insist that
$W_{\pmb{t}}=X$ for every tagged partition in $T$.   We shall therefore
need to add something to the definition of the filter $\Cal F$ on $T$
beyond what is possible in the language of 481E.   In the examples
below, the extra condition will always be of the following form.   There
will be a collection $\frak R$ }%end of dvro
of {\bf residual families}
$\Cal R\subseteq\Cal PX$.   It will help to have a phrase corresponding
to the phrase `$\delta$-fine':  if $\Cal R\subseteq\Cal PX$, and
$\pmb{t}$ is a tagged partition on $X$, \cmmnt{I will say that}
$\pmb{t}$ is {\bf $\Cal R$-filling} if
$X\setminus W_{\pmb{t}}\in\Cal R$.   Now, given a family $\frak R$ of
residual sets, and a family $\Delta$ of gauges on $X$, we can seek to
define a filter $\Cal F(T,\Delta,\frak R)$ on $T$ as that generated by
sets of the form $T_{\delta}$, for $\delta\in\Delta$, and $T'_{\Cal R}$,
for $\Cal R\in\frak R$, where

\Centerline{$T'_{\Cal R}
=\{\pmb{t}:\pmb{t}\in T$ is $\Cal R$-filling$\}.$}

\noindent When there is such a filter\cmmnt{, that is, the family
$\{T_{\delta}:\delta\in\Delta\}\cup\{T'_{\Cal R}:\Cal R\in\frak R\}$ has
the finite
intersection property}, I will say that $T$ is {\bf compatible} with
$\Delta$ and $\frak R$.

\cmmnt{It is important here to note that we shall {\it not} suppose
that, for a typical residual family
$\Cal R\in\frak R$, subsets of members of $\Cal R$ again belong to
$\Cal R$;  there will frequently be a restriction on the `shape' of
members of $\Cal R$ as well as on their size.   On the other hand, it
will usually be helpful to arrange that $\frak R$ is a filter base, so
that (if $\Delta$ is also
downwards-directed, and neither $\Delta$ nor $\frak R$ is empty)
we need only show that $T_{\delta}\cap T'_{\Cal R}$ is always
non-empty, and
$\{T_{\delta\Cal R}:\delta\in\Delta,\,\Cal R\in\frak R\}$ will be a base
for $\Cal F(T,\Delta,\frak R)$.}

\cmmnt{If the filter $\Cal F$ is defined as in 481Ea, with no mention
of a family $\frak R$, we can still bring the construction into the
framework considered here by setting $\frak R=\emptyset$.   If it is
convenient to define $T$ in terms which do not impose any requirement on
the sets $W_{\pmb{t}}$, but nevertheless we wish to restrict attention
to sums $S_{\pmb{t}}(f,\nu)$ for which the tagged partition covers the
whole space $X$, we can do so by setting $\frak R=\{\{\emptyset\}\}$.}

\leader{481G}{Subdivisions}\cmmnt{ When we come to analyse the
properties of integrals constructed by the method of 481C, there is an
important approach which depends on the following combination of features.}   I will say that
$(X,T,\Delta,\frak R)$ is a {\bf tagged-partition structure allowing
subdivisions}, {\bf witnessed by} $\Cal C$, if

\inset{(i) $X$ is a set.

(ii) $\Delta$ is a non-empty downwards-directed family of gauges on $X$.

(iii)($\alpha$) $\frak R$ is a non-empty downwards-directed collection
of families of subsets of $X$, all containing $\emptyset$;

\quad($\beta$) for every $\Cal R\in\frak R$ there is an
$\Cal R'\in\frak R$ such that $A\cup B\in\Cal R$ whenever $A$,
$B\in\Cal R'$ are disjoint.

(iv) $\Cal C$ is a family of subsets of $X$ such that
whenever $C$, $C'\in\Cal C$ then
$C\cap C'\in\Cal C$ and $C\setminus C'$ is expressible as the union of a
disjoint finite subset of $\Cal C$.

(v) Whenever $\Cal C_0\subseteq\Cal C$ is finite and $\Cal R\in\frak R$,
there is a finite set $\Cal C_1\subseteq\Cal C$, including $\Cal C_0$,
such that $X\setminus\bigcup\Cal C_1\in\Cal R$.

(vi) $T\subseteq[X\times\Cal C]^{<\omega}$ is\cmmnt{ (in the language
of 481Ba)} a non-empty straightforward set of tagged partitions on $X$.

(vii) Whenever $C\in\Cal C$, $\delta\in\Delta$ and
$\Cal R\in\frak R$ there is a $\delta$-fine tagged partition
$\pmb{t}\in T$ such that $W_{\pmb{t}}\subseteq C$ and
$C\setminus W_{\pmb{t}}\in\Cal R$.}

\leader{481H}{Remarks}\cmmnt{ {\bf (a)} Conditions (ii) and
(iii-$\alpha$) of 481G are included primarily for convenience, since
starting from any $\Delta$ and $\frak R$ we can find
non-empty directed sets
leading to the same filter $\Cal F(T,\Delta,\frak R)$.   (iii-$\beta$),
on the other hand, is saying something new.

\medskip

}\cmmnt{{\bf (b)} It is important to note, in
(vii) of 481G, that the tags of $\pmb{t}$ there are {\it not} required
to belong to the set $C$.

\medskip

}\cmmnt{{\bf (c)} All the applications below will fall into one
of two classes.   In one type, the residual families $\Cal R\in\frak R$
will be families of `small' sets, in some recognisably measure-theoretic
sense, and, in particular, we shall have subsets of members of any
$\Cal R$ belonging to
$\Cal R$.   In the other type, (vii) of 481G will be true because we can
always find $\pmb{t}\in T$ such that $W_{\pmb{t}}=C$.

\medskip

} {\bf (d)}\cmmnt{ The following elementary fact got left out of
\S136 and Chapter 31.}   Let $\frak A$ be a Boolean algebra and
$C\subseteq\frak A$.   Set

\Centerline{$E
=\{\sup C_0:C_0\subseteq C$ is finite and disjoint$\}$.}

\noindent If $c\Bcap c'$ and $c\Bsetminus c'$ belong to $E$ for all $c$,
$c'\in C$, then $E$ is a subring of $\frak A$.
\prooflet{\Prf\ Write $\Cal D$ for the family of finite disjoint subsets
of $C$.   (i) If $C_0$, $C_1\in\Cal D$, then for $c\in C_0$,
$c'\in C_1$ there is a $D_{cc'}\in\Cal D$ with supremum
$c\Bcap c'$.   Now $D=\bigcup_{c\in C_0,c'\in C_1}D_{cc'}$ belongs to
$\Cal D$ and has supremum $(\sup C_0)\Bcap(\sup C_1)$.   Thus
$e\Bcap e'\in E$ for all $e$, $e'\in E$.   (ii) Of course
$e\Bcup e'\in E$ whenever $e$, $e'\in E$ and $e\Bcap e'=0$.
(iii) Again suppose that $C_0$, $C_1\in\Cal D$.   Then
$c\Bsetminus c'\in E$ for all
$c\in C_0$, $c'\in C_1$.   By (i), $c\Bsetminus\sup C_1\in E$ for every
$c\in C_0$;  by (ii), $(\sup C_0)\Bsetminus(\sup C_1)\in E$.   Thus
$e\Bsetminus e'\in E$ for all $e$, $e'\in E$.   (iv) Putting (ii) and
(iii) together, $e\Bsymmdiff e'\in E$ for all $e$, $e'\in E$.   (v) As
$0=\sup\emptyset$ belongs to $E$, $E$ is a subring of $\frak A$.\ \Qed}

In particular, if $\Cal C\subseteq\Cal PX$ has the properties in (iv) of
481G, then

\Centerline{$\Cal E
=\{\bigcup\Cal C_0:\Cal C_0\subseteq\Cal C$ is finite and disjoint$\}$}

\noindent is a ring of subsets of $X$.

\spheader 481He Suppose that $X$ is a set and that
$\frak R\subseteq\Cal P\Cal PX$ satisfies (iii) of 481G.   Then for
every $\Cal R\in\frak R$ there is a non-increasing sequence
$\sequencen{\Cal R_n}$ in
$\frak R$ such that $\bigcup_{i\le n}A_i\in\Cal R$ whenever
$A_i\in\Cal R_i$ for $i\le n$ and $\langle A_i\rangle_{i\le n}$ is
disjoint.   \prooflet{\Prf\ Take
$\Cal R_0\in\frak R$ such that $\Cal R_0\subseteq\Cal R$ and
$A\cup B\in\Cal R$ for all disjoint $A$, $B\in\Cal R_0$;  similarly, for
$n\in\Bbb N$, choose
$\Cal R_{n+1}\in\frak R$ such that $\Cal R_{n+1}\subseteq\Cal R_n$ and
$A\cup B\in\Cal R_n$ for all disjoint $A$, $B\in\Cal R_{n+1}$.   Now,
given that $A_i\in\Cal R_i$ for $i\le n$ and
$\langle A_i\rangle_{i\le n}$ is disjoint, we see by downwards induction
on $m$ that $\bigcup_{m<i\le n}A_i\in\Cal R_m$ for each $m\le n$, so
that $\bigcup_{i\le n}A_i\in\Cal R$.\ \Qed}

\spheader 481Hf If $(X,T,\Delta,\frak R)$ is a tagged-partition
structure allowing subdivisions, then $T$ is compatible with $\Delta$
and $\frak R$\cmmnt{ in the sense of 481F}.
\prooflet{\Prf\ $\emptyset\in T$ so $T$ is not empty.   Take
$\delta\in\Delta$ and $\Cal R\in\frak R$.   Let $\sequence{i}{\Cal R_i}$
be a sequence in $\frak R$ such that $\bigcup_{i\le n}A_i\in\Cal R$
whenever $A_i\in\Cal R_i$ for $i\le n$ and $\langle A_i\rangle_{i\le n}$
is disjoint ((e) above).   There is a finite
set $\Cal C_1\subseteq\Cal C$ such that
$X\setminus\bigcup\Cal C_1\in\Cal R_0$, by 481G(iv).   By (d), there is
a disjoint family $\Cal C_0\subseteq\Cal C$ such that
$\bigcup\Cal C_0=\bigcup\Cal C_1$;  enumerate $\Cal C_0$ as
$\ofamily{i}{n}{C_i}$.   For each $i<n$, there is a $\delta$-fine
$\pmb{t}_i\in T$ such that $W_{\pmb{t}_i}\subseteq C_i$ and
$C_i\setminus W_{\pmb{t}_i}\in\Cal R_{i+1}$, by 481G(vii).   Set
$\pmb{t}=\bigcup_{i<n}\pmb{t}_i$;  then $\pmb{t}\in T$ is $\delta$-fine,
and

\Centerline{$X\setminus W_{\pmb{t}}
=(X\setminus\bigcup\Cal C_1)
  \cup\bigcup_{i<n}(C_i\setminus W_{\pmb{t}_i})
\in\Cal R$}

\noindent by the choice of $\sequence{i}{\Cal R_i}$.   Thus we have a
$\delta$-fine $\Cal R$-filling member of $T$;  as $\delta$ and $\Cal R$
are arbitrary, $T$ is compatible with $\Delta$ and $\frak R$.\ \Qed
}%end of prooflet

\cmmnt{\spheader 481Hg For basic results which depend on `subdivisions'
as described in 481G(vii), see 482A-482B below.   A hypothesis
asserting the existence of a different
sort of subdivision appears in 482G(iv).}

\leader{481I}{}\cmmnt{ I now run through some simple examples of these
constructions, limiting myself for the moment to the definitions, the
proofs that $T$ is compatible with $\Delta$ and $\frak R$, and (when
appropriate) the proofs that the structures allow subdivisions.

\medskip

\noindent}{\bf The proper Riemann integral} Fix a
non-empty closed interval $X=[a,b]\subseteq\Bbb R$.   Write
$\Cal C$ for the set of all intervals (open, closed or
half-open, and allowing the empty set to count as an interval)
included in $[a,b]$, and set
$Q=\{(x,C):C\in C,\,x\in\overline{C}\}$;  let $T$ be the straightforward
set of tagged partitions generated by $Q$.   Let $\Delta$ be the set of
uniform metric gauges on $X$, and $\frak R=\{\{\emptyset\}\}$.   Then
$(X,T,\Delta,\frak R)$ is a
tagged-partition structure allowing subdivisions, witnessed by $\Cal C$.
If $a<b$, then $\Delta$ is not countably full.

\proof{ (i), (iii) and (vi) of 481G are trivial and (ii), (iv) and (v)
are elementary.   As for (vii), given $\eta>0$ and $C\in\Cal C$, take a
disjoint family $\familyiI{C_i}$ of non-empty intervals of length less than
$2\eta$ covering $C$, and $x_i$ to be the midpoint of $C_i$ for
$i\in I$; then $\pmb{t}=\{(x_i,C_i):i\in I\}$ belongs to
$T$ and is $\delta_{\eta}$-fine, in the language of 481E, and
$W_{\pmb{t}}=C$.

Of course (apart from the trivial case $a=b$)
$\Delta$ is not countably full, since if we take $\delta_n$ to be
the gauge $\{(x,C):|x-y|<2^{-n}$ for every $y\in C\}$ and any unbounded
function $\phi:[a,b]\to\Bbb N$, there is no $\delta\in\Delta$ such that
$(x,C)\in\delta_{\phi(x)}$ whenever $(x,C)\in\Delta$.
}%end of proof of 481I

\leader{481J}{\bf The Henstock integral on a bounded
interval}\cmmnt{ ({\smc Henstock 63})} Take $X$, $\Cal C$, $T$
and $\frak R$ as in 481I.   This time, let $\Delta$ be the set of
{\it all} neighbourhood gauges on $[a,b]$.   Then
$(X,T,\Delta,\frak R)$ is a tagged-partition structure allowing
subdivisions, witnessed by $\Cal C$, and $\Delta$ is countably full.

\proof{ Again, only 481G(vii) needs more than a moment's consideration.
Take any
$C\in\Cal C$.   If $C=\emptyset$, then $\pmb{t}=\emptyset$ will suffice.
Otherwise, set $a_0=\inf C$, $b_0=\sup C$ and let $T_0$ be the
family of $\delta$-fine partitions $\pmb{t}\in T$ such that
$W_{\pmb{t}}$ is a relatively closed initial subinterval of $C$, that
is, is of the
form $C\cap[a_0,y_{\pmb{t}}]$ for some $y_{\pmb{t}}\in[a_0,b_0]$.   Set
$A=\{y_{\pmb{t}}:\pmb{t}\in T_0\}$.   I have to show that there is a
$\pmb{t}\in T_0$ such that $W_{\pmb{t}}=C$, that is, that $b_0\in A$.

Observe that there is an $\eta_0>0$ such that
$(a_0,A)\in\delta$ whenever
$A\subseteq[a,b]\cap[a_0-\eta_0,a_0+\eta_0]$, and now
$\{(a_0,[a_0,a_0+\eta_0]\cap C)\}$ belongs to $T_0$, so
$\min(a_0+\eta_0,b_0)\in A$ and $A$ is a non-empty subset of
$[a_0,b_0]$.
It follows that $c=\sup A$ is defined in $[a_0,b_0]$.   Let $\eta>0$ be
such that $(c,A)\in\delta$ whenever
$A\subseteq[a,b]\cap[c-\eta,c+\eta]$.   There is some
$\pmb{t}\in T_0$ such that $y_{\pmb{t}}\ge c-\eta$.   If
$y_{\pmb{t}}=b_0$, we can stop.   Otherwise, set
$C'=C\cap\ocint{y_{\pmb{t}},c+\eta}$.   Then $(c,C')\in\delta$ and
$c\in\overline{C'}$ and $C'\cap W_{\pmb{t}}$ is empty, so
$\pmb{t}'=\pmb{t}\cup\{(c,C')\}$ belongs to $T_0$ and
$y_{\pmb{t}'}=\min(c+\eta,b_0)$.   Since $y_{\pmb{t}'}\le c$, this shows
that $y_{\pmb{t}'}=c=b_0$ and again $b_0\in A$, as required.

$\Delta$ is full just because it is the family of neighbourhood gauges.
}%end of proof of 481J
% maybe {\smc Cousin 1895} relevant if I can be bothered to check

\leader{481K}{The Henstock integral on $\Bbb R$} This time, set
$X=\Bbb R$ and let $\Cal C$ be the family of all bounded
intervals in $\Bbb R$.   Let $T$ be the straightforward set of
tagged partitions generated by
$\{(x,C):C\in\Cal C$, $x\in\overline{C}\}$.   Following 481J, let $\Delta$ be the set of all neighbourhood
gauges on $\Bbb R$.   This time, set
$\frak R=\{\Cal R_{ab}:a\le b\in\Bbb R\}$, where
$\Cal R_{ab}=\{\Bbb R\setminus[c,d]:c\le a,\,d\ge b\}\cup\{\emptyset\}$.
Then $(X,T,\Delta,\frak R)$ is a tagged-partition structure allowing
subdivisions, witnessed by $\Cal C$.

\proof{ This time we should perhaps take a moment to look at
(iii) of 481G.   But all we need to note is that
$\Cal R_{ab}\cap\Cal R_{a'b'}=\Cal R_{\min(a,a'),\max(b,b')}$, and that
any two members of $\Cal R_{ab}$ have non-empty intersection.
Conditions (i), (ii), (iv), (v) and (vi) of 481G
are again elementary, so once more we are left with (vii).   But this
can be dealt with by exactly the same argument as in 481J.
}%end of proof of 481K

\leader{481L}{The symmetric Riemann-complete integral}\cmmnt{ (cf.\
{\smc Carrington 72}, chap.\ 3)} Again take $X=\Bbb R$, and $\Cal C$ the
set of all
bounded intervals in $\Bbb R$.   This time, take $T$ to be the
straightforward set of tagged partitions generated by the set
of pairs $(x,C)$ where $C\in\Cal C\setminus\{\emptyset\}$ and $x$ is the
{\it midpoint} of
$C$.   As in 481K, take $\Delta$ to be the set of all neighbourhood
gauges on
$\Bbb R$;  but this time take $\frak R=\{\Cal R'_a:a\ge 0\}$, where
$\Cal R'_a=\{\Bbb R\setminus[-c,c]:c\ge a\}\cup\{\emptyset\}$.   Then
$T$ is compatible with $\Delta$ and $\frak R$.

\proof{ Take $\delta\in\Delta$ and $\Cal R\in\frak R$.   For $x\ge 0$,
let $\theta(x)>0$ be such that
$(x,D)\in\delta$ whenever $D\subseteq[x-\theta(x),x+\theta(x)]$
and $(-x,D)\in\delta$ whenever
$D\subseteq[-x-\theta(x),-x+\theta(x)]$.   Write $A$ for the set of
those $a>0$ such that there is a finite sequence $(a_0,\ldots,a_n)$
such that $0<a_0<a_1<\ldots<a_n=a$, $a_0\le\theta(0)$ and
$a_{i+1}-a_i\le 2\theta(\bover12(a_i+a_{i+1}))$ for $i<n$.

\Quer\ Suppose, if possible, that $A$ is bounded above.   Then
$c=\inf(\coint{0,\infty}\setminus\overline{A})$ is defined in
$\coint{0,\infty}$.   Observe that if $0<a\le\theta(0)$, then the
one-term sequence $\fraction{a}$ witnesses that $a\in A$.
So $c\ge\theta(0)>0$.   Now there
must be $u$, $v$ such that $c<u<v<\min(c+\theta(c),2c)$ and
$\ooint{u,v}\cap A=\emptyset$;  on the other hand, the interval
$\ooint{2c-v,2c-u}$ must contain a point $x$ of $A$.   Set $y=2c-x$.
Then we can find $a_0<\ldots<a_n=x$ such that $0<a_0\le\theta(0)$ and
$a_{i+1}-a_i\le 2\theta(\bover12(a_i+a_{i+1}))$ for $i<n$;  setting
$a_{n+1}=y$, we see that $\langle a_i\rangle_{i\le n+1}$ witnesses that
$y\in A$, though $y\in\ooint{u,v}$.\ \Bang\

This contradiction shows that $A$ is unbounded above.   So now suppose
that $\Cal R=\Cal R_a$ where $a\ge 0$.   Take $a_0,\ldots,a_n$ such that
$0<a_0<\ldots<a_n$, $0<a_0\le\theta(0)$ and
$a_{i+1}-a_i\le 2\theta(\bover12(a_i+a_{i+1}))$ for $i<n$, and
$a_n\ge a$.
For $i<n$, set $x_i=\bover12(a_i+a_{i+1})$, $C_i=\ocint{a_i,a_{i+1}}$,
$x'_i=-x_i$, $C'_i=\coint{-a_{i+1},a_i}$.   Then $x_i$, $x'_i$ are the
midpoints of $C_i$, $C'_i$ and (by the choice of the function $\theta$)
$(x_i,C_i)\in\delta$, $(x'_i,C'_i)\in\delta$ for $i<n$.   So
if we set

\Centerline{$\pmb{t}
=\{(x_i,C_i):i<n\}\cup\{(x'_i,C'_i):i<n\}\cup\{(0,[-a_0,a_0])\}$}

\noindent we shall obtain a $\delta$-fine $\Cal R$-filling member of
$T$.

As $\Delta$ and $\frak R$ are both downwards-directed, this is enough to
show that $T$ is compatible with $\Delta$ and $\frak R$.
}%end of proof of 481L

\leader{481M}{The McShane integral on an
interval}\cmmnt{ ({\smc McShane 73})} As in 481J, take
$X=[a,b]$ and let $\Cal C$ be the family of subintervals of
$[a,b]$.   This time, take $T$ to be the straightforward set of tagged
partitions generated by $Q=X\times\Cal C$\cmmnt{, so that {\it no}
condition is imposed relating the tags to their associated intervals}.
As in 481J, let $\Delta$ be the set of all neighbourhood gauges on $X$,
and $\frak R=\{\{\emptyset\}\}$.   \cmmnt{Proceed as before.   Since
the only change is that $Q$ and $T$ have been enlarged,}
$(X,T,\Delta,\frak R)$ is\cmmnt{ still} a tagged-partition structure
allowing subdivisions, witnessed by $\Cal C$.

\leader{481N}{The McShane integral on a topological
space}\cmmnt{ ({\smc Fremlin 95})} Now let
\ifdim\pagewidth=390pt\penalty-1000\fi
$(X,\frak T,\Sigma,\mu)$ be any effectively locally finite
$\tau$-additive topological measure space, and take
$\Cal C=\{E:E\in\Sigma,\,\mu E<\infty\}$,
$Q=X\times\Cal C$;  let $T$ be the straightforward set of tagged
partitions generated by $Q$.   Again let
$\Delta$ be the set of all neighbourhood gauges on $X$.   \cmmnt{This
time, define $\frak R$ as follows.}   For any set $E\in\Sigma$ of finite
measure and $\eta>0$, let $\Cal R_{E\eta}$ be the set
$\{F:F\in\Sigma,\,\mu(F\cap E)\le\eta\}$, and set
$\frak R=\{\Cal R_{E\eta}:\mu E<\infty,\,\eta>0\}$.   Then
$(X,T,\Delta,\frak R)$ is a tagged-partition structure allowing
subdivisions, witnessed by $\Cal C$.

\proof{ As usual, everything is elementary except perhaps 481G(vii).
But if $C\in\Cal C$, $\delta\in\Delta$, $E\in\Sigma$, $\mu E<\infty$ and
$\eta>0$, take for each $x\in X$ an open set $G_x$ containing $x$ such
that $(x,A)\in\delta$ whenever $A\subseteq G_x$.
$\{G_x:x\in X\}$ is an open cover of $X$, so by 414Ea there is a
finite family $\ofamily{i}{n}{x_i}$ in $X$ such that
$\mu(E\cap C\setminus\bigcup_{i<n}G_{x_i})\le\eta$;  setting
$C_i=C\cap G_{x_i}\setminus\bigcup_{j<i}G_{x_j}$ for
$i<n$, we get a $\delta$-fine tagged partition
$\pmb{t}=\{(x_i,C_i):i<n\}$ such that
$C\setminus W_{\pmb{t}}\in\Cal R_{E\eta}$.
}%end of proof of 481N

\leader{481O}{Convex partitions in $\BbbR^r$} Fix $r\ge 1$.   Let us say
that a {\bf convex polytope} in $\BbbR^r$ is a non-empty bounded set
expressible as the intersection of finitely many open or closed
half-spaces;  let
$\Cal C$ be the family of convex polytopes in $X=\BbbR^r$, and $T$ the
straightforward set of tagged partitions generated by
$\{(x,C):x\in\overline{C}\}$.   Let $\Delta$ be the set of
neighbourhood gauges on $\BbbR^r$.   For $a\ge 0$, let $\Cal C_a$ be the
set of closed convex polytopes $C\subseteq\BbbR^r$ such that, for some
$b\ge a$, $B(0,b)\subseteq C\subseteq B(0,2b)$, where $B(0,b)$ is the
ordinary Euclidean ball with centre $0$ and radius $b$;  set
$\Cal R_a=\{\BbbR^r\setminus C:C\in\Cal C_a\}\cup\{\emptyset\}$, and
$\frak R=\{\Cal R_a:a\ge 0\}$.   Then $(X,T,\Delta,\frak R)$ is a
tagged-partition structure allowing subdivisions, witnessed by $\Cal C$.

%what about Saks-regular partitions?  see \S475
%note that $\frak R$ looks just peculiar at the moment

\proof{ As usual, only 481G(vii) requires thought.

\medskip

{\bf (a)} We need a geometrical fact:  if $C\in\Cal C$,
$x\in\overline{C}$ and $y\in C$, then $\alpha y+(1-\alpha)x\in C$ for
every $\alpha\in\ocint{0,1}$.   \Prf\ The family of sets
$C\subseteq\BbbR^r$ with this property is closed under finite
intersections and contains all half-spaces.\ \QeD\   It follows that if
$C_1$, $C_2\in\Cal C$ are not disjoint, then
$\overline{C_1\cap C_2}=\overline{C}_1\cap\overline{C}_2$.

\medskip

{\bf (b)} Write $\Cal D\subseteq\Cal C$ for the family of products of
non-empty bounded intervals in $\Bbb R$.   The next step is to show that
if $D\in\Cal D$ and $\delta\in\Delta$, then there is a
$\delta$-fine tagged partition $\pmb{t}\in T$ such that $W_{\pmb{t}}=D$
and $\pmb{t}\subseteq\BbbR^r\times\Cal D$.
\Prf\ Induce on $r$.   For $r=1$ this is just 481J again.   For the
inductive step to $r+1$, suppose that
$D\subseteq\BbbR^{r+1}$ is a product of bounded intervals and that
$\delta$ is a neighbourhood gauge on $\BbbR^{r+1}$.   Identifying
$\BbbR^{r+1}$ with $\BbbR^r\times\Bbb R$, express $D$ as $D'\times L$,
where $D'\subseteq\BbbR^r$ is a product of bounded intervals and
$L\subseteq\Bbb R$ is a bounded interval.   For $y\in\BbbR^r$,
$\alpha\in\Bbb R$ let $G(y,\alpha)$, $H(y,\alpha)$ be open sets
containing $y$, $\alpha$ respectively such that
$((y,\alpha),A)\in\delta$ whenever
$A\subseteq G(y,\alpha)\times H(y,\alpha)$.
For $y\in\BbbR^r$, set $\delta_y=\{(\alpha,A):\alpha\in\Bbb
R,\,A\subseteq H(y,\alpha)\}$;  then $\delta_y$ is a neighbourhood gauge
on $\Bbb R$.

By the one-dimensional case there is a $\delta_y$-fine tagged partition
$\pmb{s}_y\in T_1$ such that $W_{\pmb{s}_y}=L$, where I write $T_1$ for
the set of tagged partitions used in 481K.
Set

\Centerline{$\delta'
=\{(y,A):y\in\BbbR^r,\,A\subseteq G(y,\alpha)$ for every
$(\alpha,F)\in\pmb{s}_y\}$.   }

$\delta'$ is a neighbourhood gauge on $\BbbR^r$.   By the inductive
hypothesis, there is a $\delta'$-fine tagged partition $\pmb{u}\in T_r$
such that $W_{\pmb{u}}=D'$, where here $T_r$ is the set of tagged
partitions on $\BbbR^r$ corresponding to the $r$-dimensional version of
this result.   Consider the family

\Centerline{$\pmb{t}
=\{((y,\alpha),E\times F):(y,E)\in\pmb{u},\,(\alpha,F)\in\pmb{s}_y\}$.}

\noindent For $(y,E)\in\pmb{u}$, $(\alpha,F)\in\pmb{s}_y$, we have

\Centerline{$y\in\overline{E}$,
\quad$E\subseteq G(y,\alpha)$,
\quad$\alpha\in\overline{F}$,
\quad$F\subseteq H(y,\alpha)$,}

\noindent so

\Centerline{$(y,\alpha)\in\overline{E\times F}$,
\quad$E\times F\subseteq G(y,\alpha)\times H(y,\alpha)$,}

\noindent and $((y,\alpha),E\times F)\in\delta$.
If $((y,\alpha),E\times F)$, $((y',\alpha'),E'\times F')$ are distinct
members of $\pmb{t}$, then either $(y,E)\ne(y',E')$ so
$E\cap E'=\emptyset$ and $(E\times F)\cap(E'\times F')$ is empty, or
$y=y'$ and $(\alpha,F)$, $(\alpha',F')$ are distinct members of
$\pmb{s}_y$, so that $F\cap F'=\emptyset$ and again $E\times F$,
$E'\times F'$ are disjoint.    Thus $\pmb{t}$ is a $\delta$-fine member
of $T_{r+1}$.   Finally,

\Centerline{$W_{\pmb{t}}
=\bigcup_{(y,E)\in\pmb{u}}\bigcup_{(\alpha,F)\in\pmb{s}_y}
   E\times F
=\bigcup_{(y,E)\in\pmb{u}}E\times L=D'\times L=D$.}

\noindent So the induction proceeds.\ \Qed

\medskip

{\bf (c)} Now suppose that $C_0$ is an arbitrary member of $\Cal C$ and
that $\delta$ is a neighbourhood gauge on $\BbbR^r$.   Set

\Centerline{$\delta'=\delta\cap\{(x,A):$ either $x\in\overline{C}_0$ or
$A\cap\overline{C}_0=\emptyset\}$.}

\noindent Then $\delta'$ is a neighbourhood gauge on $\BbbR^r$, being
the intersection of $\delta$ with the neighbourhood gauge associated
with the family $\family{x}{\BbbR^r}{U_x}$, where $U_x=\BbbR^r$ if
$x\in\overline{C}_0$, $\BbbR^r\setminus\overline{C}_0$ otherwise.   Let
$D\in\Cal D$ be such that $C_0\subseteq D$.   By (b), there is a
$\delta'$-fine tagged partition $\pmb{t}\in T$ such that
$W_{\pmb{t}}=D$.   Set
$\pmb{s}=\{(x,C\cap C_0):(x,C)\in\pmb{t},\,C\cap C_0\ne\emptyset\}$.
Since $\pmb{t}\subseteq\delta'$, $x\in\overline{C}_0$ whenever
$(x,C)\in\pmb{t}$ and
$C\cap C_0\ne\emptyset$.   By (a), $x\in\overline{C\cap C_0}$ for all
such pairs $(x,C)$;  and of course $(x,C\cap C_0)\in\delta$ for every
$(x,C)\in\pmb{t}$.   So $\pmb{s}$ belongs to $T$, and
$W_{\pmb{s}}=W_{\pmb{t}}\cap C_0=C_0$.   As $C_0$ and $\delta$ are
arbitrary, 481G(vii) is satisfied.
}%end of proof of 481O

\leader{481P}{Box products}\cmmnt{ (cf. {\smc Muldowney 87}, Prop.\
1)} Let $\familyiI{(X_i,\frak T_i)}$ be a non-empty family of non-empty
compact metrizable spaces with product $(X,\frak T)$.   Set
$\pi_i(x)=x(i)$ for $x\in X$ and $i\in I$.   For each $i\in I$, let
$\Cal C_i\subseteq\Cal PX_i$ be such that ($\alpha$) whenever $E$,
$E'\in\Cal C_i$ then
$E\cap E'\in\Cal C_i$ and $E\setminus E'$ is expressible as the union of
a disjoint finite subset of $\Cal C_i$ ($\beta$) $\Cal C_i$ includes a
base for $\frak T_i$.

Let $\Cal C$ be the set of subsets of $X$ of the form

\Centerline{$C=\{X\cap\bigcap_{i\in J}\pi_i^{-1}[E_i]:
J\in[I]^{<\omega},\,E_i\in\Cal C_i$ for every $i\in J\}$,}

\noindent and let $T$ be the straightforward set of tagged partitions
generated by $\{(x,C):C\in\Cal C,\,x\in\overline{C}\}$.   Let
$\Delta$ be the set of those neighbourhood gauges $\delta$ on $X$
defined by families $\family{x}{X}{G_x}$ of open sets such that, for
some countable $J\subseteq I$, every $G_x$ is determined by coordinates
in $J$\cmmnt{ (definition:  254M)}.   Then
$(X,T,\Delta,\{\{\emptyset\}\})$ is a tagged-partition structure
allowing subdivisions, witnessed by $\Cal C$.    $\Delta$ is countably
full;  $\Delta$ is full iff $I'=\{i:\#(X_i)>1\}$ is countable.

\proof{ Conditions (i), (iii) and (vi) of 481G are trivial, and (ii),
(iv) and (v) are elementary;  so we are left with (vii), as usual.
\Quer\ Suppose, if possible, that $C\in\Cal C$ and $\delta\in\Delta$ are
such that there is no $\delta$-fine $\pmb{t}\in T$ with $W_{\pmb{t}}=C$.
Let $\family{x}{X}{G_x}$ be the family of open sets determining
$\delta$, and $J\subseteq I$ a non-empty countable set such that $G_x$
is determined by coordinates in $J$ for every $x\in X$.   For $i\in J$,
let $\Cal C'_i\subseteq\Cal C_i$ be a countable base for $\frak T_i$
(4A2P(a-iii)), and take a sequence $\sequencen{(i_n,E_n)}$ running over
$\{(i,E):i\in J,\,E\in\Cal C'_i\}$.

Write $\Cal D=\{W_{\pmb{t}}:\pmb{t}\in T$ is $\delta$-fine$\}$.
Note that if $D_1$, $D_2\in\Cal D$ are disjoint then
$D_1\cup D_2\in\Cal D$.   So if
$D\in\Cal C\setminus\Cal D$ and $C\in\Cal C$, there must be some
$D'\in\Cal C\setminus\Cal D$ such that either $D'\subseteq D\cap C$ or
$D'\subseteq D\setminus C$, just because $\Cal C$ satisfies 481G(iv).
Now choose $\sequencen{C_n}$ inductively so that $C_0=C$ and

\Centerline{$C_n\in\Cal C\setminus\Cal D$,}

\Centerline{either $C_{n+1}\subseteq C_n\cap\pi_{i_n}^{-1}[E_n]$ or
$C_{n+1}\subseteq C_n\setminus\pi_{i_n}^{-1}[E_n]$}

\noindent for every $n\in\Bbb N$.   Because $X$, being a product of
compact spaces, is compact, there is an
$x\in\bigcap_{n\in\Bbb N}\overline{C_n}$.   We know that $G_x$ is
determined by coordinates in $J$, so
$G_x=\tilde\pi^{-1}[\tilde\pi[G_x]]$, where $\tilde\pi$ is the canonical
map from $X$ onto $Y=\prod_{i\in J}X_i$.   $V=\tilde\pi[G_x]$ is open,
so there must be a finite set $K\subseteq J$ and a family
$\family{i}{K}{V_i}$ such that $x(i)\in V_i\in\frak T_i$ for every
$i\in K$ and $\{y:y\in Y,\,y(i)\in V_i$ for every $i\in K\}$ is included
in $V$.   This means that
$\{z:z\in X,\,z(i)\subseteq V_i$ for every $i\in K\}$ is included in
$G_x$.   Now, for each $i\in K$, there is some $m\in\Bbb N$ such that
$i=i_m$ and
$x(i)\in E_m\subseteq V_i$.   Because $x\in\overline{C}_{m+1}$,
$\pi_{i_m}^{-1}[E_m]$ cannot be disjoint from $C_{m+1}$, and
$C_{m+1}\subseteq\pi_{i_m}^{-1}[E_m]\subseteq\pi_i^{-1}[V_i]$.

But this means that, for any $n$ large enough, $C_n\subseteq G_x$ and
$\pmb{t}=\{(x,C_n)\}$ is a $\delta$-fine member of $T$ with
$W_{\pmb{t}}=C_n$;  contradicting the requirement that
$C_n\notin\Cal D$.\ \Bang

This contradiction shows that 481G(vii) also is satisfied.

To see that $\Delta$ is countably full, note that if $\sequencen{\delta_n}$
is a sequence in $\Delta$, we have for each $n\in\Bbb N$ a countable set
$J_n\subseteq I$ and a family $\family{x}{X}{G_{nx}}$ of open sets, all
determined by coordinates in $J_n$, such
that $x\in G_{nx}$ and $(x,C)\in\delta_n$ whenever $x\in X$ and
$C\subseteq G_{nx}$.   Now, given $\phi:X\to\Bbb N$, set
$\delta=\{(x,C):x\in X$, $C\subseteq G_{\phi(x),x}\}$, and observe that
$\delta\in\Delta$ and that $(x,C)\in\delta_{\phi(x)}$ whenever
$(x,C)\in\Delta$.

If $I'$ is countable, then $\Delta$ is the set of all neighbourhood gauges
on $X$, so is full.   If $I'$ is uncountable, then for $j\in I'$ and
$x\in X$ choose a proper open subset $H_{jx}$ of $X_j$ containing
$x(j)$ and set $G_{jx}=\{y:y\in X$, $y(j)\in H_{jx}\}$.   For $j\in I'$ set
$\delta_j=\{(x,C):C\subseteq G_{jx}\}\in\Delta$.   Let $\phi:X\to I'$ be
any function such that $\phi[X]$ is uncountable;
then there is no $\delta\in\Delta$ such that
$(x,C)\in\delta_{\phi(x)}$ whenever $(x,C)\in\delta$, so
$\family{x}{X}{\delta_{\phi(x)}}$ witnesses that $\Delta$ is not full.
}%end of proof of 481P

\leader{481Q}{The approximately continuous Henstock
integral}\cmmnt{ ({\smc Gordon 94}, chap.\ 16)} Let $\mu$ be
Leb\discretionary{-}{}{}esgue
measure on $\Bbb R$.   As in 481K, let $\Cal C$ be the family of
non-empty bounded intervals in $\Bbb R$, $T$
the straightforward set of tagged partitions generated by
$\{(x,C):C\in\Cal C$, $x\in\overline{C}\}$, and
$\frak R=\{\Cal R_{ab}:a,\,b\in\Bbb R\}$, where
$\Cal R_{ab}=\{\Bbb R\setminus[c,d]:c\le a,\,d\ge b\}\cup\{\emptyset\}$
for $a$, $b\in\Bbb R$.

\cmmnt{This time, define gauges as follows.}   Let $\Epsilon$ be the
set of families $\pmb{e}=\family{x}{\Bbb R}{E_x}$ where every $E_x$ is a
measurable set containing $x$ such that $x$ is a density point of
$E_x$\cmmnt{ (definition:  223B)}.   For
$\pmb{e}=\family{x}{\Bbb R}{E_x}\in\Epsilon$, set

\Centerline{$\delta_{\pmb{e}}=\{(x,C):C\in\Cal C,
  \,x\in\overline{C},\,\inf C\in E_x$ and $\sup C\in E_x\}$.}

\noindent Set $\Delta=\{\delta_{\pmb{e}}:\pmb{e}\in\Epsilon\}$.   Then
$(X,T,\Delta,\frak R)$ is a tagged-partition structure allowing
subdivisions, witnessed by $\Cal C$, and $\Delta$ is full.

\proof{{\bf (a)} Turning to 481G, we find, as usual, that most of the
conditions are satisfied for elementary reasons.   Since we have here a
new kind of gauge, we had better check 481G(ii);  but if
$\pmb{e}=\family{x}{\Bbb R}{E_x}$ and
$\pmb{e}'=\family{x}{\Bbb R}{E'_x}$ both belong to $\Epsilon$, so does
$\pmb{e}\wedge\pmb{e}'=\family{x}{\Bbb R}{E_x\cap E'_x}$, because

$$\eqalign{\liminf_{\eta\downarrow 0}\Bover1{2\eta}
&\mu([x-\eta,x+\eta]\cap E_x\cap E'_x)\cr
&\ge\lim_{\eta\downarrow 0}\Bover1{2\eta}
  \bigl(\mu([x-\eta,x+\eta]\cap E_x)
  +\mu([x-\eta,x+\eta]\cap E'_x)-2\eta\bigr)
=1\cr}$$

\noindent for every $x$;  and now
$\delta_{\pmb{e}}\cap\delta_{\pmb{e}'}=\delta_{\pmb{e}\wedge\pmb{e}'}$
belongs to $\Delta$.   Everything else we have done before, except, of
course, (vii).

\medskip

{\bf (b)} So take any $\delta\in\Delta$;  express $\delta$ as
$\delta_{\pmb{e}}$, where
$\pmb{e}=\family{x}{\Bbb R}{E_x}\in\Epsilon$.   For $x$, $y\in\Bbb R$,
write $x\frown y$ if $x\le y$ and
$E_x\cap E_y\cap[x,y]\ne\emptyset$;  note that we always have
$x\frown x$.   For $x\in\Bbb R$, let $\eta_x>0$ be such that
$\mu(E_x\cap[x-\eta,x+\eta])\ge\bover53\eta$ whenever
$0\le\eta\le\eta_x$.

Fix $a<b$ in $\Bbb R$ for the moment.   Say that a finite string
$(x_0,\ldots,x_n)$ is `acceptable' if
$a\le x_0\frown\ldots\frown x_n\le b$ and
$\mu([x_0,x_n]\cap\bigcup_{i<n}E_{x_i}^+)\ge\bover12(x_n-x_0)$, where
$E_x^+=E_x\cap\coint{x,\infty}$ for $x\in\Bbb R$.   Observe that if
$(x_0,\ldots,x_m)$ and $(x_m,x_{m+1},\ldots,x_n)$ are both acceptable,
so is $(x_0,\ldots,x_n)$.   For $x\in[a,b]$, set

\Centerline{$h(x)=\sup\{x_n:(x,x_1,\ldots,x_n)$ is acceptable$\}$;}

\noindent this is defined in $[x,b]$ because the string $(x)$ is
acceptable.   If $a\le x<b$, then $(x,y)$ is acceptable whenever
$y\in E_x$ and $0\le y\le\min(b,x+\eta_x)$, so $h(x)>x$.
Now choose sequences $\sequence{i}{x_i}$, $\sequence{k}{n_k}$
inductively, as follows.   $n_0=0$ and $x_0=a$.   Given that
$x_i\frown x_{i+1}$ for $i<n_k$ and that $(x_{n_j},\ldots,x_{n_k})$ is
acceptable for every $j\le k$, let $n_{k+1}>n_k$,
$(x_{n_k+1},\ldots,x_{n_{k+1}})$ be such that
$(x_{n_k},\ldots,x_{n_{k+1}})$ is acceptable and
$x_{n_{k+1}}\ge\bover12(x_{n_k}+h(x_{n_k}))$;  then
$(x_{n_j},\ldots,x_{n_{k+1}})$ is acceptable for any $j\le k+1$;
continue.

At the end of the induction, set

\Centerline{$c=\sup_{i\in\Bbb N}x_i=\sup_{k\in\Bbb N}x_{n_k}$.}

\noindent Then there are infinitely many $i$ such that $x_i\frown c$.
\Prf\ For any $k\in\Bbb N$,

$$\eqalignno{\mu([x_{n_k},c]\cap\bigcup_{i\ge n_k}E_{x_i}^+)
&=\lim_{l\to\infty}
\mu\bigl([x_{n_k},x_{n_l}]\cap\bigcup_{n_k\le i<n_l}E_{x_i}^+\bigr)\cr
&\ge\lim_{l\to\infty}\Bover12(x_{n_l}-x_{n_k})\cr
\displaycause{because $(x_{n_k},\ldots,x_{n_l})$ is always an acceptable
string}
&=\Bover12(c-x_{n_k}).\cr}$$

\noindent But this means that if we take $k$ so large that
$x_{n_k}\ge c-\eta_c$, so that
$\mu(E_c\cap[x_{n_k},c])\ge\bover23(c-x_{n_k})$, there must be some
$z\in E_c\cap\bigcup_{i\ge n_k}E_i^+\cap[x_{n_k},c]$;  and if $i\ge n_k$
is such that
$z\in E_i^+$, then $z$ witnesses that $x_i\frown c$.   As $k$ is
arbitrarily large, we have the result.\ \Qed

\Quer\ If $c<b$, take any $y\in E_c$ such that $c<y\le\min(c+\eta_c,b)$.
Take $k\in\Bbb N$ such that
$x_{n_k}\ge c-\bover13(y-c)$, and $j\ge n_k$ such that $x_j\frown c$.
In this case, $(x_{n_k},x_{n_k+1},\ldots,x_j,c,y)$ is an acceptable
string, because

\Centerline{$\mu E_c^+\cap[x_{n_k},y]\ge\mu E_c\cap[c,y]
\ge\bover23(y-c)\ge\bover12(y-x_{n_k})$.}

\noindent But this means that $h(x_{n_k})\ge y$, so that

\Centerline{$x_{n_{k+1}}\le c<\Bover12(x_{n_k}+h(x_{n_k}))$,}

\noindent contrary to the choice of
$x_{n_k+1},\ldots,x_{n_{k+1}}$.\ \Bang

Thus $c=b$.   We therefore have a $j\in\Bbb N$ such that $x_j\frown b$,
and $a=x_0\frown\ldots\frown x_j\frown b$.

\medskip

{\bf (c)} Now suppose that $C\in\Cal C$.   Set $a=\inf C$ and
$b=\sup C$.   If $a=b$, then $\pmb{t}=\{(a,C)\}\in T$ and
$W_{\pmb{t}}=C$.   Otherwise, (b) tells us that we have $x_0,\ldots,x_n$
such that $a=x_0\frown\ldots\frown x_n=b$.   Choose
$a_i\in[x_{i-1},x_i]\cap E_{x_{i-1}}\cap E_{x_i}$ for $1\le i\le n$.
Set $C_i=\coint{a_i,a_{i+1}}$ for $1\le i<n$, $C_0=\coint{a,a_1}$,
$C_n=[a_n,b]$;  set $I=\{i:i\le n,\,C\cap C_i\ne\emptyset\}$;  and
check that $(x_i,C\cap C_i)\in\delta$ for $i\in I$, so that
$\pmb{t}=\{(x_i,C\cap C_i):i\in I\}$ is a $\delta$-fine member of $T$
with $W_{\pmb{t}}=C$.   As $C$ and $\delta$ are arbitrary, 481G(vii) is
satisfied.

\medskip

{\bf (d)} $\Delta$ is full.   \Prf\ Let $\family{x}{\Bbb R}{\delta'_x}$
be a
family in $\Delta$.   For each $x\in X$, there is a measurable set $E_x$
such that $x$ is a density point of $E_x$ and $(x,C)\in\delta'_x$ whenever
$C\in\Cal C$, $x\in\overline{C}$ and both $\inf C$, $\sup C$ belong to
$E_x$.   Set $\pmb{e}=\family{x}{\Bbb R}{E_x}\in\Epsilon$;
then $(x,C)\in\delta'_x$ whenever $(x,C)\in\delta_{\pmb{e}}$.\ \Qed
}%end of proof of 481Q

\exercises{\leader{481X}{Basic exercises (a)}
%\spheader 481Xa
Let $X$, $\Cal C$, $T$ and $\Cal F$ be as in 481C.   Show that if
$f:X\to\Bbb R$, $\mu:\Cal C\to\Bbb R$ and $\nu:\Cal C\to\Bbb R$ are
functions, then $I_{\mu+\nu}(f)=I_{\mu}(f)+I_{\nu}(f)$ whenever the
right-hand side is defined.
%481C

\sqheader 481Xb Let $I$ be any set.   Set
$T=\{\{(i,\{i\}):i\in J\}:J\in[I]^{<\omega}\}$,
$\delta=\{(i,\{i\}):i\in I\}$, $\Delta=\{\delta\}$.   For
$J\in[I]^{<\omega}$ set $\Cal R_J
=\{I\setminus K:J\subseteq K\in[I]^{<\omega}\}
  \cup\{\emptyset\}$;  set
$\frak R=\{\Cal R_J:J\in[I]^{<\omega}\}$.   Show that
$(I,T,\Delta,\frak R)$ is a tagged-partition structure
allowing subdivisions, witnessed by $[I]^{<\omega}$, and that $\Delta$ is
full.   Let
$\nu:[I]^{<\omega}\to\Bbb R$ be any additive functional.   Show that,
for a function $f:I\to\Bbb R$,
$\lim_{\pmb{t}\to\Cal F(T,\Delta,\frak R)}S_{\pmb{t}}(f,\nu)
=\sum_{i\in I}f(i)\nu(\{i\})$ if either exists in $\Bbb R$.
%481G

\sqheader 481Xc
Set $T=\{\{(n,\{n\}):n\in I\}:I\in[\Bbb Z]^{<\omega}\}$,
$\delta=\{(n,\{n\}):n\in\Bbb Z\}$, $\Delta=\{\delta\}$.   For
$I\in[\Bbb Z]^{<\omega}$ and $m$, $n\in\Bbb N$ set
$R_{mn}=\Bbb Z\setminus\{i:-m\le i\le n\}$,
$\Cal R'_n=\{R_{kl}:k,\,l\ge n\}\cup\{\emptyset\}$,
$\Cal R''_n=\{R_{kk}:k\ge n\}\cup\{\emptyset\}$,
$\frak R'=\{\Cal R'_n:n\in\Bbb N\}$,
$\frak R''=\{\Cal R''_n:n\in\Bbb N\}$.   Show that
$(\Bbb Z,T,\Delta,\frak R')$ and
$(\Bbb Z,T,\Delta,\frak R'')$ are tagged-partition structures
allowing subdivisions, witnessed by $[\Bbb Z]^{<\omega}$, and that $\Delta$
is full.   Let $\mu$ be
counting measure on $\Bbb Z$.   Show that, for a function
$f:\Bbb Z\to\Bbb R$,
(i) $\lim_{\pmb{t}\to\Cal F(T,\Delta,\frak R')}S_{\pmb{t}}(f,\nu)
=\lim_{m,n\to\infty}\sum_{i=-m}^nf(i)$ if either is defined in $\Bbb R$
(ii) $\lim_{\pmb{t}\to\Cal F(T,\Delta,\frak R'')}S_{\pmb{t}}(f,\mu)
=\lim_{n\to\infty}\sum_{i=-n}^nf(i)$ if either is defined in $\Bbb R$.
%481G

\spheader 481Xd Set $X=\Bbb N\cup\{\infty\}$, and let $T$ be the
straightforward set of tagged partitions generated by
$\{(n,\{n\}):n\in\Bbb N\}\cup\{(\infty,X\setminus n):n\in\Bbb N\}$
(interpreting a member of $\Bbb N$ as the set of its predecessors).
For $n\in\Bbb N$ set $\delta_n
=\{(i,\{i\}):i\in\Bbb N\}\cup\{(\infty,A):A\subseteq X\setminus n\}$;
set $\Delta=\{\delta_n:n\in\Bbb N\}$.   Show that
$(X,T,\Delta,\{\{\emptyset\}\})$ is a tagged-partition structure
allowing subdivisions, witnessed by
$\Cal C=[\Bbb N]^{<\omega}\cup\{X\setminus I:I\in[\Bbb N]^{<\omega}\}$, and
that $\Delta$ is full.
Let $h:\Bbb N\to\Bbb R$ be any function, and define
$\nu:\Cal C\to\Bbb R$ by setting $\nu I=\sum_{i\in I}h(i)$,
$\nu(X\setminus I)=-\nu I$ for $I\in[\Bbb N]^{<\omega}$.   Let
$f:X\to\Bbb R$ be any function such that $f(\infty)=0$.   Show that
$\lim_{\pmb{t}\to\Cal F(T,\Delta,\{\{\emptyset\}\})}S_{\pmb{t}}(f,\mu)
=\lim_{n\to\infty}\sum_{i=0}^nf(i)h(i)$ if either is defined in
$\Bbb R$.
%481G

\sqheader 481Xe Take $X$, $T$, $\Delta$ and $\frak R$ as in 481I.   Show
that if $\mu$ is Lebesgue measure on $[a,b]$ then the gauge integral
$I_{\mu}=\lim_{\pmb{t}\to\Cal F(T,\Delta,\frak R)}S_{\pmb{t}}(.,\mu)$ is
the ordinary Riemann integral $\Rint_a^b$ as described in
134K.   \Hint{show first that they agree on step-functions.}
%481I

\sqheader 481Xf Let $(X,\Sigma,\mu)$ be a semi-finite measure space, and
$\Sigma^f$ the family of measurable sets of finite measure.   Let
$T$ be the straightforward set of tagged partitions generated by
$\{(x,E):x\in E\in\Sigma^f\}$.   For $E\in\Sigma^f$ and $\epsilon>0$ set
$\Cal R_{E\epsilon}=\{F:F\in\Sigma,\,\mu(E\setminus F)\le\epsilon\}$;
set $\frak R=\{\Cal R_{E\epsilon}:E\in\Sigma^f,\,\epsilon>0\}$.   Let
$\frak E$ be the family of countable partitions of $X$ into measurable
sets, and set
$\delta_{\Cal E}=\bigcup_{E\in\Cal E}\{(x,A):x\in E,\,A\subseteq E\}$
for $\Cal E\in\frak E$,
$\Delta=\{\delta_{\Cal E}:\Cal E\in\frak E\}$.   Show that
$(X,T,\Delta,\frak R)$ is a
tagged-partition structure allowing subdivisions, witnessed by
$\Sigma^f$.      In what circumstances is
$\Delta$ full or countably full?
Show that, for a function $f:X\to\Bbb R$,
$\int fd\mu=\lim_{\pmb{t}\to\Cal F(T,\Delta,\frak R)}S_{\pmb{t}}(f,\mu)$
if either is defined in $\Bbb R$.   \Hint{when showing that if
$I_{\mu}(f)$ is defined then $f$ is $\mu$-virtually measurable, you will
need 413G or something similar;  compare 482E.}
%481N

\spheader 481Xg Let $(X,\Sigma,\mu)$ be a totally finite measure space,
and $T$ the straightforward set of tagged partitions generated by
$\{(x,E):x\in E\in\Sigma\}$.   Let $\frak E$ be the family of finite
partitions of $X$ into measurable sets, and set
$\delta_{\Cal E}=\bigcup_{E\in\Cal E}\{(x,A):x\in E,\,A\subseteq E\}$
for $\Cal E\in\frak E$, $\Delta=\{\delta_{\Cal E}:\Cal E\in\frak E\}$.
Show that $(X,T,\Delta,\{\{\emptyset\}\})$ is a
tagged-partition structure allowing subdivisions, witnessed by $\Sigma$.
In what circumstances is
$\Delta$ full or countably full?
Show that, for a
function $f:X\to\Bbb R$,
$I_{\mu}(f)=\lim_{\pmb{t}\to\Cal F(T,\Delta,\frak R)}S_{\pmb{t}}(f,\mu)$
is defined iff $f\in\eusm L^{\infty}(\mu)$ (definition:  243A), and
that then $I_{\mu}(f)=\int fd\mu$.
%481Xf, 481N

\spheader 481Xh Let $X$ be a zero-dimensional compact Hausdorff space
and $\Cal E$ the algebra of open-and-closed subsets of $X$.   Let $T$ be
the straightforward set of tagged partitions generated by
$\{(x,E):x\in E\in\Cal E\}$.   Let
$\Delta$ be the set of all neighbourhood gauges on $X$.   Show that
$(X,T,\Delta,\{\{\emptyset\}\})$ is a tagged-partition structure
allowing subdivisions, witnessed by $\Cal E$.   Now let
$\nu:\Cal E\to\Bbb R$ be an additive functional, and set $I_{\nu}(f)
=\lim_{\pmb{t}\to\Cal F(T,\Delta,\{\{\emptyset\}\})}
S_{\pmb{t}}(f,\nu)$ when $f:X\to\Bbb R$ is such that the limit is defined.   (i) Show that $I_{\nu}(\chi E)=\nu E$ for every $E\in\Cal E$.
(ii) Show that if $\nu$ is bounded then $I_{\nu}(f)$ is defined for every
$f\in C(X)$, and is equal to $\dashint fd\nu$ as defined in 363L, if we
identify $X$ with the Stone space of $\frak A$ and $C(X)$ with
$L^{\infty}(\frak A)$.
%+

\spheader 481Xi Let $X$ be a set, $\Delta$ a set of gauges on $X$,
$\frak R$ a collection of families of subsets of $X$, and $T$ a set of
tagged partitions on $X$ which is compatible with $\Delta$ and
$\frak R$.   Let $H\subseteq X$ be such that there is a
$\tilde\delta\in\Delta$ such that $H\cap A=\emptyset$ whenever
$x\in X\setminus H$ and $(x,A)\in\tilde\delta$, and set
$\delta_H=\{(x,A\cap H):x\in H,\,(x,A)\in\delta\}$ for
$\delta\in\Delta$, $\Delta_H=\{\delta_H:\delta\in\Delta\}$,
$\frak R_H=\{\{R\cap H:R\in\Cal R\}:\Cal R\in\frak R\}$,
$T_H=\{\{(x,C\cap H):(x,C)\in\pmb{t},\,x\in H\}:\pmb{t}\in T\}$.    Show
that $T_H$ is compatible with $\Delta_H$ and $\frak R_H$.
%+ subspaces

\spheader 481Xk\dvAnew{2012} Let $X$ be a set, $\Sigma$ an algebra of
subsets of $X$, and $\nu:\Sigma\to\coint{0,\infty}$ an additive functional.
Set $Q=\{(x,C):x\in C\in\Sigma\}$ and let $T$ be the straightforward set of
tagged partitions generated by $Q$.   Let $\Bbb E$ be the set of disjoint
families $\Cal E\subseteq\Sigma$ such that
$\sum_{E\in\Cal E}\nu E=\nu X$, and
$\Delta=\{\delta_{\Cal E}:\Cal E\in\Bbb E\}$, where

\Centerline{$\delta_{\Cal E}
=\{(x,C):(x,C)\in Q$ and there is an $E\in\Cal E$ such that
$C\subseteq E\}$}

\noindent for $\Cal E\in\Bbb E$.
Set $\frak R=\{\Cal R_{\epsilon}:\epsilon>0\}$ where
$\Cal R_{\epsilon}=\{E:E\in\Sigma$, $\nu E\le\epsilon\}$ for $\epsilon>0$.
Show that $(X,T,\Delta,\frak R)$ is a tagged-partition structure allowing
subdivisions, witnessed by $\Sigma$.

\leader{481Y}{Further exercises (a)}
%\spheader 481Ya
Suppose that $[a,b]$, $\Cal C$, $T$ and $\Delta$ are as in 481J.   Let
$T'\subseteq[[a,b]\times\Cal C]^{<\omega}$ be the set of tagged
partitions $\pmb{t}=\{(x_i,[a_i,a_{i+1}]):i<n\}$ where
$a=a_0\le x_0\le a_1\le x_2\le a_2\le\ldots\le x_{n-1}\le a_n=b$.
Show that $T'$, as well as $T$, is compatible with $\Delta$ in the sense
of 481Ea;  let $\Cal F'$, $\Cal F$ be the corresponding filters on $T'$
and $T$.   Show that if $\nu:\Cal C\to\Bbb R$ is a functional which is
additive in the sense that $\nu(C\cup C')=\nu C+\nu C'$ whenever $C$,
$C'$ are disjoint members of $\Cal C$ with union in $\Cal C$, and if
$\nu\{x\}=0$ for every $x\in[a,b]$ and $f:[a,b]\to\Bbb R$ is any
function, then $I'_{\nu}(f)=I_{\nu}(f)$ if either is defined, where

\Centerline{$I_{\nu}(f)=\lim_{\pmb{t}\to\Cal F}S_{\pmb{t}}(f,\nu)$,
\quad$I'_{\nu}(f)=\lim_{\pmb{t}\to\Cal F'}S_{\pmb{t}}(f,\nu)$}

\noindent are the gauge integrals associated with $(T,\Cal F)$ and
$(T',\Cal F')$.
%481J

\spheader 481Yb Let us say that a family $\frak R$ of residual families
is `the simple residual structure complementary to
$\Cal H\subseteq\Cal PX$' if
$\frak R=\{\Cal R_H:H\in\Cal H\}$, where
$\Cal R_H=\{X\setminus H':H\subseteq H'\in\Cal H\}\cup\{\emptyset\}$ for
each $H\in\Cal H$.   Suppose that, for each member $i$ of a non-empty
finite set $I$, $(X_i,T_i,\Delta_i,\frak R_i)$ is a tagged-partition
structure allowing subdivisions, witnessed by an upwards-directed family
$\Cal C_i\subseteq\Cal PX_i$, where $X_i$ is a topological space,
$\Delta_i$ is the set of all neighbourhood gauges on $X_i$, and
$\frak R_i$ is the simple residual structure complementary to
$\Cal C_i$.
Set $X=\prod_{i\in I}X_i$ and let $\Delta$ be the set of neighbourhood
gauges on $X$;   let $\Cal C$ be
$\{\prod_{i\in I}C_i:C_i\in\Cal C_i$ for each $i\in I\}$, and $\frak R$
the simple residual structure based on $\Cal C$;  and let $T$ be the
straightforward set of tagged partitions generated by
$\{(\familyiI{x_i},\prod_{i\in I}C_i):\{(x_i,C_i)\}\in T_i$ for every
$i\in I\}$.
Show that $(X,T,\Delta,\frak R)$ is a tagged-partition structure
allowing subdivisions, witnessed by $\Cal C$.
%481O 481P

\spheader 481Yc Give an example to show that, in 481Xi,
$(X,T,\Delta,\frak R)$ can be a tagged-partition structure allowing
subdivisions, while $(H,T_H,\Delta_H,\frak R_H)$ is not.
%481Xi mt48bits
}%end of exercises

\endnotes{
\Notesheader{481} In the examples above I have tried to give an idea of
the potential versatility of the ideas here.   Further examples may be
found in {\smc Henstock 91}.   The goal of 481A-481F is the formula
$I_{\nu}(f)
=\lim_{\pmb{t}\to\Cal F(T,\Delta,\frak R)}S_{\pmb{t}}(f,\nu)$ (481C, 481E);
the elaborate notation reflects the variety of the applications.   One
of these is a one-step definition of the ordinary integral (481Xf).   In
\S483 I will show that the Henstock integral (481J-481K) properly
extends the Lebesgue integral on $\Bbb R$.   In 481Xc I show how
adjusting $\frak R$ can change the class of integrable functions;  in
481Xd I show how a similar effect can sometimes be achieved by adding a
point at infinity and adjusting $T$ and $\Delta$.   As will become
apparent in later sections, one of the great strengths of gauge
integrals is their ability to incorporate special limiting processes.
Another is the fact that we don't need to assume that the functionals
$\nu$ are countably additive;  see 481Xd.   In the formulae of this
section, I don't even ask for finite additivity;  but of course the
functional $I_{\nu}$ is likely to have a rather small domain if $\nu$
behaves too erratically.

`Gauges', as I describe them here, have moved rather briskly forward
from the metric gauges $\delta_h$ (481Eb), which have sufficed for most
of the gauge integrals so far described.   But the generalization
affects only the notation, and makes it clear why so much of the theory
of the ordinary Henstock integral applies equally well to the
`approximately continuous Henstock integral' (481Q), for instance.
You will observe that the sets $Q$ of 481Ba are `gauges' in the
wide sense used here.   But (as the examples of this section show
clearly) we generally use them in a different way.

In ordinary measure theory, we have a fairly straightforward theory of
subspaces (\S214) and a rather deeper theory of product spaces
(chap.\ 25).   For gauge integrals, there are significant difficulties
in the theory of subspaces, some of which will appear in the next
section (see 482G-482H).   For closed subspaces, something can be done,
as in 481Xi;  but the procedure suggested there may lose some
essential element of the original tagged-partition structure (481Yc).
For products of gauge integrals, we do have a reasonably satisfying
version of Fubini's theorem (482M);  I offer 481O and 481P as
alternative approaches.   However, the example of the Pfeffer integral
(\S484) shows that other constructions may be more effective tools
for geometric measure theory.

You will note the concentration on `neighbourhood gauges' (481Eb)
in the work above.   This is partly because they are `full' in the sense of
481Ec.   As
will appear repeatedly in the next section, this flexibility in
constructing gauges is just what one needs when proving that functions
are gauge-integrable.

While I have used such phrases as `Henstock integral', `symmetric
Riemann-complete integral' above, I have not in fact discussed
integrals here, except in the exercises;  in most of the examples in
481I-481Q there is no mention of any functional $\nu$ from which a gauge
integral $I_{\nu}$ can be defined.   The essence of the method is that
we can set up a tagged-partition structure quite independently of any
set function, and it turns out that the properties of a gauge integral
depend more on this structure than on the measure involved.
}%end of notes

\discrpage


\frfilename{mt482.tex}
\versiondate{11.5.10/14.3.11}
\copyrightdate{2001}

\def\closure{\mathop{\text{cl}}}
\def\Eta{\text{H}}

\def\chaptername{Gauge integrals}
\def\sectionname{General theory}

\newsection{482}

I turn now to results which can be applied to a wide variety of
tagged-partition structures.   The first step is a `Saks-Henstock'
lemma
(482B), a fundamental property of tagged-partition structures allowing
subdivisions.   In order to relate gauge integrals to the ordinary
integrals treated elsewhere in this treatise, we need to know when
gauge-integrable functions are measurable (482E) and when integrable
functions are gauge-integrable (482F).   There are significant
difficulties when we come to interpret gauge integrals over subspaces,
but I give a partial result in 482G.   482I, 482K and 482M
are gauge-integral versions of the Fundamental Theorem of Calculus,
B.Levi's theorem and Fubini's theorem, while 482H is a limit theorem of a
new kind, corresponding to classical improper integrals.

Henstock's integral (481J-481K) remains the most important example and
the natural test case for the ideas here;  I will give the details in
the next section, and you may wish to take the two sections in
parallel.

\leader{482A}{Lemma} Suppose that $(X,T,\Delta,\frak R)$ is a
tagged-partition structure allowing subdivisions\cmmnt{ (481G)},
witnessed by $\Cal C\subseteq\Cal PX$.

(a) Whenever $\delta\in\Delta$, $\Cal R\in\frak R$ and $E$ belongs to
the subalgebra of $\Cal PX$ generated by $\Cal C$, there is a
$\delta$-fine $\pmb{s}\in T$ such that $W_{\pmb{s}}\subseteq E$ and
$E\setminus W_{\pmb{s}}\in\Cal R$.

(b) Whenever $\delta\in\Delta$, $\Cal R\in\frak R$ and $\pmb{t}\in T$
is $\delta$-fine, there is a $\delta$-fine $\Cal R$-filling 
$\pmb{t}'\in T$ including $\pmb{t}$.

(c) Suppose that $f:X\to\Bbb R$, $\nu:\Cal C\to\Bbb R$,
$\delta\in\Delta$, $\Cal R\in\frak R$ and $\epsilon\ge 0$ are such
that
$|S_{\pmb{t}}(f,\nu)-S_{\pmb{t}'}(f,\nu)|\le\epsilon$ whenever
$\pmb{t}$, $\pmb{t}'\in T$ are $\delta$-fine and $\Cal R$-filling.
Then

\quad(i) $|S_{\pmb{t}}(f,\nu)-S_{\pmb{t}'}(f,\nu)|\le\epsilon$
whenever
$\pmb{t}$, $\pmb{t}'\in T$ are $\delta$-fine and
$W_{\pmb{t}}=W_{\pmb{t}'}$;

\quad(ii) whenever $\pmb{t}\in T$ is $\delta$-fine, and
$\delta'\in\Delta$, there is a $\delta'$-fine $\pmb{s}\in T$ such that
$W_{\pmb{s}}\subseteq W_{\pmb{t}}$ and
$|S_{\pmb{s}}(f,\nu)-S_{\pmb{t}}(f,\nu)|\le\epsilon$.

(d) Suppose that $f:X\to\Bbb R$ and
$\nu:\Cal C\to\Bbb R$ are such that
$I_{\nu}(f)
=\lim_{\pmb{t}\to\Cal F(T,\Delta,\frak R)}S_{\pmb{t}}(f,\nu)$
is defined\cmmnt{, where $\Cal F(T,\Delta,\frak R)$ is the filter
described in 481F}.   Then for any $\epsilon>0$ there is a
$\delta\in\Delta$
such that $S_{\pmb{t}}(f,\nu)\le I_{\nu}(f)+\epsilon$ for every
$\delta$-fine $\pmb{t}\in T$.

(e)\dvAnew{2011} Suppose that $f:X\to\Bbb R$ and
$\nu:\Cal C\to\Bbb R$ are such that $I_{\nu}(f)
=\lim_{\pmb{t}\to\Cal F(T,\Delta,\frak R)}S_{\pmb{t}}(f,\nu)$
is defined.   Then for any $\epsilon>0$ there is a $\delta\in\Delta$ 
such that $|S_{\pmb{t}}(f,\nu)|\le\epsilon$ whenever $\pmb{t}\in T$ is
$\delta$-fine and $W_{\pmb{t}}=\emptyset$.

\proof{{\bf (a)} By 481He, there is a non-increasing sequence
$\sequence{k}{\Cal R_k}$ in $\frak R$ such
that $\bigcup_{i\le k}A_i\in\Cal R$ whenever $A_i\in\Cal R_i$ for
every $i\le k$ and $\langle A_i\rangle_{i\le k}$ is disjoint.   Let
$\Cal C_0$ be a finite subset of $\Cal C$ such that $E$
belongs to the subalgebra of $\Cal PX$ generated by $\Cal C_0$, and
let $\Cal C_1\supseteq\Cal C_0$ be a finite subset of $\Cal C$ such that
$X\setminus W\in\Cal R_0$, where $W=\bigcup\Cal C_1$ (481G(v)).   Then
either $E\subseteq W$ or $E\supseteq X\setminus W$.   In either case,
$E\cap W$ belongs to the ring generated by $\Cal C$, so is expressible
as $\bigcup_{i<n}C_i$ where $\ofamily{i}{n}{C_i}$ is a disjoint family
in $\Cal C$ (481Hd).

For each $i<n$, let $\pmb{s}_i$ be a $\delta$-fine member of $T$ such
that $W_{\pmb{s}_i}\subseteq C_i$ and
$C_i\setminus W_{\pmb{s}_i}\in\Cal R_{i+1}$ (481G(vii)).   Set
$\pmb{s}=\bigcup_{i<n}\pmb{s}_i$.   Because
$\ofamily{i}{n}{W_{\pmb{s}_i}}$ is disjoint and $T$ is a
straightforward
set of tagged partitions, $\pmb{s}\in T$;  $\pmb{s}$ is $\delta$-fine
because every $\pmb{s}_i$ is;
$W_{\pmb{s}}=\bigcup_{i<n}W_{\pmb{s}_i}$
is included in $E$;  and $E\setminus W_{\pmb{s}}$ is either
$\bigcup_{i<n}C_i\setminus W_{\pmb{s}_i}$ or
$(X\setminus W)\cup\bigcup_{i<n}(C_i\setminus W_{\pmb{s}_i})$, and in
either case belongs to $\Cal R$, by the choice of
$\sequencen{\Cal R_n}$.

\medskip

{\bf (b)} Set $E=X\setminus W_{\pmb{t}}$.   By (a), there is a
$\delta$-fine $\pmb{s}\in T$ such that $W_{\pmb{s}}\subseteq E$ and
$E\setminus W_{\pmb{s}}\in\Cal R$.   Set
$\pmb{t}'=\pmb{t}\cup\pmb{s}$;
this works.

\medskip

{\bf (c)(i)} As in (b), there is a $\delta$-fine $\pmb{s}\in T$ such
that $W_{\pmb{s}}\cap W_{\pmb{t}}=\emptyset$ and
$\pmb{t}\cup\pmb{s}$ is $\Cal R$-filling.   Now
$W_{\pmb{t}\cup\pmb{s}}=W_{{\pmb{t}'}\cup\pmb{s}}$,
so ${\pmb{t}'}\cup\pmb{s}$ also is $\Cal R$-filling, and

\Centerline{$|S_{\pmb{t}}(f,\nu)-S_{\pmb{t}'}(f,\nu)|
=|S_{\pmb{t}\cup\pmb{s}}(f,\nu)
  -S_{\pmb{t}'\cup\pmb{s}}(f,\nu)|
\le\epsilon$.}

\medskip

\quad{\bf (ii)} Replacing $\delta'$ by a lower bound of
$\{\delta,\delta'\}$ in $\Delta$ if necessary, we may suppose that
$\delta'\subseteq\delta$.   Enumerate $\pmb{t}$ as
$\ofamily{i}{n}{(x_i,C_i)}$.   Let $\sequence{k}{\Cal R_k}$ be a
sequence in $\frak R$ such that $\bigcup_{i\le k}A_i\in\Cal R$
whenever
$\langle A_i\rangle_{i\le k}$ is disjoint and $A_i\in\Cal R_i$ for
every
$i\le k$.   For each $i<n$,
let $\pmb{s}_i$ be a $\delta'$-fine member of $T$ such that
$W_{\pmb{s}_i}\subseteq C_i$ and
$C_i\setminus W_{\pmb{s}_i}\in\Cal R_{i+1}$, and set
$\pmb{s}=\bigcup_{i<n}\pmb{s}_i$, so that $\pmb{s}\in T$ is
$\delta'$-fine.   By (a), there is a $\delta$-fine $\pmb{u}\in T$ such
that $W_{\pmb{u}}\cap W_{\pmb{t}}=\emptyset$ and
$X\setminus(W_{\pmb{t}}\cup W_{\pmb{u}})\in\Cal R_0$.   Set
$\pmb{t}'=\pmb{t}\cup\pmb{u}$, $\pmb{s}'=\pmb{s}\cup\pmb{u}$;
then $\pmb{t}'$ and $\pmb{s}'$ are $\delta$-fine and
$\Cal R$-filling, because

\Centerline{$X\setminus W_{\pmb{s}'}
=(X\setminus(W_{\pmb{t}}\cup W_{\pmb{u}}))
  \cup\bigcup_{i<n}(C_i\setminus W_{\pmb{s}_i})
\in\Cal R$,}

\noindent by the choice of $\sequence{k}{\Cal R_k}$.   So

\Centerline{$|S_{\pmb{t}}(f,\nu)-S_{\pmb{s}}(f,\nu)|
=|S_{\pmb{t}'}(f,\nu)-S_{\pmb{s}'}(f,\nu)|\le\epsilon$,}

\noindent as required.

\medskip

{\bf (d)} There are $\delta\in\Delta$ and $\Cal R\in\frak R$ such that
$|S_{\pmb{t}}(f,\nu)-I_{\nu}(f)|\le\epsilon$ whenever $\pmb{t}\in T$
is
$\delta$-fine and $\Cal R$-filling.   If $\pmb{t}\in T$ is an
arbitrary
$\delta$-fine tagged partition, there is a $\delta$-fine
$\Cal R$-filling $\pmb{t}'\supseteq\pmb{t}$, by (b), so

\Centerline{$S_{\pmb{t}}(f,\nu)\le S_{\pmb{t}'}(f,\nu)
\le I_{\nu}(f)+\epsilon$,}

\noindent as claimed.

\medskip

{\bf (e)} Let $\delta\in\Delta$, $\Cal R\in\frak R$ be such that
$|S_{\pmb{s}}(f,\nu)-I_{\nu}(f)|\le\bover12\epsilon$ whenever 
$\pmb{s}\in T$ is $\delta$-fine and $\Cal R$-filling.   If $\pmb{t}\in T$
is $\delta$-fine and $W_{\pmb{t}}=\emptyset$, take any
$\delta$-fine $\Cal R$-filling $\pmb{s}\in T$, and consider
$\pmb{s}'=\pmb{s}\setminus\pmb{t}$, $\pmb{s}''=\pmb{s}\cup\pmb{t}$.
Because $W_{\pmb{s}}\cap W_{\pmb{t}}=\emptyset$, both $\pmb{s}'$ and
$\pmb{s}''$ belong to $T$;  both are $\delta$-fine;  and because
$W_{\pmb{s}'}=W_{\pmb{s}''}=W_{\pmb{s}}$, both are $\Cal R$-filling.
So

\Centerline{$|S_{\pmb{t}}(f,\nu)|
=|S_{\pmb{s}''}(f,\nu)-S_{\pmb{s}'}(f,\nu)|
\le|S_{\pmb{s}''}(f,\nu)-I_{\nu}(f)|
+|S_{\pmb{s}'}(f,\nu)-I_{\nu}(f)|
\le\epsilon$,}

\noindent as required.
}%end of proof of 482A

\leader{482B}{Saks-Henstock Lemma} Let $(X,T,\Delta,\frak R)$ be a
tagged-partition structure allowing subdivisions, witnessed by $\Cal
C$,
and $f:X\to\Bbb R$, $\nu:\Cal C\to\Bbb R$ functions such that
$I_{\nu}(f)=\lim_{\pmb{t}\to\Cal F(T,\Delta,\frak
R)}S_{\pmb{t}}(f,\nu)$
is defined.   Let $\Cal E$ be the algebra of subsets of $X$ generated
by
$\Cal C$.   Then there is a unique additive functional
$F:\Cal E\to\Bbb R$ such that for every $\epsilon>0$ there are
$\delta\in\Delta$ and $\Cal R\in\frak R$ such that

\quad($\alpha$) $\sum_{(x,C)\in\pmb{t}}|F(C)-f(x)\nu C|\le\epsilon$
for
every $\delta$-fine $\pmb{t}\in T$,

\quad($\beta$) $|F(E)|\le\epsilon$ whenever $E\in\Cal E\cap\Cal R$.

\noindent Moreover, $F(X)=I_{\nu}(f)$.

\proof{{\bf (a)} For $E\in\Cal E$, write $T_E$ for the set of those
$\pmb{t}\in T$ such that, for every $(x,C)\in\pmb{t}$, either
$C\subseteq E$ or $C\cap E=\emptyset$.   For any $\delta\in\Delta$,
$\Cal R\in\frak R$ and finite $\Cal D\subseteq\Cal E$ there is a
$\delta$-fine $\pmb{t}\in\bigcap_{E\in\Cal D}T_E$ such that
$E\setminus W_{\pmb{t}}\in\Cal R$ for every $E\in\Cal D$.   \Prf\ Let
$\sequencen{\Cal R_n}$ be a sequence in
$\frak R$ such that whenever $A_i\in\Cal R_i$ for $i\le n$ and
$\langle A_i\rangle_{i\le n}$ is disjoint then
$\bigcup_{i\le n}A_i\in\Cal R$ (481He again).   Let $\Cal E_0$ be the
subalgebra of $\Cal E$ generated by $\Cal D$, and enumerate the atoms
of $\Cal E_0$ as $\ofamily{i}{n}{E_i}$.   By 482Aa, there is for each
$i<n$ a $\delta$-fine $\pmb{s}_i\in T$ such that
$W_{\pmb{s}_i}\subseteq E_i$
and $E_i\setminus W_{\pmb{s}_i}\in\Cal R_i$.   Set
$\pmb{t}=\bigcup_{i<n}\pmb{s}_i$.   If $E\in\Cal D$ then
$E=\bigcup_{i\in J}E_i$ for some $J\subseteq n$.    For any
$(x,C)\in\pmb{t}$, there is some $i<n$ such that $C\subseteq E_i$, so
that $C\subseteq E$ if $i\in J$, $C\cap E=\emptyset$ otherwise;  thus
$\pmb{t}\in T_E$.   Moreover,
$E\setminus W_{\pmb{t}}=\bigcup_{i\in J}(E_i\setminus W_{\pmb{s}_i})$
belongs to $\Cal R$.\ \Qed

\medskip

{\bf (b)} We therefore have a filter $\Cal F^*$ on $T$ generated by
sets
of the form

\Centerline{$T_{E\delta\Cal R}
=\{\pmb{t}:\pmb{t}\in T_E$ is $\delta$-fine,
  $E\setminus W_{\pmb{t}}\in\Cal R\}$}

\noindent as $\delta$ runs over $\Delta$, $\Cal R$ runs over $\frak R$
and $E$ runs over $\Cal E$.   For $\pmb{t}\in T$, $E\subseteq X$ set
$\pmb{t}_E=\{(x,C):(x,C)\in\pmb{t},\,C\subseteq E\}$.   Now
$F(E)=\lim_{\pmb{t}\to\Cal F^*}S_{\pmb{t}_E}(f,\nu)$ is defined for
every $E\in\Cal E$.
\Prf\ For any $\epsilon>0$, there are $\delta\in\Delta$,
$\Cal R\in\frak R$ such that
$|I_{\nu}(f)-S_{\pmb{t}}(f,\nu)|\le\epsilon$ for every $\delta$-fine
$\Cal R$-filling $\pmb{t}\in T$.   Let $\Cal R'\in\frak R$ be such
that
$A\cup B\in\Cal R$ for all disjoint $A$, $B\in\Cal R'$.   If
$\pmb{t}$,
$\pmb{t}'$ belong to
$T_{E,\delta,\Cal R'}=T_{X\setminus E,\delta,\Cal R'}$, then set

\Centerline{$\pmb{s}=\{(x,C):(x,C)\in\pmb{t}',\,C\subseteq E\}
  \cup\{(x,C):(x,C)\in\pmb{t},\,C\cap E=\emptyset\}$.}

\noindent Then $\pmb{s}\in T_E$ is $\delta$-fine, and also
$E\setminus W_{\pmb{s}}=E\setminus W_{\pmb{t}'}$,
$(X\setminus E)\setminus W_{\pmb{s}}
=(X\setminus E)\setminus W_{\pmb{t}}$ both belong to $\Cal R'$;  so
their union $X\setminus W_{\pmb{s}}$ belongs to $\Cal R$, and
$\pmb{s}$
is $\Cal R$-filling.   Accordingly

$$\eqalign{|S_{\pmb{t}_E}(f,\nu)-S_{\pmb{t}'_E}(f,\nu)|
&=|S_{\pmb{t}}(f,\nu)-S_{\pmb{s}}(f,\nu)|\cr
&\le|S_{\pmb{t}}(f,\nu)-I_{\nu}(f)|+|I_{\nu}(f)-S_{\pmb{s}}(f,\nu)|
\le 2\epsilon.\cr}$$

\allowmorestretch{468}{
\noindent As $\epsilon$ is arbitrary, this is enough to show that
$\liminf_{\pmb{t}\to\Cal F^*}S_{\pmb{t}_E}(f,\nu)
=\limsup_{\pmb{t}\to\Cal F^*}S_{\pmb{t}_E}(f,\nu)$, so that the limit
$\lim_{\pmb{t}\to\Cal F^*}S_{\pmb{t}_E}(f,\nu)$ is defined (2A3Sf).\
\Qed
}%end of allowmorestretch

\medskip

{\bf (c)} $F(\emptyset)=0$.   \Prf\ Let $\epsilon>0$.   
By 482Ae, there is a
$\delta\in\Delta$ such that $|S_{\pmb{t}}(f,\nu)|\le\epsilon$ whenever
$\pmb{t}\in T$ is $\delta$-fine and $W_{\pmb{t}}=\emptyset$.   Since
$\{\pmb{t}:\pmb{t}$ is $\delta$-fine$\}$ belongs to $\Cal F^*$,

\Centerline{$|F(\emptyset)|
=|\lim_{\pmb{t}\to\Cal F^*}S_{\pmb{t}_{\emptyset}}(f,\nu)|\le\epsilon$;}

\noindent as $\epsilon$ is arbitrary, $F(\emptyset)=0$.\ \Qed

If $E$, $E'\in\Cal E$, then

\Centerline{$S_{\pmb{t}_{E\cup E'}}(f,\nu)+S_{\pmb{t}_{E\cap E'}}(f,\nu)
=S_{\pmb{t}_E}(f,\nu)+S_{\pmb{t}_{E'}}(f,\nu)$}

\noindent for every $\pmb{t}\in T_E\cap T_{E'}$;  as $T_E\cap T_{E'}$
belongs to $\Cal F^*$,

\Centerline{$F(E\cup E')+F(E\cap E')=F(E)+F(E')$.}

\noindent Since $F(\emptyset)=0$, $F(E\cup E')=F(E)+F(E')$ whenever
$E\cap E'=\emptyset$, and $F$ is additive.

\medskip

{\bf (d)} Now suppose that $\epsilon>0$.   Let $\delta\in\Delta$,
$\Cal R^*\in\frak R$ be such that
$|I_{\nu}(f)-S_{\pmb{t}}(f,\nu)|\le\bover14\epsilon$ for every
$\delta$-fine, $\Cal R^*$-filling $\pmb{t}\in T$.   Let
$\Cal R\in\frak R$ be such that $A\cup B\in\Cal R^*$ for all disjoint
$A$, $B\in\Cal R$.

\medskip

\quad{\bf (i)} If $\pmb{t}\in T$ is $\delta$-fine, then
$|F(W_{\pmb{t}})-S_{\pmb{t}}(f,\nu)|\le\bover12\epsilon$.   \Prf\ For
any $\eta>0$, there is a $\delta$-fine $\pmb{s}\in T$ such that

\inset{$|I_{\nu}(f)-S_{\pmb{s}}(f,\nu)|\le\eta$,

for every $(x,C)\in\pmb{s}$, either $C\subseteq W_{\pmb{t}}$ or
$C\cap W_{\pmb{t}}=\emptyset$,

$(X\setminus W_{\pmb{t}})\setminus W_{\pmb{s}}\in\Cal R$,
$W_{\pmb{t}}\setminus W_{\pmb{s}}\in\Cal R$,

$|F(W_{\pmb{t}})-\sum_{(x,C)\in\pmb{s},C\subseteq W_{\pmb{t}}}f(x)\nu
C|
\le\eta$}

\noindent because the set of $\pmb{s}$ with these properties belongs
to
$\Cal F^*$.   Now, setting
$\pmb{s}_1=\{(x,C):(x,C)\in\pmb{s},\,C\subseteq W_{\pmb{t}}\}$ and
$\pmb{t}'=\pmb{t}\cup(\pmb{s}\setminus\pmb{s}_1)$,
$\pmb{t}'$ is $\delta$-fine and $\Cal R^*$-filling, like $\pmb{s}$, so

$$\eqalign{|F(W_{\pmb{t}})-S_{\pmb{t}}(f,\nu)|
&\le|F(W_{\pmb{t}})-S_{\pmb{s}_1}(f,\nu)|
  +|S_{\pmb{s}_1}(f,\nu)-S_{\pmb{t}}(f,\nu)|\cr
&\le\eta+|S_{\pmb{s}}(f,\nu)-S_{\pmb{t}'}(f,\nu)|\cr
&\le\eta+|S_{\pmb{s}}(f,\nu)-I_{\nu}(f)|
  +|I_{\nu}(f)-S_{\pmb{t}'}(f,\nu)|
\le\eta+\Bover12\epsilon.\cr}$$

\noindent As $\eta$ is arbitrary we have the result.\ \Qed

\medskip

\quad{\bf (ii)} So if $\pmb{t}\in T$ is $\delta$-fine,
$\sum_{(x,C)\in\pmb{t}}|F(C)-f(x)\nu C|\le\epsilon$.   \Prf\ Set
$\pmb{t}'=\{(x,C):(x,C)\in\pmb{t},\,F(C)\le f(x)\nu C\}$,
$\pmb{t}''=\pmb{t}\setminus\pmb{t}''$.   Then both $\pmb{t}'$ and
$\pmb{t}''$ are $\delta$-fine, so

$$\eqalign{\sum_{(x,C)\in\pmb{t}}&|F(C)-f(x)\nu C|\cr
&=|\sum_{(x,C)\in\pmb{t}'}F(C)-f(x)\nu C|
  +|\sum_{(x,C)\in\pmb{t}''}F(C)-f(x)\nu C|
\le\epsilon.  \text{ \Qed}\cr}$$

\medskip

\quad{\bf (iii)} If $E\in\Cal E\cap\Cal R$, then
$|F(E)|\le\epsilon$.   \Prf\ Let $\Cal R'\in\frak R$ be such that
$A\cup B\in\Cal R$ whenever $A$, $B\in\Cal R'$ are disjoint.   Let
$\pmb{t}$ be such that

\inset{$\pmb{t}\in T_E$ is $\delta$-fine,

$E\setminus W_{\pmb{t}}$ and
$(X\setminus E)\setminus W_{\pmb{t}}$ both belong to $\Cal R'$,

$|F(E)-S_{\pmb{t}_E}(f,\nu)|\le\bover12\epsilon$;}

\noindent once again, the set of candidates belongs to $\Cal F^*$, so
is
not empty.   Then $\pmb{t}$ and
$\pmb{t}_{X\setminus E}$ are both
$\Cal R^*$-filling and $\delta$-fine, so

\Centerline{$|F(E)|\le\Bover12\epsilon+|S_{\pmb{t}_E}(f,\nu)|
=\Bover12\epsilon
  +|S_{\pmb{t}}(f,\nu)-S_{\pmb{t}_{X\setminus E}}(f,\nu)|
\le\epsilon$. \Qed}

\noindent As $\epsilon$ is arbitrary, this shows that $F$ has all the
required properties.

\medskip

{\bf (e)} I have still to show that $F$ is unique.   Suppose that
$F':\Cal E\to\Bbb R$ is another functional with the same properties,
and
take $E\in\Cal E$ and $\epsilon>0$.   Then there are $\delta$,
$\delta'\in\Delta$ and $\Cal R$, $\Cal R'\in\frak R$ such that

\inset{$\sum_{(x,C)\in\pmb{t}}|F(C)-f(x)\nu C|\le\epsilon$ for every
$\delta$-fine $\pmb{t}\in T$,

$\sum_{(x,C)\in\pmb{t}}|F'(C)-f(x)\nu C|\le\epsilon$ for every
$\delta'$-fine $\pmb{t}\in T$,

$|F(R)|\le\epsilon$ whenever $R\in\Cal E\cap\Cal R$,

$|F'(R)|\le\epsilon$ whenever $R\in\Cal E\cap\Cal R'$.}

\noindent Now taking $\delta''\in\Delta$ such that
$\delta''\subseteq\delta\cap\delta'$, and $\Cal R''\in\frak R$ such
that
$\Cal R''\subseteq\Cal R\cap\Cal R'$, there is a $\delta''$-fine
$\pmb{t}\in T$ such that $E'=W_{\pmb{t}}$ is included in $E$ and
$E\setminus E'\in\Cal R''$.   In this case

$$\eqalignno{|F(E)-F'(E)|
&\le|F(E\setminus E')|+\sum_{(x,C)\in\pmb{t}}|F(C)-F'(C)|
  +|F'(E\setminus E')|\cr
\displaycause{because $F$ and $F'$ are both additive}
&\le 2\epsilon+\sum_{(x,C)\in\pmb{t}}|F(C)-f(x)\nu C|
   +\sum_{(x,C)\in\pmb{t}}|F'(C)-f(x)\nu C|
\le 4\epsilon.\cr}$$

\noindent As $\epsilon$ and $E$ are arbitrary, $F=F'$, as required.

\medskip

{\bf (f)} Finally, to calculate $F(X)$, take any $\epsilon>0$.   Let
$\delta\in\Delta$ and $\Cal R\in\frak R$ be such that
$\sum_{(x,C)\in\pmb{t}}|F(C)-f(x)\nu C|\le\epsilon$ for every
$\delta$-fine $\pmb{t}\in T$
and $|F(E)|\le\epsilon$ whenever $E\in\Cal E\cap\Cal R$.   
Let $\pmb{t}$ be any
$\delta$-fine $\Cal R$-filling member of $T$ such that
$|S_{\pmb{t}}(f,\nu)-I_{\nu}(f)|\le\epsilon$.   Then, because $F$ is
additive,

$$\eqalign{|F(X)-I_{\nu}(f)|
&\le|F(X)-F(W_{\pmb{t}})|
  +|\sum_{(x,C)\in\pmb{t}}F(C)-f(x)\nu C|
+|S_{\pmb{t}}(f,\nu)-I_{\nu}(f)|\cr
&\le 3\epsilon.\cr}$$

\noindent As $\epsilon$ is arbitrary, $F(X)=I_{\nu}(f)$.
}%end of proof of 482B

\leader{482C}{Definition} In the context of 482B, I will call the
function $F$ the {\bf Saks-Henstock indefinite integral} of
$f$\cmmnt{;  of course it depends on the whole structure
$(X,T,\Delta,\frak R,\Cal C,f,\nu)$ and not just on $(X,f,\nu)$}.
\cmmnt{You should {\it not} take it for granted that
$F(E)=I_{\nu}(f\times\chi E)$ (482Ya); but see 482G.
}%end of comment

\leader{482D}{}\cmmnt{ The Saks-Henstock lemma characterizes the
gauge integral, as follows.

\medskip

\noindent}{\bf Theorem} Let $(X,T,\Delta,\frak R)$ be a
tagged-partition structure allowing subdivisions, witnessed by $\Cal C$,
and $\nu:\Cal C\to\Bbb R$ any function.   Let $\Cal E$ be the algebra of
subsets of $X$ generated by $\Cal C$.   If $f:X\to\Bbb R$ is any
function, then the following are equiveridical:

\inset{(i)
$I_{\nu}(f)
=\lim_{\pmb{t}\to\Cal F(T,\Delta,\frak R)}S_{\pmb{t}}(f,\nu)$
is defined in $\Bbb R$;

(ii) there is an additive functional $F:\Cal E\to\Bbb R$ such that

\quad($\alpha$) for every $\epsilon>0$ there is a $\delta\in\Delta$
such that $\sum_{(x,C)\in\pmb{t}}|F(C)-f(x)\mu C|\le\epsilon$ for
every $\delta$-fine $\pmb{t}\in T$,

\quad($\beta$) for every $\epsilon>0$ there is an $\Cal R\in\frak R$
such that $|F(E)|\le\epsilon$ for every $E\in\Cal E\cap\Cal R$;

(iii)\dvAnew{2011} 
there is an additive functional $F:\Cal E\to\Bbb R$ such that

\quad($\alpha$) for every $\epsilon>0$ there is a $\delta\in\Delta$
such that $|F(W_{\pmb{t}})-\sum_{(x,C)\in\pmb{t}}f(x)\mu C|\le\epsilon$ 
for every $\delta$-fine $\pmb{t}\in T$,

\quad($\beta$) for every $\epsilon>0$ there is an $\Cal R\in\frak R$
such that $|F(E)|\le\epsilon$ for every $E\in\Cal E\cap\Cal R$.}

\noindent In this case, $F(X)=I_{\nu}(f)$.

\proof{ (i)$\Rightarrow$(ii) is just 482B above, and (ii)$\Rightarrow$(iii)
is elementary, because $F(W_{\pmb{t}})=\sum_{(x,C)\in\pmb{t}}F(C)$ whenever
$F:\Cal E\to\Bbb R$ is additive and $\pmb{t}\in T$;  so let us assume
(iii) and seek to prove (i).    Given $\epsilon>0$, take $\delta\in\Delta$
and $\Cal R\in\frak R$ such that ($\alpha$) and ($\beta$) of (iii)
are satisfied.   Let $\pmb{t}\in T$ be $\delta$-fine and $\Cal R$-filling.
Then

\Centerline{$|F(X)-S_{\pmb{t}}(f,\mu)|
\le|F(X\setminus W_{\pmb{t}})|
  +|F(W_{\pmb{t}})-\sum_{(x,C)\in\pmb{t}}f(x)\mu C|
\le 2\epsilon$.}

\noindent As $\epsilon$ is arbitrary, $I_{\nu}(f)$ is defined and
equal to $F(X)$.
}%end of proof of 482D

\leader{482E}{Theorem} Let $(X,\rho)$ be a metric space and $\mu$ a
complete locally determined measure on $X$ with domain $\Sigma$.   Let
$\Cal C$, $Q$, $T$, $\Delta$ and $\frak R$ be such that

\inset{(i) $\Cal C\subseteq\Sigma$ and $\mu C$ is finite for every
$C\in\Cal C$;

(ii) $Q\subseteq X\times\Cal C$, and for each $C\in\Cal C$, $(x,C)\in Q$
for almost every $x\in C$;

(iii) $T$ is the straightforward set of tagged partitions generated by
$Q$;

(iv) $\Delta$ is a downwards-directed family of gauges
on $X$ containing all the uniform metric gauges;

(v) if $\delta\in\Delta$, there are a negligible set $F\subseteq X$ and
a neighbourhood gauge $\delta_0$ on $X$ such that
$\delta\supseteq\delta_0\setminus(F\times\Cal PX)$;

(vi) $\frak R$ is a downwards-directed collection of families of
subsets
of $X$ such that whenever $E\in\Sigma$, $\mu E<\infty$ and
$\epsilon>0$,
there is an $\Cal R\in\frak R$ such that $\mu^*(E\cap R)\le\epsilon$
for every $R\in\Cal R$;

(vii) $T$ is compatible with $\Delta$ and $\frak R$.}

\noindent Let $f:X\to\Bbb R$ be any function such that
$I_{\mu}(f)=\lim_{\pmb{t}\to\Cal F(T,\Delta,\frak
R)}S_{\pmb{t}}(f,\mu)$
is defined.   Then $f$ is $\Sigma$-measurable.

\proof{ \Quer\ Suppose, if possible, otherwise.

Because $\mu$ is complete and locally determined, there are a
measurable
set $E$ of non-zero finite measure and $\alpha<\beta$ in $\Bbb R$ such
that

\Centerline{$\mu^*\{x:x\in E,\,f(x)\le\alpha\}
=\mu^*\{x:x\in E,\,f(x)\ge\beta\}=\mu E$}

\noindent (413G).   Let $\epsilon>0$ be such that
$(\beta-\alpha)(\mu E-3\epsilon)>2\epsilon$.
Let $\delta\in\Delta$, $\Cal R\in\frak R$ be such that
$|S_{\pmb{t}}(f,\mu)-I_{\nu}(f)|\le\epsilon$ whenever $\pmb{t}\in T$ is
$\delta$-fine and $\Cal R$-filling.   By (v), there are a negligible set
$F\subseteq X$ and a family $\family{x}{X}{G_x}$
of open sets such that $x\in G_x$ for every $x\in X$ and
$\delta\supseteq\{(x,C):x\in X\setminus F$, $C\subseteq G_x\}$.
For $m\ge 1$, set

\Centerline{$A_m
=\{x:x\in E\setminus F,\,f(x)\le\alpha,\,U_{1/m}(x)\subseteq G_x\}$,}

\noindent writing $U_{1/m}(x)$ for $\{y:\rho(y,x)<\bover1m\}$,

\Centerline{$B_m
=\{x:x\in E\setminus F,\,f(x)\ge\beta,\,U_{1/m}(x)\subseteq G_x\}$.}

\noindent Then there is some $m\ge 1$ such that
$\mu^*A_m\ge\mu E-\epsilon$ and $\mu^*B_m\ge\mu E-\epsilon$.   By (iv),
there is a $\delta'\in\Delta$ such that

\Centerline{$\delta'
\subseteq\delta\cap\{(x,C):x\in C\subseteq U_{1/3m}(x)\}$.}

\noindent By (vi), there is an $\Cal R'\in\frak R$
such that $\Cal R'\subseteq\Cal R$
and $\mu^*(R\cap E)\le\epsilon$ for every $R\in\Cal R'$.

Let $\pmb{t}$ be any $\delta'$-fine
$\Cal R'$-filling member of $T$.   Enumerate $\pmb{t}$ as
$\ofamily{i}{n}{(x_i,C_i)}$.   Set

\Centerline{$J=\{i:i<n,\,C_i\cap A_m$ is negligible$\}$,
\quad$J'=\{i:i<n,\,C_i\cap B_m$ is negligible$\}$.}

\noindent Then

\Centerline{$\mu(E\cap\bigcup_{i\in J}C_i)\le\mu_*(E\setminus A_m)
=\mu E-\mu^*(E\cap A_m)\le\epsilon$,}

\noindent and similarly $\mu(E\cap\bigcup_{i\in J'}C_i)\le\epsilon$.
Also, because $X\setminus\bigcup_{i<n}C_i=X\setminus W_{\pmb{t}}$
belongs to $\Cal R'$,
$\mu(E\setminus\bigcup_{i<n}C_i)\le\epsilon$.   So, setting
$K=n\setminus(J\cup J')$, $\sum_{i\in K}\mu C_i\ge\mu E-3\epsilon$.

For $i\in K$, $\mu^*(C_i\cap A_m)>0$, while
$\{x:x\in C_i,\,(x,C_i)\notin Q\}$ is negligible, by (ii), so we can find
$x'_i\in C_i\cap A_m$ such that $(x'_i,C_i)\in Q$;  similarly, there
is an $x_i''\in C_i\cap B_m$ such that $(x_i'',C_i)\in Q$.   For other
$i<n$, set $x'_i=x_i''=x_i$.  Now $\pmb{s}=\{(x_i',C_i):i<n\}$ and
$\pmb{s}'=\{(x_i'',C_i):i<n\}$ belong to $T$.   Of course they are
$\Cal R'$-filling, therefore $\Cal R$-filling, because $\pmb{t}$ is.
We also see that, because
$(x_i,C_i)\in\delta'$, the diameter of $C_i$ is at
most $\bover{2}{3m}$ for each $i<n$, so that $C_i\subseteq G_{x'_i}$;  as
also $x'_i\in A_m\subseteq X\setminus F$, $(x'_i,C_i)\in\delta$,
for each $i\in K$.
But since surely $(x_i,C_i)\in\delta'\subseteq\delta$ for
$i\in n\setminus K$, this means that $\pmb{s}$ is $\delta$-fine.
Similarly, $\pmb{s}'$ is $\delta$-fine.

We must therefore have

\Centerline{$|S_{\pmb{s}'}(f,\mu)-S_{\pmb{s}}(f,\mu)|
\le|S_{\pmb{s}'}(f,\mu)-I_{\mu}(f)|+|S_{\pmb{s}}(f,\mu)-I_{\mu}(f)|
\le 2\epsilon$.}

\noindent But

$$\eqalign{S_{\pmb{s}'}(f,\mu)-S_{\pmb{s}}(f,\mu)
&=\sum_{i\in K}(f(x_i'')-f(x_i'))\mu C_i\cr
&\ge(\beta-\alpha)\sum_{i\in K}\mu C_i
\ge(\beta-\alpha)(\mu E-3\epsilon)
>2\epsilon\cr}$$

\noindent by the choice of $\epsilon$.\ \Bang

So we have the result.
}%end of proof of 482E

\leader{482F}{Proposition} Let $X$, $\Sigma$, $\mu$, $\frak T$, $T$,
$\Delta$ and $\frak R$ be such that

\inset{(i) $(X,\Sigma,\mu)$ is a measure space;

(ii) $\frak T$ is a topology on $X$ such that $\mu$ is inner regular
with respect to the closed sets and outer regular with respect to the
open sets;

(iii) $T\subseteq[X\times\Sigma]^{<\omega}$ is a set of
tagged partitions such that $C\cap C'$ is empty whenever $(x,C)$,
$(x',C')$ are distinct members of any $\pmb{t}\in T$;

(iv) $\Delta$ is a set of gauges on $X$ containing every neighbourhood
gauge on $X$;

(v) $\frak R$ is a collection of families of subsets of $X$ such that
whenever $\mu E<\infty$ and $\epsilon>0$ there is an $\Cal R\in\frak
R$
such that $\mu^*(E\cap R)\le\epsilon$ for every $R\in\Cal R$;

(vi) $T$ is compatible with $\Delta$ and $\frak R$.}

\noindent Then
$I_{\mu}(f)=\lim_{\pmb{t}\to\Cal F(T,\Delta,\frak
R)}S_{\pmb{t}}(f,\mu)$
is defined and equal to $\int fd\mu$ for every $\mu$-integrable
function
$f:X\to\Bbb R$.

\proof{{\bf (a)} It is worth noting straight away that we can replace
$(X,\Sigma,\mu)$ by its completion $(X,\hat\Sigma,\hat\mu)$.   \Prf\
We need to check that $\hat\mu$ is inner and outer regular.   But inner
regularity is 412Ha, and outer regularity is equally elementary:  if
$\hat\mu E<\gamma$, there is an $E'\in\Sigma$ such that
$E\subseteq E'$
and $\mu E'=\hat\mu E$ (212C), and now there is an open set
$G\in\Sigma$
such that $E'\subseteq G$ and $\mu G\le\gamma$, so that $E\subseteq G$
and $\hat\mu G\le\gamma$.   Since we are not changing $T$ or $\Delta$
or
$\frak R$, $I_{\hat\mu}(f)=I_{\mu}(f)$ if either is defined;  while
also
$\int fd\mu=\int fd\hat\mu$ if either is defined, by 212Fb.\ \Qed

So let us suppose that $\mu$ is actually complete.

\medskip

{\bf (b)} In this case, $f$ is measurable.   Suppose to begin with
that
it is non-negative.   Let $\epsilon>0$.   For $m\in\Bbb Z$, set
$E_m=\{x:x\in X,\,(1+\epsilon)^m\le f(x)<(1+\epsilon)^{m+1}\}$.   Then
$E_m$ is measurable and has finite measure, so there is a measurable
open set $G_m\supseteq E_m$ such that
$(1+\epsilon)^{m+1}\mu(G_m\setminus E_m)\le 2^{-|m|}\epsilon$.

Take a set $H_0$ of finite measure and $\eta_0>0$ such that
$\int_Efd\mu\le\epsilon$ whenever $E\in\Sigma$ and
$\mu(E\cap H_0)\le 2\eta_0$ (225A);  replacing $H_0$ by
$\{x:x\in H_0,\,f(x)>0\}$ if necessary, we may suppose that
$H_0\subseteq\bigcup_{m\in\Bbb Z}E_m$.   Let $F\subseteq H_0$ be a
closed set such that $\mu(H_0\setminus F)\le\eta_0$.

Define $\family{x}{X}{V_x}$ by setting $V_x=G_m$ if $m\in\Bbb Z$ and
$x\in E_m$, $V_x=X\setminus F$ if $f(x)=0$.   Let $\delta\in\Delta$ be
the corresponding neighbourhood gauge
$\{(x,C):x\in X,\,C\subseteq V_x\}$.
Let $\Cal R\in\frak R$ be such that $\mu^*(R\cap H_0)\le\eta_0$ for
every $R\in\Cal R$.

Suppose that $\pmb{t}$ is any $\delta$-fine $\Cal R$-filling member of
$T$.   Enumerate $\pmb{t}$ as $\ofamily{i}{n}{(x_i,C_i)}$.   For each
$m\in\Bbb Z$, set $J_m=\{i:i<n,\,x_i\in E_m\}$.   Then
$C_i\subseteq V_{x_i}\subseteq G_m$ for every $i\in J_m$, so

$$\eqalign{S_{\pmb{t}}(f,\mu)
&=\sum_{i<n}f(x_i)\mu C_i
=\sum_{m\in\Bbb Z}\sum_{i\in J_m}f(x_i)\mu C_i
\le\sum_{m\in\Bbb Z}(1+\epsilon)^{m+1}\mu G_m\cr
&\le(1+\epsilon)\sum_{m\in\Bbb Z}(1+\epsilon)^m\mu E_m
   +\sum_{m\in\Bbb Z}(1+\epsilon)^{m+1}\mu(G_m\setminus E_m)\cr
&\le(1+\epsilon)\int fd\mu+\sum_{m\in\Bbb Z}2^{-|m|}\epsilon
=(1+\epsilon)\int fd\mu+3\epsilon.\cr}$$

\noindent On the other hand, set $F'=F\cap\bigcup_{i<n}C_i$.   Because
$X\setminus\bigcup_{i<n}C_i\in\Cal R$,
$\mu(H_0\setminus F')\le 2\eta_0$, and

$$\eqalign{S_{\pmb{t}}(f,\mu)
&\ge\sum_{m\in\Bbb Z}\sum_{i\in J_m}f(x_i)\mu(C_i\cap F)\cr
&\ge\Bover1{1+\epsilon}\sum_{m\in\Bbb Z}\sum_{i\in J_m}
  (1+\epsilon)^{m+1}\mu(C_i\cap F)\cr
&\ge\Bover1{1+\epsilon}\int_{F'}fd\mu
\ge\Bover1{1+\epsilon}(\int fd\mu-\epsilon).\cr}$$

What this means is that

\Centerline{$\{\pmb{t}:\Bover1{1+\epsilon}(\biggerint fd\mu-\epsilon)
\le S_{\pmb{t}}(f,\nu)
\le(1+\epsilon)\int fd\mu+3\epsilon\}$}

\noindent belongs to $\Cal F(T,\Delta,\frak R)$, for any $\epsilon>0$.
So $\lim_{\pmb{t}\to\Cal F(T,\Delta,\frak R)}S_{\pmb{t}}(f,\mu)$ is
defined and equal to $\int fd\mu$.

\medskip

{\bf (c)} In general, $f$ is expressible as $f^+-f^-$ where $f^+$ and
$f^-$ are non-negative integrable functions, so

\Centerline{$I_{\nu}(f)=I_{\nu}(f^+)-I_{\nu}(f^-)=\int fd\mu$}

\noindent by 481Ca.
}%end of proof of 482F

\leader{482G}{Proposition} Let $(X,T,\Delta,\frak R)$ be a
tagged-partition structure allowing subdivisions, witnessed by $\Cal C$.
Suppose that

\inset{(i) $\frak T$ is a topology on $X$, and $\Delta$ is the set of
neighbourhood gauges on $X$;

(ii) $\nu:\Cal C\to\Bbb R$ is a function which is additive in the sense that if 
$C_0,\ldots,C_n\in\Cal C$ are disjoint and have union $C\in\Cal C$, then 
$\nu C=\sum_{i=0}^n\nu C_i$;

(iii) whenever $E\in\Cal C$ and $\epsilon>0$, there are
closed sets $F\subseteq E$, $F'\subseteq X\setminus E$ such that
$\sum_{(x,C)\in\pmb{t}}|\nu C|\le\epsilon$ whenever $\pmb{t}\in T$ and
$W_{\pmb{t}}\cap(F\cup F')=\emptyset$;

(iv) for every $E\in\Cal C$ and $x\in X$ there is a neighbourhood $G$ of
$x$ such that if $C\in\Cal C$, $C\subseteq G$ and $\{(x,C)\}\in T$, there
is a finite partition $\Cal D$ of $C$ into members of $\Cal C$, each either
included in $E$ or disjoint from $E$, such that
$\{(x,D)\}\in T$ for every $D\in\Cal D$;

(v) for every $C\in\Cal C$ and $\Cal R\in\frak R$, there is an 
$\Cal R'\in\frak R$ such that $C\cap A\in\Cal R$ whenever $A\in\Cal R'$.}

\noindent Let $f:X\to\Bbb R$ be a function such that
$I_{\nu}(f)=\lim_{\pmb{t}\to\Cal F(T,\Delta,\frak R)}S_{\pmb{t}}(f,\nu)$
is defined.   Let $\Cal E$ be the algebra of subsets of $X$ generated
by $\Cal C$, and $F:\Cal E\to\Bbb R$ the Saks-Henstock indefinite
integral of $f$.   Then
$I_{\nu}(f\times\chi E)$ is defined and equal to $F(E)$ for every
$E\in\Cal E$.

\proof{{\bf (a)} Because both $F$ and $I_{\nu}$ are additive, and
$F(X)=I_{\nu}(f)$, and either $E$ or its complement is a finite
disjoint union of members of $\Cal C$ (see 481Hd), 
it is enough to consider the case in which
$E\in\Cal C$.  

\medskip

{\bf (b)} Let $\epsilon>0$.   For each $x\in X$ let $G_x$ be an
open set containing $x$ such that whenever
$C\in\Cal C$, $C\subseteq G$ and $\{(x,C)\}\in T$, there
is a finite partition $\Cal D$ of $C$ into members of $\Cal C$ such that
$\{(x,D)\}\in T$ for every $D\in\Cal D$ and every member of $\Cal D$ is
either included in $E$ or disjoint from $E$.
For each $n\in\Bbb N$, let $F_n\subseteq E$, $F'_n\subseteq X\setminus E$
be closed sets such that 
$\sum_{(x,C)\in\pmb{t}}|\nu C|\le\Bover{2^{-n}\epsilon}{n+1}$ 
whenever $\pmb{t}\in T$ and
$W_{\pmb{t}}\cap(F_n\cup F_n')=\emptyset$;  now define $G'_x$, for 
$x\in X$, by saying that

$$\eqalign{G'_x
&=G_x\setminus F'_n\text{ if }x\in E\text{ and }n\le|f(x)|<n+1,\cr
&=G_x\setminus F_n\text{ if }x\in X\setminus E
   \text{ and }n\le|f(x)|<n+1.\cr}$$

\noindent Let $\delta_0\in\Delta$ be the neighbourhood gauge defined by
the family $\family{x}{X}{G'_x}$.   Let $\delta\in\Delta$ and
$\Cal R_1\in\frak R$ be such that $\delta\subseteq\delta_0$,
$\sum_{(x,C)\in\pmb{t}}|F(C)-f(x)\nu C|\le\epsilon$ for every
$\delta$-fine $\pmb{t}\in T$, and $|F(E)|\le\epsilon$ for every
$E\in\Cal E\cap\Cal R_1$.   Let
$\Cal R\in\frak R$ be such that $R\cap E\in\Cal R_1$ whenever
$R\in\Cal R$.

\medskip

{\bf (c)} As in the proof of 482B,
let $T_E$ be the set of those $\pmb{t}\in T$ such that, for
each $(x,C)\in\pmb{t}$, either $C\subseteq E$ or $C\cap E=\emptyset$.
The key to the proof is the following fact:  if $\pmb{t}\in T$ is
$\delta$-fine, then there is a $\delta$-fine
$\pmb{s}\in T_E$ such that $W_{\pmb{s}}=W_{\pmb{t}}$ and
$S_{\pmb{s}}(g,\nu)=S_{\pmb{t}}(g,\nu)$ for every $g:X\to\Bbb R$.
\Prf\ For each $(x,C)\in\pmb{t}$, we know that 
$C\subseteq G'_x\subseteq G_x$, because
$\delta\subseteq\delta_0$.   Let $\Cal D_{(x,C)}$ be a 
finite partition of
$C$ into members of $\Cal C$, each either included in $E$ or disjoint from
$E$, such that $\{(x,D)\}\in T$ for every $D\in\Cal D_{(x,C)}$.   
Then $\pmb{s}=\{(x,D):(x,C)\in\pmb{t}$, $D\in\Cal D_{(x,C)}\}$ belongs to
$T_E$.   Because
$\delta$ is a neighbourhood gauge, $(x,D)\in\delta$ whenever
$(x,C)\in\pmb{t}$ and $D\in\Cal D_{(x,C)}$, so $\pmb{s}$ is $\delta$-fine.

If $g:X\to\Bbb R$ is any function,

$$\eqalignno{S_{\pmb{s}}(g,\nu)
&=\sum_{(x,C)\in\pmb{t}}\sum_{D\in\Cal D_{(x,C)}}g(x)\nu D\cr
&=\sum_{(x,C)\in\pmb{t}}g(x)\sum_{D\in\Cal D_{(x,C)}}\nu D
=\sum_{(x,C)\in\pmb{t}}g(x)\nu C\cr
\displaycause{because $\nu$ is additive}
&=S_{\pmb{t}}(g,\nu).  \text{ \Qed}\cr}$$

\medskip

{\bf (d)} Now suppose that $\pmb{t}\in T$ is $\delta$-fine and 
$\Cal R$-filling.   Let $\pmb{s}\in T_E$ be as in (c), and set

\Centerline{$\pmb{s}^*=\{(x,D):(x,D)\in\pmb{s}$, $x\in E$, 
  $D\subseteq E\}$,}
  
\Centerline{$\pmb{s}'=\{(x,D):(x,D)\in\pmb{s}$, $x\notin E$, 
  $D\subseteq E\}$,}

\Centerline{$\pmb{s}''=\{(x,D):(x,D)\in\pmb{s}$, $x\in E$, 
  $D\cap E=\emptyset\}$.}

\noindent Because $\pmb{s}\in T_E$, 

\Centerline{$W_{\pmb{s}^*\cup\pmb{s}'}
=E\cap W_{\pmb{s}}=E\cap W_{\pmb{t}}$}

\noindent and
$E\setminus W_{\pmb{s}^*\cup\pmb{s}'}=E\setminus W_{\pmb{t}}$ belongs to 
$\Cal R_1$, by the choice of $\Cal R$.   Accordingly

\Centerline{$|F(E)-S_{\pmb{s}^*\cup\pmb{s}'}(f,\nu)|
\le|F(E)-F(W_{\pmb{s}^*\cup\pmb{s}'})|
   +|F(W_{\pmb{s}^*\cup\pmb{s}'})-S_{\pmb{s}^*\cup\pmb{s}'}(f,\nu)|
\le 2\epsilon$}

\noindent because $\pmb{s}^*\cup\pmb{s}'\subseteq\pmb{s}$ is $\delta$-fine.

For $n\in\Bbb N$ set

\Centerline{$\pmb{s}'_n=\{(x,D):(x,D)\in\pmb{s}'$, $n\le|f(x)|<n+1\}$,}

\Centerline{$\pmb{s}''_n=\{(x,D):(x,D)\in\pmb{s}''$, $n\le|f(x)|<n+1\}$.}

\noindent Then $W_{\pmb{s}'_n}\subseteq E\setminus F_n$.
\Prf\ If $(x,D)\in\pmb{s}'_n$, there is a $C\in\Cal C$ such that 
$D\subseteq E\cap C$ and $(x,C)\in\pmb{t}$, while $x\notin E$,
so that $C\subseteq G'_x$
and $C\cap F_n=\emptyset$.\ \QeD\  Similarly, 
$W_{\pmb{s}''_n}\subseteq(X\setminus E)\setminus F'_n$.   Thus
$W_{\pmb{s}'_n\cup\pmb{s}''_n}$ is disjoint from $F_n\cup F_n'$ and

$$\eqalign{|S_{\pmb{s}'_n}(f,\nu)-S_{\pmb{s}''_n}(f,\nu)|
&=|\sum_{(x,D)\in\pmb{s}'_n}f(x_i)\nu D
  -\sum_{(x,D)\in\pmb{s}''_n}f(x_i)\nu D|\cr
&\le\sum_{(x,D)\in\pmb{s}'_n\cup\pmb{s}''_n}|f(x_i)||\nu D|\cr
&\le(n+1)\sum_{(x,D)\in\pmb{s}'_n\cup\pmb{s}''_n}|\nu D|
\le 2^{-n}\epsilon\cr}$$

\noindent by the choice of $F_n$ and $F'_n$.   	

Consequently, 

$$\eqalignno{|F(E)-S_{\pmb{t}}(f\times\chi E,\nu)|
&=|F(E)-S_{\pmb{s}}(f\times\chi E,\nu)|
=|F(E)-S_{\pmb{s}^*\cup\pmb{s}''}(f,\nu)|\cr
\displaycause{because 
$\pmb{s}^*\cup\pmb{s}''=\{(x,D):(x,D)\in\pmb{s}$, $x\in E\}$}
&\le|F(E)-S_{\pmb{s}^*\cup\pmb{s}'}(f,\nu)|
   +|S_{\pmb{s}'}(f,\nu)-S_{\pmb{s}''}(f,\nu)|\cr
\displaycause{because $\pmb{s}^*$, $\pmb{s}'$ and $\pmb{s}''$ are disjoint
subsets of $\pmb{s}$}   
&\le 2\epsilon
   +|\sum_{n=0}^{\infty}S_{\pmb{s}'_n}(f,\nu)
       -\sum_{n=0}^{\infty}S_{\pmb{s}''_n}(f,\nu)|\cr
\displaycause{the infinite
sums are well-defined because $\pmb{s}$ is finite, so that
all but finitely many terms are zero}
&\le 2\epsilon
   +\sum_{n=0}^{\infty}|S_{\pmb{s}'_n}(f,\nu)
       -S_{\pmb{s}''_n}(f,\nu)|\cr
&\le 2\epsilon
   +\sum_{n=0}^{\infty}2^{-n}\epsilon
=4\epsilon.\cr}$$

\noindent As $\epsilon$ is arbitrary, $I_{\nu}(f\times\chi E)$ is defined
and equal to $F(E)$, as required.
}%end of proof of 482G

\leader{482H}{Proposition} Suppose that $X$, $\frak T$, $\Cal C$, $\nu$,
$T$, $\Delta$ and $\frak R$ satisfy the conditions (i)-(v) of 482G, and
that $f:X\to\Bbb R$, $\sequencen{H_n}$, $H$ and $\gamma$ are such that

\inset{(vi) $\sequencen{H_n}$ is a sequence of open subsets of $X$
with union $H$,

(vii) $I_{\nu}(f\times\chi H_n)$ is defined for every $n\in\Bbb N$,

(viii) $\lim_{\pmb{t}\to\Cal F(T,\Delta,\frak R)}
I_{\nu}(f\times\chi W_{\pmb{t}\restr H})$ is defined and
equal to $\gamma$,}

\noindent where $\pmb{t}\restr H=\{(x,C):(x,C)\in\pmb{t}$, $x\in H\}$
for $\pmb{t}\in T$.   Then $I_{\nu}(f\times\chi H)$ is defined and
equal to $\gamma$.

\proof{ Let $\epsilon>0$.   For each $n\in\Bbb N$, let $F_n$ be the
Saks-Henstock indefinite integral of $f\times\chi H_n$.   Let
$\delta_n\in\Delta$ be such that

$$\eqalign{2^{-n}\epsilon
&\ge\sum_{(x,C)\in\pmb{s}}|F_n(C)-(f\times\chi H_n)(x)\nu C|\cr
&\ge|F_n(W_{\pmb{s}})-S_{\pmb{s}}(f\times\chi H_n,\nu)|\cr}$$

\noindent whenever $\pmb{s}\in T$ is $\delta_n$-fine.   Set

$$\eqalign{\tilde\delta&=\{(x,A):x\in X\setminus H,\,A\subseteq X\}\cr
&\mskip100mu
  \cup\bigcup_{n\in\Bbb N}\{(x,A):x\in H_n\setminus\bigcup_{i<n}H_i,\,
  A\subseteq H_n,\,(x,A)\in\delta_n\},\cr}$$

\noindent so that $\tilde\delta\in\Delta$.   Note that if $x\in H$ and
$C\in\Cal C$ and $(x,C)\in\tilde\delta$, then there is some $n\in\Bbb
N$
such that $x\in H_n$ and $C\subseteq H_n$, so that

\Centerline{$I_{\nu}(f\times\chi C)
=I_{\nu}((f\times\chi H_n)\times\chi C)=F_n(C)$}

\noindent is defined, by 482G;  this means that
$I_{\nu}(f\times\chi W_{\pmb{t}\restr H})$ will be defined for every
$\tilde\delta$-fine $\pmb{t}\in T$.
Let $\delta\in\Delta$, $\Cal R\in\frak R$ be
such that $|\gamma-I_{\nu}(f\times\chi W_{\pmb{t}\restr
H})|\le\epsilon$
whenever $\pmb{t}\in T$ is $\delta$-fine and $\Cal R$-filling.

Let $\pmb{t}\in T$ be
$(\delta\cap\tilde\delta)$-fine and $\Cal R$-filling.   For
$n\in\Bbb N$, set
$\pmb{t}_n=\{(x,C):(x,C)\in\pmb{t},\,x\in H_n\setminus\bigcup_{i<n}H_i\}$.
Then $\pmb{t}\restr H=\bigcup_{n\in\Bbb N}\pmb{t}_n$, and $\pmb{t}_n$ is
$\delta_n$-fine and $W_{\pmb{t}_n}\subseteq H_n$ for every $n$.   So

$$\eqalignno{|\gamma-S_{\pmb{t}}(f\times\chi H,\nu)|
&=|\gamma-\sum_{n=0}^{\infty}S_{\pmb{t}_n}(f\times\chi H_n,\nu)|\cr
&\le|\gamma-I_{\nu}(f\times\chi W_{\pmb{t}\restr H})|
  +\sum_{n=0}^{\infty}|I_{\nu}(f\times\chi W_{\pmb{t}_n})
       -S_{\pmb{t}_n}(f\times\chi H_n,\nu)|\cr
\displaycause{note that $\pmb{t}_n=\emptyset$ for all but finitely
many $n$, so that $I_{\nu}(f\times\chi W_{\pmb{t}\restr H})
=\sum_{n=0}^{\infty}I_{\nu}(f\times\chi W_{\pmb{t}_n})$}
&\le\epsilon
  +\sum_{n=0}^{\infty}|I_{\nu}(f\times\chi H_n\times\chi W_{\pmb{t}_n})
  -S_{\pmb{t}_n}(f\times\chi H_n,\nu)|\cr
\displaycause{because $\pmb{t}$ is $\delta$-fine and $\Cal R$-filling,
while $W_{\pmb{t}_n}\subseteq H_n$ for each $n$}
&=\epsilon+\sum_{n=0}^{\infty}|F_n(W_{\pmb{t}_n})
  -S_{\pmb{t}_n}(f\times\chi H_n,\nu)|\cr
\displaycause{by 482G}
&\le\epsilon+\sum_{n=0}^{\infty}2^{-n}\epsilon\cr
\displaycause{because every $\pmb{t}_n$ is $\delta_n$-fine}
&=3\epsilon.\cr}$$

\noindent As $\epsilon$ is arbitrary, $\gamma=I_{\nu}(f\times\chi H)$,
as claimed.
}%end of proof of 482H

\cmmnt{\medskip

\noindent{\bf Remark} For applications of this result see 483Bd and
483N.
}%end of comment

\leader{482I}{Integrating a \dvrocolon{derivative}}\cmmnt{ As will
appear in the next two sections, the real strength of gauge integrals is
in their power to integrate derivatives.   I give an elementary general
expression of this fact.   In the formulae below, we can think of $f$ as
a `derivative' of $F$ if $\nu=\theta$ is strictly positive and additive
and we rephrase condition (iii) as

\Centerline{`$\lim_{C\to\Cal G_x}\bover{F(C)}{\nu C}=f(x)$',}

\noindent where $\Cal G_x$ is the filter on $\Cal C$ generated by the
sets $\{C:(x,C)\in\delta\}$ as $\delta$ runs over $\Delta$.

\medskip

\noindent}{\bf Theorem} Let $X$, $\Cal C\subseteq\Cal PX$,
$\Delta\subseteq\Cal P(X\times\Cal PX)$,
$\frak R\subseteq\Cal P\Cal PX$, $T\subseteq[X\times\Cal
C]^{<\omega}$,
$f:X\to\Bbb R$, $\nu:\Cal C\to\Bbb R$, $F:\Cal C\to\Bbb R$,
$\theta:\Cal C\to[0,1]$ and $\gamma\in\Bbb R$ be such that

\inset{(i) $T$ is a straightforward set of tagged partitions which is
compatible with $\Delta$ and $\frak R$,

(ii) $\Delta$ is a full set of gauges on $X$,

(iii) for every $x\in X$ and $\epsilon>0$ there is a $\delta\in\Delta$
such that $|f(x)\nu C-F(C)|\le\epsilon\theta C$ whenever
$(x,C)\in\delta$,

(iv) $\sum_{i=0}^n\theta C_i\le 1$ whenever $C_0,\ldots,C_n\in\Cal C$
are disjoint,

(v) for every $\epsilon>0$ there is an $\Cal R\in\frak R$ such that
$|\gamma-\sum_{C\in\Cal C_0}F(C)|\le\epsilon$ whenever
$\Cal C_0\subseteq\Cal C$ is a finite disjoint set and
$X\setminus\bigcup\Cal C_0\in\Cal R$.}

\noindent Then
$I_{\nu}(f)=\lim_{\pmb{t}\to\Cal F(T,\Delta,\frak
R)}S_{\pmb{t}}(f,\nu)$
is defined and equal to $\gamma$.

\proof{ Let $\epsilon>0$.   For each $x\in X$ let $\delta_x\in\Delta$
be
such that
$|f(x)\nu C-F(C)|\le\epsilon\theta C$ whenever $(x,C)\in\delta_x$.
Because $\Delta$ is full, there is a $\delta\in\Delta$ such that
$(x,C)\in\delta_x$ whenever $(x,C)\in\delta$.   Let $\Cal R\in\frak R$
be as in (v).   If $\pmb{t}\in T$ is $\delta$-fine and
$\Cal R$-filling, then

$$\eqalignno{|S_{\pmb{t}}(f,\nu)-\gamma|
&\le|\gamma-\sum_{(x,C)\in\pmb{t}}F(C)|
  +\sum_{(x,C)\in\pmb{t}}|f(x)\nu C-F(C)|\cr
&\le\epsilon+\sum_{(x,C)\in\pmb{t}}\epsilon\theta C\cr
\displaycause{because $X\setminus\bigcup_{(x,C)\in\pmb{t}}C\in\Cal R$,
while $(x,C)\in\delta_x$ whenever $(x,C)\in\pmb{t}$}
&\le 2\epsilon\cr}$$

\noindent by condition (iv).   As $\epsilon$ is arbitrary, we have the
result.
}%end of proof of 482I

\leader{482J}{Definition}
Let $X$ be a set, $\Cal C$ a family of subsets of $X$,
$T\subseteq[X\times\Cal C]^{<\omega}$ a family of tagged partitions,
$\nu:\Cal C\to\coint{0,\infty}$ a function, and $\Delta$ a family of
gauges on $X$.   I will say that $\nu$ is {\bf moderated} (with respect
to $T$ and $\Delta$) if there are a $\delta\in\Delta$ and a function
$h:X\to\ooint{0,\infty}$ such that $S_{\pmb{t}}(h,\nu)\le 1$ for every
$\delta$-fine $\pmb{t}\in T$.

\leader{482K}{B.Levi's theorem}\dvArevised{2010} Let $(X,T,\Delta,\frak R)$
be a
tagged-partition structure allowing subdivisions, witnessed by $\Cal C$,
such that $\Delta$ is countably full, and
$\nu:\Cal C\to\coint{0,\infty}$ a function
which is moderated with respect to $T$ and $\Delta$.

Let $\sequencen{f_n}$ be a non-decreasing sequence of functions from
$X$ to $\Bbb R$ with supremum $f:X\to\Bbb R$.   If
$\gamma=\lim_{n\to\infty}I_{\nu}(f_n)$ is defined in
$\Bbb R$, then $I_{\nu}(f)$ is defined and equal to $\gamma$.

\proof{ As in the proof of 123A, we may, replacing $\sequencen{f_n}$ by
$\sequencen{f_n-f_0}$ if necessary, suppose that $f_n(x)\ge 0$ for every
$n\in\Bbb N$ and $x\in X$.

\medskip

{\bf (a)} Take $\epsilon>0$.   Then there is a $\delta\in\Delta$ such
that $S_{\pmb{t}}(f,\nu)\le\gamma+4\epsilon$ for every $\delta$-fine
$\pmb{t}\in T$.

\noindent\Prf\ Fix a strictly positive function
$h:X\to\ooint{0,\infty}$
and a $\tilde\delta\in\Delta$ such that $S_{\pmb{t}}(h,\nu)\le 1$ for
every $\tilde\delta$-fine $\pmb{t}\in T$.
For each $n\in\Bbb N$ choose $\delta_n\in\Delta$ and
$\Cal R_n\in\frak R$ such that
$|I_{\nu}(f_n)-S_{\pmb{t}}(f,\nu)|\le 2^{-n-1}\epsilon$ for every
$\delta_n$-fine $\Cal R_n$-filling $\pmb{t}\in T$.   For each
$x\in X$, take $r_x\in\Bbb N$ such that
$f(x)\le f_{r_x}(x)+\epsilon h(x)$.   Let $\delta\in\Delta$ be such
that $(x,C)\in\tilde\delta\cap\delta_{r_x}$ for every $(x,C)\in\delta$.

Suppose that $\pmb{t}\in T$ is $\delta$-fine.   Enumerate $\pmb{t}$ as
$\ofamily{i}{n}{(x_i,C_i)}$.   Let $k\in\Bbb N$ be so large that
$S_{\pmb{t}}(f,\nu)\le S_{\pmb{t}}(f_k,\nu)+\epsilon$ and $r_{x_i}\le
k$
for every $i<n$.   For $m\le k$, set $J_m=\{i:i<n,\,r_{x_i}=m\}$.
For
each $i\in J_m$, $(x_i,C_i)\in\delta_m$.   By 482A(c-ii), there is a
$\delta_k$-fine $\pmb{s}_m\in T$ such that
$W_{\pmb{s}_m}\subseteq\bigcup_{i\in J_m}C_i$ and
$|S_{\pmb{s}_m}(f_m,\nu)-\sum_{i\in J_m}f_m(x_i)\nu C_i|
\le 2^{-m}\epsilon$.

Set $\pmb{s}=\bigcup_{m\le k}\pmb{s}_m$, so that $\pmb{s}$ is a
$\delta_k$-fine member of $T$ and

$$\eqalignno{\sum_{m=0}^k\sum_{i\in J_m}f_m(x_i)\nu C_i
&\le\sum_{m=0}^kS_{\pmb{s}_m}(f_m,\nu)+2^{-m}\epsilon
\le\sum_{m=0}^kS_{\pmb{s}_m}(f_k,\nu)+2^{-m}\epsilon\cr
&\le S_{\pmb{s}}(f_k,\nu)+2\epsilon
\le I_{\nu}(f_k)+3\epsilon\cr
\displaycause{because $\pmb{s}$ extends to a $\delta_k$-fine
$\Cal R_k$-filling member of $T$, by 482Ab}
&\le\gamma+3\epsilon.\cr}$$

Now $\pmb{t}$ is $\tilde\delta$-fine, so $S_{\pmb{t}}(h,\nu)\le 1$.
Accordingly

$$\eqalign{S_{\pmb{t}}(f,\nu)
&=\sum_{i<n}f(x_i)\nu C_i
=\sum_{m=0}^k\sum_{i\in J_m}f(x_i)\nu C_i\cr
&\le\sum_{m=0}^k\sum_{i\in J_m}(f_m(x_i)+\epsilon h(x_i))\nu C_i
\le\gamma+3\epsilon+\epsilon S_{\pmb{t}}(h,\nu)
\le\gamma+4\epsilon,\cr}$$

\noindent as required.\ \Qed

As $\epsilon$ is arbitrary,
$\limsup_{\pmb{t}\to\Cal F(T,\Delta,\frak R)}S_{\pmb{t}}(f,\nu)$ is at
most $\gamma$.

\medskip

{\bf (b)} On the other hand, given $\epsilon>0$, there is an
$n\in\Bbb N$ such that $I_{\nu}(f_n)\ge\gamma-\epsilon$.   So taking
$\delta_n\in\Delta$, $\Cal R_n\in\frak R$ as in (a) above,

\Centerline{$S_{\pmb{t}}(f,\nu)\ge S_{\pmb{t}}(f_n,\nu)
\ge I_{\nu}(f_n)-\epsilon\ge\gamma-2\epsilon$}

\noindent for every $\delta_n$-fine $\Cal R_n$-filling $\pmb{t}\in T$.
So $\liminf_{\pmb{t}\to\Cal F(T,\Delta,\frak R)}S_{\pmb{t}}(f,\nu)$ is
at
least $\gamma$, and $I_{\nu}(f)=\gamma$.
}%end of proof of 482K

\leader{482L}{Lemma} Let $X$ be a set, $\Cal C$ a family of subsets of
$X$, $\Delta$ a countably full downwards-directed set of gauges on
$X$, $\frak R\subseteq\Cal P\Cal PX$ a downwards-directed collection of
residual families, and $T\subseteq[X\times\Cal C]^{<\omega}$ a
straightforward set of
tagged partitions of $X$ compatible with $\Delta$ and $\frak R$.
Suppose further that whenever $\delta\in\Delta$,
$\Cal R\in\frak R$ and $\pmb{t}\in T$ is $\delta$-fine, there is a
$\delta$-fine $\Cal R$-filling $\pmb{t}'\in T$ including $\pmb{t}$.
\cmmnt{(For instance, $(X,T,\Delta,\frak R)$ might be a tagged-partition
structure allowing subdivisions, as in 482Ab.)}   If
$\nu:\Cal C\to\coint{0,\infty}$ and $f:X\to\coint{0,\infty}$ are such
that $I_{\nu}(f)=0$, and $g:X\to\Bbb R$ is such that $g(x)=0$ whenever
$f(x)=0$, then $I_{\nu}(g)=0$.

\proof{ Let $\epsilon>0$.   For each $n\in\Bbb N$, let
$\delta_n\in\Delta$, $\Cal R_n\in\frak R$ be such that
$S_{\pmb{t}}(f,\nu)\le 2^{-n}\epsilon$ for every $\delta_n$-fine
$\Cal R_n$-filling $\pmb{t}\in T$.   For $x\in X$, set
$\phi(x)=\min\{n:|g(x)|\le nf(x)\}$;  let $\delta\in\Delta$ be such
that
$(x,C)\in\delta_{\phi(x)}$ whenever $(x,C)\in\delta$.

Let $\pmb{t}$ be any $\delta$-fine member of $T$.   Then
$|S_{\pmb{t}}(g,\nu)|\le 2\epsilon$.   \Prf\ Enumerate $\pmb{t}$ as
$\ofamily{i}{n}{(x_i,C_i)}$.   For each $m\in\Bbb N$, set
$K_m=\{i:i<n,\,\phi(x_i)=m\}$;  then $\{(x_i,C_i):i\in K_m\}$ is a
$\delta_m$-fine member of $T$, so extends to a $\delta_m$-fine
$\Cal R_m$-filling member $\pmb{t}_m$ of $T$, and

\Centerline{$\sum_{i\in K_m}|g(x_i)\nu C_i|
\le m\sum_{i\in K_m}f(x_i)\nu C_i
\le mS_{\pmb{t}_m}(f,\nu)
\le 2^{-m}m\epsilon$.}

\noindent Summing over $m$,

\Centerline{$|S_{\pmb{t}}(g,\nu)|
\le\sum_{m=0}^{\infty}2^{-m}m\epsilon=2\epsilon$.  \Qed}

\noindent Because $T$ is compatible with $\Delta$ and $\Cal R$, this
is
enough to show that $I_{\nu}(g)$ is defined and equal to $0$.
}%end of proof of 482L

\leader{482M}{Fubini's theorem} Suppose that, for $i=1$ and $i=2$, we
have $X_i$, $\frak T_i$, $T_i$, $\Delta_i$, $\Cal C_i$ and $\nu_i$
such
that

\inset{(i) $(X_i,\frak T_i)$ is a topological space;

(ii) $\Delta_i$ is the set of neighbourhood gauges on $X_i$;

(iii) $T_i\subseteq[X_i\times\Cal C_i]^{<\omega}$ is a straightforward
set of tagged partitions, compatible with $\Delta_i$ and
$\{\{\emptyset\}\}$;

(iv) $\nu_i:\Cal C_i\to\coint{0,\infty}$ is a function;

(v) $\nu_1$ is moderated with respect to $T_1$ and $\Delta_1$;

(vi) whenever $\delta\in\Delta_1$ and $\pmb{s}\in T_1$ is
$\delta$-fine, there is a $\delta$-fine $\pmb{s}'\in T_1$, including
$\pmb{s}$, such that $W_{\pmb{s}'}=X_1$.}

\noindent Write $X$ for $X_1\times X_2$;  $\Delta$ for the set of
neighbourhood gauges on $X$;  $\Cal C$ for
$\{C\times D:C\in\Cal C_1,\,D\in\Cal C_2\}$;  $Q$ for
$\{((x,y),C\times D):\{(x,C)\}\in T_1,\,\{(y,D)\}\in T_2\}$;
$T$ for the straightforward set of tagged partitions generated by $Q$;
and set
$\nu(C\times D)=\nu_1C\cdot\nu_2D$ for $C\in\Cal C_1$, $D\in\Cal C_2$.

(a) $T$ is compatible with $\Delta$ and $\{\{\emptyset\}\}$.

(b) Let $I_{\nu_1}$, $I_{\nu_2}$ and $I_{\nu}$ be the gauge integrals
defined by these structures\cmmnt{ as in 481C-481F}.   Suppose that
$f:X\to\Bbb R$ is such that $I_{\nu}(f)$ is defined.   Set
$f_x(y)=f(x,y)$ for $x\in X_1$, $y\in X_2$.   Let $g:X_1\to\Bbb R$ be
any function such that $g(x)=I_{\nu_2}(f_x)$ whenever this is defined.
Then $I_{\nu_1}(g)$ is defined and equal to $I_{\nu}(f)$.

\proof{{\bf (a)} Let $\delta\in\Delta$;  we seek a tagged partition
$\pmb{u}\in T$ such that $W_{\pmb{u}}=X$.   Let
$\family{(x,y)}{X}{V_{xy}}$ be the family of open sets in $X$ defining
$\delta$;  choose open sets $G_{xy}\subseteq X_1$,
$H_{xy}\subseteq X_2$
such that $(x,y)\in G_{xy}\times H_{xy}\subseteq V_{xy}$ for all
$x\in X_1$, $y\in X_2$.   For each $x\in X_1$, let $\delta_x$ be the
neighbourhood gauge on $X_2$ defined from the family
$\family{y}{X_2}{H_{xy}}$.   Then there is a $\delta_x$-fine tagged
partition $\pmb{t}_x\in T_2$ such that $W_{\pmb{t}_x}=X_2$.   Set
$G_x=X_1\cap\bigcap_{(y,D)\in\pmb{t}_x}G_{xy}$.

The family $\family{x}{X_1}{G_x}$ defines a neighbourhood gauge
$\delta^*$ on $X_1$, and there is a $\delta^*$-fine $\pmb{s}\in T_1$
such that $W_{\pmb{s}}=X_1$. Now consider

\Centerline{$\pmb{u}
=\{((x,y),C\times D):(x,C)\in\pmb{s},\,(y,D)\in\pmb{t}_x\}$.}

\noindent Then it is easy to check (just as in part (b) of the proof
of 481O) that $\pmb{u}$ is a $\delta$-fine member of $T$ with
$W_{\pmb{u}}=X$.

\medskip

{\bf (b)(i)} Set $A=\{x:x\in X_1,\,I_{\nu_2}(f_x)$ is defined$\}$.
Let $h:X_1\to\Bbb R$ be any function such that $h(x)=0$ for every
$x\in A$.   For $x\in X_1$, set

\Centerline{$h_0(x)
=\inf(\{1\}\cup\{\sup_{\pmb{t},\pmb{t}'\in F}S_{\pmb{t}}(f_x,\nu_2)
  -S_{\pmb{t}'}(f_x,\nu_2):
F\in\Cal F(T_2,\Delta_2,\{\{\emptyset\}\})\})$.}

\noindent (Thus
$I_{\nu_2}(f_x)$ is defined iff $h_0(x)=0$.)   Then
$I_{\nu_1}(h_0)=0$.
\Prf\ Let $\epsilon>0$.   Then there is a $\delta\in\Delta$ such that
$S_{\pmb{u}}(f,\nu)-S_{\pmb{u}'}(f,\nu)\le\epsilon$ whenever
$\pmb{u}$,
$\pmb{u}'\in T$ are $\delta$-fine and $W_{\pmb{u}}=W_{\pmb{u}'}=X$.
Define $\family{(x,y)}{X}{V_{xy}}$, $\family{(x,y)}{X}{G_{xy}}$,
$\family{(x,y)}{X}{H_{xy}}$ and $\family{x}{X_1}{\delta_x}$ from
$\delta$ as in (a) above.   For each $x\in X_1$, we can find
$\delta_x$-fine partitions $\pmb{t}_x$, $\pmb{t}'_x\in T_2$ such that
$W_{\pmb{t}_x}=W_{\pmb{t}'_x}=X_2$ and
$S_{\pmb{t}_x}(f_x,\nu_2)-S_{\pmb{t}'_x}(f_x,\nu_2)\ge\bover12h_0(x)$.
Set $G_x=X_1\cap\bigcap_{(y,D)\in\pmb{t}_x\cup\pmb{t}'_x}G_{xy}$.

Let $\delta^*$ be the neighbourhood gauge on $X_1$ defined from
$\family{x}{X_1}{G_x}$.   Let $\pmb{s}$ be any $\delta^*$-fine member
of $T_1$ with $W_{\pmb{s}}=X_1$.   Set

\Centerline{$\pmb{u}
=\{((x,y),C\times D):(x,C)\in\pmb{s},\,(y,D)\in\pmb{t}_x\}$,}

\Centerline{$\pmb{u}'
=\{((x,y),C\times D):(x,C)\in\pmb{s},\,(y,D)\in\pmb{t}'_x\}$,}

\noindent Then $\pmb{u}$ and $\pmb{u}'$ are $\delta$-fine members of
$T$ with $W_{\pmb{u}}=W_{\pmb{u}'}=X$, so

$$\eqalign{S_{\pmb{s}}(h_0,\nu_1)
&=\sum_{(x,C)\in\pmb{s}}h_0(x)\nu_1(C)
\le 2\sum_{(x,C)\in\pmb{s}}
  (S_{\pmb{t}_x}(f_x,\nu_2)-S_{\pmb{t}'_x}(f_x,\nu_2))\nu_1(C)\cr
&=2\bigl(\sum_{\Atop{(x,C)\in\pmb{s}}{(y,D)\in\pmb{t}_x}}
      f(x,y)\nu_1(C)\nu_2(D)
  -\sum_{\Atop{(x,C)\in\pmb{s}}{(y,D)\in\pmb{t}'_x}}
      f(x,y)\nu_1(C)\nu_2(D)\bigr)\cr
&=2\bigl(S_{\pmb{u}}(f,\nu)-S_{\pmb{u}'}(f,\nu)\bigr)
\le 2\epsilon.\cr}$$

\noindent As $\epsilon$ is arbitrary, $I_{\nu_1}(h_0)=0$.\ \Qed

By 482L, $I_{\nu_1}(h)$ also is zero.

\medskip

\quad{\bf (ii)} Again take any $\epsilon>0$, and let $\delta\in\Delta$
be such that $|S_{\pmb{u}}(f,\nu)-I_{\nu}(f)|\le\epsilon$ for every
$\delta$-fine $\pmb{u}\in T$ such that $W_{\pmb{u}}=X$.
Define $\family{(x,y)}{X}{V_{xy}}$, $\family{(x,y)}{X}{G_{xy}}$,
$\family{(x,y)}{X}{H_{xy}}$ and $\family{x}{X_1}{\delta_x}$ from
$\delta$ as in (a) and (i) above.   Let $\tilde\delta\in\Delta_1$,
$\tilde h:X_1\to\ooint{0,\infty}$ be such that
$S_{\pmb{s}}(\tilde h,\nu_1)\le 1$ for every $\tilde\delta$-fine
$\pmb{s}\in T_1$.

For $x\in A$, let $\pmb{t}_x\in T_2$ be $\delta_x$-fine and
such that $W_{\pmb{t}_x}=X_2$ and
$|S_{\pmb{t}_x}(f_x,\nu_2)-I_{\nu_2}(f_x)|\le\epsilon\tilde h(x)$; for
$x\in X_1\setminus A$, let $\pmb{t}_x$ be any $\delta_x$-fine member
of
$T_2$ such that $W_{\pmb{t}_x}=X_2$.   Set
$G_x=X_1\cap\bigcap_{(y,D)\in\pmb{t}_x}G_{xy}$ for every $x\in X_1$.
Let $\delta^*$ be the neighbourhood gauge on $X_1$ defined by
$\family{x}{X_1}{G_x}$.

Set $h_1(x)=g(x)-S_{\pmb{t}_x}(f_x,\nu_2)$ for $x\in X_1\setminus A$,
$0$ for $x\in A$.   Then $I_{\nu_1}(|h_1|)=0$, by (a).   Let
$\delta'\in\Delta_1$ be such that
$\delta'\subseteq\delta^*\cap\tilde\delta$ and
$S_{\pmb{s}}(|h_1|,\nu_1)\le\epsilon$ for every $\delta'$-fine
$\pmb{s}\in T_1$ such that $W_{\pmb{s}}=X_1$.

Now suppose that $\pmb{s}\in T_1$ is $\delta'$-fine and that
$W_{\pmb{s}}=X_1$.   Set

\Centerline{$\pmb{u}=\{((x,y),C\times
D):(x,C)\in\pmb{s},\,(y,D)\in\pmb{t}_x\}$,}

\noindent so that $\pmb{u}\in T$ is $\delta$-fine and $W_{\pmb{u}}=X$.
Then

$$\eqalign{|S_{\pmb{s}}(g,\nu_1)-I_{\nu}(f)|
&\le|S_{\pmb{u}}(f,\nu)-I_{\nu}(f)|
  +|S_{\pmb{s}}(g,\nu_1)-S_{\pmb{u}}(f,\nu)|\cr
&\le\epsilon+\bigl|\sum_{(x,C)\in\pmb{s}}
  \bigl(g(x)-\sum_{(y,D)\in\pmb{t}_x}f(x,y)\nu_2D\bigr)\nu_1C\bigr|\cr
&=\epsilon+\bigl|\sum_{(x,C)\in\pmb{s}}
  \bigl(g(x)-S_{\pmb{t}_x}(f_x,\nu_2)\bigr)\nu_1C\bigr|\cr
&\le\epsilon+\sum_{(x,C)\in\pmb{s}}|h_1(x)|\nu_1C\cr
&\qquad\qquad+\sum_{\Atop{(x,C)\in\pmb{s}}{x\in A}}
     |g(x)-\sum_{(y,D)\in\pmb{t}_x}f(x,y)\nu_2D|\nu_1C\cr
&\le 2\epsilon
  +\sum_{\Atop{(x,C)\in\pmb{s}}{x\in A}}\epsilon\tilde h(x)\nu_1(C)
\le 2\epsilon
  +\epsilon S_{\pmb{s}}(\tilde h,\nu_1)
\le 3\epsilon.\cr}$$

\noindent As $\epsilon$ is arbitrary, $I_{\nu_1}(g)$ is defined and
equal to $I_{\nu}(f)$, as claimed.
}%end of proof of 482M

\exercises{\leader{482X}{Basic exercises (a)}
%\spheader 482Xa
Let $(X,T,\Delta,\frak R)$ be a tagged-partition structure allowing
subdivisions, witnessed by $\Cal C$, and $\Cal E$ the algebra of
subsets
of $X$ generated by $\Cal C$.   Write $\eusm I$ for the set of pairs
$(f,\nu)$ such that $f:X\to\Bbb R$ and $\nu:\Cal C\to\Bbb R$ are
functions and
$I_{\nu}(f)=\lim_{\pmb{t}\to\Cal F(T,\Delta,\frak R)}S_{\pmb{t}}(f,\nu)$
is defined;  for $(f,\nu)\in\eusm I$, let $F_{f\nu}:\Cal E\to\Bbb R$ be
the corresponding Saks-Henstock indefinite integral.   Show that
$(f,\nu)\mapsto F_{f\nu}$ is bilinear in the sense that

\Centerline{$F_{f+g,\nu}=F_{f\nu}+F_{g\nu}$,
\quad$F_{\alpha f,\nu}=F_{f,\alpha\nu}=\alpha F_{f\nu}$,
\quad$F_{f,\mu+\nu}=F_{f\mu}+F_{f\nu}$}

\noindent whenever $(f,\nu)$, $(g,\nu)$ and $(f,\mu)$ belong to
$\eusm I$ and $\alpha\in\Bbb R$.
%482B

\spheader 482Xb Let $(X,T,\Delta,\frak R)$ be a tagged-partition
structure allowing subdivisions, witnessed by $\Cal C$, and $\Cal E$
the
algebra of subsets of $X$ generated by $\Cal C$.   Let $f:X\to\Bbb R$
and $\nu:\Cal C\to\coint{0,\infty}$ be functions such that
$I_{\nu}(f)=\lim_{\pmb{t}\to\Cal F(T,\Delta,\frak
R)}S_{\pmb{t}}(f,\nu)$
is defined, and let $F:\Cal E\to\Bbb R$ be the corresponding
Saks-Henstock indefinite integral.   Show that $F$ is non-negative iff
$I_{\nu}(f^-)=0$, where $f^-(x)=\max(0,-f(x))$ for every $x\in X$.
%482B

\sqheader 482Xc
Let $X$ be a zero-dimensional compact Hausdorff space, $\Cal E$ the
algebra of open-and-closed subsets of $X$,
$Q=\{(x,E):x\in E\in\Cal E\}$, $T$ the straightforward set of tagged
partitions generated by $Q$, $\Delta$ the set of neighbourhood gauges
on $X$ and $\nu:\Cal E\to\Bbb R$ a
non-negative additive functional.   Let $\mu$ be the corresponding Radon
measure on $X$ (416Qa) and $I_{\nu}$ the gauge integral defined by
$(X,T,\Delta,\{\{\emptyset\}\})$ (cf.\ 481Xh).   Show that, for
$f:X\to\Bbb R$, $I_{\nu}(f)=\int fd\mu$ if either is defined in
$\Bbb R$.   \Hint{if $f$ is measurable but not $\mu$-integrable, take
$x_0$ such that $f$ is not integrable on any neighbourhood of $x_0$.
Given $\delta\in\Delta$, fix a $\delta$-fine partition containing
$(x_0,V_0)$ for some $V_0$;  now replace $(x_0,V_0)$ by refinements
$\{(x_0,V'_0),(x_1,V_1),\ldots,(x_n,V_n)\}$, where
$\sum_{i=1}^nf(x_i)\nu V_i$ is large, to show that
$S_{\pmb{t}}(f,\nu)$
cannot be controlled by $\delta$.}
%481Xh, 482E, 482F

\sqheader 482Xd Let $(X,\frak T,\Sigma,\mu)$ be an effectively locally
finite $\tau$-additive topological measure space in which $\mu$ is
inner regular with respect to the closed sets and outer regular with respect
to the open sets (see 412W).   Let $T$ be the straightforward set of
tagged partitions generated by $X\times\{E:\mu E<\infty\}$, $\Delta$
the set of neighbourhood gauges on $X$, and
$\frak R=\{\Cal R_{E\eta}:\mu E<\infty,\,\eta>0\}$, where
$\Cal R_{E\eta}=\{F:\mu(F\cap E)\le\eta\}$, as in N.   Show that if
$I_{\mu}$ is the associated gauge integral, and $f:X\to\Bbb R$ is a
function, then $I_{\mu}(f)=\int fd\mu$ if either is defined in $\Bbb R$.
%482E, 482F, 482Xc

\spheader 482Xe Let $\Cal C$ be the set of non-empty subintervals of
$X=[0,1]$, $T$ the straightforward tagged-partition structure
generated
by $[0,1]\times\Cal C$, and $\Delta$ the set of neighbourhood gauges
on $[0,1]$, as in 481M.   Let $\mu$ be any Radon measure on $[0,1]$, and
$I_{\mu}$ the gauge integral defined from $\mu$ and the
tagged-partition
structure $([0,1],T,\Delta,\{\{\emptyset\}\})$.   Show that, for any
$f:[0,1]\to\Bbb R$, $I_{\mu}(f)=\int fd\mu$ if either is defined in
$\Bbb R$.
%482E, 482F, 482Xc

\spheader 482Xf Let $\Cal C$ be the set of non-empty subintervals of
$[0,1]$, $T$ the straightforward tagged-partition structure generated
by $\{(x,C):C\in\Cal C,\,x\in\overline{C}\}$, and $\Delta$ the set of
neighbourhood gauges on $X$, as in 481J.   Let $\Cal E$ be the ring
of subsets of $[0,1]$ generated by $\Cal C$,
$\nu:\Cal E\to\Bbb R$ a bounded additive functional, and
$I_{\nu}$ the gauge integral defined from $\nu$ and
$(X,T,\Delta,\{\{\emptyset\}\})$.   Show that
$I_{\nu}(\chi\coint{a,b})=\lim_{x\uparrow a,y\uparrow b}\nu([x,y])$
whenever $0<a<b\le 1$.
%482G

\spheader 482Xg Let $\Cal C$ be the set of non-empty subintervals of
$X=[0,1]$, $T$ the straightforward tagged-partition structure
generated
by $\{(x,C):C\in\Cal C,\,x\in\overline{C}\}$, and $\Delta$ the set of
uniform metric gauges on $[0,1]$, as in 481I.   Let $\mu$ be the Dirac
measure on $[0,1]$ concentrated at $\bover12$, and let
$I_{\mu}
=\lim_{\pmb{t}\to\Cal F(T,\Delta,\{\{\emptyset\}\})}S_{\pmb{t}}(\,,\mu)$
be the corresponding gauge integral.   Show that $I_{\mu}(\chi[0,1])$
is
defined but that $I_{\mu}(\chi[0,\bover12])$ is not.
%482G

\sqheader 482Xh(i) Show that the McShane integral on an interval
$[a,b]$ as described in 481M coincides with the Lebesgue integral on
$[a,b]$.   (ii) Show that if $(X,\frak T,\Sigma,\mu)$ is a quasi-Radon
measure space and $\mu$ is outer regular with respect to the open sets
then the McShane integral as described in 481N coincides with the
usual integral.
%482G

\spheader 482Xi Explain how the results in 481Xb-481Xc can be regarded
as special cases of 482H.

\spheader 482Xj Let $(X,\frak T)$ be a topological space, $\Cal C$ a
ring of subsets of $X$, $T\subseteq[X\times\Cal C]^{<\omega}$ a
straightforward set of tagged partitions,
$\nu:\Cal C\to\coint{0,\infty}$ an additive function, and $\Delta$ the
family of neighbourhood gauges on $X$.   Suppose that there is a
sequence $\sequencen{G_n}$ of open sets, covering $X$, such that
$\sup\{\nu C:C\in\Cal C,\,C\subseteq G_n\}$ is finite for every
$n\in\Bbb N$.   Show that $\nu$ is moderated with respect to $T$ and
$\Delta$.
%482J

\spheader 482Xk Let $(X,\frak T,\Sigma,\mu)$ be a quasi-Radon measure
space.   Let $T$ be the straightforward tagged-partition structure
generated by
$\{(x,E):\mu E<\infty,\,x\in E\}$ and $\Delta$ the set of all
neighbourhood gauges on $X$.   Show that $\mu$ is moderated with
respect
to $T$ and $\Delta$ iff there is a sequence of open sets of finite
measure covering $X$.
%482J

\spheader 482Xl Let $r\ge 1$ be an integer, and $\mu$ a Radon measure
on $\BbbR^r$.   Let $Q$ be the set of pairs $(x,C)$ where
$x\in\BbbR^r$ and $C$ is a closed ball with centre $x$, and $T$ the
straightforward set of tagged partitions generated by $Q$.   Let
$\Delta$ be the set of neighbourhood gauges on $\BbbR^r$, and
$\frak R=\{\Cal R_{E\eta}:\mu E<\infty,\,\eta>0\}$, where
$\Cal R_{E\eta}=\{F:\mu(F\cap E)\le\eta\}$, as in 482Xd.   (i) Show
that $T$ is compatible with $\Delta$ and $\frak R$.   \Hint{472C.}
(ii) Show that if
$I_{\mu}$ is the associated gauge integral, and $f:\BbbR^r\to\Bbb R$
is a function, then $I_{\mu}(f)=\int fd\mu$ if either is defined in
$\Bbb R$.
%482?

\spheader 482Xm\dvAnew{2012}
Let $(X,T,\Delta,\frak R)$, $\Sigma$ and $\nu$ be as in 481Xj, so that
$(X,T,\Delta,\frak R)$ is a tagged-partition structure allowing
subdivisions, witnessed by an algebra $\Sigma$ of subsets of $X$, and
$\nu:\Sigma\to\coint{0,\infty}$ is additive.   Let $I_{\nu}$ be the
corresponding gauge integral, and $V\subseteq\BbbR^X$ its domain.   
(i) Show that $\chi E\in V$ and $I_{\nu}(\chi E)=\nu E$
for every $E\in\Sigma$.   (ii) Show that if $f\in\BbbR^X$ then $f\in V$ iff
for every $\epsilon>0$ there is a disjoint family $\Cal E\subseteq\Sigma$
such that $\sum_{E\in\Cal E}\nu E=\nu X$ and
$\sum_{E\in\Cal E}\nu E\cdot\sup_{x,y\in E}|f(x)-f(y)|\le\epsilon$.
(iii) Show that if
$\sequencen{f_n}$ is a non-decreasing sequence in $V$ with supremum 
$f\in\BbbR^X$, and $\gamma=\sup_{n\in\Bbb N}I_{\nu}(f_n)$ is finite, then
$I_{\nu}(f)$ is defined and equal to $\gamma$.
(iv) Show that $I_{\nu}$ extends $\dashint d\nu$ as described in
363L, if we identify $L^{\infty}(\Sigma)$ with a space $\eusm L^{\infty}$
of functions as in 363H.   (v) Show that if $\Sigma$ is a $\sigma$-algebra
of sets then $I_{\nu}$ extends $\dashint d\nu$ as described in
364Xj.


\leader{482Y}{Further exercises (a)}\dvAnew{2010}
%\spheader 482Ya
Let $X$ be the interval $[0,1]$,
$\Cal C$ the family of subintervals of $X$,
$Q$ the set $\{(x,C):C\in\Cal C$, $x\in\overline{\interior C}\}$, $T$ the
straightforward set of tagged partitions generated by $Q$, $\Delta$ the set
of neighbourhood gauges on $X$, and $\frak R$ the singleton
$\{[X]^{<\omega}\}$.   Show that $(X,T,\Delta,\frak R)$ is a
tagged-partition structure allowing subdivisions, witnessed by $\Cal C$.
For $C\in\Cal C$ set $\nu C=1$ if $0\in\overline{\interior C}$, $0$
otherwise, and let $f$ be $\chi\{0\}$.   Show that
$I_{\nu}(f)=\lim_{\pmb{t}\to\Cal F(T,\Delta,\frak R)}S_{\pmb{t}}(f,\nu)$
is defined and equal
to $1$.   Let $F$ be the Saks-Henstock indefinite integral of $f$.
Show that $F(\ocint{0,1})=1$.
%482B

\spheader 482Yb
Set $X=\Bbb R$ and let $\Cal C$ be the family of
non-empty bounded intervals in $X$;  set
$Q=\{(x,C):C\in\Cal C,\,x=\inf C\}$, and let $T$ be the
straightforward
set of tagged partitions generated by $Q$.   Let $\frak S$ be the
Sorgenfrey right-facing topology on $X$ (415Xc), and $\Delta$ the set of
neighbourhood gauges for $\frak S$.   Set
$\Cal R_n=\{E:E\in\Sigma,\,\mu([-n,n]\cap E)\le 2^{-n}\}$ for
$n\in\Bbb N$, where $\mu$ is Lebesgue measure on $\Bbb R$ and $\Sigma$
its domain, and write $\frak R=\{\Cal R_n:n\in\Bbb N\}$.   Show that
$(X,T,\Delta,\frak R)$ is a tagged-partition structure allowing
subdivisions.   Show that if $f:X\to\Bbb R$ is such that $I_{\mu}(f)$
is defined, then $f$ is Lebesgue measurable.
%482E

\spheader 482Yc Give an example of $X$, $\frak T$, $\Sigma$, $\mu$,
$T$, $\Delta$, $\Cal C$, $f$ and $C$ such that
$(X,\frak T,\Sigma,\mu)$ is a compact metrizable Radon probability
space, $\Delta$ is the set of all neighbourhood gauges on $X$,
$(X,T,\Delta,\{\{\emptyset\}\})$ is a tagged-partition structure
allowing subdivisions, witnessed by
$\Cal C$, $f:X\to\Bbb R$ is a function such that $I_{\mu}(f)
=\lim_{\pmb{t}\to\Cal F(T,\Delta,\{\{\emptyset\}\})}S_{\pmb{t}}(f,\mu)$
is defined, $C\in\Cal C$ is a closed set with negligible boundary, and
$I_{\mu}(f\times\chi C)$ is not defined.
%482G

\spheader 482Yd Suppose that, for $i=1$ and $i=2$, we have a
tagged-partition structure $(X_i,T_i,\Delta_i,\frak R_i)$ allowing
subdivisions, witnessed by a ring $\Cal C_i\subseteq\Cal PX_i$, where
$X_i$ is a topological space, $\Delta_i$ is the set of all neighbourhood
gauges on $X_i$, and $\frak R_i$ is the simple residual structure
complementary to $\Cal C_i$, as in 481Yb.   Set
$X=X_1\times X_2$ and let $\Delta$ be the set of neighbourhood gauges
on $X$;  set $\Cal C=\{C\times D:C\in\Cal C_1,\,D\in\Cal C_2\}$;  let
$\frak R$ be the simple residual structure on $X$ complementary to
$\Cal C$;  and let $T$ be the straightforward tagged-partition
structure generated by
$\{((x,y),C\times D):\{(x,C)\}\in T_1,\,\{(y,D)\}\in T_2\}$.   For each
$i$, let $\nu_i:\Cal C_i\to\coint{0,\infty}$ be a function moderated
with respect to $T_i$ and $\Delta_i$, and define
$\nu:\Cal C\to\coint{0,\infty}$ by setting
$\nu(C\times D)=\nu_1C\cdot\nu_2D$ for $C\in\Cal C_1$, $D\in\Cal C_2$.
Show that $T$ is compatible with $\Delta$ and $\frak R$.   Let
$I_{\nu_1}$, $I_{\nu_2}$, $I_{\nu}$ be the gauge integrals defined by
these structures.   Suppose that $f:X\to\Bbb R$ is such that
$I_{\nu}(f)$ is defined.   Set $f_x(y)=f(x,y)$ for $x\in X_1$,
$y\in X_2$.   Let $g:X_1\to\Bbb R$ be any function such that
$g(x)=I_{\nu_2}(f_x)$ whenever this is defined.   Show that
$I_{\nu_1}(g)$ is defined and equal to $I_{\nu}(f)$.
%482M
}%end of exercises

\endnotes{
\Notesheader{482}
In 482E, 482F and 482G the long lists of conditions reflect the variety
of possible applications of these arguments.   The price to be paid for
the versatility of the constructions here is a theory which is rather
weak in the absence of special hypotheses.
As everywhere in this book, I try to set ideas out in maximal convenient
generality;   you may feel that in this section the generality is
becoming inconvenient;  but the theory of gauge integrals has not, to my
eye, matured to the point that we can classify the systems here even as
provisionally as I set out to classify topological measure spaces in
Chapters 41 and 43.

Enthusiasts for gauge integrals offer two substantial arguments for
taking them seriously, apart from the universal argument in pure
mathematics, that these structures offer new patterns for our delight
and new challenges to our ingenuity.   First, they say, gauge integrals
integrate more functions than Lebesgue-type integrals, and it is the
business of a theory of integration to integrate as many functions as
possible;  and secondly, gauge integrals offer an easier path to the
principal theorems.   I have to say that I think the first argument is
sounder than the second.   It is quite true that the Henstock integral
on $\Bbb R$ (481K) can be rigorously defined in fewer words, and with
fewer concepts, than the Lebesgue integral.   The style of Chapters 11
and 12 is supposed to be better adapted to the novice than the style
of this chapter, but you will have no difficulty in putting the ideas of
481A, 481C, 481J and 481K together into an elementary definition of an
integral for real functions in which the only non-trivial argument is
that establishing the existence of enough tagged partitions (481J),
corresponding I suppose to Proposition 114D.   But the path I took in
defining the integral in \S122, though arduous at that point, made (I
hope) the convergence theorems of \S123 reasonably natural;  the proof
of 482K, on the other hand, makes significant demands on our
technique.
Furthermore, the particular clarity of the one-dimensional Henstock
integral is not repeated in higher dimensions.   Fubini's theorem,
with
exact statement and full proof, even for products of Lebesgue measures
on Euclidean spaces, is a lot to expect of an undergraduate;  but
Lebesgue measure on $\BbbR^r$ makes sense in a way that it is quite
hard
to repeat with gauge integrals.   (For instance, Lebesgue measure is
invariant under isometries;  this is not particularly easy to
prove -- see 263A -- but at least it is true;  if we want a gauge
integral which
is invariant under isometries, then we have to use a construction such
as 481O, which does not directly match any natural general definition
of `product gauge integral' along the lines of 481P, 482M or 482Yd.)

In my view, a stronger argument for taking gauge integrals seriously
is their `power', that is, their ability to provide us with integrals of
many functions in consistent ways.   482E, 482F and 482I give us an
idea
of what to expect.   If we start from a measure space $(X,\Sigma,\mu)$
and build a gauge integral $I_{\mu}$ from a set
$T\subseteq[X\times\Sigma]^{<\omega}$ of tagged partitions, then we
can hope that integrable functions will be gauge-integrable, with the
right
integrals (482F);  while gauge-integrable functions will be measurable
(482E).   What this means is that for {\it non-negative} functions,
the integrals will coincide.   Any `new' gauge-integrable functions $f$
will be such that $\int f^+=\int f^-=\infty$;  the gauge integral will
offer
a process for cancelling the divergent parts of these integrals.   On
the other hand, we can hope for a large class of gauge-integrable
derivatives.   In the next two sections, I will explain how this works
in the Henstock and Pfeffer
integrals.   For simple examples calling for such procedures, see the
formulae of \S\S282 and 283;  for radical applications of the idea, see
{\smc Muldowney 87}.

Against this, gauge integrals are not effective in `general' measure
spaces, and cannot be, because there is no structure in an abstract
measure space $(X,\Sigma,\mu)$ which allows us to cancel an infinite
integral $\int f^+=\int_Ff$ against
$\int f^-=\int_{X\setminus F}f$.   Put another way, if a
tagged-partition structure is invariant under all automorphisms of the
structure $(X,\Sigma,\mu)$, as in 481Xf-481Xg, we cannot expect
anything
better than the standard integral.   In order to get something new, the
most important step seems to be the specification of a family
$\Cal C$ of `regular' sets, preliminary to describing a set $T$ of
tagged partitions.   To get a `powerful' gauge integral, we want a fine
filter on $T$, corresponding to a small set $\Cal C$ and a large set of
gauges.   The residual
families of 481F are generally introduced just to ensure `compatibility'
in the sense described there;  as a rule, we try to keep them simple.
But even if we take the set of all neighbourhood gauges, as in the
Henstock integral, this is not enough unless we also sharply restrict
both the family $\Cal C$ and the permissible tags
(482Xc-482Xe).  %482Xc 482Xd 482Xe
The most successful restrictions, so far, have been `geometric', as in
481J and 481O, and 484F below.   Further limitations on admissible
pairs $(x,C)$,
as in 481L and 481Q, in which $\Cal C$ remains the set of intervals, but
fewer tags are permitted, also lead to very interesting results.

Another limitation in the scope of gauge integrals is the difficulty
they have in dealing with spaces of infinite measure.   Of course we
expect to have to specify a limiting procedure if we are to calculate
$I_{\mu}(f)$ from sums $S_{\pmb{t}}(f,\mu)$ which necessarily consider
only sets of finite measure, and this is one of the functions of the
collections $\frak R$ of residual families.   But this is not yet
enough.   In B.Levi's theorem (482K) we already need to suppose that our
set-function $\nu$ is `moderated' in order to determine how closely
$f_n(x)$ needs to approximate each $f(x)$.   The condition

\Centerline{$S_{\pmb{t}}(h,\nu)\le 1$ for every $\delta$-fine
$\pmb{t}$}

\noindent of 482J is very close to saying that $I_{\mu}(h)\le 1$.   But
it is {\it not} the same as saying that $\mu$ is $\sigma$-finite;  it
suggests rather that $X$ should be covered by a sequence of open sets
of finite measure (482Xk).

Because gauge integrals are not absolute -- that is, we can have
$I_{\nu}(f)$ defined and finite while $I_{\nu}(|f|)$ is not -- we are
bound to have difficulties with integrals $\int_Hf$, even if we
interpret these in the simplest way, as $I_{\nu}(f\times\chi H)$, so
that we do not need a theory of subspaces as developed in \S214.
482G(iii)-(v) are an
attempt to find a reasonably versatile sufficient set of conditions.
The `multiplier problem', for a given gauge integral $I_{\nu}$, is the
problem of characterizing the functions $g$ such that
$I_{\nu}(f\times g)$ is defined whenever $I_{\nu}(f)$ is defined, and
even for some intensively studied integrals remains challenging.   In
484L I will give an important case which is not covered by 482G.

One of the striking features of gauge integrals is that there is no
need
to assume that the set-function $\nu$ is countably additive.   We can
achieve countable additivity of the integral -- in the form of
B.Levi's
theorem, for instance -- by requiring only that the set of gauges
should
be `countably full' (482K, 482L;  contrast 482Xg).   If we watch our
definitions carefully, we can make this match the rules for Stieltjes
integrals (114Xa, 482Xf).   In 481Db I have already remarked on the
potential use of gauge integrals in vector integration.

It is important to recognise that a value $F(E)$ of a Saks-Henstock
indefinite integral (482B-482C) cannot be
identified with either $I_{\nu}(f\times\chi E)$ or with
$I_{\nu\restrp\Cal PE}(f\restr E)$, because in the formula
$F(E)=\lim_{\pmb{t}\to\Cal F^*}S_{\pmb{t}_E}(f,\nu)$ used in the proof of
482B the tags of the
partitions $\pmb{t}_E$ need not lie in $E$.   (See 482Ya.)
The idea of 482G is
to impose sufficient conditions to ensure that the contributions of
`straddling' elements $(x,C)$, where either $x\in E$ and
$C\not\subseteq E$ or $x\notin E$ and $C\cap E\ne\emptyset$, are
insignificant.   To achieve this we seem to need both a regularity
condition on the functional $\nu$ (condition (iii) of 482G)
and a geometric condition on the set $\Cal C$ underlying $T$ (482G(iv)).
As usual,
the regularity condition required is closer to {\it outer} than to
{\it inner} regularity, in contexts in which there is a distinction.

I am not sure that I have the `right' version of Proposition 482E.
The hypothesis there is that we have a metric space.   But in the
principal
non-metrizable cases the result is still valid (482Xc-482Xd, 482Yb),
and the same happens in 482Xl, where condition 482E(ii) is not
satisfied.
Proposition 482H is a `new' limit theorem;  it shows that certain improper
integrals from the classical theory can be represented as gauge integrals.
The hypotheses seem, from where we
are standing at the moment, to be exceedingly restrictive.   In the
leading examples in \S483, however, the central requirement 482H(viii) is
satisfied for straightforward geometric reasons.

Gauge integrals insist on finite functions defined everywhere.   But
since we have an effective theory of negligible sets (482L), we can
easily get a consistent theory of integration for functions which are
defined and real-valued almost everywhere if we say that

\Centerline{$I_{\nu}(f)=I_{\nu}(g)$ whenever $g:X\to\Bbb R$ extends
$f\restr f^{-1}[\Bbb R]$}

\noindent whenever $I_{\nu}(g)=I_{\nu}(g')$ for all such extensions.
}%end of notes

\discrpage

\frfilename{mt483.tex}
\versiondate{6.9.10}
\copyrightdate{2000}

\def\chaptername{Gauge integrals}
\def\sectionname{The Henstock integral}
\def\FSH{F^{_{\text{SH}}}}

\newsection{483}

I come now to the original gauge integral, the `Henstock integral' for
real functions.   The first step is to check that the results of \S482
can be applied to show that this is an extension of both the Lebesgue
integral and the improper Riemann integral (483B), coinciding with
the Lebesgue integral for non-negative functions (483C).   It turns out
that any Henstock integrable function can be approximated in a strong
sense by a sequence of Lebesgue integrable functions (483G).   The
Henstock integral can be identified with the Perron and special Denjoy
integrals (483J, 483N\cmmnt{, 483Yh}).
Much of the rest of the section is concerned
with indefinite Henstock integrals.   Some of the results of \S482 on
tagged-partition structures allowing subdivisions condense into a
particularly strong Saks-Henstock lemma (483F).   If $f$ is Henstock
integrable, it is equal almost everywhere to the derivative of its
indefinite Henstock integral (483I).   
Finally, indefinite Henstock integrals can be characterized as 
continuous ACG$_*$ functions (483R).

\leader{483A}{Definition} The following notation will apply throughout
the section.   Let $\Cal C$ be the family of non-empty bounded intervals
in $\Bbb R$, and let
$T\subseteq[\Bbb R\times\Cal C]^{<\omega}$ be the straightforward set of
tagged partitions generated by
$\{(x,C):C\in\Cal C$, $x\in\overline{C}\}$.   Let $\Delta$ be the set of
all neighbourhood gauges on $\Bbb R$.    Set
$\frak R=\{\Cal R_{ab}:a\le b\in\Bbb R\}$, where
$\Cal R_{ab}=\{\Bbb R\setminus[c,d]:c\le a,\,d\ge b\}\cup\{\emptyset\}$.
Then $(\Bbb R,T,\Delta,\frak R)$ is a tagged-partition structure
allowing subdivisions\cmmnt{ (481K)}, so $T$ is compatible with
$\Delta$ and $\frak R$\cmmnt{ (481Hf)}.   The {\bf Henstock integral}
is the gauge integral defined\cmmnt{ by the process of 481E-481F} from
$(\Bbb R,T,\Delta,\frak R)$ and one-dimensional
Lebesgue measure $\mu$.   For a function
$f:\Bbb R\to\Bbb R$ I will say that $f$ is {\bf Henstock integrable},
and that $\Heint f=\gamma$, if
$\lim_{\pmb{t}\to\Cal F(T,\Delta,\frak R)}S_{\pmb{t}}(f,\mu)$ is defined
and equal to $\gamma\in\Bbb R$.   For $\alpha$,
$\beta\in[-\infty,\infty]$ I will write $\Heint_{\alpha}^{\beta}f$ for
$\Heint f\times\chi\ooint{\alpha,\beta}$ if this is defined in $\Bbb R$.
I will use the symbol $\int$ for the ordinary integral, so that
$\int fd\mu$ is the Lebesgue integral of $f$.

\leader{483B}{}\cmmnt{ Tracing through the theorems of \S482, we have
the following.

\medskip

\noindent}{\bf Theorem} (a) Every Henstock integrable function on
$\Bbb R$ is Lebesgue measurable.

(b) Every Lebesgue integrable function $f:\Bbb R\to\Bbb R$ is Henstock
integrable, with the same integral.

(c) If $f$ is Henstock integrable so is $f\times\chi C$ for every
interval $C\subseteq\Bbb R$.

(d) Suppose that $f:\Bbb R\to\Bbb R$ is any function, and
$-\infty\le\alpha<\beta\le\infty$.   Then

\Centerline{$\biggerHeint_{\alpha}^{\beta}f
=\lim_{a\downarrow\alpha}\biggerHeint_a^{\beta}f
=\lim_{b\uparrow\beta}\biggerHeint_{\alpha}^bf
=\lim_{a\downarrow\alpha,b\uparrow\beta}\biggerHeint_a^bf$}

\noindent if any of the four terms is defined in $\Bbb R$.

\proof{{\bf (a)} Apply 482E.

\medskip

{\bf (b)} Apply 482F, referring to 134F to confirm that condition
482F(ii) is satisfied.

It will be useful to note at once that this shows that
$\Heint f\times\chi\{a\}=0$ for every $f:\Bbb R\to\Bbb R$ and every
$a\in\Bbb R$.   Consequently $\Heint_a^bf=\Heint_a^cf+\Heint_c^bf$ whenever
$a\le c\le b$ and the right-hand side is defined.

\medskip

{\bf (c)-(d)} In the following argument, $f$ will always be a function from
$\Bbb R$ to itself;  when $f$ is Henstock integrable, $\FSH$ will be its
Saks-Henstock indefinite integral (482B-482C).

\medskip

\quad{\bf (i)} The first thing to check is that the conditions of 482G
are satisfied by $\Bbb R$, $T$, $\Delta$, $\frak R$, $\Cal C$ and
$\mu\restr\Cal C$.   \Prf\  482G(i) is just 481K, and 482G(ii) is trivial.
482G(iii-$\alpha$) and 482G(v) are elementary, and so is
482G(iii-$\beta$) --- if you like, this is a special case of
412W(b-iii).   As for 482G(iv), if
$E\in\Cal C$ is a singleton $\{x\}$, then whenever
$(x,C)\in\Cal C$ we can express $C$ as a union of one or more of the
sets $C\cap\ooint{-\infty,x}$, $C\cap\{x\}$ and $C\cap\ooint{x,\infty}$,
and for any non-empty $C'$ of these we have $\{(x,C')\}\in T$ and either
$C'\subseteq E$ or
$C'\cap E=\emptyset$.   Otherwise, let $\eta>0$ be half the length of
$E$, and let $\delta$ be the uniform metric gauge
$\{(x,A):A\subseteq\ooint{x-\eta,x+\eta}\}$.    Then if $x\in\partial E$
and $(x,C)\in T\cap\delta$, we can again express $C$ as a union
of one or more of the sets
$C\cap\ooint{-\infty,x}$, $C\cap\{x\}$ and $C\cap\ooint{x,\infty}$, and
these will witness that 482G(iv) is satisfied.\ \Qed

\medskip

\quad{\bf (ii)} Now suppose that $f$ is Henstock integrable.   Then
$\Heint_a^bf$ is defined whenever $a\le b$ in $\Bbb R$.
\Prf\ 482G tells us that $\Heint f\times\chi C$ is
defined and equal to $\FSH(C)$ for every $C\in\Cal C$;  in particular,
$\Heint_a^bf=\Heint f\times\chi\ooint{a,b}$ is defined whenever $a<b$
in $\Bbb R$.\ \QeD\   Note that because
$\Heint f\times\chi\{c\}=0$ for every $c$,
$\Heint f\times\chi C=\Heint_{\inf C}^{\sup C}f$ whenever $C\in\Cal C$ is
non-empty.

This proves (c) for bounded intervals;  we shall come to unbounded
intervals in (vii) below.

\medskip

\quad{\bf (iii)} If $f$ is Henstock integrable, then
$\lim_{a\to-\infty,b\to\infty}\Heint_a^bf$ is defined and equal to
$\Heint f$.   \Prf\ Given
$\epsilon>0$, there is an $\Cal R\in\frak R$ such that
$|\FSH(\Bbb R\setminus C)|\le\epsilon$ whenever $C\in\Cal C$ and
$\Bbb R\setminus C\in\Cal R$;  that is, there are $a_0\le b_0$ such that

\Centerline{$|\biggerHeint f-\biggerHeint_a^bf|
=|\FSH(\Bbb R\setminus\ooint{a,b})|=|\FSH(\Bbb R\setminus[a,b])|\le\epsilon$}

\noindent whenever $a\le a_0\le b_0\le b$.
As $\epsilon$ is arbitrary, $\lim_{a\to-\infty,b\to\infty}\Heint_a^bf$
is defined and equal to $\Heint f$.\ \Qed

\medskip

\quad{\bf (iv)} It follows that if $f$ is Henstock integrable and
$c\in\Bbb R$, $\lim_{b\to\infty}\Heint_c^bf$ is defined.   \Prf\ Let
$\epsilon>0$.   Then there are $a_0\le c$, $b_0\ge c$ such that
$|\Heint_a^bf-\Heint f|\le\epsilon$ whenever $a\le a_0$ and $b\ge b_0$.
But this means that

\Centerline{$|\biggerHeint_c^bf-\biggerHeint_c^{b'}f|
=|\biggerHeint_{a_0}^bf-\biggerHeint_{a_0}^{b'}f|\le 2\epsilon$}

\noindent whenever $b$, $b'\ge b_0$.   As $\epsilon$ is arbitrary,
$\lim_{b\to\infty}\int_c^bf$ is defined.\ \Qed

Similarly, $\lim_{a\to-\infty}\Heint_a^cf$ is defined.

\medskip

\quad{\bf (v)} Moreover, if $f$ is Henstock integrable and
$a<b$ in $\Bbb R$, then $\lim_{c\uparrow b}\Heint_a^cf$ is defined and
equal to $\int_a^bf$.   \Prf\ Let $\epsilon>0$.   Then
there is a $\delta\in\Delta$ such that
$\sum_{(x,C)\in\pmb{t}}|\FSH(C)-f(x)\mu C|\le\epsilon$ whenever
$\pmb{t}\in T$ is $\delta$-fine.   Let $\eta>0$ be such that
$\eta|f(b)|\le\epsilon$ and $(b,\coint{c,b})\in\delta$ whenever
$b-\eta\le c<b$.   Then whenever $\max(a,b-\eta)\le c<b$,

\Centerline{$\biggerHeint_a^bf-\biggerHeint_a^cf
=\FSH(\coint{c,b})
\le|\FSH(\coint{c,b})-f(b)\mu\coint{c,b}|+|f(b)|\mu\coint{c,b}
\le 2\epsilon$.}

\noindent As $\epsilon$ is arbitrary,
$\lim_{c\uparrow b}\Heint_a^cf=\Heint_a^bf$.\ \Qed

Similarly, $\lim_{c\downarrow a}\Heint_c^bf$ is defined and equal to
$\Heint_a^bf$.

\medskip

\quad{\bf (vi)} Now for a much larger step.
If $-\infty\le\alpha<\beta\le\infty$ and $f:\Bbb R\to\Bbb R$ is such that
$\lim_{a\downarrow\alpha,b\uparrow\beta}\Heint_a^bf$ is defined and equal
to $\gamma$, then $\Heint_{\alpha}^{\beta}f$ is defined and equal to
$\gamma$.
\Prf\ I seek to apply 482H, with $H=\ooint{\alpha,\beta}$ and
$H_n=\ooint{a_n,b_n}$, where $\sequencen{a_n}$ is a strictly decreasing
sequence with limit $\alpha$, $\sequencen{b_n}$ is a strictly increasing
sequence with limit $\beta$, and $a_0<b_0$.
We have already seen that the conditions of
482G are satisfied;  482H(vi) is elementary, and 482H(vii) is covered by
(ii) above.   So we are left with 482H(viii).   Given
$\epsilon>0$, let $m\in\Bbb N$ be such that
$|\gamma-\Heint_a^bf|\le\epsilon$ whenever $\alpha<a\le a_m$ and
$b_m\le b<\beta$.   For $x\in\Bbb R$ let $G_x$ be an open set,
containing $x$, such that

$$\eqalign{\overline{G}_x
&\subseteq H\text{ if }x\in H,\cr
&\subseteq\ooint{-\infty,a_m}\text{ if }x<a_m,\cr
&\subseteq\ooint{b_m,\infty}\text{ if }x>b_m,\cr}$$

\noindent and let $\delta\in\Delta$ be the neighbourhood gauge
corresponding to $\family{x}{\Bbb R}{G_x}$.
Now suppose that $\pmb{t}\in T$ is $\delta$-fine and
$\Cal R_{a_mb_m}$-filling.
Then $W_{\pmb{t}}$ is a closed bounded interval
including $[a_m,b_m]$.   Since $\family{(x,C)}{\pmb{t}}{C}$ is a disjoint
family of intervals, $\pmb{t}$ must have an
enumeration $\langle(x_i,C_i)\rangle_{i\le r}$ where $x\le y$ whenever
$i\le j\le r$, $x\in C_i$ and $y\in C_j$.   As $H\subseteq\Bbb R$ is an
interval, there are $i_0\le i_1$ such that

\Centerline{$\pmb{t}\restr H=\{(x_i,C_i):i\le r$, $x_i\in H\}
=\{(x_i,C_i):i_0\le i\le i_1\}$.}

\noindent Because $W_{\pmb{t}}$ is an interval, so is
$W_{\pmb{t}\restr H}=\bigcup_{i_0\le i\le i_1}C_i$;
set $a=\inf W_{\pmb{t}\restr H}$ and
$b=\sup W_{\pmb{t}\restr H}$;  then
$\Heint f\times\chi W_{\pmb{t}}=\Heint_a^bf$ (see (ii)).
Next, $\overline{W}_{\pmb{t}\restr H}\subseteq H$ (because
$\overline{G}_x\subseteq H$ if $x\in H$) and
$[a_m,b_m]\subseteq W_{\pmb{t}\restr H}$
(because $[a_m,b_m]\subseteq W_{\pmb{t}}$ and
$\overline{G}_x\cap[a_m,b_m]=\emptyset$ for $x\notin[a_m,b_m]$).
So $\alpha<a\le a_m$, $b_m\le b<\beta$, and

\Centerline{$|\gamma-\biggerHeint f\times\chi W_{\pmb{t}\restr H}|
=|\gamma-\biggerHeint_a^bf|\le\epsilon$.}

As $\epsilon$ is arbitrary, this shows that

\Centerline{$\lim_{\pmb{t}\to\Cal F(T,\Delta,\frak R)}
\biggerHeint f\times\chi W_{\pmb{t}\restr H}=\gamma$,}

\noindent as required by 482H(viii).   So
$\Heint_{\alpha}^{\beta}f=\Heint f\times\chi H$ is defined and equal to
$\gamma$, as claimed.\ \Qed

\medskip

\quad{\bf (vii)} We are now in a position to confirm that if $f$ is Henstock
integrable then $\Heint_c^{\infty}f=\lim_{b\to\infty}\Heint_c^bf$ is
defined for every $c\in\Bbb R$.
\Prf\ By (iii) and (iv),
$\lim_{a\downarrow c}\Heint_a^{c+1}f=\Heint_c^{c+1}f$ and
$\lim_{b\to\infty}\Heint_{c+1}^bf$ are both defined.   So

$$\eqalign{\lim_{a\downarrow c,b\to\infty}\Heint_a^bf
&=\lim_{a\downarrow c}\Heint_a^{c+1}f
+\lim_{b\to\infty}\Heint_{c+1}^bf\cr
&=\Heint_c^{c+1}f+\lim_{b\to\infty}\Heint_{c+1}^b
=\lim_{b\to\infty}\Heint_c^bf\cr}$$

\noindent is defined, and is equal to $\Heint_c^{\infty}f$, by (vi).\
\Qed

Similarly, $\Heint_{-\infty}^cf=\lim_{a\to-\infty}\Heint_a^cf$ is
defined.   So $\Heint f\times\chi C$ is defined for sets $C$ of the form
$\ooint{c,\infty}$ or $\ooint{-\infty,c}$, and therefore for any unbounded
interval, since the case $C=\Bbb R$ is immediate.   So the proof of (c) is
complete.

\medskip

\quad{\bf (viii)} As for (d), (vi) has already given us part of it:  if
$\lim_{a\downarrow\alpha,b\uparrow\beta}\Heint_a^bf$ is defined, this is
$\Heint_{\alpha}^{\beta}f$.   In the other direction, if
$\Heint_{\alpha}^{\beta}f$ is defined, set
$g=f\times\chi\ooint{\alpha,\beta}$, so that $g$ is Henstock integrable,
and take any
$c\in\ooint{\alpha,\beta}$.   Then $\lim_{b\uparrow\beta}\Heint_c^bg$ is
defined, and equal to $\Heint_c^{\beta}g$, by (v) if $\beta$ is finite
and by (vii) if $\beta=\infty$.   Similarly,
$\lim_{a\downarrow\alpha}\Heint_a^cg$ is defined and equal to
$\Heint_{\alpha}^cg$.   Consequently

\Centerline{$\lim_{a\downarrow\alpha,b\uparrow\beta}\biggerHeint_a^bf
=\lim_{a\downarrow\alpha,b\uparrow\beta}\biggerHeint_a^bg
=\lim_{a\downarrow\alpha}\biggerHeint_a^cg
+\lim_{b\uparrow\beta}\biggerHeint_c^bg$}

\noindent is defined, and must be equal to
$\Heint_{\alpha}^{\beta}g=\Heint_{\alpha}^{\beta}f$, while also

\Centerline{$\lim_{a\downarrow\alpha}\biggerHeint_a^{\beta}f
=\lim_{a\downarrow\alpha}\biggerHeint_a^{\beta}g
=\lim_{a\downarrow\alpha}\biggerHeint_a^cg
   +\biggerHeint_c^{\beta}g
=\biggerHeint_{\alpha}^cg+\biggerHeint_c^{\beta}g
=\biggerHeint_{\alpha}^{\beta}g
=\biggerHeint_{\alpha}^{\beta}f$,}

\noindent and similarly

\Centerline{$\lim_{b\uparrow\beta}\biggerHeint_{\alpha}^bf
=\biggerHeint_{\alpha}^{\beta}f$.}

\medskip

\quad{\bf (ix)} Finally, we need to consider the case in which we are told
that $\lim_{a\downarrow\alpha}\Heint_a^{\beta}f$ is defined.
Taking any $c\in\ooint{\alpha,\beta}$, we know that $\Heint_c^{\beta}f$
is defined, by (ii) or (vii) applied to $f\times\chi\ooint{a,\beta}$ for
some $a\le c$, and equal to
$\lim_{b\uparrow\beta}\Heint_c^bf$, by (v) or (vii).   But this means that

\Centerline{$\lim_{a\downarrow\alpha,b\uparrow\beta}\biggerHeint_a^bf
=\lim_{a\downarrow\alpha}\biggerHeint_a^cf
+\lim_{b\uparrow\beta}\biggerHeint_c^bf$}

\noindent is defined, so (vi) and (viii) tell us that
$\Heint_{\alpha}^{\beta}f$ is defined and equal to
$\lim_{a\downarrow\alpha}\Heint_a^{\beta}f$.   The same argument, suitably
inverted, deals with the case in which
$\lim_{b\uparrow \beta}\Heint_{\alpha}^bf$ is defined.
}%end of proof of 483B

\leader{483C}{Corollary} The Henstock and Lebesgue integrals agree on
non-negative functions, in the sense that if
$f:\Bbb R\to\coint{0,\infty}$ then $\Heint f=\int fd\mu$ if either is
defined in $\Bbb R$.

\proof{ If $f$ is Lebesgue integrable, it is Henstock integrable, with
the same integral, by 483Bb.   If it is Henstock integrable, then it is
measurable, by 483Ba, so that $\int fd\mu$ is defined in $[0,\infty]$;
but

$$\eqalignno{\int fd\mu
&=\sup\{\int g\,d\mu:
  g\le f\text{ is a non-negative simple function}\}\cr
\displaycause{213B}
&=\sup\{\Heint g:g\le f\text{ is a non-negative simple function}\}
\le\Heint f\cr}$$

\noindent (481Cb) is finite, so $f$ is Lebesgue integrable.
}%end of proof of 483C

\leader{483D}{Corollary} If $f:\Bbb R\to\Bbb R$ is Henstock integrable,
then
$\alpha\mapsto\Heint_{-\infty}^{\alpha}f:[\infty,\infty]\to\Bbb R$ and
$(\alpha,\beta)\mapsto\Heint_{\alpha}^{\beta}f:
[-\infty,\infty]^2\to\Bbb R$ are continuous and bounded.

\proof{ Let $F$ be the indefinite Henstock integral of $f$.   Take any
$x_0\in\Bbb R$ and $\epsilon>0$.   By 483Bd, there is an
$\eta_1>0$ such that
$|\Heint_{-\infty}^{x}f-\Heint_{-\infty}^{x_0}f|\le\epsilon$
whenever $x_0-\eta_1\le x\le x_0$.   By 483Bd again, there
is an $\eta_2>0$ such that
$|\Heint_{x}^{\infty}f-\Heint_{x_0}^{\infty}f|\le\epsilon$
whenever $x_0\le x\le x_0+\eta_2$.   But this means that
$|F(x)-F(x_0)|\le\epsilon$ whenever
$x_0-\eta_1\le x\le x_0+\eta_2$.   As $\epsilon$ is
arbitrary, $F$ is continuous at $x_0$.

We know also that $\lim_{x\to\infty}F(x)=\Heint f$ is defined
in $\Bbb R$;  while

\Centerline{$\lim_{x\to-\infty}F(x)
=\biggerHeint f-\lim_{x\to-\infty}\biggerHeint_x^{\infty}f=0$}

\noindent is also defined, by 483Bd once more.   So $F$ is
continuous at $\pm\infty$.

Now writing $G(\alpha,\beta)=\Heint_{\alpha}^{\beta}f$, we have
$G(\alpha,\beta)=F(\beta)-F(\alpha)$ if $\alpha\le\beta$ and zero if
$\beta\le\alpha$.   So $G$ also is continuous.   $F$ and $G$ are bounded
because $[-\infty,\infty]$ is compact.
}%end of proof of 483D

\leader{483E}{Definition} If $f:\Bbb R\to\Bbb R$ is Henstock integrable,
then its {\bf indefinite Henstock integral} is the function
$F:\Bbb R\to\Bbb R$ defined by saying that $F(x)=\Heint_{-\infty}^xf$
for every $x\in\Bbb R$.

\leader{483F}{}\cmmnt{ In the present context, the Saks-Henstock lemma
can be sharpened, as follows.

\medskip

\noindent}{\bf Theorem} Let $f:\Bbb R\to\Bbb R$ and
$F:\Bbb R\to\Bbb R$ be functions.   Then the following are
equiveridical:

(i) $f$ is Henstock integrable and $F$ is its indefinite Henstock
integral;

(ii)($\alpha$) $F$ is continuous,

\quad($\beta$) $\lim_{x\to-\infty}F(x)=0$ and $\lim_{x\to\infty}F(x)$ is
defined in $\Bbb R$,

\quad($\gamma$) for every $\epsilon>0$ there are a gauge
$\delta\in\Delta$ and a non-decreasing function
$\phi:\Bbb R\to[0,\epsilon]$ such that
$|f(x)(b-a)-F(b)+F(a)|\le\phi(b)-\phi(a)$ whenever $a\le x\le b$ in
$\Bbb R$ and $(x,[a,b])\in\delta$.

\proof{{\bf (i)$\Rightarrow$(ii)} ($\alpha$)-($\beta$) are covered by
483D.   As for ($\gamma$), 482G tells us that we can identify the
Saks-Henstock indefinite integral of $f$ with
$E\mapsto\Heint f\times\chi E:\Cal E\to\Bbb R$, where $\Cal E$ is the
algebra generated by $\Cal C$.   Let
$\epsilon>0$.   Then there is a $\delta\in\Delta$ such
that $\sum_{(x,C)\in\pmb{t}}|f(x)\mu C-\Heint f\times\chi C|\le\epsilon$
for every $\delta$-fine $\pmb{t}\in T$.   Set

\Centerline{$\phi(a)
=\sup_{\pmb{t}\in T\cap\delta}
\sum_{(x,C)\in\pmb{t},C\subseteq\ocint{-\infty,a}}
$$|f(x)\mu C-\biggerHeint f\times\chi C|$,}

\noindent so that $\phi:\Bbb R\to[0,\epsilon]$ is a non-decreasing
function.   Now suppose that $a\le y\le b$ and that
$(y,[a,b])\in\delta$.   In this case, whenever $\pmb{t}\in T\cap\delta$,
$\pmb{s}=\{(x,C):(x,C)\in\pmb{t},\,
C\subseteq\ocint{-\infty,a}\}\cup\{(y,\ooint{a,b})\}$ also belongs to
$T\cap\delta$.   Now $F(b)-F(a)=\Heint_a^bf$, so

$$\eqalign{\phi(b)
&\ge\sum_{(x,C)\in\pmb{s}}|f(x)\mu C-\Heint f\times\chi C|\cr
&=\sum_{(x,C)\in\pmb{t},C\subseteq\ocint{-\infty,a}}
  |f(x)\mu C-\Heint f\times\chi C|
  +|f(y)(b-a)-F(b)+F(a)|.\cr}$$

\noindent As $\pmb{t}$ is arbitrary,

\Centerline{$\phi(b)\ge\phi(a)+|f(y)(b-a)-F(b)+F(a)|$,}

\noindent that is, $|f(y)(b-a)-F(b)+F(a)|\le\phi(b)-\phi(a)$, as called
for by ($\gamma$).

\medskip

{\bf (ii)$\Rightarrow$(i)} Assume (ii).   Set
$\gamma=\lim_{x\to\infty}F(x)$.   Let $\epsilon>0$.   Let $a\le b$ be
such that $|F(x)|\le\epsilon$ for every $x\le a$ and
$|F(x)-\gamma|\le\epsilon$ for every $x\ge b$.   Let $\delta\in\Delta$,
$\phi:\Bbb R\to[0,\epsilon]$ be such that $\phi$ is non-decreasing and
$|f(x)(\beta-\alpha)-F(\beta)+F(\alpha)|\le\phi(\beta)-\phi(\alpha)$
whenever $\alpha\le x\le\beta$ and $(x,[\alpha,\beta])\in\delta$.   Let
$\delta'\in\Delta$ be such that $(x,\overline{A})\in\delta$ whenever
$(x,A)\in\delta'$.   For $C\in\Cal C$ set
$\lambda C=F(\sup C)-F(\inf C)$, $\nu C=\phi(\sup C)-\phi(\inf C)$;
then if $(x,C)\in\delta'$, $(x,[\inf C,\sup C])\in\delta$, so
$|f(x)\mu C-\lambda C|\le\nu C$.   Note that $\lambda$ and $\nu$ are
both additive in the sense that
$\lambda(C\cup C')=\lambda C+\lambda C'$, $\nu(C\cup C')=\nu C+\nu C'$
whenever $C$, $C'$ are disjoint members of $\Cal C$ such that
$C\cup C'\in\Cal C$ (cf.\ 482G(iii-$\alpha$)).

Let $\pmb{t}\in T$ be $\delta'$-fine and $\Cal R_{ab}$-filling.   Then
$W_{\pmb{t}}$ is of the form $[c,d]$ where $c\le a$ and $b\le d$.   So

$$\eqalign{|S_{\pmb{t}}(f,\mu)-\gamma|
&\le 2\epsilon+|S_{\pmb{t}}(f,\mu)-F(d)+F(c)|
=2\epsilon+|S_{\pmb{t}}(f,\mu)-\lambda[c,d]|\cr
&=2\epsilon+|\sum_{(x,C)\in\pmb{t}}f(x)\mu C-\lambda C|
\le 2\epsilon+\sum_{(x,C)\in\pmb{t}}\nu C\cr
&=2\epsilon+\nu[c,d]
\le 3\epsilon.\cr}$$

\noindent As $\epsilon$ is arbitrary, $f$ is Henstock integrable, with
integral $\gamma$.

I still have to check that $F$ is the indefinite integral of $f$.   Set
$F_1(x)=\Heint_{-\infty}^xf$ for $x\in\Bbb R$, and $G=F-F_1$.   Then
(ii) applies equally to the pair $(f,F_1)$, because
(i)$\Rightarrow$(ii).   So, given $\epsilon>0$, we have
$\delta$, $\delta_1\in\Delta$ and non-decreasing functions $\phi$,
$\phi_1:\Bbb R\to[0,\epsilon]$ such that

\inset{$|f(x)(b-a)-F(b)+F(a)|\le\phi(b)-\phi(a)$ whenever $a\le x\le b$
in $\Bbb R$ and $(x,[a,b])\in\delta$,

$|f(x)(b-a)-F_1(b)+F_1(a)|\le\phi_1(b)-\phi_1(a)$ whenever $a\le x\le b$
in $\Bbb R$ and $(x,[a,b])\in\delta_1$.}

\noindent Putting these together,

\inset{$|G(b)-G(a)|\le\psi(b)-\psi(a)$ whenever $a\le x\le b$ in
$\Bbb R$ and $(x,[a,b])\in\delta\cap\delta_1$,}

\noindent where $\psi=\phi+\phi_1$.   But if $a\le b$ in $\Bbb R$, there
are $a_0\le x_0\le a_1\le x_1\le\ldots\le x_{n-1}\le a_n$ such that
$a=a_0$, $a_n=b$ and $(x_i,[a_i,a_{i+1}])\in\delta$ for $i<n$ (481J), so
that

$$\eqalign{|G(b)-G(a)|
&\le\sum_{i=0}^{n-1}|G(a_{i+1})-G(a_i)|\cr
&\le\sum_{i=0}^{n-1}\psi(a_{i+1})-\psi(a_i)
=\psi(b)-\psi(a)\le 2\epsilon.\cr}$$

\noindent As $\epsilon$ is arbitrary, $G$ is constant.   As
$\lim_{x\to-\infty}F(x)=\lim_{x\to-\infty}F_1(x)=0$, $F=F_1$, as
required.
}%end of proof of 483F

\leader{483G}{Theorem} %\cmmnt{ (see {\smc Gordon 94}, 9.18)}
Let $f:\Bbb R\to\Bbb R$ be a Henstock integrable function.
Then there is a
countable cover $\Cal K$ of $\Bbb R$ by compact sets such that
$f\times\chi K$ is Lebesgue integrable for every $K\in\Cal K$.

\proof{{\bf (a)} For $n\in\Bbb N$ set
$E_n=\{x:|x|\le n,\,|f(x)|\le n\}$.   By 483Ba, $f$ is Lebesgue
measurable, so for each $n\in\Bbb N$ we can find a compact set
$K_n\subseteq E_n$ such that $\mu(E_n\setminus K_n)\le 2^{-n}$;  $f$ is
Lebesgue integrable over $K_n$, and
$Y=\Bbb R\setminus\bigcup_{n\in\Bbb N}K_n$ is Lebesgue negligible.

Let $F$ be the indefinite Henstock integral of $f$,
and take a gauge $\delta_0\in\Delta$ and a non-decreasing function
$\phi:\Bbb R\to[0,1]$ such that
$|f(x)(b-a)-F(b)+F(a)|\le\phi(b)-\phi(a)$ whenever
$a\le x\le b$ and $(x,[a,b])\in\delta_0$ (483F).   Because
$\Heint|f|\times\chi Y=\int_Y|f|d\mu=0$ (483Bb), there is a
$\delta_1\in\Delta$ such that $S_{\pmb{t}}(|f|\times\chi Y,\mu)\le 1$
whenever $\pmb{t}\in T$ is $\delta_1$-fine (482Ad).   For $C\in\Cal C$,
set $\lambda C=F(\sup C)-F(\inf C)$, $\nu C=\phi(\sup C)-\phi(\inf C)$.
Set

\Centerline{$D_n
=\{x:x\in E_n\cap Y,\,(x,[a,b])\in\delta_0\cap\delta_1$
whenever $x-2^{-n}\le a\le x\le b\le x+2^{-n}\}$,}

\noindent so that $\bigcup_{n\in\Bbb N}D_n=Y$;  set
$K'_n=\overline{D}_n$, so that $K'_n$ is compact and
$\bigcup_{n\in\Bbb N}K'_n\supseteq Y$.

\medskip

{\bf (b)} The point is that $f\times\chi K'_n$ is Lebesgue integrable
for each $n$.   \Prf\ For $k\in\Bbb N$, let $A_k$ be the set of points
$x\in K'_n$ such that $\ocint{x,x+2^{-k}}\cap K'_n=\emptyset$;  then
$A_k$ is finite, because $K'_n\subseteq[-n,n]$ is bounded.   Similarly,
if $A'_k=\{x:x\in K'_n,\,\coint{x-2^{-k},x}\cap K'_n=\emptyset\}$,
$A'_k$ is finite.   Set
$B=K'_n\setminus\bigcup_{k\in\Bbb N}(A_k\cup A'_k)$, so
that $K'_n\setminus B$ is countable.

Set

\Centerline{$\delta
=\delta_0\cap\delta_1\cap
\{(x,A):x\in\Bbb R,\,A\subseteq\ooint{x-2^{-n-1},x+2^{-n-1}}\}$,}

\noindent so that $\delta\in\Delta$.   Note that if $C\in\Cal C$,
$x\in B\cap\overline{C}$, $(x,C)\in\delta$ and $\mu C>0$, then
$\interior C$ meets $K'_n$ (because there are points of $K'_n$
arbitrarily close to $x$ on both sides) so $\interior C$ meets
$D_n$;  and if $y\in D_n\cap\interior C$ then
$(y,C)\in\delta_0\cap\delta_1$, because $\diam C\le 2^{-n}$.   This
means that if $\pmb{t}\in T$ is $\delta$-fine and
$\pmb{t}\subseteq B\times\Cal C$, then there is a
$\delta_0\cap\delta_1$-fine $\pmb{s}\in T$ such that
$\pmb{s}\subseteq D_n\times\Cal C$, $W_{\pmb{s}}\subseteq W_{\pmb{t}}$
and whenever $(x,C)\in\pmb{t}$ and $C$ is not a singleton, there is a
$y$ such that $(y,C)\in\pmb{s}$.   Accordingly

$$\eqalign{\sum_{(x,C)\in\pmb{t}}|\lambda C|
&\le\sum_{(y,C)\in\pmb{s}}|\lambda C|
\le\sum_{(y,C)\in\pmb{s}}|f(y)\mu C-\lambda C|
  +\sum_{(y,C)\in\pmb{s}}|f(y)|\mu C\cr
&\le\sum_{(y,C)\in\pmb{s}}\nu C+S_{\pmb{s}}(|f|\times\chi Y,\mu)
\le 2.\cr}$$

But this means that if $\pmb{t}\in T$ is $\delta$-fine,

$$\eqalignno{S_{\pmb{t}}(|f\times\chi B|,\mu)
&=\sum_{(x,C)\in\pmb{t}\restr B}|f(x)\mu C|\cr
\displaycause{where $\pmb{t}\restr B=\pmb{t}\cap(B\times\Cal C)$}
&\le\sum_{(x,C)\in\pmb{t}\restr B}|f(x)\mu C-\lambda C|
  +\sum_{(x,C)\in\pmb{t}\restr B}|\lambda C|\cr
&\le\sum_{(x,C)\in\pmb{t}\restr B}\nu C+2
\le 3.\cr}$$

\noindent It follows that if $g$ is a $\mu$-simple function and
$0\le g\le|f\times\chi B|$,

$$\eqalignno{\int g\,d\mu
&=\Heint g
\le\sup_{\pmb{t}\in T\text{ is }\delta\text{-fine}}
  S_{\pmb{t}}(g,\mu)\cr
&\le\sup_{\pmb{t}\in T\text{ is }\delta\text{-fine}}
  S_{\pmb{t}}(|f\times\chi B|,\mu)
\le 3,\cr}$$

\noindent and $|f\times\chi B|$ is $\mu$-integrable, by 213B, so
$f\times\chi B$ is $\mu$-integrable, by 122P.   As $K'_n\setminus B$ is
countable, therefore negligible, $f\times\chi K'_n$ is
$\mu$-integrable.\ \Qed

\medskip

{\bf (c)} So if we set
$\Cal K=\{K_n:n\in\Bbb N\}\cup\{K'_n:n\in\Bbb N\}$, we have a suitable
family.
}%end of proof of 483G

\leader{483H}{Upper and lower derivates:  Definition} Let
$F:\Bbb R\to\Bbb R$ be any function.   For $x\in\Bbb R$, set

\Centerline{$(\DiniD F)(x)=\limsup_{y\to x}\Bover{F(y)-F(x)}{y-x}$,
\quad$(\Dinid F)(x)=\liminf_{y\to x}\Bover{F(y)-F(x)}{y-x}$}

\noindent in $[-\infty,\infty]$\cmmnt{, that is,
$(\DiniD F)(x)=\max((\DiniD^+F)(x),(\DiniD^-F)(x))$ and
$(\Dinid F)(x)=\min((\Dinid^+F)(x),(\Dinid^-F)(x))$ as defined in 222J}.

\leader{483I}{Theorem} Suppose that $f:\Bbb R\to\Bbb R$ is Henstock
integrable, and $F$ is its indefinite Henstock integral.   Then $F'(x)$
is defined and equal to $f(x)$ for almost every $x\in\Bbb R$.

\proof{ For $n\in\Bbb N$, set
$A_n=\{x:|x|\le n,\,(\DiniD F)(x)>f(x)+2^{-n}\}$.   Then
$\mu^*A_n\le 2^{-n+1}$.   \Prf\ Let $\delta\in\Delta$ and
$\phi:\Bbb R\to[0,4^{-n}]$ be such that $\phi$ is non-decreasing and
$|f(x)(b-a)-F(b)+F(a)|\le\phi(b)-\phi(a)$ whenever $a\le x\le b$ and
$(x,[a,b])\in\delta$ (483F).   Let $\Cal I$ be the set of non-trivial
closed intervals $[a,b]\subseteq\Bbb R$ such that, for some
$x\in[a,b]\cap A_n$, $(x,[a,b])\in\delta$ and
$\Bover{F(b)-F(a)}{b-a}\ge f(x)+2^{-n}$.   By Vitali's theorem (221A) we
can find a countable disjoint family
$\Cal I_0\subseteq\Cal I$ such that $A_n\setminus\bigcup\Cal I_0$ is
negligible;  so we have a finite family $\Cal I_1\subseteq\Cal I_0$ such
that $\mu^*(A_n\setminus\bigcup\Cal I_1)\le 2^{-n}$.   Enumerate
$\Cal I_1$ as $\ofamily{i}{m}{[a_i,b_i]}$, and for each $i<m$ take
$x_i\in[a_i,b_i]\cap A_n$ such that $(x_i,[a_i,b_i])\in\delta$ and
$F(b_i)-F(a_i)\ge(b_i-a_i)(f(x_i)+2^{-n})$.   Then

\Centerline{$\phi(b_i)-\phi(a_i)\ge|f(x_i)(b_i-a_i)-F(b_i)+F(a_i)|
\ge 2^{-n}(b_i-a_i)$}

\noindent for each $i<m$, so

\Centerline{$\mu(\bigcup\Cal I_1)=\sum_{i<m}b_i-a_i
\le 2^n\sum_{i<m}\phi(b_i)-\phi(a_i)\le 2^{-n}$,}

\noindent and $\mu^*A_n\le 2^{-n+1}$.\ \Qed

Accordingly
$\{x:(\DiniD F)(x)>f(x)\}=\bigcup_{m\in\Bbb N}\bigcap_{n\ge m}A_n$
is negligible.   Similarly, or applying the argument to $-f$,
$\{x:(\Dinid F)(x)<f(x)\}$ is negligible.   So
$\DiniD F\leae f\leae\Dinid F$.
Since $\Dinid F\le\DiniD F$ everywhere, $\DiniD F\eae\Dinid F\eae f$.
But $F'(x)=f(x)$
whenever $(\DiniD F)(x)=(\Dinid F)(x)=f(x)$, so we have the result.
}%end of proof of 483I

\vleader{48pt}{483J}{Theorem} Let $f:\Bbb R\to\Bbb R$ be a function.   Then
the following are equiveridical:

(i) $f$ is Henstock integrable;

(ii) for every $\epsilon>0$ there are functions $F_1$,
$F_2:\Bbb R\to\Bbb R$, with finite limits at both $-\infty$ and
$\infty$, such that $(\DiniD F_1)(x)\le f(x)\le(\Dinid F_2)(x)$ and
$0\le F_2(x)-F_1(x)\le\epsilon$ for every $x\in\Bbb R$.

\proof{{\bf (i)$\Rightarrow$(ii)} Suppose that $f$ is Henstock
integrable and that $\epsilon>0$.   Let $F$ be the indefinite Henstock
integral of $f$.   Let $\delta\in\Delta$,
$\phi:\Bbb R\to[0,\bover12\epsilon]$ be such that $\phi$ is
non-decreasing and $|f(x)(b-a)-F(b)+F(a)|\le\phi(b)-\phi(a)$ whenever
$a\le x\le b$ and $(x,[a,b])\in\delta$ (483F).   Set $F_1=F-\phi$,
$F_2=F+\phi$;  then $F_1(x)\le F_2(x)\le F_1(x)+\epsilon$ for every
$x\in\Bbb R$, and the limits at $\pm\infty$ are defined because $F$ and
$\phi$ both have limits at both ends.   If $x\in\Bbb R$, there is an
$\eta>0$ such that $(x,A)\in\delta$ whenever
$A\subseteq[x-\eta,x+\eta]$.   So if
$x-\eta\le a\le x\le b\le x+\eta$ and $a<b$,

\Centerline{$|\Bover{F(b)-F(a)}{b-a}-f(x)|
\le\Bover{\phi(b)-\phi(a)}{b-a}$,}

\noindent and

\Centerline{$\Bover{F_1(b)-F_1(a)}{b-a}\le f(x)
\le\Bover{F_2(b)-F_2(a)}{b-a}$.}

\noindent In particular, this is true whenever $x-\eta\le a<x=b$ or
$x=a<b\le x+\eta$.   So $(\DiniD F_1)(x)\le f(x)\le(\Dinid F_2)(x)$.   As $x$ is
arbitrary, we have a suitable pair $F_1$, $F_2$.

\medskip

{\bf (ii)$\Rightarrow$(i)} Suppose that (ii) is true.   Take
any $\epsilon>0$.   Let $F_1$, $F_2:\Bbb R\to\Bbb R$ be as in the
statement of (ii).

\medskip

\quad\grheada\ We need to know that $F_2-F_1$ is non-decreasing.   \Prf\
Set $G=F_2-F_1$.   Then

$$\eqalignno{\liminf_{y\to x}\Bover{G(y)-G(x)}{y-x}
&=\liminf_{y\to x}\Bover{F_2(y)-F_2(x)}{y-x}
  -\Bover{F_1(y)-F_1(x)}{y-x}\cr
&\ge\liminf_{y\to x}\Bover{F_2(y)-F_2(x)}{y-x}
  -\limsup_{y\to x}\Bover{F_1(y)-F_1(x)}{y-x}\cr
\displaycause{2A3Sf}
&=(\Dinid F_2)(x)-(\DiniD F_1)(x)\ge 0\cr}$$

\noindent for any $x\in\Bbb R$.   \Quer\ If $a<b$ and $G(a)>G(b)$, set
$\gamma=\Bover{G(a)-G(b)}{2(b-a)}$, and choose $\sequencen{a_n}$,
$\sequencen{b_n}$ inductively as follows.   $a_0=a$ and $b_0=b$.   Given
that $a_n<b_n$ and $G(a_n)-G(b_n)>\gamma(b_n-a_n)$, set
$c=\bover12(a_n+b_n)$;  then either $G(a_n)-G(c)>\gamma(c-a_n)$ or
$G(c)-G(b_n)\ge\gamma(b_n-c)$;  in the former case, take $a_{n+1}=a_n$
and $b_{n+1}=c$;  in the latter, take $a_{n+1}=c$ and $b_{n+1}=b_n$.
Set $x=\lim_{n\to\infty}a_n=\lim_{n\to\infty}b_n$.   Then for each $n$,
either $G(a_n)-G(x)>\gamma(x-a_n)$ or $G(x)-G(b_n)>\gamma(b_n-x)$.   In
either case, we have a $y$ such that $0<|y-x|\le 2^{-n}(b-a)$ and
$\Bover{G(y)-G(x)}{y-x}<-\gamma$.   So $(\Dinid G)(x)\le-\gamma<0$, which is
impossible.\ \Bang

Thus $G$ is non-decreasing, as required.\ \Qed

\medskip

\quad\grheadb\ Let $a\le b$ be such that $|F_1(x)-F_1(a)|\le\epsilon$
whenever $x\le a$ and
$|F_1(x)-F_1(b)|\le\epsilon$ whenever $x\ge b$.   Let
$h:\Bbb R\to\ooint{0,\infty}$ be a strictly positive integrable function
such that $\int h\,d\mu\le\epsilon$.   Then $\Heint h\le\epsilon$, by
483Bb, so there is a $\delta_0\in\Delta$ such that
$S_{\pmb{t}}(h,\mu)\le 2\epsilon$ for every $\delta_0$-fine
$\pmb{t}\in T$ (482Ad).   For $x\in\Bbb R$ let $\eta_x>0$ be such that

\Centerline{$\Bover{F_1(y)-F_1(x)}{y-x}\le f(x)+h(x)$,
\quad$\Bover{F_2(y)-F_2(x)}{y-x}\ge f(x)-h(x)$}

\noindent whenever $0<|y-x|\le 2\eta_x$;  set $\delta
=\{(x,A):(x,A)\in\delta_0,\,A\subseteq\ooint{x-\eta_x,x+\eta_x}\}$, so
that $\delta\in\Delta$.   Note that if $x\in\Bbb R$ and
$x-\eta_x\le\alpha\le x\le\beta\le x+\eta_x$, then

\Centerline{$F_1(\beta)-F_1(x)\le(\beta-x)(f(x)+h(x))$,
\quad$F_1(x)-F_1(\alpha)\le(x-\alpha)(f(x)+h(x))$,}

\noindent so that $F_1(\beta)-F_1(\alpha)\le(\beta-\alpha)(f(x)+h(x))$;
and similarly
$F_2(\beta)-F_2(\alpha)\ge(\beta-\alpha)(f(x)-h(x))$.

For $C\in\Cal C$, set

\Centerline{$\lambda_1C=F_1(\sup C)-F_1(\inf C)$,
\quad$\lambda_2C=F_2(\sup C)-F_2(\inf C)$.}

\noindent Then if $C\in\Cal C$, $x\in\overline{C}$ and $(x,C)\in\delta$,

\Centerline{$\lambda_1C\le(f(x)+h(x))\mu C$,
\quad$\lambda_2C\ge(f(x)-h(x))\mu C$.}

Suppose that $\pmb{t}\in T$ is $\delta$-fine and $\Cal R_{ab}$-filling.
Then $W_{\pmb{t}}=[\alpha,\beta]$ for some $\alpha\le a$ and $\beta\ge b$,
so that

$$\eqalign{S_{\pmb{t}}(f,\mu)
&=\sum_{(x,C)\in\pmb{t}}f(x)\mu C
\le\sum_{(x,C)\in\pmb{t}}\lambda_2C+h(x)\mu C
=\lambda_2[\alpha,\beta]+S_{\pmb{t}}(h,\mu)\cr
&\le F_2(\beta)-F_2(\alpha)+2\epsilon
\le F_1(\beta)-F_1(\alpha)+3\epsilon
\le F_1(b)-F_1(a)+5\epsilon.\cr}$$

\noindent Similarly,

$$\eqalign{S_{\pmb{t}}(f,\mu)
&=\sum_{(x,C)\in\pmb{t}}f(x)\mu C
\ge\sum_{(x,C)\in\pmb{t}}\lambda_1C-h(x)\mu C\cr
&=\lambda_1[\alpha,\beta]-S_{\pmb{t}}(h,\mu)
\ge F_1(\beta)-F_1(\alpha)-2\epsilon
\ge F_1(b)-F_1(a)-4\epsilon.\cr}$$

\noindent But this means that if $\pmb{t}$, $\pmb{t}'$ are two
$\delta$-fine $\Cal R_{ab}$-filling members of $T$,
$|S_{\pmb{t}}(f,\mu)-S_{\pmb{t}'}(f,\mu)|\le 9\epsilon$.   As $\epsilon$
is arbitrary,

\Centerline{$\lim_{\pmb{t}\to\Cal F(T,\Delta,\frak R)}S_{\pmb{t}}(f,\mu)
=\biggerHeint f$}

\noindent is defined.
}%end of proof of 483J

\cmmnt{\medskip

\noindent{\bf Remark} The formulation (ii) above is a version of the
method of integration described by {\smc Perron 1914}.
}%end of comment

\leader{483K}{Proposition} Let $f:\Bbb R\to\Bbb R$ be a Henstock
integrable function, and $F$ its indefinite Henstock integral.
Then $F[E]$ is Lebesgue negligible for every Lebesgue negligible set
$E\subseteq\Bbb R$.

\proof{ Let $\epsilon>0$.   By 483C and 482Ad, as usual, together with
483F, there are a $\delta\in\Delta$ and a non-decreasing
$\phi:\Bbb R\to[0,\epsilon]$ such that

\Centerline{$S_{\pmb{t}}(|f|\times\chi E,\mu)\le\epsilon$,
\quad$|f(x)(b-a)-F(b)+F(a)|\le\phi(b)-\phi(a)$}

\noindent whenever $\pmb{t}\in T$ is $\delta$-fine, $a\le x\le b$ and
$(x,[a,b])\in\delta$.   For $n\in\Bbb N$ and $i\in\Bbb Z$, set

\Centerline{$E_{ni}=\{x:x\in E\cap\coint{2^{-n}i,2^{-n}(i+1)},
  \,(x,A)\in\delta$ whenever $A\subseteq[x-2^{-n},x+2^{-n}]\}$.}

\noindent Set
$J_n=\{i:i\in\Bbb Z,\,-4^n<i\le 4^n,\,E_{ni}\ne\emptyset\}$.
Observe that

\Centerline{$E
=\bigcup_{n\in\Bbb N}\bigcap_{m\ge n}\bigcup_{i\in J_m}E_{mi}$.}

\noindent For $i\in J_n$, take $x_{ni}$, $y_{ni}\in E_{ni}$ such that
$x_{ni}\le y_{ni}$ and

\Centerline{$\min(F(x_{ni}),F(y_{ni}))
\le\inf F[E_{ni}]+4^{-n}\epsilon$,}

\Centerline{$\max(F(x_{ni}),F(y_{ni}))
\ge\sup F[E_{ni}]-4^{-n}\epsilon$,}

\noindent so that
$\mu^*F[E_{ni}]\le|F(y_{ni})-F(x_{ni})|+2^{-2n+1}\epsilon$.   Now, for
each $i\in J_n$, $(x_{ni},[x_{ni},y_{ni}])\in\delta$, while
$[x_{ni},y_{ni}]\subseteq\coint{2^{-n}i,2^{-n}(i+1)}$, so
$\pmb{t}=\{(x_{ni},[x_{ni},y_{ni}]):i\in J_n\}$ is a $\delta$-fine
member of $T$, and

$$\eqalign{\mu^*F[\bigcup_{i\in J_n}E_{ni}]
&\le\sum_{i\in J_n}\mu^*F[E_{ni}]
\le\sum_{i\in J_n}2^{-2n+1}\epsilon+|F(y_{ni})-F(x_{ni})|\cr
&\le 4\epsilon+\sum_{i\in J_n}|f(x_{ni})(y_{ni}-x_{ni})|
  +\sum_{i\in J_n}\phi(y_{ni})-\phi(x_{ni})\cr
&\le 4\epsilon+S_{\pmb{t}}(|f|\times\chi E,\mu)+\epsilon
\le 6\epsilon.\cr}$$

Since this is true for every $n\in\Bbb N$,

$$\eqalignno{\mu^*F[E]
&=\mu^*F[\bigcup_{n\in\Bbb N}\bigcap_{m\ge n}\bigcup_{i\in J_m}E_i]\cr
&=\mu^*(\bigcup_{n\in\Bbb N}F[\bigcap_{m\ge n}\bigcup_{i\in J_m}E_i])
=\sup_{n\in\Bbb N}\mu^*F[\bigcap_{m\ge n}\bigcup_{i\in J_m}E_i]\cr
\displaycause{132Ae}
&\le 6\epsilon.\cr}$$

\noindent As $\epsilon$ is arbitrary, $F[E]$ is negligible, as claimed.
}%end of proof of 483K

\cmmnt{\medskip

\noindent{\bf Remark} Compare 225M.}

\leader{483L}{Definition} If $f:\Bbb R\to\Bbb R$ is Henstock integrable,
I write $\|f\|_H$ for $\sup_{a\le b}|\Heint_a^bf|$.   It is elementary
to check that this is a seminorm on the linear space of all Henstock
integrable functions.   \cmmnt{(It is finite-valued by 483D.)}

\leader{483M}{Proposition} (a) If $f:\Bbb R\to\Bbb R$ is Henstock
integrable, then $|\Heint f|\le\|f\|_H$, and $\|f\|_H=0$ iff $f=0$ a.e.

(b) Write $\eusm{HL}^1$ for the linear space of all Henstock integrable
real-valued functions on $\Bbb R$, and $HL^1$ for
$\{f^{\ssbullet}: f\in\eusm{HL}^1\}\subseteq L^0(\mu)$\cmmnt{ (\S241)}.
If we write $\|f^{\ssbullet}\|_H=\|f\|_H$ for every
$f\in\eusm{HL}^1$, then $HL^1$ is a normed space.   The ordinary space
$L^1(\mu)$ of equivalence classes of Lebesgue integrable functions is a
linear subspace of $HL^1$, and $\|u\|_H\le\|u\|_1$ for every
$u\in L^1(\mu)$.

(c) We have a linear operator $T:HL^1\to C_b(\Bbb R)$ defined by saying
that $T(f^{\ssbullet})$ is the indefinite Henstock integral of $f$ for
every $f\in\eusm{HL}^1$, and $\|T\|=1$.

\proof{{\bf (a)} Of course

\Centerline{$|\biggerHeint f|
=\lim_{a\to-\infty,b\to\infty}|\biggerHeint_a^bf|\le\|f\|_H$}

\noindent (using 483Bd).   Let $F$ be the indefinite Henstock integral of
$f$, so that
$F(b)-F(a)=\Heint_a^bf$ whenever $a\le b$.   If $f=0$ a.e., then
$F(x)=\int_{-\infty}^xfd\mu=0$ for every $x$, by 483Bb, so $\|f\|_H=0$.
If $\|f\|_H=0$, then $F$ is constant, so $f=F'=0$ a.e., by 483I.

\medskip

{\bf (b)} That $HL^1$ is a normed space follows immediately from (a).
(Compare the definitions
of the norms $\|\,\|_p$ on $L^p$, for $1\le p\le\infty$, in
\S\S242-244.)   By 483Bb, $L^1(\mu)\subseteq HL^1$, and

\Centerline{$\|u\|_H\le\|u^+\|_H+\|u^-\|_H=\|u^+\|_1+\|u^-\|_1=\|u\|_1$}

\noindent for every $u\in L^1(\mu)$, writing $u^+$ and $u^-$ for the
positive and negative parts of $u$, as in Chapter 24.

\medskip

{\bf (c)} If $f$, $g\in\eusm{HL}^1$ and $f^{\ssbullet}=g^{\ssbullet}$,
then $f$ and $g$ have the same indefinite Henstock integral, by 483Bb or
otherwise;  so $T$ is defined as a function from $HL^1$ to
$\BbbR^{\Bbb R}$.   By 483F, $Tu$ is continuous and bounded for every
$u\in HL^1$, and by 481Ca $T$ is linear.   If $f\in\eusm{HL}^1$ and
$Tf^{\ssbullet}=F$, then $\|f\|_H=\sup_{x,y\in\Bbb R}|F(y)-F(x)|$;
since $\lim_{x\to-\infty}F(x)=0$, $\|f\|_H\ge\|F\|_{\infty}$;  as $f$ is
arbitrary, $\|T\|\le 1$.   On the other hand, for any non-negative
Lebesgue integrable function $f$,
$\|Tf^{\ssbullet}\|_{\infty}=\|f\|_1=\|f\|_H$, so $\|T\|=1$.
}%end of proof 483M

\leader{483N}{Proposition} Suppose that $\family{m}{M}{I_m}$ is a
disjoint family of open intervals in $\Bbb R$ with union $G$, and that
$f:\Bbb R\to\Bbb R$ is a function such that $f_m=f\times\chi I_m$ is
Henstock integrable for every $m\in M$.   If
$\sum_{m\in M}\|f_m\|_H<\infty$, then $f\times\chi G$ is Henstock
integrable, and $\Heint f\times\chi G=\sum_{m\in M}\Heint f_m$.

\proof{ I seek to apply 482H again.   We have already seen, in the proof of
483Bc, that the conditions of 482G are satisfied by
$\Bbb R$, $T$, $\Delta$, $\frak R$, $\Cal C$, $\frak T$ and $\mu$.   Of
course $G=\bigcup_{m\in M}I_m$ is the union of a sequence of open sets
over which $f$ is Henstock integrable.   So we have only to check
482H(viii).

Set

\Centerline{$\delta_0
=\bigcup_{m\in M}\{(x,A):x\in I_m,\,A\subseteq I_m\}
\cup\{(x,A):x\in\Bbb R\setminus G,\,A\subseteq\Bbb R\}$,}

\noindent so that $\delta_0\in\Delta$.   For each $m\in M$ let $\FSH_m$ be
the Saks-Henstock indefinite integral of $f_m$.
Let $\epsilon>0$.   Then there is a finite set $M_0\subseteq M$ such
that $\sum_{m\in M\setminus M_0}\|f_m\|_H\le\epsilon$.   Next, there
must be $\delta_1\in\Delta$ and $\Cal R\in\frak R$ such that

\Centerline{$\sum_{m\in M_0}|$$\biggerHeint f_m-S_{\pmb{t}}(f_m,\mu)|
\le\epsilon$}

\noindent for every $\delta_1$-fine $\Cal R$-filling $\pmb{t}\in T$, and
$\delta_2\in\Delta$ such that
$\sum_{m\in M_0}|S_{\pmb{t}}(f_m,\mu)-\FSH_m(W_{\pmb{t}})|\le\epsilon$ for
every $\delta_2$-fine $\pmb{t}\in T$.

Now let $\pmb{t}\in T$ be $(\delta_0\cap\delta_1\cap\delta_2)$-fine and
$\Cal R$-filling.   For each $m\in M$ set
$\pmb{t}_m=\pmb{t}\restr I_m$, so that
$\pmb{t}\restr G=\bigcup_{m\in M}\pmb{t}_m$.   Because $W_{\pmb{t}}$ is
an interval, each $W_{\pmb{t}_m}$ must be an interval, as in part
(c)-(d)(vi) of the proof of 483B, and $W_{\pmb{t}_m}$ is a
subinterval of $I_m$ because $\pmb{t}$ is $\delta_0$-fine.   So (using
482G)

\Centerline{$|\FSH_m(W_{\pmb{t}_m})|
=|\biggerHeint f_m\times\chi W_{\pmb{t}_m}|\le\|f_m\|_H$.}

\noindent Also

$$\eqalign{\sum_{m\in M_0}
  |\Heint f_m-\Heint f\times\chi W_{\pmb{t}_m}|
&\le\sum_{m\in M_0}|\Heint f_m-S_{\pmb{t}}(f_m,\mu)|
  +\sum_{m\in M_0}|S_{\pmb{t}}(f_m,\mu)-\FSH_m(W_{\pmb{t}_m}|\cr
&\le 2\epsilon.\cr}$$

\noindent On the other hand,

\Centerline{$\sum_{m\in M\setminus M_0}
   $$|\biggerHeint f_m-\biggerHeint f\times\chi W_{\pmb{t}_m}|
\le 2$$\sum_{m\in M\setminus M_0}\|f_m\|_H
\le 2\epsilon$.}

Putting these together,

$$\eqalignno{|\Heint f\times\chi W_{\pmb{t}\restr G}
  -\sum_{m\in M}\Heint f_m|
&=|\sum_{m\in M}\Heint f\times\chi W_{\pmb{t}_m}
  -\sum_{m\in M}\Heint f_m|\cr
\displaycause{because $\pmb{t}$ is finite, so all but finitely many
terms in the sum $\sum_{m\in M}f\times\chi W_{\pmb{t}_m}$ are zero}
&\le\sum_{m\in M}|\Heint f\times\chi W_{\pmb{t}_m}-\Heint f_m|
\le 4\epsilon.\cr}$$

\noindent As $\epsilon$ is arbitrary, condition 482H(viii) is satisfied,
with

\Centerline{$\lim_{\pmb{t}\rightarrow\Cal F(T,\Delta,\frak R)}
  \biggerHeint f\times\chi W_{\pmb{t}\restr G}
=$$\sum_{m\in M}$$\biggerHeint f_m$,}

\noindent and 482H gives the result we seek.
}%end of proof of 483N

\leader{483O}{Definitions (a)} For any real-valued function $F$, write
$\omega(F)$ for
$\sup_{x,y\in\dom F}|F(x)-F(y)|$, the {\bf oscillation} of $F$.
(Interpret $\sup\emptyset$ as $0$, so that $\omega(\emptyset)=0$.)

\leaveitout{\spheader 483O\query For $A\subseteq\Bbb R$, set

\Centerline{$\Var^*_A(F)
=\sup\{\sum_{I\in\Cal I}\omega(F;\overline{I}):\Cal I$ is a disjoint
family of open intervals with endpoints in $A\}$.}

\noindent (Again, $\Var^*_{\emptyset}(F)=0$.)   \cmmnt{Note that
$\Var^*_A(F)$ is {\it not} determined by $F\restr A$.}
We say that $F$ is {\bf BV$_*$ on $A$} if $\Var^*_A(F)$ is finite.
}%end of leaveitout

\spheader 483Ob Let $F:\Bbb R\to\Bbb R$ be a function.   For
$A\subseteq\Bbb R$, we say that $F$ is {\bf AC$_*$
on $A$} if for every $\epsilon>0$ there is an $\eta>0$ such that
$\sum_{I\in\Cal I}\omega(F\restr\overline{I})\le\epsilon$ whenever $\Cal
I$
is a disjoint family of open intervals with endpoints in $A$ and
$\sum_{I\in\Cal I}\mu I\le\eta$.   \cmmnt{Note that whether $F$ is
AC$_*$ on $A$ is {\it not} determined by $F\restr A$, since it depends
on the behaviour of $F$ on intervals with endpoints in $A$.}

\spheader 483Oc\cmmnt{ Finally,} $F$ is
{\bf ACG$_*$} if it is continuous and there is a countable family
$\Cal A$ of sets, covering
$\Bbb R$, such that $F$ is AC$_*$ on every member of $\Cal A$.

\leader{483P}{Elementary results (a)(i)}
If $F$, $G:\Bbb R\to\Bbb R$ are functions and
$A\subseteq B\subseteq\Bbb R$, then
$\omega(F+G\restr A)\le\omega(F\restr A)+\omega(G\restr A)$ and
$\omega(F\restr A)\le\omega(F\restr B)$.

\medskip

\quad{\bf (ii)} If $F$ is the indefinite Henstock integral of
$f:\Bbb R\to\Bbb R$ and $C\subseteq\Bbb R$ is an interval, then
$\|f\times\chi C\|_H=\omega(F\restr C)$.

\medskip

\quad{\bf (iii)} If $F:\Bbb R\to\Bbb R$ is continuous, then
$(a,b)\mapsto\omega(F\restrp[a,b]):\Bbb R^2\to\Bbb R$ is continuous, and
$\omega(F\restr\overline{A})=\omega(F\restr A)$ for every set
$A\subseteq\Bbb R$.

\leaveitout{\spheader 483P\query
Then $\Var^*_A(F+G)\le\Var^*_A(F)+\Var^*_A(G)$ and
$\Var^*_A(F)\le\Var^*_B(F)$ if $A\subseteq B\subseteq\Bbb R$,
$\Var^*_A(F)=\omega(F\restr A)$ if $F$ is monotonic.}

\spheader 483Pb{\bf (i)}
If $F:\Bbb R\to\Bbb R$ is AC$_*$ on $A\subseteq\Bbb R$, it is AC$_*$ on
every subset of $A$.

\medskip

\quad{\bf (ii)} If $F:\Bbb R\to\Bbb R$ is continuous and is AC$_*$ on
$A\subseteq\Bbb R$, it is AC$_*$ on $\overline{A}$.   \prooflet{\Prf\
Let $\epsilon>0$.   Let $\eta>0$ be such that
$\sum_{I\in\Cal I}\omega(F\restr\overline{I})\le\epsilon$ whenever
$\Cal I$ is a disjoint family of open intervals with endpoints in $A$
and $\sum_{I\in\Cal I}\mu I\le\eta$.   Let $\Cal I$ be a disjoint family
of open intervals with endpoints in $\overline{A}$ and
$\sum_{I\in\Cal I}\mu I\le\bover12\eta$.   Let $\Cal I_0\subseteq\Cal I$
be a non-empty finite set;  then we can enumerate $\Cal I_0$ as
$\langle\ooint{a_i,b_i}\rangle_{i\le n}$ where
$a_0,b_0,\ldots,a_n,b_n\in\overline{A}$ and
$a_0\le b_0\le a_1\le b_1\le\ldots\le a_n\le b_n$.   Because
$(a,b)\mapsto\omega(F\restrp[a,b])$ is continuous, as noted in (a-ii)
above, we can find $a'_0,\ldots,b'_n\in A$ such that
$a'_0\le b'_0\le a'_1\le\ldots\le a'_n\le b'_n$,
$\sum_{i=0}^nb'_i-a'_i\le\eta$, and
$\sum_{i=0}^n|\omega(F\restrp[a'_i,b'_i])-\omega(F\restrp[a_i,b_i])|
\le\epsilon$;  so that

\Centerline{$\sum_{I\in\Cal I_0}\omega(F\restr\overline{I})
\le\sum_{i=0}^n\omega(F\restrp[a'_i,b'_i])\le 2\epsilon$.}

\noindent As $\Cal I_0$ is arbitrary,
$\sum_{I\in\Cal I}\omega(F\restr\overline{I})\le 2\epsilon$;  as
$\epsilon$
is arbitrary, $F$ is AC$_*$ on $\overline{A}$.\ \Qed}

\leader{483Q}{Lemma} Let $F:\Bbb R\to\Bbb R$ be a continuous function,
and $K\subseteq\Bbb R$ a non-empty compact set such that $F$ is AC$_*$
on $K$.   Write $\Cal I$ for the family of non-empty bounded open
intervals, disjoint from $K$, with endpoints in $K$.

(a) $\sum_{I\in\Cal I}\omega(F\restr I)$ is finite.

(b) Write $a^*$ for $\inf K=\min K$.   Then there is a Lebesgue
integrable function $g:\Bbb R\to\Bbb R$, zero off $K$, such that

\Centerline{$F(x)-F(a^*)
=\int_{a^*}^xg
  +\sum_{J\in\Cal I,J\subseteq[a^*,x]}F(\sup J)-F(\inf J)$}

\noindent for every $x\in K$.

\proof{{\bf (a)} Let $\eta>0$ be such that
$\sum_{I\in\Cal J}\omega(F\restr\overline{I})\le 1$ whenever $\Cal J$ is
a disjoint family of open intervals with endpoints in $K$ and
$\sum_{I\in\Cal J}\mu I\le\eta$.   Let
$m_0$, $m_1\in\Bbb Z$ be such that $K\subseteq[m_0\eta,m_1\eta]$.   For
integers $m$ between $m_0$ and $m_1$, let $\Cal I_m$ be the set of
intervals in $\Cal I$ included in $\ooint{m\eta,(m+1)\eta}$.   Then
$\sum_{I\in\Cal I_m}\mu I\le\eta$, so
$\sum_{I\in\Cal I_m}\omega(F\restr I)\le 1$ for each $m$.   Also every
member of $\Cal J=\Cal I\setminus\bigcup_{m_0\le m<m_1}\Cal I_m$
contains $m\eta$ for some $m$ between $m_0$ and $m_1$, so
$\#(\Cal J)\le m_1-m_0$.   Accordingly

$$\eqalign{\sum_{I\in\Cal I}\omega(F\restr\overline{I})
&\le\sum_{I\in\Cal J}\omega(F\restr\overline{I})
    +\sum_{m=m_0}^{m_1-1}
      \sum_{I\in\Cal I_m}\omega(F\restr\overline{I})\cr
&\le\sum_{I\in\Cal J}\omega(F\restr\overline{I})+m_1-m_0
<\infty\cr}$$

\noindent because $F$ is continuous, therefore bounded on every bounded
interval.

\medskip

{\bf (b)(i)} Set $b^*=\sup K=\max K$.   Define $G:[a^*,b^*]\to\Bbb R$ by
setting

$$\eqalign{G(x)&=F(x)\text{ if }x\in K,\cr
&=\Bover{F(b)(x-a)+F(a)(b-x)}{b-a}
  \text{ if }x\in\ooint{a,b}\in\Cal I.\cr}$$

\noindent Then $G$ is absolutely continuous.   \Prf\ $G$ is continuous
because $F$ is.   Let $\epsilon>0$.   Let $\eta_1>0$ be such that
$\sum_{I\in\Cal J}\omega(F\restr\overline{I})\le\epsilon$ whenever
$\Cal J$ is a disjoint family of open intervals with endpoints in $K$
and $\sum_{I\in\Cal J}\mu I\le\eta_1$.   Let
$\Cal I_0\subseteq\Cal I$ be a finite set such that
$\sum_{I\in\Cal I\setminus\Cal I_0}\omega(F\restr\overline{I})
\le\epsilon$,
and take $M>0$ such that $|F(b)-F(a)|\le M(b-a)$ whenever
$\ooint{a,b}\in\Cal I_0$;  set $\eta=\min(\eta_1,\Bover{\epsilon}M)>0$.

Let $\Cal J^*$ be the set of non-empty open subintervals $J$ of
$[a^*,b^*]$ such that either $J\cap K=\emptyset$ or both endpoints of
$J$ belong to $K$.   Let $\Cal J\subseteq\Cal J^*$ be a disjoint family
such that $\sum_{I\in\Cal J}\mu I\le\eta$.   Set
$\Cal J'=\{J:J\in\Cal J,\,J\cap K=\emptyset\}$.   Then

$$\eqalign{\sum_{J\in\Cal J\setminus\Cal J'}|G(\sup J)-G(\inf J)|
&=\sum_{J\in\Cal J\setminus\Cal J'}|F(\sup J)-F(\inf J)|\cr
&\le\sum_{J\in\Cal J\setminus\Cal J'}\omega(F\restr\overline{J})
\le\epsilon.\cr}$$

\noindent On the other hand,

$$\eqalignno{\sum_{J\in\Cal J'}|G(\sup J)-G(\inf J)|
&=\sum_{I\in\Cal I_0}\sum_{\Atop{J\in\Cal J}{J\subseteq I}}
  |G(\sup J)-G(\inf J)|\cr
&\qquad\qquad+\sum_{I\in\Cal I\setminus\Cal I_0}
        \sum_{\Atop{J\in\Cal J}{J\subseteq I}}|G(\sup J)-G(\inf J)|\cr
&\le M\sum_{I\in\Cal I_0}\sum_{\Atop{J\in\Cal J}{J\subseteq I}}\mu J
   +\sum_{I\in\Cal I\setminus\Cal I_0}|F(\sup I)-F(\inf I)|\cr
\displaycause{because $G$ is monotonic on $\overline{I}$ for each
$I\in\Cal I$}
&\le M\eta+\epsilon
\le 2\epsilon,\cr}$$

\noindent so $\sum_{J\in\Cal J}|G(\sup J)-G(\inf J)|\le 3\epsilon$.

Generally, if $J$ is any non-empty open subinterval of $[a^*,b^*]$, we
can split it into at most three intervals belonging to $\Cal J^*$.
So if $\Cal J$ is any disjoint family of non-empty open
subintervals of $[a^*,b^*]$ with $\sum_{J\in\Cal J}\mu J\le\eta$, we can
find a family $\tilde\Cal J\subseteq\Cal J^*$ with
$\sum_{J\in\tilde\Cal J}\mu J=\sum_{J\in\Cal J}\mu J$ and
$\sum_{J\in\tilde\Cal J}|G(\sup J)-G(\inf J)|
\ge\sum_{J\in\Cal J}|G(\sup J)-G(\inf J)|$.   But this means that
$\sum_{J\in\Cal J}|G(\sup J)-G(\inf J)|\le 3\epsilon$.   As $\epsilon$
is arbitrary, $G$ is absolutely continuous.\ \Qed

\medskip

\quad{\bf (ii)} By 225E, $G'$ is Lebesgue integrable and
$G(x)=G(a^*)+\int_{a^*}^xG'$ for every $x\in[a^*,b^*]$.   Set
$g(x)=G'(x)$ when $x\in K$ and $G'(x)$ is defined, $0$ for other
$x\in\Bbb R$, so that $g:\Bbb R\to\Bbb R$ is Lebesgue integrable.   Now
take any $x\in K$.   Then

$$\eqalignno{F(x)
&=G(x)
=G(a^*)+\int_{a^*}^{x}G'
=F(a^*)+\int_{a^*}^{x}g+\int_{[a^*,x]\setminus K}G'\cr
&=F(a^*)+\int_{a^*}^{x}g
  +\sum_{\Atop{I\in\Cal I}{I\subseteq[a^*,x]}}\int_IG'\cr
\displaycause{because $\Cal I$ is a disjoint countable family of
measurable sets, and
$\bigcup_{I\in\Cal I,I\subseteq[a^*,x]}I=[a^*,x]\setminus K$}
&=F(a^*)+\int_{a^*}^xg
  +\sum_{\Atop{I\in\Cal I}{I\subseteq[a^*,x]}}G(\sup I)-G(\inf I)\cr
\displaycause{note that this sum is absolutely summable}
&=F(a^*)+\int_{a^*}^{x}g
  +\sum_{\Atop{I\in\Cal I}{I\subseteq[a^*,x]}}F(\sup I)-F(\inf I)\cr
}$$

\noindent as required.

}%end of proof of 483Q

\leader{483R}{Theorem} %\cmmnt{ (see {\smc Gordon 94}, 11.14)}
Let $F:\Bbb R\to\Bbb R$ be a function.   Then $F$ is an indefinite Henstock
integral iff it is ACG$_*$,
$\lim_{x\to-\infty}F(x)=0$ and $\lim_{x\to\infty}F(x)$ is defined in
$\Bbb R$.

\proof{{\bf (a)} Suppose that $F$ is the indefinite Henstock integral of
$f:\Bbb R\to\Bbb R$.

\medskip

\quad{\bf (i)} By 483F, $F$ is continuous, with limit zero at $-\infty$
and finite at $\infty$.   So I have just to show that there is a
sequence of sets, covering $\Bbb R$, on each of which $F$ is AC$_*$.
Recall that there is a sequence $\sequence{m}{K_m}$ of compact sets,
covering $\Bbb R$, such that $f\times K_m$ is Lebesgue integrable for
every $m\in\Bbb N$ (483G).
By the arguments of (i)$\Rightarrow$(ii) in the proof of 483J, there is
a function $F_2\ge F$ such that $\Dinid F_2\ge f$ and $F_2-F$ is
non-decreasing and takes values between $0$ and $1$.   For $n\in\Bbb N$,
$j\in\Bbb Z$ set

\Centerline{$I_{nj}=[2^{-n}j,2^{-n}(j+1)]$,}

\Centerline{$B_{nj}=\{x:x\in I_{nj},\,\Bover{F_2(y)-F_2(x)}{y-x}\ge-n
  \text{ whenever }y\in I_{nj}\setminus\{x\}\}$.}

\noindent Observe that $\bigcup_{n\in\Bbb N,j\in\Bbb Z}B_{nj}=\Bbb R$,
so that $\{B_{nj}\cap K_m:m,\,n\in\Bbb N,\,j\in\Bbb Z\}$ is a countable
cover of $\Bbb R$.   It will therefore be enough to show that $F$ is
AC$_*$ on every $B_{nj}\cap K_m$.

\medskip

\quad{\bf (ii)} Fix $m$, $n\in\Bbb N$ and $j\in\Bbb Z$, and set
$A=B_{nj}\cap K_m$.   If $A=\emptyset$, then of course $F$ is AC$_*$ on
$A$;  suppose that $A$ is not empty.   Set $G(x)=F_2(x)+nx$ for
$x\in I_{nj}$, so that $F=G-(F_2-F)-H$ on $I_{nj}$, where $H(x)=nx$.
Whenever $a$, $b\in B_{nj}$ and $a\le x\le b$, then
$G(a)\le G(x)\le G(b)$, because $x\in I_{nj}$.   So if
$a_0,b_0,a_1,b_1,\ldots,a_k,b_k\in A$ and
$a_0\le b_0\le a_1\le b_1\le\ldots\le a_k\le b_k$,

\Centerline{$\sum_{i=0}^k\omega(G\restrp[a_i,b_i])
=\sum_{i=0}^kG(b_i)-G(a_i)
\le G(b_k)-G(a_0)\le\omega(G\restr I_{nj})$,}

\noindent and

$$\eqalignno{\sum_{i=0}^k\omega(F\restrp[a_i,b_i])
&\le\sum_{i=0}^k\omega(G\restrp[a_i,b_i])
  +\sum_{i=0}^k\omega(F_2-F\restrp[a_i,b_i])
  +\sum_{i=0}^k\omega(H\restrp[a_i,b_i])\cr
&\le\omega(G\restr I_{nj})+\omega(F_2-F\restr I_{nj})
  +\omega(H\restr I_{nj})\cr
\displaycause{because $F_2-F$ and $H$ are monotonic}
&\le\omega(F\restr I_{nj})+2\omega(F_2-F\restr I_{nj})
  +2\omega(H\restr I_{nj})\cr
&\le\omega(F\restr I_{nj})+2(1+n\mu I_{nj})
=M\cr}$$

\noindent say, which is finite, because $F$ is bounded.

\medskip

\quad{\bf (iii)} By 2A2I (or 4A2Rj), the open set
$\Bbb R\setminus\overline{A}$ is expressible as a countable union of
disjoint non-empty open intervals.   Two of these are unbounded;  let
$\Cal I$ be the set consisting of the rest, so that
$\overline{A}\cup\bigcup\Cal I=[a^*,b^*]$ is a closed interval included
in $I_{nj}$.   If $I$, $I'$ are distinct members of $\Cal I$ and
$\inf I\le\inf I'$, then $\sup I\le\inf I'$, because
$I\cap I'=\emptyset$, and there must be a point of $\overline{A}$ in the
interval $[\sup I,\inf I']$;  so in fact there must be a point of $A$ in
this interval, since $A$ does not meet either $I$ or $I'$.
It follows that
$\sum_{I\in\Cal I}\omega(F\restr I)\le M$.   \Prf\ If
$\Cal I_0\subseteq\Cal I$ is finite and non-empty, we can enumerate it
as $\langle I_i\rangle_{i\le k}$ where $\sup I_i\le\inf I_{i'}$ whenever
$i<i'\le k$.   We can find $a_0,\ldots,a_{k+1}\in A$ such that
$a_0\le\inf I_0$, $\sup I_i\le a_{i+1}\le\inf I_{i+1}$ for every $i<k$,
and $\sup I_k\le a_{k+1}$;  so that

\Centerline{$\sum_{I\in\Cal I_0}\omega(F\restr I)
=\sum_{i=0}^k\omega(F\restr I_i)
\le\sum_{i=0}^k\omega(F\restrp[a_i,a_{i+1}])
\le M$.}

\noindent As $\Cal I_0$ is arbitrary, this gives the result.\ \Qed

\medskip

\quad{\bf (iv)} Whenever $a^*\le x\le y\le b^*$,

\Centerline{$|F(y)-F(x)|
\le\int_{\overline{A}\cap[x,y]}|f|d\mu
  +\sum_{I\in\Cal I}\omega(F\restr I\cap\ooint{x,y})$.}

\noindent \Prf\ For each $I\in\Cal I$,

\Centerline{$\|f\times\chi(I\cap\ooint{x,y})\|_H
=\omega(F\restr I\cap\ooint{x,y})\le\omega(F\restr I)$}

\noindent (483Pa).   So

\Centerline{$\sum_{I\in\Cal I}\|f\times\chi(I\cap\ooint{x,y})\|_H
\le\sum_{I\in\Cal I}\omega(F\restr I)$}

\noindent is finite.   Writing $H=\ooint{x,y}\cap\bigcup\Cal I$,
$\Heint f\times\chi H$ is defined and equal to
$\sum_{I\in\Cal I}\Heint f\times\chi(I\cap\ooint{x,y})$, by 483N.
On the other hand,

\Centerline{$\ooint{x,y}\setminus H\subseteq\overline{A}\subseteq K_m$,}

\noindent so $f\times\chi(\ooint{x,y}\setminus H)$ is Lebesgue
integrable.   Accordingly

$$\eqalign{|F(y)-F(x)|
&=|\Heint f\times\chi\ooint{x,y}|\cr
&\le|\Heint f\times\chi H|
  +|\int f\times\chi(\ooint{x,y}\setminus H)d\mu|\cr
&\le\sum_{I\in\Cal I}|\Heint f\times\chi(I\cap\ooint{x,y})|
     +\int_{\overline{A}\cap[x,y]}|f|d\mu\cr
&\le\sum_{I\in\Cal I}\omega(F\restr I\cap\ooint{x,y})
     +\int_{\overline{A}\cap[x,y]}|f|d\mu,\cr}$$

\noindent as claimed.\ \Qed

\medskip

\quad{\bf (v)} So if $a^*\le a\le b\le b^*$,

\Centerline{$\omega(F\restrp[a,b])
\le\sum_{I\in\Cal I}\omega(F\restr I\cap[a,b])
  +\int_{\overline{A}\cap[a,b]}|f|d\mu$.}

\noindent\Prf\ We have only to observe that if $a\le x\le y\le b$, then
$\omega(F\restr I\cap\ooint{x,y})\le\omega(F\restr I\cap[a,b])$ for
every $I\in\Cal I$, and
$\int_{\overline{A}\cap\ooint{x,y}}|f|d\mu
\le\int_{\overline{A}\cap[a,b]}|f|d\mu$.\ \Qed

\medskip

\quad{\bf (vi)} Now let $\epsilon>0$.   Setting
$\tilde F(x)=\int_{a^*}^x|f|\times\chi\overline{A}\,d\mu$ for
$x\in[a^*,b^*]$, $\tilde F$ is absolutely continuous (225E), and there
is an $\eta_0>0$ such that
$\sum_{i=0}^k\tilde F(b_i)-\tilde F(a_i)\le\epsilon$ whenever
$a^*\le a_0\le b_0\le a_1\le b_1\le\ldots\le a_k\le b_k\le b^*$ and
$\sum_{i=0}^kb_i-a_i\le\eta_0$.   Take $\Cal I_0\subseteq\Cal I$ to be a
finite set such that
$\sum_{I\in\Cal I\setminus\Cal I_0}\omega(F\restr I)\le\epsilon$, and
let
$\eta>0$ be such that $\eta\le\eta_0$ and $\eta<\diam I$ for every
$I\in\Cal I_0$.

Suppose that $a_0,b_0,\ldots,a_k,b_k\in A$ are such that
$a_0\le b_0\le\ldots\le a_k\le b_k$ and $\sum_{i=0}^kb_i-a_i\le\eta$.
Then no member of $\Cal I_0$ can be included in any interval
$[a_i,b_i]$, and therefore, because no $a_i$ or $b_i$ can belong to
$\bigcup\Cal I$, no member of $\Cal I_0$ meets any $[a_i,b_i]$.   Also,
of course, $a^*\le a_0$ and $b_k\le b^*$.   We therefore have

$$\eqalignno{\sum_{i=0}^k\omega(F\restrp[a_i,b_i])
&\le\sum_{i=0}^k\bigl(\sum_{I\in\Cal I}\omega(F\restr I\cap[a_i,b_i])
  +\int_{\overline{A}\cap[a_i,b_i]}|f|d\mu\bigr)\cr
&=\sum_{i=0}^k\sum_{I\in\Cal I\setminus\Cal I_0}
   \omega(F\restr I\cap[a_i,b_i])
  +\sum_{i=0}^k\tilde F(b_i)-\tilde F(a_i)\cr
&\le\sum_{i=0}^k
  \sum_{\Atop{I\in\Cal I\setminus\Cal I_0}{I\subseteq[a_i,b_i]}}
      \omega(F\restr I)
  +\epsilon\cr
\displaycause{because if $I\in\Cal I$ meets $[a_i,b_i]$, it is included
in it}
&\le\sum_{I\in\Cal I\setminus\Cal I_0}\omega(F\restr I)+\epsilon
\le 2\epsilon.\cr}$$

\noindent As $\epsilon$ is arbitrary, $F$ is AC$_*$ on $A$.   This
completes the proof that $F$ is ACG$_*$ and therefore satisfies the
conditions given.

\medskip

{\bf (b)} Now suppose that $F$ satisfies the conditions.   Set
$F(-\infty)=0$ and $F(\infty)=\lim_{x\to\infty}F(x)$, so that
$F:[-\infty,\infty]\to\Bbb R$ is continuous.   For $x\in\Bbb R$, set
$f(x)=F'(x)$ if this is defined, $0$ otherwise.   Let $\Cal J$ be the
family of all
non-empty intervals $C\subseteq\Bbb R$ such that $\Heint f\times\chi C$
is defined and equal to $F(\sup C)-F(\inf C)$, and let $\Cal I$ be the
set of non-empty open intervals $I$ such that every non-empty
subinterval of $I$ belongs to $\Cal J$.
I seek to show that $\Bbb R$ belongs to $\Cal I$.

\medskip

\quad{\bf (i)} Of course singleton intervals belong to $\Cal J$.   If
$C_1$, $C_2\in\Cal J$ are disjoint and $C=C_1\cup C_2$ is an interval,
then

$$\eqalign{\Heint f\times\chi C
&=\Heint f\times\chi C_1+\Heint f\times\chi C_2\cr
&=F(\sup C_1)-F(\inf C_1)+F(\sup C_2)-F(\inf C_2)\cr
&=F(\sup C)-F(\inf C)\cr}$$

\noindent and $C\in\Cal J$.   If $I_1$, $I_2\in\Cal I$ and $I_1\cap I_2$
is non-empty, then $I_1\cup I_2$ is an interval;  also any subinterval
$C$ of $I_1\cup I_2$
is either included in one of the $I_j$ or is expressible as a disjoint
union $C_1\cup C_2$ where $C_j$ is a subinterval of $I_j$ for each $j$;
so $C\in\Cal J$ and $I\in\Cal I$.   If $I_1$, $I_2\in\Cal I$ and
$\sup I_1=\inf I_2$, then $I=I_1\cup I_2\cup\{\sup I_1\}\in\Cal I$,
because any subinterval of $I$ is expressible as the disjoint union of
at most three intervals in $\Cal J$.

\medskip

\quad{\bf (ii)} If $\Cal I_0\subseteq\Cal I$ is non-empty and
upwards-directed, then $\bigcup\Cal I_0\in\Cal I$.   \Prf\ This is a
consequence of 483Bd.   If we take a non-empty open subinterval $J$ of
$\bigcup\Cal I_0$ and express it as $\ooint{\alpha,\beta}$, where
$-\infty\le\alpha<\beta\le\infty$, then whenever $\alpha<a<b<\beta$ there
are members of $\Cal I_0$ containing $a$ and $b$, and therefore a member of
$\Cal I_0$ containing both, so that $[a,b]\in\Cal J$.   Accordingly
$\Heint_a^bf$ is defined and equal to $F(b)-F(a)$.   Since $F$ is
continuous, $\lim_{a\downarrow\alpha,b\uparrow\beta}\Heint_a^bf$ is defined
and equal to $F(\beta)-F(\alpha)$;  by 483Bd, $\Heint_{\alpha}^{\beta}f$ is
defined and equal to $F(\beta)-F(\alpha)$, so that $J\in\Cal J$.   I
wrote this out for open intervals, for convenience;  but any non-empty
subinterval of $\bigcup\Cal I_0$ is either a singleton or expressible as
an open interval with at most two points added, so belongs to $\Cal J$.
Accordingly $\bigcup\Cal I_0\in\Cal I$.\ \Qed

\medskip

\quad{\bf (iii)} It follows that every member of $\Cal I$ is included in
a maximal member of $\Cal I$.   Let $\Cal I^*$ be the set of maximal
members of $\Cal I$.   By (i), these are all disjoint, so no endpoint of
any member of $\Cal I^*$ can belong to $\bigcup\Cal I^*$.

\Quer\ Suppose, if possible, that $\Bbb R\notin\Cal I$.   Then
$\bigcup\Cal I=\bigcup\Cal I^*$ cannot be $\Bbb R$, and
$V=\Bbb R\setminus\bigcup\Cal I$ is a non-empty closed set.   By (i), no
two distinct members of $\Cal I^*$ can share a boundary point, so $V$
has no isolated points.

We are supposing that $F$ is ACG$_*$, so there is a countable family
$\Cal A$ of sets, covering $\Bbb R$, such that $F$ is AC$_*$ on $A$ for
every $A\in\Cal A$.   By Baire's theorem (3A3G or 4A2Ma), applied to the
locally compact Polish space $V$, $V\setminus\overline{A}$
cannot be dense in $V$ for every $A\in\Cal A$, so there are an
$A\in\Cal A$ and a bounded open interval $\tilde J$ such that
$\emptyset\ne V\cap\tilde J\subseteq\overline{A}$.   Set
$K=\overline{\tilde J\cap A}$;  by 483Pb, $F$ is AC$_*$ on $K$, and
$V\cap\tilde J\subseteq K$.   Because $V$ has no isolated points,
$V\cap\tilde J$ is infinite, so, setting $a^*=\min K$ and $b^*=\max K$,
$V\cap\ooint{a^*,b^*}$ is non-empty.

Let $\Cal I_0$ be the family of non-empty bounded open intervals, disjoint from $K$, with endpoints in $K$.   By 483Q,
$\sum_{I\in\Cal I_0}\omega(F\restr I)$ is
finite, and there is a Lebesgue integrable function $g:\Bbb R\to\Bbb R$
such that $g(x)=0$ for $x\in\Bbb R\setminus K$ and

\Centerline{$F(x)
=F(a^*)+\int_{a^*}^xg
  +\sum_{I\in\Cal I_0,I\subseteq[a^*,x]}F(\sup I)-F(\inf I)$}

\noindent for $x\in K$.   Since every member of $\Cal I_0$ is disjoint from $V$, it is
included in some member of $\Cal I^*$ and belongs to $\Cal J$, so
$F(\sup I)-F(\inf I)=\Heint f\times\chi I$ for every $I\in\Cal I_0$.
If $I\in\Cal I_0$, then

$$\eqalign{\|f\times\chi I\|_H
&=\sup_{C\in\Cal C}|\Heint f\times\chi I\times\chi C|
=\sup_{C\in\Cal C,C\subseteq I}|\Heint f\times\chi C|\cr
&=\sup_{C\in\Cal C,C\subseteq I}|F(\sup C)-F(\inf C)|
=\omega(F\restr I)\cr}$$

\noindent because every non-empty subinterval of $I$ belongs to
$\Cal J$.   So $\sum_{I\in\Cal I_0}\|f\times\chi I\|_H$ is finite, and
$f\times\chi H$ is Henstock integrable, where $H=\bigcup\Cal I_0$, by
483N.   Moreover, if $x\in K$, then

$$\eqalign{\Heint_{a^*}^xf\times\chi H
&=\Heint f\times\chi(\bigcup\{I:I\in\Cal I_0,\,I\subseteq[a^*,x]\})\cr
&=\sum_{I\in\Cal I_0,I\subseteq[a^*,x]}\Heint f\times\chi I
=\sum_{I\in\Cal I_0,I\subseteq[a^*,x]}F(\sup I)-F(\inf I).\cr}$$

\noindent But this means that if $y\in[a^*,b^*]$, and
$x=\max(K\cap[a^*,y])$, so that $\ooint{x,y}\subseteq H$, then

$$\eqalign{F(y)
&=F(x)+F(y)-F(x)\cr
&=F(a^*)
   +\int_{a^*}^xg
   +\sum_{I\in\Cal I_0,I\subseteq[a^*,x]}(F(\sup I)-F(\inf I))
   +\Heint f\times\chi\ooint{x,y}\cr
&=F(a^*)
   +\int g\times\chi(K\cap\coint{a^*,y})
   +\Heint_{a^*}^xf\times\chi H
   +\Heint f\times\chi\ooint{x,y}\cr
&=F(a^*)+\Heint_{-\infty}^yh\cr}$$

\noindent where $h=f\times\chi H+g\times\chi K$ is Henstock integrable
because $f\times\chi H$ is Henstock integrable and $g\times\chi K$ is
Lebesgue integrable.

Accordingly, if $C$ is any non-empty subinterval of $[a^*,b^*]$,
$F(\sup C)-F(\inf C)=\Heint h\times\chi C$.   But we also know that
$\bover{d}{dy}\Heint_{-\infty}^yh=h(y)$ for almost every $y$, by 483I.
So $F'(y)$ is defined and equal to $h(y)$ for almost every
$y\in[a^*,b^*]$, and $h=f$ a.e.\ on $[a^*,b^*]$.   This means that
$F(\sup C)-F(\inf C)=\Heint f\times\chi C$ for any non-empty subinterval
$C$ of $[a^*,b^*]$, and $\ooint{a^*,b^*}\in\Cal I$.   But
$\ooint{a^*,b^*}$ meets $V$, so this is impossible.\ \Bang

\medskip

\quad{\bf (iv)} This contradiction shows that $\Bbb R\in\Cal I$ and that
$F$ is an indefinite Henstock integral, as required.
}%end of proof of 483R

\exercises{\leader{483X}{Basic exercises $\pmb{>}$(a)}
%\spheader 483Xa
Let $[a,b]\subseteq\Bbb R$ be a non-empty closed
interval, and let $I_{\mu}$ be the gauge integral on $[a,b]$ defined
from Lebesgue measure and the tagged-partition structure defined in
481J.   Show that, for $f:\Bbb R\to\Bbb R$,
$I_{\mu}(f\restrp[a,b])=\Heint f\times\chi[a,b]$ if either is defined.
%483A

\sqheader 483Xb Extract ideas from the proofs of 482G and 482H to give
a direct proof of 483B(c)-(d).
%483B

\spheader 483Xc Set $f(0)=0$ and $f(x)=\bover{\sin x}{x}$ for other real
$x$.   Show that $f$ is Henstock integrable, and that
$\Heint_{\mskip4mu 0}^{\infty}f=\bover{\pi}2$.   \Hint{283Da.}
%483B

\spheader 483Xd Set $f(x)=\Bover1x\cos(\Bover1{x^2})$ for $0<x\le 1$,
$0$ for other real $x$.   Show that $f$ is Henstock integrable but not
Lebesgue integrable.   \Hint{by considering
$\Bover{d}{dx}(x^2\sin\Bover1{x^2})$, show that
$\lim_{a\downarrow 0}\int_a^1f$ is defined.}
%483B

\spheader 483Xe  Let $f:\Bbb R\to\Bbb R$ be a Henstock integrable
function.   Show that there is a finitely additive functional
$\lambda:\Cal P\Bbb R\to\Bbb R$ such that for every $\epsilon>0$ there
are a gauge $\delta\in\Delta$ and a Radon measure $\nu$ on $\Bbb R$ such
that $\nu\Bbb R\le\epsilon$ and
$|S_{\pmb{t}}(f,\mu)-\lambda W_{\pmb{t}}|\le\nu W_{\pmb{t}}$ for every
$\delta$-fine $\pmb{t}\in T$.
%483F

\sqheader 483Xf Let $f:\Bbb R\to\Bbb R$ be a Henstock integrable
function.   Show that $\bigcup\{G:G\subseteq\Bbb R$ is open,
$f$ is Lebesgue integrable over $G\}$ is dense.   \Hint{483G.}
%483G

\sqheader 483Xg Let $f:\Bbb R\to\Bbb R$ be a Henstock integrable
function and $F$ its indefinite Henstock integral.   Show that $f$ is
Lebesgue integrable iff $F$ is of bounded variation on $\Bbb R$.
\Hint{224I.}
%483I

\sqheader 483Xh Let $F:\Bbb R\to\Bbb R$ be a continuous function such
that $\lim_{x\to-\infty}F(x)=0$, $\lim_{x\to\infty}F(x)$ is defined in
$\Bbb R$, and $F'(x)$ is defined for all but countably many
$x\in\Bbb R$.   Show that $F$ is the indefinite Henstock integral of any
function $f:\Bbb R\to\Bbb R$ extending $F'$.   \Hint{in 483J, take $F_1$
and $F_2$ differing from $F$ by saltus functions.}
%483J

\spheader 483Xi Let $f:\Bbb R\to\Bbb R$ be a Henstock integrable
function, and $\sequencen{I_n}$ a disjoint sequence of intervals in
$\Bbb R$.   Show that $\lim_{n\to\infty}\|f\times\chi I_n\|_H=0$.
%483D

\spheader 483Xj Show that $HL^1$ is not a Banach space.   \Hint{there is
a continuous function which is nowhere differentiable (477K).}
%483M

\sqheader 483Xk Use 483N to replace part of the proof of 483Bd.
%483N

\spheader 483Xl Let $F:\Bbb R\to\Bbb R$ be such that $\DiniD F$ and $\Dinid F$
are both finite everywhere.   Show that
$F(b)-F(a)=\Heint\DiniD F\times\chi[a,b]$ whenever $a\le b$ in $\Bbb R$.
\Hint{$F$ is AC$_*$ on
$\{x:|F(y)-F(x)|\le n|y-x|$ whenever $|y-x|\le 2^{-n}\}$.}
%483R

\spheader 483Xm\dvAnew{2011}
For integers $r\ge 1$, write $\Cal C_r$ for the family of
subsets of $\BbbR^r$ of the form $\prod_{i<r}C_i$ where 
$C_i\subseteq\Bbb R$ is a bounded interval for each $i<r$.   Set
$Q_r=\{(x,C):C\in\Cal C_r$, $x\in\overline{C}\}$;  let $T_r$ be the
straightforward set of tagged partitions generated by $Q_r$, 
$\Delta_r$ the set
of neighbourhood gauges on $\BbbR^r$, and 
$\frak R_r=\{\Cal R_C:C\in\Cal C_r\}$
where $\Cal R_C
=\{\BbbR^r\setminus C':C\subseteq C'\in\Cal C_r\}\cup\{\emptyset\}$ for 
$C\in\Cal C_r$.   Let $\nu_r$ be the restriction of $r$-dimensional
Lebesgue measure to $\Cal C_r$.   
(i) Show that $(\BbbR^r,T_r,\Delta_r,\frak R_r)$ is
a tagged-partition structure allowing subdivisions, witnessed by 
$\Cal C_r$.   (ii) For a function $f:\BbbR^r\to\Bbb R$ write 
$\Heint f(x)dx$ for the gauge integral $I_{\nu_r}(f)$ associated with this
structure when it is defined.   Show that if $r$, $s\ge 1$ are integers,
$f:\BbbR^{r+s}\to\Bbb R$ has compact support and $\Heint f(z)dz$ is
defined, then, identifying $\BbbR^{r+s}$ with $\BbbR^r\times\BbbR^s$,
$\Heint g(x)dx$ is defined and equal to $\Heint f(x,y)d(x,y)$ whenever
$g:\BbbR^r\to\Bbb R$ is such that $g(x)=\Heint f(x,y)dy$ for every
$x\in\BbbR^r$ for which this is defined.   
%482M

\leader{483Y}{Further exercises (a)}
%\spheader 483Ya
Let us say that a {\bf Lebesgue measurable neighbourhood
gauge} on $\Bbb R$ is a neighbourhood gauge of the form
$\{(x,A):x\in\Bbb R,\,A\subseteq\ooint{x-\eta_x,x+\eta_x}\}$ where
$x\mapsto\eta_x$ is a Lebesgue measurable function from $\Bbb R$ to
$\ooint{0,\infty}$.   Let $\tilde\Delta$ be the set of Lebesgue
measurable neighbourhood gauges.   Show that the gauge integral defined
by the tagged-partition structure $(\Bbb R,T,\tilde\Delta,\frak R)$ and
$\mu$ is the Henstock integral.
%483F

\spheader 483Yb Show that if $\Delta_0\subseteq\Delta$ is any set of
cardinal at most $\frak c$, then the gauge integral defined by
$(\Bbb R,T,\Delta_0,\frak R)$ and $\mu$ does not extend the Lebesgue
integral, so is not the Henstock integral.
%483Ya, 483B

\spheader 483Yc Let $f:\Bbb R\to\Bbb R$ be a Henstock integrable
function with indefinite Henstock integral $F$, and $\nu$ a totally
finite Radon measure on $\Bbb R$.   Set $G(x)=\nu\ocint{-\infty,x}$ for
$x\in\Bbb R$.   Show that $f\times G$ is Henstock integrable, with
indefinite Henstock integral $H$, where
$H(x)=F(x)G(x)-\int_{\ocint{-\infty,x}}F\,d\nu$ for $x\in\Bbb R$.
%483F

\spheader 483Yd Let $\nu$ be any Radon measure on $\Bbb R$,
$I_{\nu}$ the gauge integral defined from $\nu$ and the
tagged-partition structure of 481K and this section, and
$f:\Bbb R\to\Bbb R$ a function.

\quad(i) Show that if $I_{\nu}(f)$ is defined, then $f$ is
$\dom\nu$-measurable.

\quad(ii) Show that if $\int fd\nu$ is
defined in $\Bbb R$, then $I_{\nu}(f)$ is defined and equal to
$\int fd\nu$.

\quad(iii) Show that if $\alpha\in\ocint{-\infty,\infty}$ then
$I_{\nu}(f\times\chi\ooint{-\infty,\alpha})
=\lim_{\beta\uparrow\alpha}
  I_{\nu}(f\times\chi\ooint{-\infty,\beta})$ if either is defined in
$\Bbb R$.

\quad(iv) Suppose that $I_{\nu}(f)$ is defined.   ($\alpha$) Let $\FSH$ be
the Saks-Henstock indefinite integral of $f$ with respect to $\nu$.   Show
that for any $\epsilon>0$ there are a Radon measure $\zeta$ on $\Bbb R$
and a $\delta\in\Delta$ such that $\zeta\Bbb R\le\delta$ and
$|\FSH(W_{\pmb{t}})-S_{\pmb{t}}(f,\nu)|\le\zeta W_{\pmb{t}}$ whenever
$\pmb{t}\in T$ is $\delta$-fine.
($\beta$) Show that there is a
countable cover of $\Bbb R$ by compact sets $K$ such that
$\int_K|f|d\nu<\infty$.
%+ 483B 483Xe 483G 483F

\spheader 483Ye Let $f:\Bbb R\to\Bbb R$ be a Henstock integrable
function, and $G:\Bbb R\to\Bbb R$ a function of bounded variation.
Show that $f\times G$ is Henstock integrable, and

\Centerline{$\biggerHeint f\times G
\le(\lim_{x\to\infty}|G(x)|+\Var_{\Bbb R}G)
  \sup_{x\in\Bbb R}|\biggerHeint_{-\infty}^xf|$.}

\noindent(Compare 224J.)
%483Yc, 483M

\spheader 483Yf Let $f:\Bbb R\to\Bbb R$ be a Henstock integrable
function and $g:\Bbb R\to\Bbb R$ a Lebesgue integrable function;  let
$F$ and $G$ be their indefinite (Henstock) integrals.   Show that
$\Heint f\times G+\int g\times F\,d\mu$ is defined and equal to
$\lim_{x\to\infty}F(x)G(x)$.
%483Ye, 483M

\spheader 483Yg Let $f:\Bbb R\to\Bbb R$ be a function.   Show that $f$
is Lebesgue integrable iff $f\times g$ is Henstock integrable for every
bounded continuous function $g:\Bbb R\to\Bbb R$.
%483Ye, 483M

\spheader 483Yh Let $U$ be a linear subspace of $\BbbR^{\Bbb R}$ and
$\phi:U\to\Bbb R$ a linear functional such that (i) $f\in U$ and
$\phi f=\int fd\mu$ for every Lebesgue integrable function
$f:\Bbb R\to\Bbb R$ (ii) $f\times\chi C\in U$ whenever $f\in U$ and
$C\in\Cal C$ (iii) whenever $f\in\BbbR^{\Bbb R}$
and $\Cal I$ is a disjoint family
of non-empty open intervals such that $f\times\chi I\in U$ for every
$I\in\Cal I$ and
$\sum_{I\in\Cal I}\sup_{C\subseteq I,C\in\Cal C}
  |\phi(f\times\chi C)|<\infty$, then $f\times\chi(\bigcup\Cal I)\in U$
and
$\phi(f\times\chi(\bigcup\Cal I))=\sum_{I\in\Cal I}\phi(f\times\chi I)$.
Show that if $f:\Bbb R\to\Bbb R$ is any Henstock integrable function,
then $f\in U$ and $\phi(f)=\Heint f$.   \Hint{use the argument of part
(b) of the proof of 483R.} 
%483R 483Xf

\spheader 483Yi ({\smc Bongiorno Piazza \& Preiss 00}) Let $\Cal C$ be the set of non-empty subintervals of a closed interval
$[a,b]\subseteq\Bbb R$, and $T$ the straightforward set of tagged
partitions generated by $[a,b]\times\Cal C$.   Let $\Delta_{[a,b]}$
be the set of neighbourhood gauges on $[a,b]$.   For $\alpha\ge 0$ set

\Centerline{$T_{\alpha}
=\{\pmb{t}:\pmb{t}\in T,\,\sum_{(x,C)\in\pmb{t}}\rho(x,C)\le\alpha\}$,}

\noindent writing $\rho(x,C)=\inf_{y\in C}|x-y|$ as usual.   Show that
$T_{\alpha}$ is compatible with $\Delta_{[a,b]}$ and
$\frak R=\{\{\emptyset\}\}$
in the sense of 481F.   Show that if $I_{\alpha}$ is the gauge integral
defined from $[a,b]$, $T_{\alpha}$, $\Delta_{[a,b]}$, $\frak R$ and
Lebesgue
measure, then $I_{\alpha}$ extends the ordinary Lebesgue integral and
$I_{\alpha}(F')=F(b)-F(a)$ whenever $F:[a,b]\to\Bbb R$ is
differentiable relative to its domain.
%+

\spheader 483Yj Let $V$ be a Banach space and $f:\Bbb R\to V$ a
function.   For $\pmb{t}\in T$, set
$S_{\pmb{t}}(f,\mu)=\sum_{(x,C)\in\pmb{t}}\mu C\cdot f(x)$.
We say that $f$ is {\bf Henstock integrable}, with {\bf Henstock integral}
$v=\Heint f\in V$, if
$v=\lim_{\pmb{t}\to\Cal F(T,\Delta,\frak R)}S_{\pmb{t}}(f,\mu)$.

\quad(i) Show that the set $\eusm{HL}^1_V$ of Henstock integrable
functions from $\Bbb R$ to $V$ is a linear subspace of $V^{\Bbb R}$
including the space $\eusm L^1_V$ of Bochner integrable functions
(253Yf), and that $\Heint:\eusm{HL}^1_V\to V$ is a linear operator
extending the Bochner integral.

\quad(ii) Show that if $f:\Bbb R\to V$ is Henstock integrable, so is
$f\times\chi C$ for every interval $C\subseteq\Bbb R$, and that
$(a,b)\mapsto\Heint f\times\chi\ooint{a,b}$ is continuous.   Set
$\|f\|_H=\sup_{C\in\Cal C}\|\Heint f\times\chi C\|$.

\quad(iii) Show that if $\Cal I$ is a disjoint family of open intervals
in $\Bbb R$, and $f:\Bbb R\to V$ is such that
$f\times\chi I\in\eusm{HL}^1_V$ for every $I\in\Cal I$ and
$\sum_{I\in\Cal I}\|f\times\chi I\|_H$ is finite, then
$\Heint f\times\chi(\bigcup\Cal I)$ is defined and equal to
$\sum_{I\in\Cal I}\Heint f\times\chi I$.

\quad(iv) Define $f:\Bbb R\to\ell^{\infty}([0,1])$ by setting
$f(x)=\chi([0,1]\cap\ocint{-\infty,x})$ for $x\in\Bbb R$.   Show that
$f$ is Henstock integrable, but that if
$F(x)=\Heint f\times\chi\ooint{-\infty,x}$ for $x\in\Bbb R$, then
$\lim_{y\to x}\Bover1{y-x}(F(y)-F(x))$ is not defined in
$\ell^{\infty}([0,1])$ for any $x\in[0,1]$.

\spheader 483Yk\dvAnew{2011}
Find a function $f:\BbbR^2\to\Bbb R$, with compact support, such that
$\Heint f$ is defined in the sense of 483Xm, but $\Heint fT$ is not
defined, where $T(x,y)=\Bover1{\sqrt2}(x+y,x-y)$ for $x$, $y\in\Bbb R$.
%483Xm

\spheader 483Yl\dvAnew{2011}
Show that for a function $g:\Bbb R\to\Bbb R$ the following are 
equiveridical:  (i) there is a function $h:\Bbb R\to\Bbb R$, of bounded 
variation, such that $g\eae h$ (ii) $g$ is a {\bf multiplier} for the 
Henstock integral, that is, $f\times g$ is Henstock integrable for every 
Henstock integrable $f:\Bbb R\to\Bbb R$.
%483Ye mt48bits
}%end of exercises

\endnotes{
\Notesheader{483} I hope that the brief account here (largely taken from
{\smc Gordon 94}) will give an idea of the extraordinary power of gauge
integrals.   While what I am calling the `Henstock integral', regarded
as a linear functional on a space of real functions, was constructed
long ago by Perron and Denjoy, the gauge integral approach makes it far
more accessible, and gives clear pathways to corresponding Stieltjes and
vector integrals (483Yd, 483Yj).

Starting from our position in the fourth volume of a book on measure
theory, it is natural to try to describe the Henstock integral in terms
of the Lebesgue integral, as in 483C (they agree on non-negative
functions) and 483Yh (offering an extension process to generate the
Henstock integral from the Lebesgue integral);  on the way, we see that
Henstock integrable functions are necessarily Lebesgue integrable over
many intervals (483Xf).   Alternatively, we can set out to understand
indefinite Henstock integrals and their derivatives, just as Lebesgue
integrable functions can be characterized as almost everywhere equal to
derivatives of absolutely continuous functions (222E, 225E), because if
$f$ is Henstock integrable then it is equal almost everywhere to the
derivative of its indefinite integral (483I).   Any differentiable
function (indeed, any continuous function differentiable except on a
countable set) is an indefinite Henstock integral (483Xh).   Recall that
the Cantor function (134H) is continuous and differentiable almost
everywhere but is not an indefinite integral, so we have to look for a
characterization which can exclude such cases.   For this we have to
work quite hard, but we find that `ACG$_*$ functions' are the
appropriate class (483R).

Gauge integrals are good at integrating derivatives (see 483Xh), but bad
at integrating over subspaces.   Even to show that
$f\times\chi\coint{0,\infty}$ is Henstock integrable whenever $f$ is
(483Bc) involves us in some unexpected manoeuvres.   I give an argument
which is designed to show off the general theory of \S482, and I
recommend you to look for short cuts (483Xb), but any method must depend
on careful examination of the exact classes $\Cal C$ and $\frak R$
chosen for the definition of the integral.   We do however have a new
kind of convergence theorem in 482H and 483N.

One of the incidental strengths of the Henstock integral is that it
includes the improper Riemann integral (483Bd, 483Xc);  so that, for
instance, Carleson's theorem (286U) can be written in the form

\Centerline{$\varhatf(y)
=\Bover1{\sqrt{2\pi}}\biggerHeint e^{-ixy}f(x)dx$
for almost every $y$ if $f:\Bbb R\to\Bbb C$ is square-integrable.}

\noindent But to represent the many expressions of the type
$\lim_{a\to\infty}\int_{-a}^af$ in \S283 (e.g., 283F, 283I and 283L)
directly in the form $I_{\mu}(f)$ we need to change $\frak R$, as in
481L or 481Xc.
}%end of notes

\discrpage

\frfilename{mt484.tex}
\versiondate{21.1.10}
\copyrightdate{2001}

\def\varinnerprod#1#2{#1\dotproduct#2}

\def\chaptername{Gauge integrals}
\def\sectionname{The Pfeffer integral}

\newsection{484}

I give brief notes on what seems at present to be the most
interesting of the multi-dimensional versions of the Henstock integral,
leading to Pfeffer's Divergence Theorem (484N).

\leader{484A}{Notation}\cmmnt{ This section will depend heavily on
Chapter 47, and will use much of the same notation.}   $r\ge 2$ will be
a fixed integer, and $\mu$ will be Lebesgue measure on
$\BbbR^r$\cmmnt{, while
$\mu_{r-1}$ is Lebesgue measure on $\BbbR^{r-1}$}.   As in
\S\S473-475, let
$\nu$ be `normalized' $(r-1)$-dimensional Hausdorff measure on
$\BbbR^r$\cmmnt{, as
described in \S265;  that is, $\nu=2^{-r+1}\beta_{r-1}\mu_{H,r-1}$,
where $\mu_{H,r-1}$ is
$(r-1)$-dimensional Hausdorff measure on $\BbbR^r$ as described in
\S264}, and

$$\eqalign{\beta_{r-1}
&=\Bover{2^{2k}k!\pi^{k-1}}{(2k)!}\text{ if }r=2k\text{ is even},\cr
&=\Bover{\pi^k}{k!}\text{ if }r=2k+1\text{ is odd}\cr}$$

\noindent\cmmnt{is} the Lebesgue measure of a ball of radius $1$ in
$\BbbR^{r-1}$\cmmnt{ (264I)}.
For this section only, let us say that a subset of $\BbbR^r$ is {\bf
thin} if it is of the form $\bigcup_{n\in\Bbb N}A_n$ where $\nu^*A_n$ is
finite for every $n$.
\cmmnt{Note that every thin set is $\mu$-negligible (471L).
For $A\subseteq\BbbR^r$, write $\partial A$ for
its ordinary topological boundary.   If $x\in\BbbR^r$ and $\epsilon>0$,
$B(x,\epsilon)$ will be the closed ball $\{y:\|y-x\|\le\epsilon\}$.
}%end of comment

I will use the term {\bf dyadic cube} for sets of the
form $\prod_{i<r}\coint{2^{-m}k_i,2^{-m}(k_i+1)}$ where $m$,
$k_0,\ldots,k_{r-1}\in\Bbb Z$;  write $\Cal D$ for the set of dyadic
cubes in $\BbbR^r$.   Note that if\cmmnt{ $D$, $D'\in\Cal D$, either
$D\subseteq D'$ or $D'\subseteq D$ or $D\cap D'=\emptyset$;
so if} $\Cal D_0\subseteq\Cal D$, the
maximal members of $\Cal D_0$ are disjoint.

It will be helpful to have an abbreviation for the following expression:
set

\Centerline{$\alpha^*
=\min(\Bover1{r^{r/2}},\Bover{2^{r-2}}{r\beta_r^{(r-1)/r}})$.}

\noindent\cmmnt{(As will become apparent, the
actual value of this
constant is of no importance;  but the strict logic of the arguments
below depends on $\alpha^*$ being small enough.)}

\cmmnt{As in \S475, I write $\intstar A$, $\clstar A$ and
$\partstar A$ for the
essential interior, essential closure and essential boundary of a set
$A\subseteq\BbbR^r$ (475B).   Recall that a set $A\subseteq\BbbR^r$ has
finite perimeter
in the sense of 474D iff $\nu(\partstar A)$ is finite, and then

\Centerline{$\nu(\partstar A)=\lambda^{\partial}_A(\BbbR^r)=\per A$}

\noindent is the perimeter of $A$ (475M);  we shall also need to
remember that $A$ is necessarily Lebesgue measurable.}%end of comment

$\Cal C$ will be the family of subsets of $\BbbR^r$ with locally finite
perimeter, and
$\Cal V$ the family of bounded sets in $\Cal C$\cmmnt{, that is, the
family of bounded sets with finite perimeter}.
%`Cacciopoli sets' in Pfeffer, Tamanini & Giacomelli
\cmmnt{Note that $\Cal C$ is an algebra of subsets of $\BbbR^r$
(475Ma), and that $\Cal V$ is an ideal in $\Cal C$.
}

\leader{484B}{Theorem}\cmmnt{ ({\smc Tamanini \& Giacomelli 89})}
%Theorem 2.3
Let $E\subseteq\BbbR^r$ be a Lebesgue measurable set of
finite measure and perimeter, and $\epsilon>0$.   Then there is a
Lebesgue measurable set $G\subseteq E$ such that $\per G\le\per E$,
$\mu(E\setminus G)\le\epsilon$ and $\clstar G=\overline{G}$.

\proof{({\smc Pfeffer 91b}) {\bf (a)} Set
$\alpha=\bover1{\epsilon}\per E$.   For measurable sets
$G\subseteq E$ set $q(G)=\per G-\alpha\mu G$.   Then there is a
self-supporting measurable set $G\subseteq E$ such that $q(G)\le q(G')$
whenever $G'\subseteq E$ is measurable.

\Prf\ Write $\Sigma$ for the family of Lebesgue measurable subsets of
$\BbbR^r$;  give $\Sigma$ the topology of convergence in measure defined
by the pseudometrics $\rho_H(G,G')=\mu((G\symmdiff G')\cap H)$ for
measurable sets $H$ of finite measure\cmmnt{ (cf.\ 474T)}.
Extend $q$ to $\Sigma$ by setting
$q(G)=\per(E\cap G)-\alpha\mu(E\cap G)$ for every $G\in\Sigma$.
Because $\per:\Sigma\to[0,\infty]$ is lower semi-continuous for the
topology of convergence in measure (474Ta), and $G\mapsto E\cap G$,
$G\mapsto\mu(E\cap G)$ are continuous, $q:\Sigma\to\coint{0,\infty}$ is
lower semi-continuous (4A2Bd).   Next,
$\Cal K=\{G:G\in\Sigma,\,\per G\le\per E\}$ is compact (474Tb), while
$\Cal L=\{G:\mu(G\setminus E)=0\}$ is closed, so there is a
$G_0\in\Cal L\cap\Cal K$ such that
$q(G_0)=\inf_{G\in\Cal L\cap\Cal K}q(G)$ (4A2Gl).   Since
$G_0\in\Cal L$,
$\per(G_0\cap E)=\per(G_0)$ and $\mu(G_0\cap E)=\mu G_0$, so we may
suppose that $G_0\subseteq E$.   Moreover, there is a self-supporting
set $G\subseteq G_0$ such that $G_0\setminus G$ is negligible (414F),
and we still have $q(G)=q(G_0)$.   Of course $q(G)\le q(E)$, just
because $E\in\Cal L\cap\Cal K$.

\Quer\ If there is a measurable set $G'\subseteq E$ such that
$q(G')<q(G)$, then

\Centerline{$\per G'=q(G')+\alpha\mu G'\le q(E)+\alpha\mu E=\per E$,}

\noindent so $G'\in\Cal K$;  but this means that $G'\in\Cal L\cap\Cal K$
and $q(G)=q(G_0)\le q(G')$.\ \BanG\  So $G$ has the required
properties.\ \Qed

\medskip

{\bf (b)} Since $q(G)\le q(E)$, we must have

\Centerline{$\per G+\alpha\mu(E\setminus G)
=q(G)+\alpha\mu E\le q(E)+\alpha\mu E=\per E=\alpha\epsilon$.}

\noindent So $\mu(E\setminus G)\le\epsilon$.

\medskip

{\bf (c)} Next, $\overline{G}\subseteq\clstar G$.   \Prf\ Let
$x\in\overline{G}$.   For every $t>0$, set
$U_t=\{y:\|y-x\|<t\}$;  then

$$\eqalignno{\per(G\cap U_t)+\per(G\setminus U_t)
&=\nu(\partstar(G\cap U_t))+\nu(\partstar(G\setminus U_t))\cr
&\le\nu(\partstar G\cap U_t)+\nu(\clstar G\cap\partial U_t)\cr
&\qquad\qquad\qquad
      +\nu(\partstar G\setminus U_t)+\nu(\clstar G\cap\partial U_t)\cr
\displaycause{475Cf, because
$\partial(\BbbR^r\setminus U_t)=\partial U_t)$}
&=\nu(\partstar G)+2\nu(\clstar G\cap\partial U_t)
=\per G+2\nu(G\cap\partial U_t)\cr}$$

\noindent for almost every $t>0$, because

\Centerline{$\int_0^{\infty}
  \nu((G\symmdiff\clstar G)\cap\partial U_t)dt
=\mu(G\symmdiff\clstar G)=0$}

\noindent (265G).   So, for almost every $t$,

$$\eqalignno{\mu(G\cap U_t)^{(r-1)/r}
&\le\per(G\cap U_t)\cr
\displaycause{474La}
&\le\per G+2\nu(G\cap\partial U_t)-\per(G\setminus U_t)\cr
&=q(G)+\alpha\mu(G\cap U_t)+2\nu(G\cap\partial U_t)-q(G\setminus U_t)\cr
&\le\alpha\mu(G\cap U_t)+2\nu(G\cap\partial U_t)\cr}$$

\noindent because $q(G)$ is minimal.

For $t>0$, set

\Centerline{$g(t)=\mu(G\cap U_t)=\int_0^t\nu(G\cap\partial U_s)ds$,}

\noindent so that

\Centerline{$g'(t)=\nu(G\cap\partial U_t)
\ge\Bover12(g(t)^{(r-1)/r}-\alpha g(t))$}

\noindent for almost every $t$.   Because $G$ is self-supporting and
$U_t$ is open and $G\cap U_t\ne\emptyset$, $g(t)>0$ for every $t>0$;
and $\lim_{t\downarrow 0}g(t)=0$.

Set

\Centerline{$h(t)=\Bover{d}{dt}g(t)^{1/r}
=\Bover{g'(t)}{rg(t)^{(r-1)/r}}
\ge\Bover1{2r}(1-\alpha g(t)^{1/r})$}

\noindent for almost every $t$.   Then

\Centerline{$\limsup_{t\downarrow 0}\Bover{g(t)^{1/r}}t
=\limsup_{t\downarrow 0}\Bover1t\int_0^th\ge\Bover1{2r}$,}

\noindent and

\Centerline{$\limsup_{t\downarrow 0}
\Bover{\mu(G\cap B(x,t)))}{\mu B(x,t))}
=\limsup_{t\downarrow 0}\Bover{g(t)}{\beta_rt^r}>0$.}

Thus $x\in\clstar G$.   As $x$ is arbitrary,
$\overline{G}\subseteq\clstar G$.\ \Qed

Since we certainly have
$\clstar G\subseteq\overline{G}$, this $G$ serves.
}%end of proof of 484B

\leader{484C}{Lemma} Let $E\in\Cal V$ and $l\in\Bbb N$ be such that
$\max(\per E,\diam E)\le l$.   Then $E$ is expressible as
$\bigcup_{i<n}E_i$ where $\ofamily{i}{n}{E_i}$ is disjoint,
$\per E_i\le 1$ for each $i<n$ and $n$ is at most
$2^r(l+1)^r(4r(2l^2+1))^{r/(r-1)}+2^{r+1}l^2$.

\proof{ For $D\in\Cal D$, write $\Cal D_D$ for
$\{D':D'\in\Cal D,\,D'\subseteq D,\,\diam D'=\bover12\diam D\}$, the
family of the $2^r$ dyadic subcubes of $D$ at the next level down.

\medskip

{\bf (a)} If $l\le 1$ the result is trivial, so let us suppose that
$l\ge 2$.   Let $m\in\Bbb N$ be minimal subject to
$4r(2l^2+1)\le 2^{m(r-1)}$, so that
$2^m\le 2(4r(2l^2+1))^{1/(r-1)}$.   Then we can cover $E$ by a family
$\Cal L_0$ of dyadic cubes of side $2^{-m}$ with

\Centerline{$\#(\Cal L_0)\le (2^ml+1)^r\le 2^{mr}(l+1)^r
\le 2^r(l+1)^r(4r(2l^2+1))^{r/(r-1)}$.}

\medskip

{\bf (b)} Let $\Cal L_1$ be the set of those $D\in\Cal D$ such that
$\lfloor 2\nu(D'\cap\partstar E)\rfloor
<\lfloor 2\nu(D\cap\partstar E)\rfloor$ for every $D'\in\Cal D_D$.
Then $\#(\Cal L_1)\le 2l^2$.   \Prf\
For $k\ge 1$, set

\Centerline{$\Cal L_1^{(k)}=\{F:D\in\Cal L_1$,
$\lfloor 2\nu(D\cap\partstar E)\rfloor=k\}$.}

\noindent If $D$, $D'\in\Cal L_1^{(k)}$ are distinct, neither can be
included in the other, so they are disjoint.   Accordingly
$k\#(\Cal L_1^{(k)})\le 2\nu(\partstar E)\le 2l$ and
$\#(\Cal L_1^{(k)})\le 2l$.   Since
$\Cal L_1=\bigcup_{1\le k\le l}\Cal L_1^{(k)}$,
$\#(\Cal L_1)\le 2l^2$.\ \Qed

\medskip

{\bf (c)} For $D\in\Cal D$, set
$\tilde D=D\setminus\bigcup\{D':D'\in\Cal L_1$, $D'\subseteq D\}$.
Then $\nu(\tilde D\cap\partstar E)\le\bover12$.   \Prf\Quer\ Otherwise,
set $j=\lfloor 2\nu(D\cap\partstar E)\rfloor\ge 1$, and
choose $\sequence{i}{D_i}$ inductively, as follows.   $D_0=D$.   Given that
$D_i\in\Cal D$, $D_i\subseteq D$ and
$\lfloor 2\nu(D_i\cap\partstar E)\rfloor=j$,
$\nu((D\setminus D_i)\cap\partstar E)<\bover12$ and
$D_i\cap\tilde D$ is non-empty, so $D_i\notin\Cal L_1$ and there must be a
$D_{i+1}\in\Cal D_{D_i}$ such that
$\lfloor 2\nu(D_{i+1}\cap\partstar E)\rfloor=j$.   Continue.   This gives
us a strictly decreasing sequence $\sequence{i}{D_i}$ in $\Cal D$ such that
$\nu(D_i\cap\partstar E)\ge\bover{j}2$ for every $i$.   But (because
$\per E$ is finite) this means that, writing $x$ for the
unique member of $\bigcap_{i\in\Bbb N}\overline{D}_i$,
$\nu\{x\}\ge\bover{j}2$, which is absurd.\ \Bang\Qed

\medskip

{\bf (d)} Set


\Centerline{$\Cal L'_1
=\{D:D\in\Cal L_1$ is included in some member of $\Cal L_0\}$,}

\Centerline{$\Cal L_2=\Cal L_0\cup\bigcup\{\Cal D_D:D\in\Cal L'_1\}$,
\quad$\Cal K=\{\tilde D:D\in\Cal L_2\}$.}

\medskip

\quad{\bf (i)}

$$\eqalign{\#(\Cal K)
&\le\#(\Cal L_2)
\le\#(\Cal L_0)+2^r\#(\Cal L_1)\cr
&\le 2^r(l+1)^r(4r(2l^2+1))^{r/(r-1)}+2^{r+1}l^2.\cr}$$

\medskip

\quad{\bf (ii)} $\bigcup\Cal K\supseteq E$.   \Prf\ If $x\in E$, there
is a smallest member $D$ of $\Cal L_2$ containing it, because
certainly $x\in\bigcup\Cal L_0$.   But now $x$ cannot belong to any
member of $\Cal L_1$ included in $D$, so $x\in\tilde D$.\ \Qed

\medskip

\quad{\bf (iii)} $\Cal K$ is disjoint.   \Prf\ If $D_1$, $D_2\in\Cal L_2$
are disjoint, then of course $\tilde D_1\cap\tilde D_2=\emptyset$.   If
$D_1\subset D_2$, then $D_1\notin\Cal L_0$, so there is a $D\in\Cal L'_1$
such that $D_1\in\Cal D_D$;  in this case $D\subseteq D_2$ so
$\tilde D_2\subseteq D_2\setminus D$ is disjoint from $D_1$.\ \Qed

\medskip

\quad{\bf (iv)} $\per(D\cap E)\le 1$ for every $D\in\Cal K$.
\Prf\ Take $D_0\in\Cal L_2$ such that $D=\tilde D_0$;  then

\Centerline{$\nu(\partial D)
\le\nu(\partial D_0)+\sum_{D'\in\Cal L'_1}\nu(\partial D')
\le 2r(2l^2+1)2^{-m(r-1)}
\le\Bover12$}

\noindent by the choice of $m$.   So

\Centerline{$\per(D\cap E)\le\nu(\partial D)+\nu(D\cap\partstar E)
\le\Bover12+\Bover12=1$}

\noindent by 475Cf.\ \Qed

\medskip

{\bf (e)} So if we take $\ofamily{i}{n}{E_i}$ to be an enumeration of
$\{E\cap D:D\in\Cal K\}$, we shall have the required result.
}%end of proof of 484C

\leader{484D}{Definitions}\cmmnt{ The gauge integrals of this section
will be based on the following residual families.}   Let $\Eta$ be the
family of strictly positive
sequences $\eta=\sequence{i}{\eta(i)}$ in $\Bbb R$.   For $\eta\in\Eta$,
write $\Cal M_{\eta}$ for the set of disjoint sequences
$\sequence{i}{E_i}$ of measurable subsets of $\BbbR^r$ such that
$\mu E_i\le\eta(i)$ and $\per E_i\le 1$ for every
$i\in\Bbb N$, and $E_i$ is empty for all but finitely many $i$.
For $\eta\in\Eta$ and $V\in\Cal V$ set

\Centerline{$\Cal R_{\eta}
=\{\bigcup_{i\in\Bbb N}E_i:\sequence{i}{E_i}\in\Cal M_{\eta}\}
\subseteq\Cal C$,
\quad$\Cal R^{(V)}_{\eta}
=\{R:R\subseteq\BbbR^r,\,R\cap V\in\Cal R_{\eta}\}$;}

\noindent
finally, set $\frak R=\{\Cal R^{(V)}_{\eta}:V\in\Cal V,\,\eta\in\Eta\}$.

\leader{484E}{Lemma} (a)(i) For every $\Cal R\in\frak R$, there is an
$\eta\in\Eta$ such that $\Cal R_{\eta}\subseteq\Cal R$.

\quad(ii) If $\Cal R\in\frak R$ and $C\in\Cal C$, there is an
$\Cal R'\in\frak R$ such that $C\cap R\in\Cal R$ whenever $R\in\Cal R'$.

(b)(i) If $\eta\in\Eta$ and $\gamma\ge 0$, there is an $\epsilon>0$ such
that $R\in\Cal R_{\eta}$ whenever $\mu R\le\epsilon$, $\diam R\le\gamma$
and $\per R\le\gamma$.

\quad(ii) If $\Cal R\in\frak R$ and $\gamma\ge 0$, there is an
$\epsilon>0$ such that $R\in\Cal R$ whenever $\mu R\le\epsilon$
and $\per R\le\gamma$.

(c) If $\Cal R\in\frak R$ there is an $\Cal R'\in\frak R$ such that
$R\cup R'\in\Cal R$ whenever $R$, $R'\in\Cal R'$ and
$R\cap R'=\emptyset$.

(d)(i) If $\eta\in\Eta$ and $A\subseteq\BbbR^r$ is a thin set, then
there is a set $\Cal D_0\subseteq\Cal D$ such that every point of $A$
belongs to the interior of $\bigcup\Cal D_1$ for some finite
$\Cal D_1\subseteq\Cal D_0$, and $\bigcup\Cal D_1\in\Cal R_{\eta}$ for
every finite set $\Cal D_1\subseteq\Cal D_0$.

\quad(ii) If $\Cal R\in\frak R$ and $A\subseteq\BbbR^r$ is a thin
set, then there is a set $\Cal D_0\subseteq\Cal D$ such that every point
of $A$ belongs to the interior of $\bigcup\Cal D_1$ for some finite set
$\Cal D_1\subseteq\Cal D_0$, and $\bigcup\Cal D_1\in\Cal R$ for every
finite set $\Cal D_1\subseteq\Cal D_0$.

\proof{{\bf (a)(i)} Express $\Cal R$ as $\Cal R^{(V)}_{\eta'}$ where
$\eta'\in\Eta$ and $V\in\Cal V$.   Let $l\in\Bbb N$ be such that
$\max(\diam V,1+\per V)\le l$, and take
$n\ge 2^r(l+1)^r(4r(2l^2+1))^{r/(r-1)}+2^{r+1}l^2$.   Set
$\eta(i)=\min\{\eta'(j):ni\le j<n(i+1)\}$ for every $i\in\Bbb N$, so
that $\eta\in\Eta$.   If $R\in\Cal R_{\eta}$, express it as
$\bigcup_{i\in\Bbb N}E_i$ where $\sequence{i}{E_i}\in\Cal M_{\eta}$.
For each $i$, $\max(\diam(E_i\cap V),\per(E_i\cap V))\le l$, so
by 484C we can express $E_i\cap V$ as $\bigcup_{ni\le j<n(i+1)}E'_j$,
where $\langle E'_j\rangle_{ni\le j<n(i+1)}$ is disjoint and
$\per E'_j\le 1$ for each $j$.   Now

\Centerline{$\mu E'_j\le\mu E_i\le\eta(i)\le\eta'(j)$}

\noindent for $ni\le j<n(i+1)$.   Also $\{j:E'_j\ne\emptyset\}$ is
finite because $\{j:E_j=\emptyset\}$ is finite.   So
$\sequence{j}{E'_j}\in\Cal M_{\eta'}$ and
$R\cap V=\bigcup_{j\in\Bbb N}E'_j$ belongs to $\Cal R_{\eta'}$, that is,
$R\in\Cal R$.   As $R$ is arbitrary, $\Cal R_{\eta}\subseteq\Cal R$.

\medskip

\quad{\bf (ii)} Express $\Cal R$ as $\Cal R^{(V)}_{\eta}$, where
$V\in\Cal V$ and $\eta\in\Eta$.   By (i), there is an $\eta'\in\Eta$
such that $\Cal R_{\eta'}\subseteq\Cal R^{(C\cap V)}_{\eta}$.   Set
$\Cal R'=\Cal R^{(V)}_{\eta'}\in\frak R$.   If $R\in\Cal R'$, then
$R\cap V\in\Cal R_{\eta'}$, so
$C\cap R\cap V\in\Cal R_{\eta}$ and $C\cap R\in\Cal R$.

\medskip

{\bf (b)(i)} Take $l\ge\gamma$ and
$n\ge 2^r(l+1)^r(4r(2l^2+1))^{r/(r-1)}+2^{r+1}l^2$, and set
$\epsilon=\min_{i<n}\eta(i)$.   If $\mu R\le\epsilon$, $\diam R\le l$
and $\per R\le l$, then $R$ is expressible as
$\bigcup_{i<n}E_i$ where $\ofamily{i}{n}{E_i}$ is disjoint and
$\per E_i\le 1$ for each $i<n$.   Since $\mu E_i\le\mu R\le\eta(i)$ for
each $i$, $R\in\Cal R_{\eta}$.

\quad{\bf (ii)} Express $\Cal R$ as $\Cal R^{(V)}_{\eta}$.   By
(i), there is an $\epsilon>0$ such that $R\in\Cal R_{\eta}$
whenever $\diam R\le\diam V$, $\per R\le\gamma+\per V$ and
$\mu R\le\epsilon$;  and this $\epsilon$ serves.

\medskip

{\bf (c)} Express $\Cal R$ as $\Cal R^{(V)}_{\eta}$.
Set $\eta'(i)=\min(\eta(2i),\eta(2i+1))$
for every $i$;  then $\eta'\in\Eta$ and if $\sequence{i}{E_i}$,
$\sequence{i}{E'_i}$ belong to $\Cal M_{\eta'}$ and have
disjoint unions, then
$(E_0,E'_0,E_1,E'_1,\ldots)\in\Cal M_{\eta}$;  this is
enough to show that $R\cup R'\in\Cal R_{\eta}$ whenever $R$,
$R'\in\Cal R_{\eta'}$ are disjoint, so that $R\cup R'\in\Cal R$ whenever
$R$, $R'\in\Cal R^{(V)}_{\eta'}$ are disjoint.

\medskip

{\bf (d)(i)}\grheada\ We can express $A$ as $\bigcup_{i\in\Bbb N}A_i$
where $\mu^*_{H,r-1}A_i<2^{-r}/2r$ for every $i$.   \Prf\ Because $A$
is thin, it is the union of a sequence of sets of finite outer measure
for $\nu$, and therefore for $\mu_{H,r-1}$.   On each of these the
subspace measure is atomless (471E, 471Dg), so that the set can be
dissected into
finitely many sets of measure less than $2^{-r}/2r$ (215D).\ \Qed

\medskip

\qquad\grheadb\ For each $i\in\Bbb N$,
we can cover $A_i$ by a sequence $\sequence{j}{A_{ij}}$ of sets such
that $\diam A_{ij}<\eta(i)$ for every $j$ and
$\sum_{j=0}^{\infty}(\diam A_{ij})^{r-1}<2^{-r}/2r$;  enlarging the
$A_{ij}$ slightly if need be, we can suppose that they are all open.
Now we can
cover each $A_{ij}$ by $2^r$ cubes $D_{ijk}\in\Cal D$ in such a way
that the side length of each $D_{ijk}$ is at most the diameter of
$A_{ij}$.

Setting $\Cal D_0=\{D_{ijk}:i$, $j\in\Bbb N$, $k<2^r\}$, we see that
every point of $A$ belongs to an open set $A_{ij}$ which is covered by a
finite subset of $\Cal D_0$.   If $\Cal D_1\subseteq\Cal D_0$ is finite,
let $\Cal D'_1$ be the family of maximal elements of $\Cal D_1$, so that
$\Cal D'_1$ is disjoint and $\bigcup\Cal D'_1=\bigcup\Cal D_1$.
Express $\Cal D'_1$ as $\{D_{ijk}:(i,j,k)\in I\}$ where
$I\subseteq\Bbb N\times\Bbb N\times 2^r$ is finite and
$\family{(i,j,k)}{I}{D_{ijk}}$ is disjoint.   Set
$I_i=\{(j,k):(i,j,k)\in I\}$ and $E_i=\bigcup_{(j,k)\in I_i}D_{ijk}$ for
$i\in\Bbb N$.   Then $\sequence{i}{E_i}$ is disjoint, $E_i=\emptyset$
for all but finitely many $i$, and for each $i\in\Bbb N$

$$\eqalign{\mu E_i
&\le\sum_{j=0}^{\infty}\sum_{k=0}^{2^r-1}\mu D_{ijk}
\le\sum_{j=0}^{\infty}2^r(\diam A_{ij})^r\cr
&\le 2^r\sum_{j=0}^{\infty}\eta(i)(\diam A_{ij})^{r-1}
\le\eta(i),\cr
\nu(\partial E_i)
&\le\sum_{(j,k)\in I_i}\nu(\partial D_{ijk})
\le\sum_{j=0}^{\infty}\sum_{k=0}^{2^r-1}\nu(\partial D_{ijk})\cr
&\le\sum_{j=0}^{\infty}2^r\cdot 2r(\diam A_{ij})^{r-1}
\le 1.\cr}$$

\noindent So $\bigcup\Cal D_1=\bigcup_{i\in\Bbb N}E_i$ belongs to
$\Cal R_{\eta}$.

\medskip

\quad{\bf (ii)}  By (a-i), there is an $\eta\in\Eta$ such that
$\Cal R_{\eta}\subseteq\Cal R$.   By (i) here, there is a
$\Cal D_0\subseteq\Cal D$ such that every point of $A$ belongs to the
interior of $\bigcup\Cal D_1$ for some finite
$\Cal D_1\subseteq\Cal D_0$, and
$\bigcup\Cal D_1\in\Cal R_{\eta}\subseteq\Cal R$ for
every finite set $\Cal D_1\subseteq\Cal D_0$.
}%end of proof of 484E

\leader{484F}{A family of tagged-partition structures} For $\alpha>0$,
let $\Cal C_{\alpha}$ be the family of those $C\in\Cal V$ such that
$\mu C\ge\alpha(\diam C)^r$ and $\alpha\per C\le(\diam C)^{r-1}$, and
let $T_{\alpha}$ be the straightforward set of tagged partitions
generated by the set

\Centerline{$\{(x,C):C\in\Cal C_{\alpha},\,x\in\clstar C\}$.}

\noindent Let $\Theta$ be the set of functions
$\theta:\BbbR^r\to\coint{0,\infty}$ such that $\{x:\theta(x)=0\}$ is
thin\cmmnt{ (definition:  484A)},
and set $\Delta=\{\delta_{\theta}:\theta\in\Theta\}$, where
$\delta_{\theta}
=\{(x,A):x\in\BbbR^r,\,\theta(x)>0,\,\|y-x\|<\theta(x)$ for every
$y\in A\}$.

Then whenever $0<\alpha<\alpha^*$, $(\BbbR^r,T_{\alpha},\Delta,\frak R)$
is a tagged-partition structure allowing subdivisions, witnessed by
$\Cal C$.

\proof{{\bf (a)} We had better look again at all the conditions in 481G.

(i) and (vi) really are trivial.
(iv) and (v) are true because $\Cal C$ is actually an algebra of sets
and $\emptyset\in\Cal R$ for every $\Cal R\in\frak R$.

For (ii), we have to observe that the
union of two thin sets is thin, so that $\theta\wedge\theta'\in\Theta$
for all $\theta$, $\theta'\in\Theta$;  since
$\delta_{\theta}\cap\delta_{\theta'}
=\delta_{\theta\wedge\theta'}$, this is all we need.

(iii)($\alpha$) follows from 484Ea:  given $V$, $V'\in\Cal V$ and
$\eta$, $\eta'\in\Eta$, take $\tilde\eta$, $\tilde\eta'\in\Eta$ such
that $\Cal R_{\tilde\eta}\subseteq\Cal R^{(V)}_{\eta}$ and
$\Cal R_{\tilde\eta'}\subseteq\Cal R^{(V')}_{\eta'}$.
Then $V\cup V'\in\Cal V$ and
$\Cal R^{(V)}_{\eta}\cap\Cal R^{(V')}_{\eta'}
\supseteq\Cal R^{(V\cup V')}_{\tilde\eta\wedge\tilde\eta'}$.
($\beta$) is just 484Ec.

\medskip

{\bf (b)} Now let us turn to 481G(vii).

\medskip

\quad{\bf (i)} Fix $C\in\Cal C$, $\delta\in\Delta$ and
$\Cal R\in\frak R$.   Express $\delta$ as $\delta_{\theta}$ where
$\theta\in\Theta$.   Let $\Cal R'\in\frak R$ be such that
$A\cup A'\in\Cal R$ whenever $A$, $A'\in\Cal R'$ are disjoint.   Express
$\Cal R'$ as $\Cal R^{(V)}_{\eta}$ where $V\in\Cal V$ and $\eta\in\Eta$.
By 484E(b-ii), there is an $\epsilon>0$ such that $R\in\Cal R'$ whenever
$\per R\le 2\per(C\cap V)$ and $\mu R\le\epsilon$.

By 484B, there is an $E\subseteq C\cap V$ such that
$\per E\le\per(C\cap V)$,
$\mu((C\cap V)\setminus E)\le\epsilon$ and $\clstar E=\overline{E}$.
In this case,

\Centerline{$\per(C\cap V\setminus E)\le\per(C\cap V)+\per E
\le 2\per(C\cap V)$,}

\noindent so $C\cap V\setminus E\in\Cal R'$, by the choice of
$\epsilon$.   By
484E(a-ii), there is an $\Cal R''\in\frak R$ such that
$E\cap R\in\Cal R'$ whenever $R\in\Cal R''$.

Now consider

\Centerline{$A=\{x:\theta(x)=0\}\cup\partstar E
   \cup\{x:\limsup_{\zeta\downarrow 0}
     \sup_{x\in G,0<\diam G\le\zeta}
     \Bover{\nu^*(G\cap\partstar E)}{(\diam G)^{r-1}}>0\}$.}

Because $\per E<\infty$  and

\Centerline{$\{x:x\in\BbbR^r\setminus\partstar E,\,
  \limsup_{\zeta\downarrow 0}
     \sup_{x\in G,0<\diam G\le\zeta}
     \Bover{\nu^*(G\cap\partstar E)}{(\diam G)^{r-1}}>0\}$}

\noindent is $\nu$-negligible (471Pc), $A$ is thin.   By 484E(d-ii),
there is a set $\Cal D_0\subseteq\Cal D$ such that

\Centerline{$A\subseteq\bigcup\{\interior(\bigcup\Cal D_1):
  \Cal D_1\in[\Cal D_0]^{<\omega}\}$,
\quad $\bigcup\Cal D_1\in\Cal R''$ for every
  $\Cal D_1\in[\Cal D_0]^{<\omega}$.}

\medskip

\quad{\bf (ii)} Write $T'$ for the set of those $\delta$-fine
$\pmb{t}\in T_{\alpha}$ such that every
member of $\pmb{t}$ is of the form $(x,D\cap E)$ for some $D\in\Cal D$.
If $x\in\intstar E\setminus A$, there is an $h(x)>0$
such that $h(x)<\theta(x)$ and $\{(x,D\cap E)\}\in T'$ whenever
$D\in\Cal D$, $x\in\overline{D}$ and $\diam D\le h(x)$.
\Prf\ Let $\epsilon_1>0$ be such that $r^{r/2}\epsilon_1\le\bover12$ and

\Centerline{$\alpha\le\Bover{1-r^{r/2}\epsilon_1}{r^{r/2}}$,
\quad$\Bover1{\alpha}\ge(2r+r^{r/2}\epsilon_1)\cdot\Bover1{2^{r-1}}
  \bigl(\Bover{\beta_r}{1-r^{r/2}\epsilon_1}\bigr)^{(r-1)/r}$;}

\noindent this is where we need to know that $\alpha<\alpha^*$.
Because $x\notin A$, $\theta(x)>0$;  let $h(x)\in\ooint{0,\theta(x)}$ be
such that ($\alpha$)
$\nu^*(D\cap\partstar E)\le\epsilon_1(\diam D)^{r-1}$ whenever
$x\in\overline{D}$ and $\diam D\le h(x)$ ($\beta$)
$\mu(B(x,t)\setminus E)\le\epsilon_1t^r$ whenever
$0\le t\le h(x)$.   Now suppose that $D\in\Cal D$ and
$x\in\overline{D}$ and $\diam D\le h(x)$.   Then, writing $\gamma$ for
the side length of $D$,

$$\eqalign{\mu(D\cap E)
&\ge\mu D-\mu(B(x,\diam D)\setminus E)
\ge\gamma^r-\epsilon_1(\diam D)^r\cr
&=\gamma^r(1-r^{r/2}\epsilon_1)
\ge\alpha\gamma^rr^{r/2}
=\alpha(\diam D)^r
\ge\alpha\diam(D\cap E)^r.\cr}$$

\noindent Using 264H, we see also that

\Centerline{$\diam(D\cap E)
\ge\bigl(\Bover{2^r}{\beta_r}\mu(D\cap E)\bigr)^{1/r}
\ge 2\gamma\bigl(\Bover{1-r^{r/2}\epsilon_1}{\beta_r}\bigr)^{1/r}$.}

\noindent Next,

$$\eqalignno{\per(D\cap E)
&\le\nu(\partial D)+\nu(D\cap\partstar E)\cr
\displaycause{475Cf}
&\le 2r\gamma^{r-1}+\epsilon_1(\diam D)^{r-1}
\le\gamma^{r-1}(2r+r^{r/2}\epsilon_1)\cr
&\le(2r+r^{r/2}\epsilon_1)\cdot\Bover1{2^{r-1}}
  \bigl(\Bover{\beta_r}{1-\epsilon_1r^{r/2}}\bigr)^{(r-1)/r}
  \diam(D\cap E)^{r-1}\cr
&\le\Bover1{\alpha}\diam(D\cap E)^{r-1}.\cr}$$

\noindent So $D\cap E\in\Cal C_{\alpha}$.   Also, for every $s>0$, there
is a $D'\in\Cal D$ such that
$D'\subseteq D$ and $x\in\overline{D'}$ and $\diam D'\le s$.
In this case

\Centerline{$\mu(B(x,\diam D')\setminus E)
\le\epsilon_1(\diam D')^r=\epsilon_1r^{r/2}\mu D'\le\Bover12\mu D'$,}

\noindent so

$$\eqalign{\mu(D\cap E\cap B(x,\diam D'))
&\ge\mu D'-\mu(B(x,\diam D')\setminus E)\cr
&\ge\Bover12\mu D'
=\Bover1{2\beta_rr^{r/2}}\mu B(x,\diam D').\cr}$$

\noindent As $\diam D'$ is arbitrarily small, $x\in\clstar(D\cap E)$ and
$\pmb{t}=\{(x,D\cap E)\}\in T_{\alpha}$.   Finally, since
$\diam D<\theta(x)$, $(x,D\cap E)\in\delta$ and $\pmb{t}\in T'$.\ \Qed

\medskip

\quad{\bf (iii)} Let $\Cal H$ be the set of those $H\subseteq\BbbR^r$
such that
$W_{\pmb{t}}\subseteq E\cap H\subseteq W_{\pmb{t}}\cup\bigcup\Cal D_1$
for some $\pmb{t}\in T'$ and finite
$\Cal D_1\subseteq\Cal D_0$.
Then $H\cup H'\in\Cal H$ whenever $H$, $H'\in\Cal H$ are disjoint.
If $\sequencen{D_n}$ is any strictly decreasing sequence in $\Cal D$,
then some $D_n$ belongs to $\Cal H$.   \Prf\
Let $x$ be the unique point of $\bigcap_{n\in\Bbb N}\overline{D_n}$.

\qquad{\bf case 1} If $x\in A$, then there is a finite subset $\Cal D_1$
of $\Cal D_0$ whose union is a neighbourhood of $x$, and therefore
includes $D_n$ for some $n$;  so $\pmb{t}=\emptyset$ and
$\Cal D_1$ witness that $D_n\in\Cal H$.

\qquad{\bf case 2} If $x\in E\setminus A$, then $x\in\intstar E$, so
$h(x)>0$ and there is some $n\in\Bbb N$ such that
$\diam D_n\le h(x)$.   In this case $\pmb{t}=\{(x,D_n\cap E)\}$ belongs
to $T'$, by the choice of $h(x)$, so that $\pmb{t}$ and $\emptyset$
witness that $D_n\in\Cal H$.

\qquad{\bf case 3} Finally, if $x\notin E\cup A$, then
$x\notin\clstar E$ so $x\notin\overline{E}$ and there is some $n$ such
that $D_n\cap E=\emptyset$, in which case $\pmb{t}=\Cal D_1=\emptyset$
witness that $D_n\in\Cal H$.\ \Qed

\medskip

\quad{\bf (iv)} In fact $\BbbR^r\in\Cal H$.   \Prf\Quer\ Otherwise,
because $E\subseteq V$ is bounded, it can be covered by a finite
disjoint family in $\Cal D$, and there must be some
$D_0\in\Cal D\setminus\Cal H$.   Now we can find
$\langle D_n\rangle_{n\ge 1}$ in $\Cal D\setminus\Cal H$ such that
$D_n\subseteq D_{n-1}$ and $\diam D_n=\bover12\diam D_{n-1}$ for every
$n$.   But this contradicts (iii).\Bang\Qed

\medskip

\quad{\bf (v)} We therefore have a $\pmb{t}\in T'$ and a
finite set $\Cal D_1\subseteq\Cal D_0$ such that
$W_{\pmb{t}}\subseteq E\subseteq W_{\pmb{t}}\cup\bigcup\Cal D_1$.
Now we can find $\pmb{t}'\subseteq\pmb{t}$ and
$\Cal D'_1\subseteq\Cal D_1$ such that
$W_{\pmb{t}'}\cap\bigcup\Cal D'_1=\emptyset$ and
$E\subseteq W_{\pmb{t}'}\cup\bigcup\Cal D'_1$.
\Prf\ Express $\pmb{t}$ as $\familyiI{(x_i,D_i\cap E)}$ where
$D_i\in\Cal D$ for each $i$.   Then
$E\subseteq\bigcup_{i\in I}D_i\cup\bigcup\Cal D_1$.   Set
$\Cal D_1'=\{D:D\in\Cal D_1$, $D\not\subseteq D_i$ for every $i\in I\}$,
$J=\{i:i\in I$, $D_i\not\subseteq D$ for every $D\in\Cal D'_1\}$,
$\pmb{t}'=\{(x_i,D_i\cap E):i\in J\}$.\ \Qed

By the choice of $\Cal D_0$, $\bigcup\Cal D'_1\in\Cal R''$;  by the
choice of $\Cal R''$,
$E\setminus W_{\pmb{t}'}=E\cap\bigcup\Cal D'_1$ belongs to $\Cal R'$.
But we know also that $C\cap V\setminus E\in\Cal R'$, that is,
$C\setminus E\in\Cal R'$, because $\Cal R'=\Cal R^{(V)}_{\eta}$.   By
the choice of $\Cal R'$, $C\setminus W_{\pmb{t}'}\in\Cal R$.   And
$\pmb{t}'\in T'$ is a $\delta$-fine member of $T_{\alpha}$.   As $C$,
$\delta$ and $\Cal R$ are
arbitrary, 481G(vii) is satisfied, and the proof is complete.
}%end of proof of 484F

\leader{484G}{The Pfeffer integral (a)} For
$\alpha\in\ooint{0,\alpha^*}$, write $I_{\alpha}$ for the linear
functional defined by setting

\Centerline{$I_{\alpha}(f)
=\lim_{\pmb{t}\to\Cal F(T_{\alpha},\Delta,\frak R)}
  S_{\pmb{t}}(f,\mu)$}

\noindent whenever $f:\BbbR^r\to\Bbb R$ is such that the limit is
defined.   (See 481F for the notation $\Cal F(T_{\alpha},\Delta,\frak R)$.)
Then if $0<\beta\le\alpha<\alpha^*$ and $I_{\beta}(f)$ is defined, so
is $I_{\alpha}(f)$, and the
two are equal.   \prooflet{\Prf\ All we have to observe is that
$\Cal C_{\alpha}\subseteq\Cal C_{\beta}$ so that
$T_{\alpha}\subseteq T_{\beta}$, while
$\Cal F(T_{\alpha},\Delta,\frak R)$ is just
$\{A\cap T_{\alpha}:A\in\Cal F(T_{\beta},\Delta,\frak R)\}$.\ \Qed}

\spheader 484Gb Let $f:\BbbR^r\to\Bbb R$ be a function.   I will say
that it
is {\bf Pfeffer integrable}, with {\bf Pfeffer integral} $\Pfint f$, if

\Centerline{$\biggerPfint f
=\lim_{\alpha\downarrow 0}I_{\alpha}(f)$}

\noindent is defined;  that is to say, if $I_{\alpha}(f)$ is
defined whenever $0<\alpha<\alpha^*$.

\leader{484H}{}\cmmnt{ The first step is to work through the results
of \S482 to see which ideas apply directly to the limit integral
$\Pfint\,$.

\medskip

\noindent}{\bf Proposition} (a) The domain of $\Pfint\,$ is a linear
space of functions, and $\Pfint\,$ is a positive linear functional.

(b) If $f$, $g:\BbbR^r\to\Bbb R$ are such that $|f|\le g$ and
$\Pfint g=0$, then $\Pfint f$ is defined and equal to $0$.

(c) If $f:\BbbR^r\to\Bbb R$ is Pfeffer
integrable, then there is a unique additive functional
$F:\Cal C\to\Bbb R$ such that whenever $\epsilon>0$ and
$0<\alpha<\alpha^*$ there are
$\delta\in\Delta$ and $\Cal R\in\frak R$ such that

\Centerline{$\sum_{(x,C)\in\pmb{t}}|F(C)-f(x)\mu C|\le\epsilon$ for
every $\delta$-fine $\pmb{t}\in T_{\alpha}$,}

\Centerline{$|F(E)|\le\epsilon$ whenever $E\in\Cal C\cap\Cal R$.}

\noindent Moreover, $F(\BbbR^r)=\Pfint f$.

(d) Every Pfeffer integrable function is Lebesgue measurable.

(e) Every Lebesgue integrable function is Pfeffer integrable, with the
same integral.

(f) A non-negative function is Pfeffer integrable iff it is Lebesgue
integrable.

\proof{{\bf (a)-(b)} Immediate from 481C.

\medskip

{\bf (c)} For each $\alpha\in\ooint{0,\alpha^*}$ let $F_{\alpha}$ be the
Saks-Henstock indefinite integral corresponding to the the structure
$(\BbbR^r,T_{\alpha},\Delta,\penalty-100\frak R,\mu)$.
Then all the $F_{\alpha}$
coincide.   \Prf\ Suppose that $0<\beta\le\alpha<\alpha^*$.
Then, for any $\epsilon>0$, there are $\delta\in\Delta$,
$\Cal R\in\frak R$ such that

\Centerline{$\sum_{(x,C)\in\pmb{t}}|F_{\beta}(C)-f(x)\mu C|
\le\epsilon$ for every $\delta$-fine $\pmb{t}\in T_{\beta}$,}

\Centerline{$|F_{\beta}(E)|\le\epsilon$
whenever $E\in\Cal R$.}

\noindent Since $T_{\alpha}\subseteq T_{\beta}$, this
means that

\Centerline{$\sum_{(x,C)\in\pmb{t}}|F_{\beta}(C)-f(x)\mu C|
\le\epsilon$
for every $\delta$-fine $\pmb{t}\in T_{\alpha}$.}

\noindent And this works for any $\epsilon>0$.   By the uniqueness
assertion in 482B, $F_{\beta}$ must be exactly the same as
$F_{\alpha}$.\ \Qed

So we have a single functional $F$;  and 482B also tells us that

\Centerline{$F(\BbbR^r)=I_{\alpha}(f)=\biggerPfint f$}

\noindent for every $\alpha$.

\medskip

{\bf (d)} In fact if there is any $\alpha$ such that
$I_{\alpha}(f)$ is defined, $f$ must be Lebesgue measurable.
\Prf\ We have only to check that the conditions of 482E are satisfied by
$\mu$, $\Cal C_{\alpha}$, $\{(x,C):C\in\Cal C_{\alpha}$, $x\in\clstar C\}$,
$T_{\alpha}$, $\Delta$ and $\frak R$.    (i), (iii) and (v) are built
into the definitions above, and (iv) and (vii) are covered by 484F.
482E(ii) is true because $C\setminus\clstar C$ is negligible
for every $C\in\Cal C$ (475Cg).

As for 482E(vi), this is true because if
$\mu E<\infty$ and
$\epsilon>0$, there are $n\in\Bbb N$ and $\eta\in\Eta$ such that
$\mu(E\setminus B(\tbf{0},n))\le\bover12\epsilon$ and
$\sum_{i=0}^{\infty}\eta(i)\le\bover12\epsilon$, so that

\Centerline{$\mu(E\cap R)
\le\mu(E\setminus B(\tbf{0},n))+\mu(R\cap B(\tbf{0},n))
\le\epsilon$}

\noindent for every $R\in\Cal R^{(B(\tbf{0},n))}_{\eta}$.\ \Qed

\medskip

{\bf (e)} This time, we have to check that the conditions of 482F are
satisfied by $T_{\alpha}$, $\Delta$ and $\frak R$ whenever
$0<\alpha<\alpha^*$.   \Prf\ Of course $\mu$
is inner regular with respect to the closed sets and outer regular with
respect to the open sets (134F).   Condition 482F(v) just repeats
482E(v), verified in (d) above.\ \Qed

\medskip

{\bf (f)} If $f\ge 0$ is integrable in the ordinary sense, then it is
Pfeffer integrable, by (e).   If it is Pfeffer integrable, then it is
measurable;  but also $\int g\,d\mu=\Pfint g\le\Pfint f$ for every
simple function $g\le f$, so $f$ is integrable (213B).
}%end of proof of 484H

\leader{484I}{Definition} If $f:\BbbR^r\to\Bbb R$ is Pfeffer
integrable, I will call the function $F:\Cal C\to\Bbb R$ defined in
484Hc the {\bf Saks-Henstock indefinite integral} of $f$.

\leader{484J}{}\cmmnt{ In fact 484Hc characterizes the Pfeffer
integral, just as the Saks-Henstock lemma can be used to define general
gauge integrals based on tagged-partition structures allowing
subdivisions.

\medskip

\noindent}{\bf Proposition} Suppose that $f:\BbbR^r\to\Bbb R$ and
$F:\Cal C\to\Bbb R$ are such that

\inset{(i) $F$ is additive,

(ii) whenever $0<\alpha<\alpha^*$ and $\epsilon>0$ there is a
$\delta\in\Delta$ such that
$\sum_{(x,C)\in\pmb{t}}|F(C)-f(x)\mu C|\le\epsilon$ for every
$\delta$-fine $\pmb{t}\in T_{\alpha}$,

(iii) for every $\epsilon>0$ there is an $\Cal R\in\frak R$ such that
$|F(E)|\le\epsilon$ for every $E\in\Cal C\cap\Cal R$.}

\noindent Then $f$ is Pfeffer integrable and $F$ is the Saks-Henstock
indefinite integral of $f$.

\proof{ By 482D, the gauge integral $I_{\alpha}(f)$ is defined and
equal to $F(\BbbR^r)$ for every $\alpha\in\ooint{0,\alpha^*}$.   So $f$
is Pfeffer integrable.   Now 484Hc tells us that $F$ must be its
Saks-Henstock indefinite integral.
}%end of proof of 484J

\leader{484K}{Lemma} Suppose that $\alpha>0$ and $0<\alpha'
<\alpha\min(\bover12,2^{r-1}(\Bover{\alpha}{2\beta_r})^{(r-1)/r})$.
If $E\in\Cal C$ is such that $E\subseteq\clstar E$, then there is a
$\delta\in\Delta$ such that $\{(x,C\cap E)\}\in T_{\alpha'}$ whenever
$(x,C)\in\delta$, $x\in E$ and $\{(x,C)\}\in T_{\alpha}$.

\proof{ Take $\epsilon>0$ such that
$\beta_r\epsilon\le\bover12\alpha$ and

\Centerline{$\Bover1{\alpha}+2^{r-1}\epsilon
\le\Bover{2^{r-1}}{\alpha'}
  \bigl(\Bover{\alpha}{2\beta_r}\bigr)^{(r-1)/r}$.}

\noindent Set

\Centerline{$A=\partstar E
   \cup\{x:\lim_{\zeta\downarrow 0}
     \sup_{x\in G,0<\diam G\le\zeta}
     \Bover{\nu^*(G\cap\partstar E)}{(\diam G)^{r-1}}>0\}$,}

\noindent so that $A$ is a thin set, as in (b-i) of the proof of 484F.
(Of course $\partstar E$ is thin because
$\nu(\partstar E\cap B(\tbf{0},n))$ is finite for every $n\in\Bbb N$.)

For $x\in E\setminus A$, we have $x\in\intstar E$ (because
$E\subseteq\clstar E$), so there is a $\theta(x)>0$ such that

\Centerline{$\mu(B(x,\zeta)\setminus E)\le\epsilon\mu B(x,\zeta)$,
\quad$\nu(\partstar E\cap B(x,\zeta))\le\epsilon(2\zeta)^{r-1}$}

\noindent whenever $0<\zeta\le 2\theta(x)$.   If we set $\theta(x)=0$
for $x\in E\cap A$ and $\theta(x)=1$ for $x\in\BbbR^r\setminus E$, then
$\theta\in\Theta$ and $\delta_{\theta}\in\Delta$.

Now suppose that $x\in E$, $(x,C)\in\delta_{\theta}$ and
$\{(x,C)\}\in T_{\alpha}$, that is, that $C\in\Cal C_{\alpha}$ and
$x\in(E\cap\clstar C)\setminus A$ and $\|x-y\|<\theta(x)$ for every
$y\in C$.   Set $\gamma=\diam C\le 2\theta(x)$.   Then

\Centerline{$\mu(C\setminus E)\le\mu(B(x,\gamma)\setminus E)
\le\epsilon\mu B(x,\gamma)=\beta_r\epsilon\gamma^r$,}

\noindent so

$$\eqalign{\mu(C\cap E)
&\ge\mu C-\beta_r\epsilon\gamma^r
\ge(\alpha-\beta_r\epsilon)\gamma^r\cr
&\ge\Bover12\alpha\gamma^r
\ge\alpha'\gamma^r
\ge\alpha'\diam(C\cap E)^r.\cr}$$

\noindent Next,

$$\eqalign{\per(C\cap E)
&=\nu(\partstar(C\cap E))
\le\nu(B(x,\gamma)\cap(\partstar C\cup\partstar E))\cr
&\le\nu(\partstar C)+\nu(B(x,\gamma)\cap\partstar E)
\le\Bover1{\alpha}\gamma^{r-1}+\epsilon(2\gamma)^{r-1}
=(\Bover1{\alpha}+2^{r-1}\epsilon)\gamma^{r-1}.\cr}$$

\noindent Moreover,

\Centerline{$2^{-r}\beta_r\diam(C\cap E)^r
\ge\mu(C\cap E)
\ge\Bover12\alpha\gamma^r$}

\noindent (264H), so
$\diam(C\cap E)\ge 2(\Bover{\alpha}{2\beta_r})^{1/r}\gamma$
and

$$\eqalign{\per(C\cap E)
&\le(\Bover1{\alpha}+2^{r-1}\epsilon)\cdot\Bover1{2^{r-1}}
  \bigl(\Bover{2\beta_r}{\alpha}\bigr)^{(r-1)/r}\diam(C\cap E)^{r-1}
\le\Bover1{\alpha'}\diam(C\cap E)^{r-1}.\cr}$$

\noindent Putting these together, we see that $C\in\Cal C_{\alpha'}$.

Finally, because $x\in\clstar C\cap\intstar E\subseteq\clstar(C\cap E)$
(475Ce), $\{(x,C\cap E)\}\in T_{\alpha'}$.
}%end of proof of 484K

\leader{484L}{Proposition} Suppose that $f:\BbbR^r\to\BbbR$ is Pfeffer
integrable, and that $F:\Cal C\to\Bbb R$ is its Saks-Henstock indefinite
integral.   Then
$\Pfint f\times\chi E$ is defined and equal to $F(E)$ for every
$E\in\Cal C$.

\proof{{\bf (a)} To begin with (down to the end of (d) below),
suppose that $E\in\Cal C$ is such that
$\intstar E\subseteq E\subseteq\clstar E$.   For $C\in\Cal C$ set
$F_1(C)=F(C\cap E)$.   I seek to show that $F_1$ satisfies the
conditions of 484J.

Of course $F_1:\Cal C\to\Bbb R$ is additive.   If $\epsilon>0$, there is
an $\Cal R\in\frak R$ such that $|F(G)|\le\epsilon$ whenever
$G\in\Cal R$, by 484H;  now there is an $\Cal R'\in\frak R$ such that
$G\cap E\in\Cal R$ for every $G\in\Cal R'$ (484E(a-ii)), so that
$|F_1(G)|\le\epsilon$ for every $G\in\Cal C\cap\Cal R'$.   Thus $F_1$
satisfies (iii) of 484J.

\medskip

{\bf (b)} Take $\alpha\in\ooint{0,\alpha^*}$ and $\epsilon>0$.   Take
$\alpha'$ such that $0<\alpha'
<\alpha\min(\bover12,2^{r-1}(\Bover{\alpha}{2\beta_r})^{(r-1)/r})$.
Applying 484K to $E$ and its complement, and appealing to the definition
of $F$, we see that there is a $\delta\in\Delta$ such that

\inset{($\alpha$) $\{(x,C\cap E)\}\in T_{\alpha'}$ whenever
$(x,C)\in\delta$, $x\in E$ and $\{(x,C)\}\in T_{\alpha}$,

($\beta$) $\{(x,C\setminus E)\}\in T_{\alpha'}$ whenever
$(x,C)\in\delta$,
$x\in\BbbR^r\setminus E$ and $\{(x,C)\}\in T_{\alpha}$,

($\gamma$) $\sum_{(x,C)\in\pmb{t}}|F(C)-f(x)\mu C|\le\epsilon$ for every
$\delta$-fine $\pmb{t}\in T_{\alpha'}$.}

\noindent (For ($\alpha$), we need to know that $E\subseteq\clstar E$
and for ($\beta$) we need $\intstar E\subseteq E$.)   Next, choose for
each $n\in\Bbb N$ closed sets
$H_n\subseteq E$, $H'_n\subseteq\BbbR^r\setminus E$ such that
$\mu(E\setminus H_n)\le 2^{-n}\epsilon$ and
$\mu((\BbbR^r\setminus E)\setminus H'_n)\le 2^{-n}\epsilon$.   Define
$\theta:\BbbR^r\to\ooint{0,\infty}$ by setting

$$\eqalign{\theta(x)
&=\min(1,\Bover12\rho(x,H'_n))\text{ if }x\in E
  \text{ and }n\le|f(x)|<n+1,\cr
&=\min(1,\Bover12\rho(x,H_n))\text{ if }x\in\BbbR^r\setminus E
  \text{ and }n\le|f(x)|<n+1,\cr}$$

\noindent writing $\rho(x,A)=\inf_{y\in A}\|x-y\|$ if
$A\subseteq\BbbR^r$ is non-empty, $\infty$ if $A=\emptyset$.   Then
$\delta_{\theta}\in\Delta$.   Let $\Cal R'\in\frak R$ be such that
$A\cap E\in\Cal R$ whenever $A\in\Cal R'$.

\medskip

{\bf (c)} Write $f_E$ for $f\times\chi E$.   Then
$\sum_{(x,C)\in\pmb{t}}|F_1(C)-f_E(x)\mu(C)|\le 11\epsilon$ whenever
$\pmb{t}\in T_{\alpha}$ is $(\delta\cap\delta_{\theta})$-fine.   \Prf\ Set

\Centerline{$\pmb{t}'=\{(x,C\cap E):(x,C)\in\pmb{t},\,x\in E\}$.}

\noindent By clause ($\alpha$) of the choice of $\delta$,
$\pmb{t}'\in T_{\alpha'}$, and of course it is $\delta$-fine.   So

\Centerline{$\sum_{(x,C)\in\pmb{t},x\in E}|F(C\cap E)-f(x)\mu(C\cap E)|
\le\epsilon$}

\noindent by clause ($\gamma$) of the choice of $\delta$.   Next,

$$\eqalignno{\sum_{(x,C)\in\pmb{t},x\in E}|f(x)\mu(C\setminus E)|
&=\sum_{n=0}^{\infty}\sum_{\Atop{(x,C)\in\pmb{t},x\in E}
  {n\le|f(x)|<n+1}}
   |f(x)|\mu(C\setminus E)\cr
&\le\sum_{n=0}^{\infty}(n+1)\mu((\BbbR^r\setminus E)\setminus H'_n)\cr
\displaycause{because $\diam C\le\theta(x)$, so $C\cap H'_n=\emptyset$
whenever $(x,C)\in\pmb{t}$, $x\in E$ and $n\le|f(x)|<n+1$}
&\le\sum_{n=0}^{\infty}2^{-n}(n+1)\epsilon
=4\epsilon,\cr}$$

\noindent and

\Centerline{$\sum_{(x,C)\in\pmb{t},x\in E}|F(C\cap E)-f(x)\mu C|
\le 5\epsilon$.}

\noindent Similarly,

\Centerline{$\sum_{(x,C)\in\pmb{t},x\notin E}
  |F(C\setminus E)-f(x)\mu C)|
\le 5\epsilon$.}

\noindent But as

\Centerline{$\sum_{(x,C)\in\pmb{t},x\notin E}|F(C)-f(x)\mu C|
\le\epsilon$}

\noindent (because surely $\pmb{t}$ itself belongs to $T_{\alpha'}$),
we have

\Centerline{$\sum_{(x,C)\in\pmb{t},x\notin E}|F(C\cap E)|
\le 6\epsilon$.}

Putting these together,

$$\eqalign{\sum_{(x,C)\in\pmb{t}}|F_1(C)-f_E(x)\mu C|
&=\sum_{\Atop{(x,C)\in\pmb{t}}{x\in E}}
      |F(C\cap E)-f(x)\mu C|
   +\sum_{\Atop{(x,C)\in\pmb{t}}{x\notin E}}
      |F(C\cap E)|\cr
&\le 5\epsilon+6\epsilon
=11\epsilon.  \text{ \Qed}\cr}$$

\medskip

{\bf (d)} As $\alpha$ and $\epsilon$ are arbitrary, condition (ii) of
484J is satisfied by $F_1$ and $f_E$, so
$\Pfint f_E=F_1(\BbbR^r)=F(E)$.

\medskip

{\bf (e)} This completes the proof when
$\intstar E\subseteq E\subseteq\clstar E$.   For a general set
$E\in\Cal C$, set $E_1=(E\cup\intstar E)\cap\clstar E$.   Then
$E\symmdiff E_1$ is negligible, so

\Centerline{$\intstar E_1=\intstar E\subseteq E_1
\subseteq\clstar E=\clstar E_1$.}

\noindent Also $\Pfint f\times\chi(E\setminus E_1)
=\int f\times\chi(E\setminus E_1)d\mu$,
$\Pfint f\times\chi(E_1\setminus E)
=\int f\times\chi(E\setminus E_1)d\mu$ are both zero, and

\Centerline{$\biggerPfint f\times\chi E=\Pfint f\times\chi E_1
=F(E_1)=F(E)$.}

\noindent (To see that $F(E_1)=F(E)$, note that $E\setminus E_1$ and
$E_1\setminus E$, being negligible sets, have empty essential boundary
and zero perimeter, so belong to every member of $\frak R$, by
484E(b-ii), or otherwise.)
}%end of proof of 484L

\leader{484M}{Lemma} Let $G$, $H\in\Cal C$ be disjoint and
$\phi:\BbbR^r\to\BbbR^r$ a continuous function.   If {\it either}
$G\cup H$ is bounded {\it or} $\phi$ has compact support,

\Centerline{$\int_{\partstar(G\cup H)}\varinnerprod{\phi}
  {\psi_{G\cup H}}\,d\nu
=\int_{\partstar G}\varinnerprod{\phi}
  {\psi_G}\,d\nu
+\int_{\partstar H}\varinnerprod{\phi}
  {\psi_G}\,d\nu$,}

\noindent where $\psi_G$, $\psi_H$ and $\psi_{G\cup H}$ are the
canonical outward-normal functions\cmmnt{ (474G)}.

\proof{{\bf (a)} Suppose first that $\phi$ is a Lipschitz function with
compact support.   Then 475N tells us that

$$\eqalign{\int_{\partstar(G\cup H)}
  \varinnerprod{\phi}{\psi_{G\cup H}}\,d\nu
&=\int_{G\cup H}\diverg\phi\,d\mu
=\int_G\diverg\phi\,d\mu+\int_H\diverg\phi\,d\mu\cr
&=\int_{\partstar G}\varinnerprod{\phi}
  {\psi_G}\,d\nu
+\int_{\partstar H}\varinnerprod{\phi}
  {\psi_G}\,d\nu.\cr}$$

\noindent (Recall from 474R that we can identify canonical outward-normal
functions with Federer exterior normals, as in the statement of 475N.)

\medskip

{\bf (b)} Now suppose that $\phi$ is a continuous function with compact
support.   Let $\sequencen{\tilde h_n}$ be the smoothing sequence of
473E.   Then all the functions $\phi*\tilde h_n$ are Lipschitz and
$\sequencen{\phi*\tilde h_n}$ converges uniformly to $\phi$ (473Df,
473Ed).   So

$$\eqalign{\int_{\partstar(G\cup H)}\varinnerprod{\phi}
  {\psi_{G\cup H}}\,d\nu
&=\lim_{n\to\infty}\int_{\partstar(G\cup H)}
  \varinnerprod{(\phi*\tilde h_n)}{\psi_{G\cup H}}\,d\nu\cr
&=\lim_{n\to\infty}\int_{\partstar G}
  \varinnerprod{(\phi*\tilde h_n)}{\psi_G}\,d\nu
+\lim_{n\to\infty}\int_{\partstar H}
  \varinnerprod{(\phi*\tilde h_n)}{\psi_H}\,d\nu\cr
&=\int_{\partstar G}\varinnerprod{\phi}
  {\psi_G}\,d\nu
+\int_{\partstar H}\varinnerprod{\phi}
  {\psi_G}\,d\nu.\cr}$$

\medskip

{\bf (c)} If, on the other hand, $\phi$ is continuous and $G$ and $H$
are bounded, then we can find a continuous function $\tilde\phi$ with
compact support agreeing with $\phi$ on $\overline{G\cup H}$ (4A2G(e-i),
or otherwise);  applying (b) to $\tilde\phi$, we get the required result
for $\phi$.
}%end of proof of 484M

\leader{484N}{Pfeffer's Divergence Theorem}  Let
$E\subseteq\BbbR^r$ be a set with locally finite perimeter, and
$\phi:\BbbR^r\to\BbbR^r$ a continuous function with compact support such
that $\{x:x\in\BbbR^r,\,\phi$ is not differentiable at $x\}$ is thin.
Let $v_x$ be the Federer exterior normal to $E$ at any point $x$ where
the normal exists.   Then $\Pfint\diverg\phi\times\chi E$ is defined and
equal to $\int_{\partstar E}\varinnerprod{\phi(x)}{v_x}\nu(dx)$.

\proof{{\bf (a)} Let $n$ be such that $\phi(x)=\tbf{0}$ for
$\|x\|\ge n$.   For $C\in\Cal C$, set
$F(C)=\int_{\partstar C}\varinnerprod{\phi}{\psi_C}\nu(dx)$, where
$\psi_C$ is the canonical outward-normal function;  recall that
$\psi_E(x)=v_x$ for $\nu$-almost every $x\in\partstar E$ (474R, 475D).
By 484M, $F$ is additive.

\medskip

{\bf (b)} If $0<\alpha<\alpha^*$ and $\epsilon>0$ and $x\in\BbbR^r$ is
such that $\phi$ is differentiable at $x$, there is a $\gamma>0$ such
that $|F(C)-\diverg\phi(x)\mu C|\le\epsilon\mu C$ whenever
$C\in\Cal C_{\alpha}$, $x\in\overline{C}$ and $\diam C\le\gamma$.
\Prf\ Let $T$ be the derivative of $\phi$ at $x$.   Let $\gamma>0$ be
such that $\|\phi(y)-\phi(x)-T(y-x)\|\le\alpha^2\epsilon\|y-x\|$
whenever $\|y-x\|\le\gamma$.   Let $\tilde\phi:\BbbR^r\to\BbbR^r$ be a
Lipschitz function with compact support such that
$\tilde\phi(y)=\phi(x)+(y-x)$ whenever $\|y-x\|\le\gamma$ (473Cf).
If $C\in\Cal C_{\alpha}$ has diameter at most $\gamma$ and
$x\in\overline{C}$, then

$$\eqalignno{|F(C)-\diverg\phi(x)\mu C|
&=|\int_{\partstar C}\varinnerprod{\phi}{\psi_C}\nu(dx)
  -\int_C\diverg\tilde\phi\,d\mu|\cr
\displaycause{because $T$ is the derivative of $\tilde\phi$ everywhere
on $B(x,\gamma)$, so $\diverg\tilde\phi(y)=\diverg\phi(x)$ for every
$y\in C$}
&=|\int_{\partstar C}\varinnerprod{(\phi-\tilde\phi)}{\psi_C}\nu(dx)|\cr
\displaycause{applying the Divergence Theorem 475N to $\tilde\phi$}
&\le\nu(\partstar C)\sup_{y\in C}\|\phi(y)-\tilde\phi(y)\|\cr
&\le\alpha^2\epsilon\diam C\per C
\le\alpha\epsilon(\diam C)^r
\le\epsilon\mu C\cr}$$

\noindent because $C\in\Cal C_{\alpha}$.\ \Qed

\medskip

{\bf (c)} If $\epsilon>0$ and $\alpha\in\ooint{0,\alpha^*}$, there is a
$\delta\in\Delta$ such that
$\sum_{(x,C)\in\pmb{t}}|F(C)-\diverg\phi(x)\mu C|\le\epsilon$ whenever
$\pmb{t}\in T_{\alpha}$ is $\delta$-fine.
\Prf\ Let $\zeta>0$ be such that $\zeta\mu B(\tbf{0},n+2)\le\epsilon$.
Set $A=\{x:x\in\BbbR^r,\,\phi$ is not differentiable at $x\}$, and for
$x\in\BbbR^r\setminus A$ let $\theta(x)\in\ocint{0,\bover12}$ be such
that $|F(C)-\diverg\phi(x)\mu C|\le\zeta\mu C$ whenever
$C\in\Cal C_{\alpha}$, $x\in\overline{C}$ and $\diam C\le\theta(x)$;
for $x\in A$ set $\theta(x)=0$.
Now suppose that $\pmb{t}\in T_{\alpha}$ is $\delta_{\theta}$-fine.
Then

$$\eqalignno{\sum_{(x,C)\in\pmb{t}}|F(C)-\diverg\phi(x)\mu C|
&=\sum_{\Atop{(x,C)\in\pmb{t}}{x\in B(\tbf{0},n+1)}}
|F(C)-\diverg\phi(x)\mu C|\cr
\displaycause{because $\diam C\le 1$ whenever $(x,C)\in\pmb{t}$, so
if $\|x\|>n+1$ then $F(C)=\diverg\phi(x)=0$}
&\le\sum_{\Atop{(x,C)\in\pmb{t}}{x\in B(\tbf{0},n+1)}}\zeta\mu C
\le\zeta\mu B(\tbf{0},n+2)
\le\epsilon. \text{ \Qed}\cr}$$

\medskip

{\bf (d)} Because $\phi$ is a continuous function with compact support,
it is uniformly continuous (apply 4A2Jf to each of the coordinates of
$\phi$).   For $\zeta>0$, let $\gamma(\zeta)>0$ be such that
$\|\phi(x)-\phi(y)\|\le\zeta$ whenever $\|x-y\|\le\gamma(\zeta)$.

If $C\in\Cal C$ and $\per C\le 1$ and $\mu C\le\zeta\gamma(\zeta)$,
where $\zeta>0$, then
$|F(C)|\le r\zeta(2\|\phi\|_{\infty}+\bover12)$, writing
$\|\phi\|_{\infty}$ for $\sup_{x\in\BbbR^r}\|\phi(x)\|$.
\Prf\ For $1\le i\le r$, let $\phi_i:\BbbR^r\to\Bbb R$ be the $i$th
component of $\phi$, and $v_i$ the $i$th unit vector
$(0,\ldots,1,\ldots,0)$;  write

\Centerline{$\alpha_i
=\int_{\partstar C}\phi_i(x)\innerprod{v_i}{\psi_E(x)}\nu(dx)$,}

\noindent so that $F(C)=\sum_{i=1}^r\alpha_i$.   I start by examining
$\alpha_r$.   By 475O, we have sequences $\sequencen{H_n}$,
$\sequencen{g_n}$ and $\sequencen{g'_n}$ such that

\inset{(i) for each $n\in\Bbb N$, $H_n$ is a Lebesgue measurable subset
of $\BbbR^{r-1}$, and $g_n$, $g'_n:H_n\to[-\infty,\infty]$ are Lebesgue
measurable functions such that $g_n(u)<g'_n(u)$ for every $u\in H_n$;

(ii) if $m$, $n\in\Bbb N$ then $g_m(u)\ne g'_n(u)$ for every
$u\in H_m\cap H_n$;

(iii) $\sum_{n=0}^{\infty}\int_{H_n}g'_n-g_nd\mu_{r-1}=\mu C$;

(iv)

\Centerline{$\alpha_r
=\sum_{n=0}^{\infty}\int_{H_n}\phi_r(u,g'_n(u))-\phi_r(u,g_n(u))
  \mu_{r-1}(du)$,}

\noindent where we interpret $\phi_r(u,\infty)$ and $\phi_r(u,-\infty)$
as $0$ if necessary;

(v) for $\mu_{r-1}$-almost every $u\in\BbbR^{r-1}$,

$$\eqalign{\{t:(u,t)\in\partstar C\}
&=\{g_n(u):n\in\Bbb N,\,u\in H_n,\,g_n(u)\ne-\infty\}\cr
&\hskip8em
  \cup\{g'_n(u):n\in\Bbb N,\,u\in H_n,\,g'_n(u)\ne\infty\}.\cr}$$
}%end of inset

\noindent From (iii) we see that $g'_n$ and $g_n$ are both finite almost
everywhere on $H_n$, for every $n$.   Consequently, by (v) and 475H,

\Centerline{$2\sum_{n=0}^{\infty}\mu_{r-1}H_n
=\int\#(\{t:(u,t)\in\partstar C\})\mu_{r-1}(du)
\le\nu(\partstar C)\le 1$.}

\noindent For each $n$, set

\Centerline{$H'_n=\{u:u\in H_n,\,g'_n(u)-g_n(u)>\gamma(\zeta)\}$.}

\noindent Then $\gamma(\zeta)\sum_{n=0}^{\infty}\mu H'_n\le\mu C$ so
$\sum_{n=0}^{\infty}\mu H'_n\le\zeta$ and

$$|\sum_{n=0}^{\infty}\int_{H'_n}
  \phi_r(u,g'_n(u))-\phi_r(u,g_n(u))\mu_{r-1}(du)|
\le 2\|\phi\|_{\infty}\sum_{n=0}^{\infty}\mu H'_n
\le 2\zeta\|\phi\|_{\infty}.$$

\noindent On the other hand, for $n\in\Bbb N$ and
$u\in H_n\setminus H'_n$,
$|\phi_r(u,g'_n(u))-\phi_r(u,g_n(u))|\le\zeta$, so

$$|\sum_{n=0}^{\infty}\int_{H_n\setminus H'_n}
  \phi_r(u,g'_n(u))-\phi_r(u,g_n(u))\mu_{r-1}(du)|
\le\sum_{n=0}^{\infty}\zeta\mu_{r-1}(H_n\setminus H'_n)
\le\Bover12\zeta.$$

\noindent Putting these together,

\Centerline{$\alpha_r\le 2\zeta\|\phi\|_{\infty}+\Bover12\zeta$.}

\noindent But of course the same arguments apply to all the $\alpha_i$,
so

\Centerline{$|F(C)|\le\sum_{i=1}^r|\alpha_i|
\le r\zeta(2\|\phi\|_{\infty}+\Bover12)$,}

\noindent as claimed.\ \Qed

\medskip

{\bf (e)} If $\epsilon>0$ there is an $\Cal R\in\frak R$ such that
$|F(E)|\le\epsilon$ for every $E\in\Cal C\cap\Cal R$.   \Prf\ Let
$\sequence{i}{\epsilon_i}$ be a sequence of strictly positive real
numbers such that
$r(2\|\phi\|_{\infty}+\bover12)\sum_{i=0}^{\infty}\epsilon_i
\le\epsilon$.
For each $i\in\Bbb N$, set $\eta(i)=\epsilon_i\gamma(\epsilon_i)>0$.
Set $V=B(\tbf{0},n+1)$, and take any
$E\in\Cal C\cap\Cal R^{(V)}_{\eta}$.   Then $F(E\setminus V)=0$, so
$F(E)=F(E\cap V)$, while $E\cap V\in\Cal R_{\eta}$.   Express $E\cap V$
as $\bigcup_{i\le n}E_i$, where $\langle E_i\rangle_{i\le n}$ is
disjoint, $\per E_i\le 1$ and $\mu E_i\le\eta(i)$ for each $i$.
Then $|F(E_i)|\le r\epsilon_i(2\|\phi\|_{\infty}+\bover12)$ for each
$i$, by (d), so

\Centerline{$|F(E)|=|F(E\cap V)|\le\sum_{i=0}^n|F(E_i)|\le\epsilon$,}

\noindent as required.\ \Qed

\medskip

{\bf (f)} By 484J, $\diverg\phi$ is Pfeffer integrable.   Moreover, by
the uniqueness assertion in 484Hc, its Saks-Henstock indefinite integral
is just the function $F$ here.   By 484L,
$F(E)=\Pfint\diverg\phi\times\chi E$ for every $E\in\Cal C$, as required.
}%end of proof of 484N

\leader{484O}{Differentiating the indefinite integral:  Theorem} Let
$f:\BbbR^r\to\Bbb R$ be a Pfeffer integrable function, and $F$ its
Saks-Henstock indefinite integral.   Then whenever $0<\alpha<\alpha^*$,

$$\eqalign{f(x)
&=\lim_{\zeta\downarrow 0}\sup\{\Bover{F(C)}{\mu C}:
  C\in\Cal C_{\alpha},\,x\in\overline{C},\,0<\diam C\le\zeta\}\cr
&=\lim_{\zeta\downarrow 0}\inf\{\Bover{F(C)}{\mu C}:
  C\in\Cal C_{\alpha},\,x\in\overline{C},\,0<\diam C\le\zeta\}\cr}$$

\noindent for $\mu$-almost every $x\in\BbbR^r$.

\proof{{\bf (a)} It will be useful to know the following:  if
$C\in\Cal C_{\alpha}$, $\diam C>0$, $x\in\overline{C}$ and $\epsilon>0$,
then for any sufficiently small $\zeta>0$,
$C\cup B(x,\zeta)\in\Cal C_{\alpha/2}$ and
$|F(C\cup B(x,\zeta))-F(C)|\le\epsilon$.   \Prf\ Let $\Cal R\in\frak R$
be such that $|F(R)|\le\epsilon$ whenever $R\in\Cal C\cap\Cal R$, let
$\Cal R'\in\frak R$ be such that $(\BbbR^r\setminus C)\cap R\in\Cal R$
whenever $R\in\Cal R'$ (484E(a-ii)), and let $\eta\in\Eta$ be such that
$\Cal R_{\eta}\subseteq\Cal R'$ (484E(a-i)).
Then for all sufficiently small $\zeta>0$, we shall have
$\per B(x,\zeta)\le 1$ and $\mu B(x,\zeta)\le\eta(0)$, so that
$B(x,\zeta)\in\Cal R_{\eta}$, $B(x,\zeta)\setminus C\in\Cal R$ and

\Centerline{$|F(C\cup B(x,\zeta))-F(C)|
=|F(B(x,\zeta)\setminus C)|\le\epsilon$.}

\noindent Next, for all sufficiently small $\zeta>0$,

$$\eqalignno{\mu(C\cup B(x,\zeta))
&\ge\mu C
\ge\alpha(\diam C)^r\cr
&\ge\Bover{\alpha}2(\zeta+\diam C)^r
\ge\Bover{\alpha}2\diam(C\cup B(x,\zeta))^r\cr
\noalign{\noindent (because $x\in\overline{C}$) and}
\per(C\cup B(x,\zeta))
&\le\per C+\per B(x,\zeta)
\le\Bover1{\alpha}(\diam C)^{r-1}+\per B(x,\zeta)\cr
&\le\Bover2{\alpha}(\diam C)^{r-1}
\le\Bover2{\alpha}\diam(C\cup B(x,\zeta))^{r-1},\cr}$$

\noindent so that $C\cup B(x,\zeta)\in\Cal C_{\alpha/2}$.\ \Qed

\medskip

{\bf (b)} For $x\in\Bbb R$, set

\Centerline{$g(x)=\lim_{\zeta\downarrow 0}
  \sup\{\Bover{F(C)}{\mu C}:C\in\Cal C_{\alpha},\,
     x\in\overline{C},\,\diam C\le\zeta\}$.}

\noindent \Quer\ Suppose, if possible, that there are rational numbers
$q<q'$ and $n\in\Bbb N$ such that
$A=\{x:\|x\|\le n,\,f(x)\le q<q'<g(x)\}$ is not $\mu$-negligible.
Set

\Centerline{$\epsilon=\Bover{(q'-q)\alpha}{4\beta_r}\mu^*A
>0$.}

\noindent Let $\theta\in\Theta$ be such that

\Centerline{$\sum_{(x,C)\in\pmb{t}}|F(C)-f(x)\mu C|
\le\epsilon$}

\noindent for every $\delta_{\theta}$-fine $\pmb{t}\in T_{\alpha/2}$.
Let $\Cal I$ be the family of all balls $B(x,\zeta)$ where $x\in A$,
$0<\zeta\le\theta(x)$ and there is a $C\in\Cal C_{\alpha}$ such that
$x\in\overline{C}$,
$\diam C=\zeta$ and $\Bover{F(C)}{\mu C}>q'$.   Then every member of
$A_1=A\setminus\theta^{-1}[\{0\}]$ is the centre of arbitrarily small
members of $\Cal I$, so by 472C there is a countable disjoint family
$\Cal J_0\subseteq\Cal I$ such that

\Centerline{$\mu(\bigcup\Cal J_0)>\Bover12\mu^*A_1
=\Bover12\mu^*A$.}

\noindent There is therefore a finite family $\Cal J_1\subseteq\Cal J_0$
such that $\mu(\bigcup\Cal J_1)>\Bover12\mu^*A$;  enumerate
$\Cal J_1$ as $\langle B(x_i,\zeta_i)\rangle_{i\le n}$ where, for each
$i\le n$, $x_i\in A$, $0<\zeta_i\le\theta_i(x)$ and there is a
$C_i\in\Cal C_{\alpha}$ such that $x_i\in\overline{C_i}$,
$\diam C_i=\zeta_i$ and $F(C_i)>q'\mu C_i$.   By (a), we can enlarge
$C_i$ by adding a sufficiently small ball around $x_i$ to form a
$C'_i\in\Cal C_{\alpha/2}$ such that $x_i\in\interior C'_i$,
$C'_i\subseteq B(x_i,\zeta_i)$ and $F(C'_i)\ge q'\mu C'_i$.

Consider $\pmb{t}=\{(x_i,C'_i):i\le n\}$.   Then, because the balls
$B(x_i,\zeta_i)$ are disjoint, and
$x_i\in\interior C'_i\subseteq\clstar C'_i$ for every $i$, $\pmb{t}$ is
a $\delta_{\theta}$-fine member of $T_{\alpha/2}$.   So
$\sum_{i=0}^nF(C'_i)\le\epsilon+\sum_{i=0}^nf(x_i)\mu C'_i$.   But as
$F(C'_i)\ge q'\mu C'_i$ and $f(x_i)\le q$ for every $i$, this means that
$(q'-q)\sum_{i=0}^n\mu C'_i\le\epsilon$.

But now remember that $\diam C'_i\ge\diam C_i=\zeta_i$ and that
$C'_i\in\Cal C_{\alpha/2}$ for each $i$.   This means that

\Centerline{$\mu C'_i
\ge\Bover{\alpha}2\zeta_i^r
\ge\Bover{\alpha}{2\beta_r}\mu B(x_i,\zeta_i)$}

\noindent for each $i$, and

$$\eqalign{\epsilon
&\ge(q'-q)\sum_{i=0}^n\mu C'_i
\ge\Bover{(q'-q)\alpha}{2\beta_r}\sum_{i=0}^n\mu B(x_i,\zeta_i)\cr
&>\Bover{(q'-q)\alpha}{4\beta_r}\mu^*A
=\epsilon\cr}$$

\noindent which is absurd.\ \Bang

\medskip

{\bf (c)} Since $q$, $q'$ and $n$ are arbitrary, this means that
$g\leae f$.   Similarly (or applying (b) to $-f$ and $-F$)

\Centerline{$f(x)
\le\lim_{\zeta\downarrow 0}\inf\{\Bover{F(C)}{\mu C}:
  C\in\Cal C_{\alpha},\,x\in\overline{C},\,0<\diam C\le\zeta\}$}

\noindent for almost all $x$, as required.
}%end of proof of 484O

\vleader{72pt}{484P}{Lemma} Let $\phi:\BbbR^r\to\BbbR^r$ be an injective
Lipschitz function, and $H$ the set of points at which it is
differentiable;  for $x\in H$, write $T(x)$ for the derivative of $\phi$
at $x$ and $J(x)$ for $|\det T(x)|$.
Then, for $\mu$-almost every $x\in\BbbR^r$,

$$\eqalign{J(x)
&=\lim_{\zeta\downarrow 0}\sup\{\Bover{\mu\phi[C]}{\mu C}:
  C\in\Cal C_{\alpha},\,x\in\overline{C},\,0<\diam C\le\zeta\}\cr
&=\lim_{\zeta\downarrow 0}\inf\{\Bover{\mu\phi[C]}{\mu C}:
  C\in\Cal C_{\alpha},\,x\in\overline{C},\,0<\diam C\le\zeta\}\cr}$$

\noindent for every $\alpha>0$.

\proof{ By Rademacher's theorem (262Q), $H$ is conegligible.   Let $H'$
be the Lebesgue set of $J$, so that $H'$ also is conegligible (261E).
Take any $x\in H'$ and $\epsilon>0$.   Then there is a $\zeta_0>0$ such
that $\int_{B(x,\zeta)}|J(y)-J(x)|\mu(dy)\le\epsilon\mu B(x,\zeta)$ for
every $\zeta\in[0,\zeta_0]$.   Now suppose that $C\in\Cal C_{\alpha}$,
$x\in\overline{C}$ and $0<\diam C\le\zeta_0$.   Then
$\mu(C\setminus H)=0$ so $\mu\phi[C\setminus H]=0$ (262D), and

\Centerline{$\mu\phi[C]=\mu\phi[C\cap H]=\int_{C\cap H}J\,d\mu$}

\noindent (263D(iv)).   So

$$\eqalign{|\mu\phi[C]-J(x)\mu C|
&=|\int_{C\cap H}J\,d\mu-J(x)\mu(C\cap H)|
\le\int_{C\cap H}|J(y)-J(x)|d\mu\cr
&\le\int_{B(x,\diam C)}|J(y)-J(x)|d\mu
\le\epsilon\mu B(x,\diam C)\cr
&=\beta_r\epsilon(\diam C)^r
\le\Bover{\beta_r\epsilon}{\alpha}\mu C.\cr}$$

\noindent Thus
$|\Bover{\mu\phi[C]}{\mu C}-J(x)|\le\Bover{\beta_r}{\alpha}\epsilon$
whenever $C\in\Cal C_{\alpha}$, $x\in\overline{C}$ and
$0<\diam C\le\zeta_0$;  as $\epsilon$ is arbitrary,

$$\eqalign{J(x)
&=\lim_{\zeta\downarrow 0}\sup\{\Bover{\mu\phi[C]}{\mu C}:
  C\in\Cal C_{\alpha},\,x\in\overline{C},\,0<\diam C\le\zeta\}\cr
&=\lim_{\zeta\downarrow 0}\inf\{\Bover{\mu\phi[C]}{\mu C}:
  C\in\Cal C_{\alpha},\,x\in\overline{C},\,0<\diam C\le\zeta\}.\cr}$$

\noindent And this is true for $\mu$-almost every $x$.
}%end of proof of 484P

\leader{484Q}{Definition} If $(X,\rho)$ and $(Y,\sigma)$ are metric
spaces, a function $\phi:X\to Y$ is a {\bf lipeomorphism} if it is
bijective and both $\phi$ and $\phi^{-1}$ are Lipschitz.   \cmmnt{Of
course a lipeomorphism is a homeomorphism.}

\leader{484R}{Lemma} Let $\phi:\BbbR^r\to\BbbR^r$ be a lipeomorphism.

(a) For any set $A\subseteq\BbbR^r$,

\Centerline{$\clstar(\phi[A])=\phi[\clstar A]$,
\quad$\intstar(\phi[A])=\phi[\intstar A]$,
\quad$\partstar(\phi[A])=\phi[\partstar A]$.}

(b) $\phi[C]\in\Cal C$ for every $C\in\Cal C$, and $\phi[V]\in\Cal V$
for every $V\in\Cal V$.

(c) For any $\alpha>0$ there is an $\alpha'\in\ocint{0,\alpha}$ such
that $\phi[C]\in\Cal C_{\alpha'}$ for every $C\in\Cal C_{\alpha}$ and
$\{(\phi(x),\phi[C]):(x,C)\in\pmb{t}\}$ belongs to $T_{\alpha'}$ for
every $\pmb{t}\in T_{\alpha}$.

(d) For any $\Cal R\in\frak R$ there is an $\Cal R'\in\frak R$ such that
$\phi[R]\in\Cal R$ for every $R\in\Cal R'$.

(e) $\theta\phi:\BbbR^r\to\coint{0,\infty}$ belongs to $\Theta$ for
every $\theta\in\Theta$.

\proof{ Let $\gamma$ be so large that it is a Lipschitz constant for
both $\phi$ and $\phi^{-1}$.   Observe that in this case

\Centerline{$\phi^{-1}[B(\phi(x),\Bover{\zeta}{\gamma})]
\subseteq B(x,\zeta)$,
\quad$\phi[B(x,\zeta)]\supseteq B(\phi(x),\Bover{\zeta}{\gamma})$}

\noindent for every $x\in\BbbR^r$ and $\zeta\ge 0$, while

\Centerline{$\mu^*A=\mu^*\phi^{-1}[\phi[A]]\le\gamma^r\mu^*\phi[A]$,
\quad$\nu^*A\le\gamma^{r-1}\nu^*\phi[A]$}

\noindent for every $A\subseteq\BbbR^r$ (471J).

\medskip

{\bf (a)} If $A\subseteq\BbbR^r$ and $x\in\clstar A$, set

\Centerline{$\epsilon
=\Bover12\limsup_{\zeta\downarrow 0}
   \Bover{\mu^*(B(x,\zeta)\cap A)}{\mu B(x,\zeta)}>0$.}

\noindent Take any $\zeta_0>0$.   Then there is a $\zeta$ such that
$0<\zeta\le\zeta_0$ and
$\mu^*(B(x,\bover{\zeta}{\gamma})\cap A)
\ge\epsilon\mu B(x,\bover{\zeta}{\gamma})$, so that

$$\eqalign{\mu^*(B(\phi(x),\zeta)\cap\phi[A])
&\ge\mu^*\phi[B(x,\Bover{\zeta}{\gamma})\cap A]
\ge\Bover1{\gamma^r}\mu^*(B(x,\Bover{\zeta}{\gamma})\cap A)\cr
&\ge\Bover{\epsilon}{\gamma^r}\mu B(x,\Bover{\zeta}{\gamma})
=\Bover{\epsilon}{\gamma^{2r}}\mu B(x,\zeta).\cr}$$

\noindent As $\zeta_0$ is arbitrary,

\Centerline{$\limsup_{\zeta\downarrow 0}
   \Bover{\mu^*(B(\phi(x),\zeta)\cap\phi[A]}{\mu B(\phi(x),\zeta)}
\ge\Bover{\epsilon}{\gamma^{2r}}>0$,}

\noindent and $\phi(x)\in\clstar(\phi[A])$.

This shows that $\phi[\clstar A]\subseteq\clstar(\phi[A])$.   The same
argument applies to $\phi^{-1}$ and $\phi[A]$, so that $\phi[\clstar A]$
must be equal to $\clstar(\phi[A])$.   Taking complements,
$\phi[\intstar A]=\intstar(\phi[A])$, so that
$\phi[\partstar A]=\partstar(\phi[A])$.

\medskip

{\bf (b)} Take $C\in\Cal C$.   Then, for any $n\in\Bbb N$,
$\phi^{-1}[B(\tbf{0},n)]$ is bounded, so is included in $B(\tbf{0},m)$
for some $m$.   Now

$$\eqalign{\nu(\partstar\phi[C]\cap B(\tbf{0},n))
&=\nu(\phi[\partstar C]\cap B(\tbf{0},n))\cr
&\le\nu(\phi[\partstar C\cap B(\tbf{0},m)])
\le\gamma^{r-1}\nu(\partstar C\cap B(\tbf{0},m))\cr}$$

\noindent is finite.   This shows that $\phi[C]$ has
locally finite perimeter and belongs to $\Cal C$.   Since $\phi[V]$ is
bounded whenever $V$ is bounded, $\phi[V]\in\Cal V$ whenever
$V\in\Cal V$.

\medskip

{\bf (c)} Set $\alpha'=\gamma^{-2r}\alpha$.   Note that as $\gamma^2$ is
a Lipschitz constant for the identity map, $\gamma\ge 1$, and
$\alpha'\le\min(\alpha,\gamma^{2-2r}\alpha)$.   If
$C\in\Cal C_{\alpha}$, then

$$\eqalign{\mu\phi[C]
&\ge\Bover1{\gamma^r}\mu C
\ge\Bover{\alpha}{\gamma^r}(\diam C)^r
\ge\alpha\diam\phi[C])^r
\ge\alpha'(\diam\phi[C])^r,\cr
\cr
\per\phi[C]
&=\nu(\partstar(\phi[C]))
=\nu(\phi[\partstar C])
\le\gamma^{r-1}\nu(\partstar C)\cr
&\le\Bover{\gamma^{r-1}}{\alpha}(\diam C)^{r-1}
\le\Bover{\gamma^{r-1}}{\alpha}(\gamma\diam\phi[C])^{r-1}
\le\Bover1{\alpha'}(\diam\phi[C])^{r-1}.\cr}$$

\noindent So $C\in\Cal C_{\alpha'}$.

If now $\pmb{t}\in T_{\alpha}$, then, for any $(x,C)\in\pmb{t}$,
$\phi[C]\in\Cal C_{\alpha'}$ and $\phi(x)\in\clstar\phi[C]$;  also,
because $\phi$ is injective, $\langle\phi[C]\rangle_{(x,C)\in\pmb{t}}$
is disjoint, so $\{(\phi(x),\phi[C]):(x,C)\in\pmb{t}\}\in T_{\alpha'}$.

\medskip

{\bf (d)} Express $\Cal R$ as $\Cal R^{(V)}_{\eta}$ where $V\in\Cal V$
and $\eta\in\Eta$, so that $R\in\Cal R$ whenever $R\cap V\in\Cal R$.
By 484Ec and 481He, there is a sequence
$\sequence{i}{\Cal Q_i}$ in $\frak R$ such that
$\bigcup_{i\le n}A_i\in\Cal R$ whenever $n\in\Bbb N$,
$\langle A_i\rangle_{i\le n}$ is disjoint and $A_i\in\Cal Q_i$ for every
$i$.   By 484E(b-ii), there is an $\eta'\in\Eta$ such that
$R\in\Cal Q_i$ whenever $i\in\Bbb N$ and $R$ is such that
$\mu R\le\gamma^r\eta'(i)$ and  $\per R\le\gamma^{r-1}$.   Try
$\Cal R'=\Cal R^{(\phi^{-1}[V])}_{\eta'}\in\frak R$.   If $R\in\Cal R'$,
we can express $R\cap\phi^{-1}[V]$ as $\bigcup_{i\le n}E_i$ where
$\per E_i\le 1$ and $\mu E_i\le\eta'_i$ for each $i\le n$, and
$\langle E_i\rangle_{i\le n}$ is disjoint.   So
$\phi[R]\cap V=\bigcup_{i\le n}\phi[E_i]$ and
$\langle\phi[E_i]\rangle_{i\le n}$ is disjoint.   Now, for each $i$,

\Centerline{$\mu\phi[E_i]\le\gamma^r\mu E_i\le\gamma^r\eta'(i)$,
\quad$\per\phi[E_i]\le\gamma^{r-1}\per E_i\le\gamma^{r-1}$,}

\noindent so $\phi[E_i]\in\Cal Q_i$.   By the choice of
$\sequence{i}{\Cal Q_i}$, $\phi[R]\cap V\in\Cal R$ and
$\phi[R]\in\Cal R$.   So $\Cal R'$ has the property we need.

\medskip

{\bf (e)} We have only to observe that if $A$ is the thin set
$\theta^{-1}[\{0\}]$, then $(\theta\phi)^{-1}[\{0\}]=\phi^{-1}[A]$ is
also thin.   \Prf\ If $A=\bigcup_{n\in\Bbb N}A_n$ where $\nu^*A_n$ is
finite for every $n$, then
$\phi^{-1}[A]=\bigcup_{n\in\Bbb N}\phi^{-1}[A_n]$, while
$\nu^*\phi^{-1}[A_n]\le\gamma^{r-1}\nu^*A_n$ is finite for
every $n\in\Bbb N$.\ \Qed
}%end of proof of 484R

\leader{484S}{Theorem} Let $\phi:\BbbR^r\to\BbbR^r$ be a lipeomorphism.
Let $H$ be the set of points at which $\phi$ is differentiable.   For
$x\in H$, write $T(x)$ for the derivative of $\phi$ at $x$;  set
$J(x)=|\det T(x)|$ for $x\in H$, $0$ for $x\in\BbbR^r\setminus H$.
Then, for any function $f:\BbbR^r\to\BbbR^r$,

\Centerline{$\biggerPfint f=\Pfint J\times f\phi$}

\noindent if either is defined in $\Bbb R$.

\proof{{\bf (a)} Let $H'\subseteq H$ be a conegligible set such that

$$\eqalign{J(x)
&=\lim_{\zeta\downarrow 0}\sup\{\Bover{\mu\phi[C]}{\mu C}:
  C\in\Cal C_{\alpha},\,x\in\overline{C},\,0<\diam C\le\zeta\}\cr
&=\lim_{\zeta\downarrow 0}\inf\{\Bover{\mu\phi[C]}{\mu C}:
  C\in\Cal C_{\alpha},\,x\in\overline{C},\,0<\diam C\le\zeta\}\cr}$$

\noindent for every $\alpha>0$ and every $x\in H'$ (484P).
To begin with (down to the end of (c)), suppose that $f$ is Pfeffer
integrable
and that $f\phi(x)=0$ for every $x\in\BbbR^r\setminus H'$.   Let $F$ be
the Saks-Henstock indefinite integral of $f$, and define
$G:\Cal C\to\Bbb R$ by setting $G(C)=F(\phi[C])$ for every $C\in\Cal C$
(using 484Rb to see that this is well-defined).

\medskip

{\bf (b)} $G$ and $J\times f\phi$ satisfy the conditions of 484J.

\medskip

\Prf{\bf (i)} Of course $G$ is additive, because $F$ is.

\medskip

\quad{\bf (ii)} Suppose that $0<\alpha<\alpha^*$ and $\epsilon>0$.   Let
$\alpha'\in\ooint{0,\alpha^*}$ be such that
$\{(\phi(x),\phi[C]):(x,C)\in\pmb{t}\}\in T_{\alpha'}$ whenever
$\pmb{t}\in T_{\alpha}$ (484Rc).   Let $\theta_1\in\Theta$ be such that
$\sum_{(x,C)\in\pmb{t}}|F(C)-f(x)\mu C|\le\bover12\epsilon$ for every
$\delta_{\theta_1}$-fine $\pmb{t}\in T_{\alpha'}$.   Let
$\theta_2:\BbbR^r\to\ocint{0,1}$ be such that whenever $x\in H'$,
$n\le\|x\|+|f\phi(x)|<n+1$, $C\in\Cal C_{\alpha'}$, $x\in\overline{C}$
and $\diam C\le 2\theta_2(x)$ then

\Centerline{$|\mu\phi[C]-J(x)\mu C|
\le\Bover{\epsilon\mu C}{2^{n+2}\beta_r(n+2)^r(n+1)}$.}

\noindent Set
$\theta(x)=\min(\Bover1{\gamma}\theta_1\phi(x),\theta_2(x))$ for
$x\in\BbbR^r$, where $\gamma>0$ is a Lipschitz constant for $\phi$, so
that $\theta\in\Theta$ (484Re).

If $\pmb{t}\in T_{\alpha}$ is $\delta_{\theta}$-fine, set
$\pmb{t}'=\{(\phi(x),\phi[C]):(x,C)\in\pmb{t}\}$.   Then
$\pmb{t}'\in T_{\alpha'}$, by the choice of $\alpha'$.   If
$(x,C)\in\pmb{t}$, then $\theta(x)>0$ so $\theta_1\phi(x)>0$;  also, for
any $y\in\phi[C]$,

\Centerline{$\|\phi(x)-y\|\le\gamma\|x-\phi^{-1}(y)\|
<\gamma\theta(x)\le\theta_1\phi(x)$.}

\noindent This shows that $\pmb{t}'$ is $\delta_{\theta_1}$-fine.   We
therefore have

$$\eqalignno{\sum_{(x,C)\in\pmb{t}}|G(C)-J(x)f(\phi(x))\mu C|
&\le\sum_{(x,C)\in\pmb{t}}|F(\phi[C])-f(\phi(x))\mu\phi[C]|\cr
&\hskip 5em +\sum_{(x,C)\in\pmb{t}}|f(\phi(x))||\mu\phi[C]-J(x)\mu C|\cr
&\le\sum_{(x,C)\in\pmb{t}'}|F(C)-f(x)\mu C|\cr
&\hskip 5em
  +\sum_{\Atop{(x,C)\in\pmb{t}}{x\in H'}}
      |f(\phi(x))||\mu\phi[C]-J(x)\mu C|\cr
\displaycause{because if $x\notin H'$ then $f(\phi(x))=0$}
&\le\Bover12\epsilon+\sum_{n=0}^{\infty}
 \sum_{\Atop{(x,C)\in\pmb{t},x\in H'}{n\le\|x\|+|f(\phi(x))|<n+1}}
   \Bover{(n+1)\epsilon\mu C}{2^{n+2}\beta_r(n+2)^r(n+1)}\cr
&\le\Bover12\epsilon+\sum_{n=0}^{\infty}
   \Bover{\epsilon\mu B(\tbf{0},n+2)}
     {2^{n+2}\beta_r(n+2)^r}\cr
\displaycause{remembering that $\theta_2(x)\le 1$, so
$C\subseteq B(\tbf{0},n+2)$ whenever $(x,C)\in\pmb{t}$ and $\|x\|<n+1$}
&=\epsilon.\cr}$$

\noindent As $\pmb{t}$ is arbitrary, this shows that $G$ and
$J\times f\phi$ satisfy (ii) of 484J.

\medskip

\quad{\bf (iii)} Given $\epsilon>0$, there is an $\Cal R\in\frak R$ such
that $|F(C)|\le\epsilon$ for every $C\in\Cal C\cap\Cal R$.   Now by
484Rd there is an $\Cal R'\in\frak R$ such that $\phi[R]\in\Cal R$ for
every $R\in\Cal R'$, so that $|G(C)|\le\epsilon$ for every
$C\in\Cal C\cap\Cal R'$.    Thus $G$ satisfies (iii) of 484J.\ \Qed

\medskip

{\bf (c)} This shows that $J\times f\phi$ is Pfeffer integrable, with
Saks-Henstock indefinite integral $G$;  so, in particular,

\Centerline{$\biggerPfint f\times\phi=G(\BbbR^r)=F(\BbbR^r)=\Pfint f$.}

\medskip

{\bf (d)} Now suppose that $f$ is an arbitrary Pfeffer integrable
function.   In this case set $f_1=f\times\chi\phi[H']$.   Because
$\BbbR^r\setminus H'$ is $\mu$-negligible, so is
$\phi[\BbbR^r\setminus H']$, and $f_1=f\,\,\mu$-a.e.   Also, of course,
$f\phi=f_1\phi\,\,\mu$-a.e.   Because the Pfeffer integral extends the
Lebesgue integral (484He),

\Centerline{$\biggerPfint J\times f\phi=\Pfint J\times f_1\phi
=\Pfint f_1=\Pfint f$.}

\medskip

{\bf (e)} All this has been on the assumption that $f$ is Pfeffer
integrable.   If $g=J\times f\phi$ is Pfeffer integrable, consider
$\tilde J\times g\phi^{-1}$, where $\tilde J(x)=|\det\tilde T(x)|$
whenever the derivative $\tilde T(x)$ of $\phi^{-1}$ at $x$ is defined,
and otherwise is zero.   Now

\Centerline{$\tilde J(x)g(\phi^{-1}(x))
=\tilde J(x)\cdot J(\phi^{-1}(x))f(x)$}

\noindent for every $x$.   But, for $\mu$-almost every $x$,

\Centerline{$\tilde J(x)J(\phi^{-1}(x))
=|\det\tilde T(x)||\det T(\phi^{-1}(x))|
=|\det\tilde T(x)T(\phi^{-1}(x))|
=1$}

\noindent because $\tilde T(x)T(\phi^{-1}(x))$ is (whenever it is
defined) the derivative at $\phi^{-1}(x)$ of the identity function
$\phi^{-1}\phi$, by 473Bc.   (I see that we need to know that
$\{x:\phi$ is differentiable at $\phi^{-1}(x)\}=\phi[H]$ is
conegligible.)   So $\tilde J\times g\phi^{-1}=f\,\mu$-a.e., and
$f$ is Pfeffer integrable.   This completes the proof.
}%end of proof of 484S

\exercises{\leader{484X}{Basic exercises $\pmb{>}$(a)}
%\spheader 484Xa
Show that for every $\Cal R\in\frak R$ there are
$\eta\in\Eta$ and $n\in\Bbb N$ such that
$\Cal R^{(B(\tbf{0},n))}_{\eta}\subseteq\Cal R$.
%484D

\sqheader 484Xb ({\smc Pfeffer 91a}) For $\alpha>0$ let
$\Cal C'_{\alpha}$ be the family of bounded Lebesgue measurable sets $C$
such that
$\mu C\ge\alpha\diam C\per C$.   Show that
$\Cal C_{\sqrt{\alpha}}\subseteq\Cal C'_{\alpha}
\subseteq\Cal C_{\min(\alpha,\alpha^r)}$.
\Hint{474La.}   %if E bdd (\mu E)^{(r-1)/r}\le\per E
%484F

\sqheader 484Xc For $\alpha>0$, let $\Cal C''_{\alpha}$ be the family of
bounded convex sets $C\subseteq\BbbR^r$ such that
$\mu C\ge\alpha(\diam C)^r$.   Show that if $0<\alpha<1/2r$
then $\Cal C''_{\alpha}\subseteq\Cal C_{\alpha}$ ({\it hint\/}: 475T)
and $T_{\alpha}\cap[\BbbR^r\times\Cal C''_{\alpha}]^{<\omega}$
is compatible with $\Delta$ and $\frak R$.
\Hint{use the argument of 484F, but in part (b) take $C=\BbbR^r$, $E=V$
a union of members of $\Cal D$.}
%484F

\spheader 484Xd Describe a suitable filter $\Cal F$ to express the
Pfeffer integral directly in the form considered in 481C.
%484G

\spheader 484Xe Let $f:\BbbR^r\to\Bbb R$ be a Pfeffer integrable
function.   Show that there is some $n\in\Bbb N$ such that
$\int_{\Bbb R^r\setminus B(\tbf{0},n)}|f|d\mu$ is finite.
%484G

\spheader 484Xf\dvAnew{2010} (Here take $r=2$.)
Let $\sequencen{\delta_n}$ be a strictly
decreasing summable sequence in $\ocint{0,1}$.   Define
$f:\BbbR^2\to\Bbb R$ by saying that
$f(x)=\Bover{(-1)^n}{(n+1)(\delta_n^2-\delta_{n+1}^2)}$ if $n\in\Bbb N$ and
$\delta_{n+1}\le\|x\|<\delta_n$, $0$ otherwise.   Show that
$\lim_{\delta\downarrow 0}\int_{\BbbR^2\setminus B(\tbf{0},\delta)}fd\mu$
is defined, but that $f$ is not Pfeffer integrable.
\Hint{484J.}
%484J

\spheader 484Xg\dvAnew{2010} (Again take $r=2$.) Show that there are a
Lebesgue integrable $f_1:\Bbb R\to\Bbb R$ and a Henstock integrable
$f_2:\Bbb R\to\Bbb R$, both with bounded support, such that
$(\xi_1,\xi_2)\mapsto f_1(\xi_1)f_2(\xi_2):\BbbR^2\to\Bbb R$ is not Pfeffer
integrable.
%484Xf 484J mt48bits

\spheader 484Xh Let $E\subseteq\BbbR^r$ be a bounded set with finite
perimeter, and $\phi:\BbbR^r\to\BbbR^r$ a differentiable function.   Let
$v_x$ be the Federer exterior normal to $E$ at any point $x$ where the
normal exists.   Show that $\Pfint\diverg\phi\times\chi E$ is defined
and equal to $\int_{\partstar E}\varinnerprod{\phi(x)}{v_x}\nu(dx)$.
%484N

\spheader 484Xi\dvAnew{2010} Show that there is a Lipschitz function
$f:\BbbR^r\to[0,1]$ such that $\BbbR^r\setminus\dom f'$ is not thin.
\Hint{there is a Lipschitz function $f:\Bbb R\to[0,1]$ not differentiable
at any point of the Cantor set.}
%484N

\leader{484Y}{Further exercises (a)}
%\spheader 484Ya
Let $E\subseteq\BbbR^r$ be any Lebesgue measurable set, and
$\epsilon>0$.   Show that there is a Lebesgue measurable set
$G\subseteq E$ such that $\per G\le\per E$,
$\mu(E\setminus G)\le\epsilon$ and $\clstar G=\overline{G}$.
%484B

\spheader 484Yb
Give an example of a compact set $K\subseteq\BbbR^2$
with zero one-dimensional Hausdorff measure such that whenever
$\theta:K\to\ooint{0,\infty}$ is a strictly positive function, and
$\gamma\in\Bbb R$, there is a disjoint family
$\langle B(x_i,\zeta_i)\rangle_{i\le n}$ of balls such that $x_i\in K$
and $\zeta_i\le\theta(x_i)$ for every $i$, while
$\per(\bigcup_{i\le n}B(x_i,\zeta_i))\ge\gamma$.
%notes %mt48bits
}%end of exercises

\endnotes{
\Notesheader{484}
Listing the properties of the Pfeffer integral as developed above, we have

\inset{expected relations with Lebesgue measure and integration
(484Hd-484Hf);

Saks-Henstock indefinite integrals (484H-484J);  %484Hc 484I 484J

integration over suitable subsets (484L);

a divergence theorem (484N);

a density theorem (484O);

a change-of-variable theorem for lipeomorphisms (484S).}

\noindent The results on indefinite integrals and integration over
subsets are restricted in comparison with
what we have for the Lebesgue integral, since we can deal only with sets
with locally finite perimeter;  and 484S is similarly narrower in scope
than 263D(v).   Pfeffer's Divergence Theorem, on the other hand,
certainly applies to many functions $\phi$ for which $\diverg\phi$ is not
Lebesgue integrable, though it does not entirely cover 475N (see 484Xi).
In comparison with the one-dimensional case, the
Pfeffer integral does not share the most basic property of the special
Denjoy integral (483Bd, 484Xf), but 484N is a step
towards the Perron integral (483J).   484O is a satisfactory rendering of
the idea in 483I, and even for Lebesgue integrable functions adds something
to 261C.
Throughout, I have written on the assumption that $r\ge 2$.
It would be possible to work through the same arguments
with $r=1$, but in this case we should find that `thin' sets became
countable, therefore easily controllable by neighbourhood gauges, making
the methods here inappropriate.

The whole point of `gauge integrals' is that we have an enormous amount
of freedom within the framework of \S\S481-482.   There is a
corresponding difficulty in making definitive choices.   The essential
ideology of the Pfeffer integral is that we take an intersection
of a family of gauge integrals, each determined by a family
$\Cal C_{\alpha}$ of sets which are `Saks regular' in the sense that
their measures, perimeters and diameters are linked (compare 484Xb).
Shrinking
$\Cal C_{\alpha}$ and $T_{\alpha}$, while leaving $\Delta$ and $\frak R$
unchanged, of course leads to a more `powerful' integral (supposing, at
least, that we do not go so far that $T_{\alpha}$ is no longer
compatible with $\Delta$ and $\frak R$), so that Pfeffer's Divergence
Theorem will remain true.   One possibility is to turn to convex sets
(484Xc), though we could not then expect invariance under lipeomorphisms.

The family $\Delta=\{\delta_{\theta}:\theta\in\Theta\}$ of gauges is
designed to permit the exclusion of tags from thin sets;  apart from this
refinement, we are looking at neighbourhood gauges, just as with the
Henstock integral.   This feature, or something like it,
seems to be essential when we come to
the identification $F(E)=\Pfint f\times\chi E$ in 484L, which is demanded
by the formula in the target theorem 484N.   In order to make our
families $T_{\alpha}$ compatible in the sense of 481F, we are
then forced to allow non-trivial residual families;
with some effort (484C, 484Ed), we can
get tagged-partition structures allowing subdivisions (484F).
Note that this is one of the
cases in which our residual families $\Cal R^{(V)}_{\eta}$ are defined by
`shape' as well as by `size'.   In the indefinite-integral characterization
of the Pfeffer integral (484J), we certainly cannot demand `for every
$\epsilon>0$ there is an $\Cal R\in\frak R$ such that $|F(E)|\le\epsilon$
whenever $E\in\Cal C$ is included in a member of $\Cal R$', since all
small balls belong to $\Cal R$, and we should immediately be driven to the
Lebesgue integral.   However I use the construction
$\Cal R^{(V)}_{\eta}=\{R:R\cap V\in\Cal R_{\eta}\}$ (484D) as a quick
method of eliminating any difficulties at infinity (484Xe).   We do not
of course need to look at arbitrary sets $V\in\Cal V$ here (484Xa).

Observe that 484B can be thought of as a refinement of 475I.
As usual, the elaborate formula in the statement of 484C is there only
to emphasize that we have a bound depending only on $l$ and $r$.
Note that 484S depends much more on the fact that the Pfeffer integral can
be characterized in the language of 484J, than on the exact choices made
in forming $\frak R$ and the $\Cal C_{\alpha}$.   For a
discussion of integrals {\it defined} by Saks-Henstock lemmas, see
{\smc Pfeffer 01}.

It would be agreeable to be able to think of the Pfeffer integral as a
product in some sense, so we naturally look for Fubini-type theorems.
I give 484Xg to indicate one of the obstacles.
}%end of notes

\discrversionA{Any
hope of a form of Fubini theorem?  or other relationship between
$\Pfint_{r+s}$ ,  $\Pfint_r$  and  $\Pfint_s$ ?\query
see 484Xg
}{}

\discrpage

\references{
\frfilename{mt4r.tex}
\versiondate{9.2.13}
\copyrightdate{2002}

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

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

\Loadcyrillic

\Centerline{\bf References for Volume 4}


\medskip%A

Aldaz J.M.\ [95] `On compactness and Loeb measures', Proc.\ Amer.\
Math.\ Soc.\ 123 (1995) 173-175.
\cmmnt{[439Xa.]}

Aldaz J.M.\ [97] `Borel extensions of Baire measures', Fund.\ Math.\ 154
(1997) 275-293.
\cmmnt{[434Yn.]}

Aldaz J.M.\ \& Render H.\ [00] `Borel measure extensions of measures
defined on sub-$\sigma$-algebras', Advances in Math.\ 150 (2000)
233-263.
\cmmnt{[432D.]}

Alexandroff P.S.\ \& Urysohn P.S.\ [1929] {\it  M\'emoire sur les espaces
topologiques compacts},
Verhandelingen Akad.\ Wetensch.\ Amsterdam 14 (1929) 1-96.
\cmmnt{[419L.]}

Anderson I.\ [87] {\it Combinatorics of Finite Sets.}  Oxford U.P.,
1987.
\cmmnt{[4A4N.]}

Andretta A.\ \& Camerlo R.\ [13] `The descriptive set theory of the
Lebesgue density theorem', Advances in Math.\ 234 (2013).
%{\tt http://arxiv.org/abs/1105.3355}.
\cmmnt{[475Yg.]}

Arkhangel'skii A.V.\ [92] {\it Topological Function Spaces.}  Kluwer,
1992.
\cmmnt{[\S462 {\it intro.}, \S467 {\it notes\/}.]}

\leaveitout{
Armitage D.H.\ \& Gardiner S.J.\ [01] {\it Classical Potential Theory.}
Springer, 2001.
478Yd}

Aronov B., Basu S., Pach J.\ \& Sharir M.\ [03] (eds.)
{\it Discrete and Computational Geometry;  the Goodman-Pollack
Festschrift,}  Springer, 2003.

Asanov M.O.\ \& Velichko N.V.\ [81] `Compact sets in $C_p(X)$', Comm.\
Math.\ Helv.\ 22 (1981) 255-266.
\cmmnt{[462Ya.]}

Austin T.\ [10a] `On the norm convergence of nonconventional ergodic
averages', Ergodic Theory and Dynamical Systems 30 (2010) 321-338.
%({\tt http://{\penalty-100}arxiv.org/abs/{\penalty-100}0805.0320}).
\cmmnt{[497M.]}

Austin T.\ [10b] `Deducing the multidimensional Szemer\'edi theorem from
the infinitary hypergraph removal lemma',
J.\ d'Analyse Math.\ 111 (2010) 131-150.
%({\tt http://arxiv.org{\bsp}abs/0808.2267}).
\cmmnt{[497M.]}

\leaveitout{
Avil\'es A., Plebanek G.\ \& Rodr\'{\i}guez J.\ [p11] `Measurability in
$C(2^{\kappa})$ and Kunen cardinals', 2011.
%material on  C(\{0,1\}^{\kappa}) ;  Baire and Borel \sigma-algebras
%in weak and pointwise topologies
}

\leaveitout{Axler S. \& Ramey W.\ [95]
`Harmonic polynomials and Dirichlet-type problems',
Proc.\ Amer.\  Math.\ Soc.\ 123 (1995) 3765-3773.
http://www.axler.net/HarmonicPolynomials.html
478Yd}

\medskip%B

Balogh Z.\ [96] `A small Dowker space in ZFC', Proc.\ Amer.\ Math.\
Soc.\ 124 (1996) 2555-2560.
\cmmnt{[439O.]}

Banach S.\ [1923] `Sur le probl\`eme de la mesure',
Fundamenta Math.\ 4 (1923) 7-33.
\cmmnt{[449O.]}

Becker H.\ \& Kechris A.S.\ [96] {\it The descriptive set theory of
Polish group actions.}  Cambridge U.P., 1996 (London Math.\ Soc.\
Lecture Note Series 232).  \cmmnt{[424H, 448P, \S448 {\it notes\/}.]}

Bellow A.\ {\it see} Ionescu Tulcea A.

Bellow A.\ \& K\"olzow D.\ [76] {\it Measure Theory, Oberwolfach 1975.}
Springer, 1976 (Lecture Notes in Mathematics 541).
%4r

Bergelson V.\ [85] `Sets of recurrence of $\Bbb Z^m$-actions and properties of
sets of differences in $\Bbb Z^m$',
J.\ London Math.\ Soc.\ (2) 31 (1985) 295-304.
%http://people.dm.unipi.it/dinasso/ULTRA2015/BUMN/Bergelson%20-%20Sets%20of%20recurrence%20of%20Zm-actions%20and%20properties%20of%20sets%20of%20differences%20in%20Zm%20(1985).pdf
\cmmnt{[491Yt.]}

Bergman G.M.\ [06] `Generating infinite symmetric groups', Bull.\ London
Math.\ Soc.\ 38 (2006) 429-440.
%http://math.berkeley.edu/~gbergman/papers/
%({\tt http:/{\bsp}arxiv.org{\bsp}abs{\bsp}math{\bsp}0401304v2}).
\cmmnt{[\S494 {\it notes}.]}

Billingsley P.\ [99] {\it Convergence of Probability Measures.}  Wiley,
1999.
\cmmnt{[\S4A3 {\it notes}.]}
%second edition ISBN 0-471-19745-9

Blackwell D.\ [56] `On a class of probability spaces',
pp.\ 1-6 in {\smc Neyman 56}.
\cmmnt{[419K, 452P.]}

Bochner S.\ [1933] `Monotone Funktionen, Stieltjessche Integrale, und
harmonische Analyse', Math.\ Ann.\ 108 (1933) 378-410.
\cmmnt{[445N.]}

\indexheader{Bogachev}
Bogachev V.I.\ [07] {\it Measure theory.}  Springer, 2007.
\cmmnt{[437Q, 437Xs, 437Yo, 437Yu, 457K.]}

Bollob\'as B.\ [79] {\it Graph Theory.}   Springer, 1979.
\cmmnt{[4A1G, 4A4N.]}

Bongiorno B., Piazza L.di \ \& Preiss D.\ [00]
`A constructive minimal integral which includes Lebesgue integrable
functions and derivatives', J.\ London Math.\ Soc.\ (2) 62 (2000) 117-126.
\cmmnt{[483Yi.]}

Borodulin-Nadzieja P.\ \& Plebanek G.\ [05] `On compactness of measures on
Polish spaces', Illinois J.\ Math.\ 49 (2005) 531-545.
\cmmnt{[451L.]}

Bourbaki N.\ [65] {\it Int\'egration,} chaps.~1-4.   Hermann, 1965
(Actualit\'es Scientifiques et Industrielles 1175).
\cmmnt{[\S416 {\it notes\/}, \S436 {\it notes\/}.]}

Bourbaki N.\ [66] {\it Espaces Vectoriels Topologiques}, chaps.~1-2,
$2^{\text{e}}$ \'ed.
Hermann, 1966 (Actualit\'es Scientifiques et Industrielles 1189).
\cmmnt{[4A2J.]}
%Store A QA 37.B6 no 7792 0070 5

Bourbaki N.\ [69] {\it Int\'egration,} chap.~9.   Hermann, 1969
(Actualit\'es Scientifiques et Industrielles 1343).
\cmmnt{[\S436 {\it notes.}]}

Bourbaki N.\ [87] {\it Topological Vector Spaces.}  Springer, 1987.
\cmmnt{[\S461 {\it notes\/}, \S4A4.]}

\indexheader{Bourgain}
Bourgain J., Fremlin D.H.\ \& Talagrand M.\ [78] `Pointwise compact
sets of Baire-measurable functions', Amer.\ J.\ Math.\ 100 (1978)
845-886.
\cmmnt{[462Ye.]}

\indexheader{Brodski\v{\i}}
Brodski\v{\i} M.L.\ [1949] `On some properties of sets of positive measure',
Uspehi Matem.\ Nauk (N.S.) 4 (1949) 136-138.
\cmmnt{[498B.]}

Brook R.B.\ [70] `A construction of the greatest ambit',
Math.\ Systems Theory 4 (1970) 243-248.
\cmmnt{[449D.]}

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Burke M.R., Macheras N.D.\ \& Strauss W.\ [p21] `The strong marginal lifting
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%bms5s.pdf

\medskip%C

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\cmmnt{[481L.]}

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% 978-9810225339  HA33.4.C5  479Xt

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\medskip%D

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Dubins L.E.\ \& Freedman D.A.\ [79]
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\cmmnt{[\S4A4.]}

Du Plessis, N.\ [70] {\it Introduction to Potential Theory.}  Oliver \&
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%used for \S478

\medskip%E

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Emerson W.\ \& Greenleaf F.\ [67] `Covering properties and F{\o}lner
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Evans L.C.\ \& Gariepy R.F.\ [92] {\it Measure Theory and Fine
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\medskip%F

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\S443 {\it notes\/}, Chap.\ 47 {\it intro.}, 471Yk,
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Fernique X.\ [97] {\it Fonctions al\'eatoires gaussiennes,
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Folland G.B.\ [95] {\it A Course in Abstract Harmonic Analysis.}   CRC
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\indexheader{F{\o}lner}
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Fremlin D.H.\ [n05] `Strictly localizable Borel measures', note of 7.5.05
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\cmmnt{[434Yr.]}

Fremlin D.H.\ [n15] `The Sorgenfrey line is not a Prokhorov space',
note of 5.8.15
({\tt http:/{\bsp}www.\penalty-100essex.ac.uk{\bsp}maths/people{\bsp}fremlin/preprints.htm}).
\cmmnt{[439Yk.]}

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\cmmnt{[\S464.]}  %{\it 464C}, 464I, 464M, 464N, \S464 {\it notes\/}

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%HA33.4.F7

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%8 (1961) 747-750.
9 (1961) 721-726.   %Paulo Monteiro, email 8.9.22
\cmmnt{[422F.]}

Frol\'\i k Z.\ [88] (ed) {\it General Topology and its Relations to
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\cmmnt{[\S497 {\it intro.}, \S497 {\it notes\/}.]}

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{\it Recurrence in Ergodic Theory and Combinatorial Number Theory.}
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\cmmnt{[\S497 {\it intro.}, 497N.]}
% QA614

Furstenberg H.\ \& Katznelson Y.\ [85]
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\cmmnt{[\S497 {\it intro.}]}

\medskip%G

Gaal S.A.\ [64] {\it Point Set Topology.}  Academic, 1964.
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\cmmnt{[438M.]}

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\cmmnt{[434G.]}

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\cmmnt{[481Q, \S483 {\it notes\/}.]}

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\cmmnt{[\S449 {\it notes.}]}

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\cmmnt{[\S462 {\it notes\/}.]}

\indexheader{Grzegorek}
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\cmmnt{[439F, \S439 {\it notes\/}.]}

\medskip%H

\indexheader{Halmos}
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\cmmnt{[438J.]}

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\cmmnt{[416N.]}

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\cmmnt{[Chap.\ 48 {\it intro.}, 481J.]}

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\cmmnt{[\S481 {\it notes\/}.]}

Herer W.\ \& Christensen J.P.R.\ [75] `On the existence of pathological submeasures and the construction of exotic topological groups', Math.\ Ann.\ 213 (1975) 203-210.
\cmmnt{[\S493 {\it notes\/}.]}

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\cmmnt{[445N.]}

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\cmmnt{[\S442 {\it notes\/}, \S444 {\it intro.}, \S444 {\it notes\/},
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`Symmetric measures on Cartesian products',
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\cmmnt{[459Xe.]}

%Holick\'y P.\ \& Spurn\'y J.\ [03] `Perfect images of absolute Souslin
%and absolute Borel Tychonoff spaces', Topology and its Appl.\ 131 (2003)
%281-294.
%\cmmnt{[434Yr.]}

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\cmmnt{[471R, 471S.]}

\medskip%I

\indexheader{Ionescu Tulcea}
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\cmmnt{[463C.]}

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\cmmnt{[\S453 {\it intro.}]}

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\medskip%J

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\cmmnt{[\S4A4.]}

\indexheader{Jayne}
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\cmmnt{[422Yc.]}

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\indexheader{Jech}
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%ISBN 3540440852
\cmmnt{[\S4A1.]}

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Judah H.\ [93] (ed.) {\it Proceedings of the Bar-Ilan Conference on Set
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\medskip%K

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\cmmnt{[445U.]}

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\cmmnt{[\S446 {\it notes.}]}

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%Kechris A.S.\ [p09] {\it Global Aspects of Ergodic Group Actions.}
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\cmmnt{[491Z.]}

Kunen K.\ [80] {\it Set Theory.}  North-Holland, 1980.
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%4r

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\cmmnt{[Chap.\ 48 {\it intro.}]}

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\medskip%L

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\cmmnt{[414Ye.]}

\medskip%M

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\cmmnt{[435C.]}

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%QA311.M2 9520 6456 1

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\cmmnt{[418T.]}

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  % one m in sym\'etrique

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\cmmnt{[492G, \S492 {\it notes\/}.]}

\leaveitout{Mohammed D.\ \& Saadi K.\ [p09#]
`Probl\`emes de mesurabilit\'es sur
l'espace des fonctions continues sur un compact', 2009.
\cmmnt{[466Y?.]}
%refereeing for Canad Math Bull;  rejected
}

Molt\'o A., Orihuela J., Troyanski S.\ \& Valdivia M.\ [09] {\it A
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\cmmnt{[\S467 {\it notes.}]}

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\cmmnt{[\S446 {\it notes.}]}

\indexheader{Moran}
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\cmmnt{[439P.]}

Moran W.\ [69] `Measures and mappings on topological spaces', Proc.\
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\cmmnt{[435Xk.]}

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\cmmnt{[438J.]}

\indexheader{Moschovakis}
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%HA33.4.W5

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\medskip%S

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\indexheader{Talagrand} %? before \medskip to act correctly

\medskip%T

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%465Xm

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%used in chap 41

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\medskip%U

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\medskip%V

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\medskip%W

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  % printer's error "2" for "1" at head of paper refers to part, not
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%\medskip%X\medskip%Y\medskip%Z

%Zindulka O.\ [p98] `Killing residual measures', preprint, 1998.

\frnewpage
}%end of references


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

\end



