% The following TeXfile comprises part of

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

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

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

%        %   \amstexsupportedfalse

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

%        \input amstex
%        \documentstyle{amsppt}

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

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

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

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

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

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

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

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


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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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


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

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


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

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

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

   \Centerline{#1}

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

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

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

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

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

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

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

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

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

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

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

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

%End of lines for missing AmSTeX

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

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

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

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

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


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

%hyphenations
\hyphenation{Archi-me-dean}

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

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

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

%Headline assignments

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

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

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

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

\newdimen\pagetotalplus
\newdimen\pagetotalminus

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

  \Centerline{#2}

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

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

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

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

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

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

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

\frfilename{mt2a.tex} 
\versiondate{3.8.15} 
 
\def\RoverC{\hbox{\biggerscriptfonts${{\Bbb R}\atop{\Bbb C}}$}} 
 
\def\chaptername{Appendix} 
\def\sectionname{Introduction} 
 
\gdef\topparagraph{} 
\gdef\bottomparagraph{Appendix to Vol.\ 2 {\it intro.}} 
 
\centerline{\bf Appendix to Volume 2} 
 
\medskip 
 
\centerline{\bf Useful Facts} 
 
\medskip 
 
In the course of writing this volume, I have found that a considerable 
number of 
concepts and facts from various branches of mathematics are necessary to 
us.   Nearly all of them are embedded in important and well-established 
theories for which many excellent textbooks are available and which I 
very much hope that you will one day study in depth.   Nevertheless, I 
am reluctant to send you off immediately to courses in 
general topology, functional analysis and set theory, as if these were 
essential prerequisites for our work here, along with real analysis and 
basic linear algebra.   For this reason I have written this 
Appendix, setting out those results which we actually need at some point 
in this volume.   The great majority of them really are  
elementary -- indeed, some are so 
elementary that they are not always spelt out in detail in orthodox 
treatments of their subjects. 
 
While I do not put this book forward as the proper place to learn any of 
these topics, I have tried to set them out in a way that you will find 
easy to integrate into regular approaches.   I do not expect anybody to 
read systematically through this work, and I hope that the references 
given in the main chapters of this volume will be adequate to guide you 
to the particular items you need. 
 
\discrpage 
 
\frfilename{mt2a1.tex} 
\versiondate{20.1.13} 
\copyrightdate{1996} 
      
\def\chaptername{Appendix} 
\def\sectionname{Set theory} 
      
\def\Enderton{{\smc Enderton 77}} 
\def\Halmos{{\smc Halmos 60}} 
\def\Henle{{\smc Henle 86}} 
\def\Krivine{{\smc Krivine 71}} 
\def\Lipschutz{{\smc Lipschutz 64}} 
\def\Roitman{{\smc Roitman 90}} 
      
\newsection{2A1} 
      
Especially for the examples in Chapter 21, we need some non-trivial set 
theory, which is best approached through the standard theory of 
cardinals and ordinals;  and elsewhere in this volume I make use of 
Zorn's Lemma. 
Here I give a very brief outline of the results involved, largely 
omitting proofs. 
Most of this material should be in any sound introduction to set theory. 
The references I give are to books which happen to have come my way and 
which I can recommend as reasonably suitable for beginners. 
      
I do not discuss axiom systems or logical foundations.   The set theory 
I employ is `naive' in the sense that I rely on my understanding of 
the collective experience of the last hundred years, rather than on any 
attempt at formal description, to distinguish legitimate from unsafe 
arguments.   There are, however, points in Volume 5 at which such a 
relaxed philosophy becomes inappropriate, and I therefore use arguments 
which can, I believe, be translated into standard 
Zermelo-Fraenkel set theory without new ideas being invoked. 
      
Although in this volume I use the axiom of choice without scruple 
whenever appropriate, I will divide this section into two parts, 
starting with ideas and results not dependent on the axiom of choice 
(2A1A-2A1I) and continuing with the remainder 
(2A1J-2A1P).   I believe that even at this level it helps us to 
understand the nature of the arguments better if we maintain a degree of 
separation. 
      
\leader{2A1A}{Ordered sets (a)} Recall that a {\bf partially ordered 
set} is a set $P$ together with a relation $\le$ on $P$ such that 
      
\inset{if $p\le q$ and $q\le r$ then $p\le r$, 
      
$p\le p$ for every $p\in P$, 
      
if $p\le q$ and $q\le p$ then $p=q$.} 
      
\noindent In this context, I will write $p\ge q$ to mean $q\le p$, and 
$p<q$ or $q>p$ to mean `$p\le q$ and $p\ne q$'.   $\le$ is a 
{\bf partial order} on $P$. 
      
\header{2A1A}{\bf (b)} Let $(P,\le)$ be a partially ordered set, and 
$A\subseteq P$.   A {\bf maximal} element of $A$ 
is a $p\in A$ such that $p\not<a$ for any $a\in A$.   Note that $A$ may 
have more than one maximal element.  An {\bf upper bound} for $A$ is a 
$p\in P$ such that $a\le p$ for every $a\in A$;  a {\bf supremum} or 
{\bf least upper bound} is an upper bound $p$ such that $p\le q$ for 
every upper bound $q$ of $A$.   There can be at most one 
such\cmmnt{, because 
if $p$, $p'$ are both least upper bounds then $p\le p'$ and $p'\le p$}. 
Accordingly we may safely write $p=\sup A$ if $p$ is the least upper 
bound of $A$. 
      
Similarly, a {\bf minimal} element of $A$ is a $p\in A$ such that 
$p\not>a$ for every $a\in A$;  a {\bf lower bound} of $A$ is a $p\in P$ 
such that $p\le a$ for every $a\in A$;  and 
$\inf A=p$ means that 
      
\Centerline{for every $q\in P$, $p\ge q\iff a\ge q\Forall a\in A$.} 
      
A subset $A$ of $P$ is {\bf order-bounded} if it has both an upper bound 
and a lower bound. 
      
A subset $A$ of $P$ is {\bf upwards-directed} if for any $p$, $p'\in A$ 
there is a $q\in A$ such that $p\le q$ and $p'\le q$\cmmnt{;  that is, 
if any non-empty finite subset of $A$ has an upper bound in $A$}. 
Similarly, $A\subseteq P$ is {\bf downwards-directed} if for any $p$, 
$p'\in A$ there is a $q\in A$ such that $q\le p$ and $q\le p'$\cmmnt{; 
that is, if any non-empty finite subset of $A$ has a lower bound in 
$A$}. 
      
\cmmnt{It is sometimes convenient to adapt the notation for closed 
intervals to arbitrary partially ordered sets:}  $[p,q]$ will be 
$\{r:p\le r\le q\}$. 
 
\wheader{2A1A}{0}{0}{0}{24pt} 
      
\header{2A1Ac}{\bf (c)} A {\bf totally ordered set} is a partially 
ordered set $(P,\le)$ such that 
      
\inset{for any $p$, $q\in P$, either $p\le q$ or $q\le p$.} 
      
\noindent $\le$ is then a {\bf total} or {\bf linear} order on $P$.    
 
In any 
totally ordered set we have a {\bf median function}:  for $p$, $q$,  
$r\in P$ set  
 
$$\eqalign{\med(p,q,r) 
&=\max(\min(p,q),\min(p,r),\min(q,r))\cr 
&=\min(\max(p,q),\max(p,r),\max(q,r))\dvro{.}{,}\cr}$$ 
 
\cmmnt{\noindent so that $\med(p,q,r)=q$ if $p\le q\le r$.} 
      
\header{2A1Ad}{\bf (d)} A {\bf lattice} is a partially ordered set 
$(P,\le)$ such that 
      
\inset{for any $p$, $q\in P$, $p\vee q=\sup\{p,q\}$ and 
$p\wedge q=\inf\{p,q\}$ are defined in $P$.} 
      
\header{2A1Ae}{\bf (e)} A {\bf well-ordered set} is a totally ordered 
set $(P,\le)$ such that\cmmnt{ $\inf A$ exists and belongs to $A$ for 
every non-empty set $A\subseteq P$;  that is,} every non-empty subset of 
$P$ has a least element.   In this case 
$\le$ is a {\bf well-ordering} of $P$. 
      
\leader{2A1B}{Transfinite Recursion:  Theorem}  Let $(P,\le)$ be a 
well-ordered set and $X$ any class.   For $p\in P$ write $L_p$ for the 
set $\{q:q\in P,\,q<p\}$ and $X^{L_p}$ for the class of all functions 
from $L_p$ to 
$X$.   Let $F:\bigcup_{p\in P}X^{L_p}\to X$ be any function.   Then 
there is a unique function $f:P\to X$ such that $f(p)=F(f\restr L_p)$ 
for every $p\in P$. 
      
\proof{ There are versions of this result in \Enderton\ (p.\ 175) and 
\Halmos\ (\S18).   Nevertheless I write out a proof, since it seems to 
me that most elementary books on set theory do not give it its proper 
place at the very beginning of the theory of well-ordered sets. 
      
\medskip 
      
{\bf (a)} Let $\Phi$ be the class of all functions $\phi$ such that 
      
\quad ($\alpha$) $\dom\phi$ is a subset of $P$, and 
$L_p\subseteq\dom\phi$ for every $p\in\dom\phi$; 
      
\quad ($\beta$) $\phi(p)\in X$ for every $p\in\dom\phi$, and 
$\phi(p)=F(\phi\restr L_p)$ for every $p\in\dom\phi$. 
      
\medskip 
      
{\bf (b)} If $\phi$, $\psi\in\Phi$ then $\phi$ and $\psi$ agree on 
$\dom\phi\cap\dom\psi$.   \Prf\Quer\ If not, then 
$A=\{q:q\in\dom\phi\cap\dom\psi,\,\phi(q)\ne\psi(q)\}$ is non-empty. 
Because $P$ is well-ordered, $A$ has a least element $p$ say.   Now 
$L_p\subseteq\dom\phi\cap\dom\psi$ and $L_p\cap A=\emptyset$, so 
      
\Centerline{$\phi(p)=F(\phi\restr L_p)=F(\psi\restr L_p)=\psi(p)$,} 
      
\noindent which is impossible.\ \Bang\Qed 
      
\medskip 
      
{\bf (c)} It follows that $\Phi$ is a set, since the function 
$\phi\mapsto\dom\phi$ is an injective function from $\Phi$ to  $\Cal 
PP$, and its inverse is a surjection from a subset of $\Cal PP$ onto 
$\Phi$.    We can therefore, without inhibitions, define a function $f$ 
by writing 
      
\Centerline{$\dom f=\bigcup_{\phi\in\Phi}\dom\phi$, 
\quad$f(p)=\phi(p)$ whenever $\phi\in\Phi$ and $p\in\dom\phi$.} 
      
\noindent (If you think that a function $\phi$ is just the set of 
ordered pairs $\{(p,\phi(p)):p\in\dom\phi\}$, then $f$ becomes 
$\bigcup\Phi$.)   Then $f\in\Phi$.   \Prf\ Of course $f$ is a function 
from a subset of $P$ to $X$.   If $p\in\dom f$, then there is a 
$\phi\in\Phi$ such that $p\in\dom\phi$, in which case 
      
\Centerline{$L_p\subseteq\dom\phi\subseteq\dom f$, 
\quad$f(p)=\phi(p)=F(\phi\restr L_p)=F(f\restr L_p)$. \Qed} 
      
\medskip 
      
{\bf (d)} $f$ is defined everywhere in $P$.   \Prf\Quer\  Otherwise, 
$P\setminus\dom f$ is non-empty and has a least element $r$ say.   Now 
$L_r\subseteq\dom f$.   Define a function $\psi$ by saying that 
$\dom\psi=\{r\}\cup\dom f$, $\psi(p)=f(p)$ for $p\in\dom f$ and 
$\psi(r)=F(f\restr L_r)$.   Then $\psi\in\Phi$, because if 
$p\in\dom\psi$ 
      
\inset{{\it either} $p\in\dom f$ so 
$L_p\subseteq\dom f\subseteq\dom\psi$ and 
      
\Centerline{$\psi(p)=f(p)=F(f\restr L_p)=F(\psi\restr L_p)$}} 
      
\inset{{\it or} $p=r$ so $L_p=L_r\subseteq\dom f\subseteq\dom\psi$ and 
      
\Centerline{$\psi(p)=F(f\restr L_r)=F(\psi\restr L_r)$.}} 
      
\noindent Accordingly $\psi\in \Phi$ and 
$r\in\dom\psi\subseteq\dom f$.\ \Bang\Qed 
      
\medskip 
      
{\bf (e)} Thus $f:P\to X$ is a function such that $f(p)=F(f\restr L_p)$ 
for every $p$.   To see that $f$ is unique, observe that any function of 
this type must belong to $\Phi$, so must agree with $f$ on their common 
domain, which is the whole of $P$. 
}%end of proof of 2A1B 
      
\cmmnt{\medskip 
      
\noindent{\bf Remark} If you have been taught to distinguish between the 
words `set' and `class', you will observe that my naive set theory 
is a relatively tolerant one in that it is willing to allow class 
variables in its theorems. 
} 
      
\vleader{48pt}{2A1C}{Ordinals} An {\bf ordinal} (sometimes called a `von 
Neumann ordinal') is a set $\xi$ such that 
      
\inset{if $\eta\in\xi$ then $\eta$ is a set and $\eta\not\in\eta$, 
      
if $\eta\in\zeta\in\xi$ then $\eta\in\xi$, 
      
writing `$\eta\le\zeta$' to mean `$\eta\in\zeta$ or 
$\eta=\zeta$', $(\xi,\le)$ is well-ordered\dvro{.}{}} 
      
\prooflet{ 
\noindent(\Enderton, p.\ 191;  \Halmos, \S19;  \Henle, p.\ 27; \Krivine, 
p.\ 24;  \Roitman, 3.2.8.\cmmnt{ Of course many set 
theories do not allow sets to belong to themselves, and/or take it for 
granted that every object of discussion is a set, but I prefer not to 
take a view on such points in general.}) 
} 
      
\leader{2A1D}{Basic facts about ordinals (a)} If $\xi$ is an ordinal, 
then every member of $\xi$ is an ordinal. \prooflet{(\Enderton, p.\ 192; 
\Henle, 6.4; 
\Krivine, p.\ 14;  \Roitman, 3.2.10.)} 
      
\header{2A1Db}{\bf (b)} If $\xi$, $\eta$ are ordinals then either 
$\xi\in\eta$ or $\xi=\eta$ or $\eta\in\xi$ (and no two of these can 
occur together).   \prooflet{(\Enderton, 
p.\ 192;  \Henle, 6.4;  \Krivine, p.\ 14;  \Lipschutz, 11.12;  \Roitman, 
3.2.13.)}   It is customary\cmmnt{, in this case,} to 
write $\eta<\xi$ if $\eta\in\xi$ and $\eta\le\xi$ if either $\eta\in\xi$ 
or $\eta=\xi$.   Note that $\eta\le\xi$ iff $\eta\subseteq\xi$. 
      
\header{2A1Dc}{\bf (c)} If $A$ is any non-empty class of ordinals, then 
there is an $\alpha\in A$ such that $\alpha\le\xi$ for every 
$\xi\in A$.   \prooflet{(\Henle, 6.7;  \Krivine, p.\ 15.)} 
      
\header{2A1Dd}{\bf (d)} If $\xi$ is an ordinal, so is $\xi\cup\{\xi\}$; 
call it `$\xi+1$'.   \cmmnt{If $\xi<\eta$ then $\xi+1\le\eta$;}  $\xi+1$ is 
the least ordinal greater than $\xi$.   \prooflet{(\Enderton, p.\ 193; 
\Henle, 6.3; 
\Krivine, p.\ 15.)}   For any ordinal $\xi$, either there is a greatest 
ordinal $\eta<\xi$, in which case $\xi=\eta+1$ and we call $\xi$ a {\bf 
successor ordinal}, or $\xi=\bigcup\xi$, in which case we call $\xi$ a 
{\bf limit ordinal}. 
      
\header{2A1De}{\bf (e)} The first few ordinals are $0=\emptyset$, 
$1=0+1=\{0\}=\{\emptyset\}$, 
$2=1+1=\{0,1\}=\{\emptyset,\{\emptyset\}\}$, 
$3=2+1=\{0,1,2\}$, $\ldots$.   The first infinite ordinal is 
$\omega=\{0,1,2,\ldots\}$, which may be identified with $\Bbb N$. 
      
\header{2A1Df}{\bf (f)} The union of any set of ordinals is an ordinal. 
\prooflet{(\Enderton, p.\ 193; \Henle, 6.8;   \Krivine, p.\ 15; 
\Roitman, 3.2.19.)} 
      
\header{2A1Dg}{\bf (g)} If $(P,\le)$ is any well-ordered set, there is a 
unique ordinal $\xi$ such that $P$ is order-isomorphic to $\xi$, and the 
order-isomorphism is unique.   \prooflet{(\Enderton, pp.\ 187-189; 
\Henle, 6.13;  \Halmos, \S20.)} 
      
\leader{2A1E}{Initial ordinals} An {\bf initial ordinal} is an ordinal 
$\kappa$ such that there is no bijection between $\kappa$ and any member 
of $\kappa$.  \prooflet{(\Enderton, p.\ 197;  \Halmos, \S25;  \Henle, 
p.\ 34;  \Krivine, p.\ 24;  \Roitman, 5.1.10, p.\ 79).} 
      
\leader{2A1F}{Basic facts about initial ordinals (a)} All finite 
ordinals, and the first infinite ordinal $\omega$, are initial ordinals. 
      
\header{2A1Fb}{\bf (b)} For every well-ordered set $P$ there is a unique 
initial ordinal $\kappa$ such that there is a bijection between $P$ and 
$\kappa$. 
      
\header{2A1Fc}{\bf (c)} For every ordinal $\xi$ there is a least initial 
ordinal greater than $\xi$.   \prooflet{(\Enderton, p.\ 195;  \Henle, 
7.2.1.)}   If 
$\kappa$ is an initial ordinal, write $\kappa^+$ for the least initial 
ordinal greater than $\kappa$.   We write $\omega_1$ for $\omega^+$, 
$\omega_2$ for $\omega_1^+$, and so on. 
      
\header{2A1Fd}{\bf (d)} For any initial ordinal $\kappa\ge\omega$ there 
is a bijection between $\kappa\times\kappa$ and $\kappa$;  consequently 
there are bijections between $\kappa$ and $\kappa^r$ for every $r\ge 1$. 
      
\leader{2A1G}{Schr\"oder-Bernstein theorem} I\cmmnt{ remind you of the 
following 
fundamental result:  i}f $X$ and $Y$ are sets and there are injections 
$f:X\to Y$, $g:Y\to X$ then there is a bijection $h:X\to Y$. 
\prooflet{(\Enderton, p.\ 147;  \Halmos, \S22;  \Henle, 7.4; 
\Lipschutz, p.\ 145;  \Roitman, 5.1.2.   \cmmnt{It is also a special 
case of 344D in Volume 3.})} 
      
\leader{2A1H}{Countable subsets of $\Cal P\Bbb N$} \cmmnt{The 
following results will be needed below. 
      
\medskip 
      
}{\bf (a)} There is a bijection between $\Cal P\Bbb N$ and $\Bbb R$. 
\cmmnt{(\Enderton, p.\ 149;  \Lipschutz, p.\ 146.)} 
      
\spheader 2A1Hb Suppose that $X$ is any set such that there is 
an injection from $X$ into $\Cal P\Bbb N$.   Let $\Cal C$ be the set of 
countable subsets of $X$.   Then there is a surjection from 
$\Cal P\Bbb N$ onto $\Cal C$.   \prooflet{\Prf\ Let $f:X\to\Cal P\Bbb N$ be an injection.   Set $f_1(x)=\{0\}\cup\{i+1:i\in f(x)\}$;  then 
$f_1:X\to\Cal P\Bbb N$ is injective and $f_1(x)\ne\emptyset$ for every 
$x\in X$.   Define $g:\Cal P\Bbb N\to\Cal PX$ by setting 
      
\Centerline{$g(A)=\{x:\exists\,n\in\Bbb N,\,f_1(x) 
=\{i:2^n(2i+1)\in A\}\}$} 
      
\noindent for each $A\subseteq\Bbb N$.    Then $g(A)$ is countable, 
since we have an injection 
      
\Centerline{$x\mapsto\min\{n:f_1(x)=\{i:2^n(2i+1)\in A\}\}$} 
      
\noindent from $g(A)$ to $\Bbb N$.   Thus $g$ is a function from $\Cal 
P\Bbb N$ to $\Cal C$.   To see that $g$ is surjective, observe that 
$\emptyset=g(\emptyset)$, while if $C\subseteq X$ is countable and not 
empty there is a surjection $h:\Bbb N\to C$;  now set 
      
\Centerline{$A=\{2^n(2i+1):n\in\Bbb N,\,i\in f_1(h(n))\}$,} 
      
\noindent and see that $g(A)=C$.\ \Qed} 
      
\spheader 2A1Hc Again suppose that $X$ is a set such that there 
is an injection from $X$ to $\Cal P\Bbb N$, and write $H$ for the set of 
functions $h$ such that $\dom h$ is a countable subset of $X$ and $h$ 
takes values in $\{0,1\}$.   Then there is a surjection from $\Cal P\Bbb 
N$ onto $H$.   \prooflet{\Prf\ Let $\Cal C$ be the set of countable 
subsets of $X$ and let $g:\Cal P\Bbb N\to\Cal C$ be a surjection, as in 
(a).   For $A\subseteq\Bbb N$ set 
      
\Centerline{$g_0(A)=g(\{i:2i\in A\})$,\quad$g_1(A)=g(\{i:2i+1\in A\})$,} 
      
\noindent so that $g_0(A)$, $g_1(A)$ are countable subsets of $X$, and 
$A\mapsto(g_0(A),g_1(A))$ is a surjection from $\Cal P\Bbb N$ onto 
$\Cal C\times\Cal C$.   Let $h_A$ be the function with domain $g_0(A)\cup g_1(A)$ such that $h_A(x)=1$ if $x\in g_1(A)$, $0$ if 
$x\in g_0(A)\setminus g_1(A)$.   Then $A\mapsto h_A$ is a surjection from $\Cal P\Bbb N$ onto $H$.\ \Qed} 
      
\leader{2A1I}{Filters} \cmmnt{I pause for a moment to discuss a 
construction which is of great value in investigating topological 
spaces, but has 
other uses, and in its nature belongs to elementary set theory (much 
more elementary, indeed, than the work above). 
      
\header{2A1Ia}}{\bf (a)} Let $X$ be a non-empty set.   A {\bf filter} on 
$X$ is a family $\Cal F$ of subsets of $X$ such that 
      
\inset{$X\in\Cal F$,\quad $\emptyset\notin\Cal F$, 
      
$E\cap F\in\Cal F$ whenever $E$, $F\in\Cal F$, 
      
$E\in\Cal F$ whenever $X\supseteq E\supseteq F\in\Cal F$.} 
      
\noindent The second condition implies (inducing on $n$) that 
$F_0\cap\ldots\cap F_n\in\Cal F$ whenever $F_0,\ldots,F_n\in\Cal F$. 
          
\header{2A1Ib}{\bf (b)} Let $X$, $Y$ be non-empty sets, $\Cal F$ a 
filter on $X$ and $f:D\to Y$ a function, where $D\in\Cal F$.   Then 
      
\Centerline{$\{E:E\subseteq Y,\,f^{-1}[E]\in\Cal F\}$} 
      
\noindent is a filter on $Y$\prooflet{ (because $f^{-1}[Y]=D$, 
$f^{-1}[\emptyset]=\emptyset$, 
$f^{-1}[E\cap F]=f^{-1}[E]\cap f^{-1}[F]$, 
$X\supseteq f^{-1}[E]\supseteq f^{-1}[F]$ whenever  
$Y\supseteq E\supseteq F$)};  I will call it $f[[\Cal F]]$,  
the {\bf image filter} of $\Cal F$ under $f$. 
      
\cmmnt{\medskip 
      
\noindent{\bf Remark} Of course there is a hidden variable in this 
notation.   Ordinarily in this book I regard a function $f$ as being 
defined by its domain $\dom f$ and its values on its domain;  that is, 
it is determined by its graph $\{(x,f(x)):x\in\dom f\}$, and indeed I 
normally do not distinguish between a function and its graph.   This 
means that when I write `$f:D\to Y$ is a function' then the class 
$D=\dom f$ can be recovered from the function, but the class $Y$ cannot; 
all I promise is that $Y$ includes the class $f[D]$ of values of $f$. 
Now in the notation $f[[\Cal F]]$ above we do actually need to know 
which set $Y$ it is to be a filter on, even though this cannot be 
discovered from knowledge of $f$ and $\Cal F$.   So you will always have 
to infer it from the context. 
} 
      
\leader{2A1J}{The Axiom of Choice}\cmmnt{ I come now to the second 
half of this 
section, in which I discuss concepts and theorems dependent on the Axiom 
of Choice.}   Let me remind you of the statement of this axiom: 
      
\inset{\hskip-20pt (AC) `whenever $I$ is a set and $\familyiI{X_i}$ 
is a family of non-empty sets indexed by $I$, there is a function $f$, 
with domain $I$, such that $f(i)\in X_i$ for every $i\in I$'.} 
      
\noindent The function $f$ is a {\bf choice function}\cmmnt{;  it picks  
out one member of each of the given family of non-empty sets  $X_i$}. 
      
\cmmnt{I believe that one's attitude to this principle is a matter for 
individual choice.   It is an indispensable foundation for very large 
parts of twentieth-century pure mathematics, including a substantial 
fraction of the present volume;  but there are also significant areas in 
which principles actually contradictory to it can be employed to 
striking effect, leading -- in my view -- to equally valid mathematics. 
(I will describe one of these in \S567 of Volume 5.) 
At present it is the case that more current mathematical activity, by 
volume, 
depends on asserting the axiom of choice than on all its rivals put 
together;  but it is a matter of judgement and taste where the most 
important, or exciting, ideas are to be found.   For the present volume 
I follow standard practice in twentieth-century abstract analysis, using 
the axiom of choice whenever necessary.} 
      
\vleader{60pt}{2A1K}{Zermelo's Well-Ordering Theorem (a)} The Axiom of  
Choice is equiveridical with each of the statements 
      
\inset{`for every set $X$ there is a well-ordering of $X$',} 
      
\inset{`for every set $X$ there is a bijection between $X$ and some 
ordinal',} 
      
\inset{`for every set $X$ there is a unique initial ordinal $\kappa$ 
such that there is a bijection between $X$ and $\kappa$.'} 

\prooflet{\noindent(\Enderton, p.\ 196 et 
seq.; \Halmos, \S17;  \Henle, 9.1-9.3;  \Krivine, p.\ 20;  \Lipschutz, 
12.1;  \Roitman, 3.6.38.)} 
      
\header{2A1Kb}{\bf (b)} When assuming the axiom of choice, \cmmnt{as I 
do 
nearly everywhere in this treatise,} I write $\#(X)$ for that initial 
ordinal $\kappa$ such that there is a bijection between $\kappa$ and 
$X$;  I call this the {\bf cardinal} of $X$. 
      
\leader{2A1L}{Fundamental consequences of the Axiom of Choice (a)} For 
any two sets $X$ and $Y$, there is a bijection between $X$ and $Y$ iff 
$\#(X)=\#(Y)$.   More generally, there is an injection from $X$ to $Y$ 
iff $\#(X)\le\#(Y)$, and a surjection from $X$ onto $Y$ iff 
either $\#(X)\ge\#(Y)>0$ or $\#(X)=\#(Y)=0$. 
      
\spheader 2A1Lb\cmmnt{ In particular,} 
$\#(\Cal P\Bbb N)=\#(\Bbb R)$; 
write $\frak c$ for this common value, the {\bf cardinal of the 
continuum}.   \cmmnt{Cantor's theorem that $\Cal P\Bbb N$ and $\Bbb R$ are uncountable becomes the result $\omega<\frak c$, that is,} 
$\omega_1\le\frak c$. 
      
\header{2A1Lc}{\bf (c)} 
If $X$ is any infinite set, and $r\ge 1$, then there is a bijection 
between $X^r$ and $X$.   \prooflet{(\Enderton, p.\ 162;  \Halmos, 
\S24.)}   \cmmnt{(I note that we need some form of the axiom of 
choice to prove the result in this generality.   But of course for most 
of the infinite sets arising naturally in mathematics -- sets like $\Bbb 
N$ and $\Cal P\Bbb R$ -- it is easy to prove the result without appeal 
to the axiom of choice.)} 
      
\header{2A1Ld}{\bf (d)} Suppose that $\kappa$ is an infinite cardinal. 
If $I$ is a set with cardinal at most $\kappa$ and  
$\langle A_i\rangle_{i\in I}$ is a family of sets with $\#(A_i)\le\kappa$  
for 
every $i\in I$, then  $\#(\bigcup_{i\in I}A_i)\le\kappa$.   Consequently 
$\#(\bigcup\Cal A)\le\kappa$ whenever $\Cal A$ is a family of sets such 
that $\#(\Cal A)\le\kappa$ and $\#(A)\le\kappa$ for every $A\in\Cal A$. 
\cmmnt{In particular,} $\omega_1$ cannot be expressed as a countable union of 
countable sets, and $\omega_2$ cannot be expressed as a countable union 
of sets with cardinal at most $\omega_1$. 
      
\header{2A1Le}{\bf (e)} Now we can rephrase 2A1Hc as:  if  
$\#(X)\le\frak c$, then $\#(H)\le\frak c$,  
where $H$ is the set of functions from a 
countable subset of $X$ to $\{0,1\}$.   \prooflet{\Prf\ For we have an 
injection from $X$ into $\Cal P\Bbb N$, and therefore a surjection from 
$\Cal P\Bbb N$ onto $H$.\ \Qed} 
      
\spheader 2A1Lf Any non-empty class of cardinals has a least 
member\cmmnt{ (by 2A1Dc)}.   
      
\leader{2A1M}{Zorn's Lemma}\cmmnt{ In 2A1K I described the 
well-ordering 
principle.}   I come now to another proposition which is equiveridical with 
the axiom of choice: 
      
\inset{`Let $(P,\le)$ be a non-empty partially ordered set such that 
every non-empty totally ordered subset of $P$ has an upper bound in $P$. 
Then $P$ has a maximal element.'} 
      
\noindent This is {\bf Zorn's Lemma}.   \prooflet{For the proof that the 
axiom of 
choice implies, and is implied by, Zorn's Lemma, see \Enderton, p.\ 151; 
\Halmos, \S16;  \Henle, 9.1-9.3;  \Roitman, 3.6.38.} 
      
\leader{2A1N}{Ultrafilters} A filter $\Cal F$ on a set $X$ is an {\bf 
ultrafilter} if for 
every $A\subseteq X$ either $A\in\Cal F$ or $X\setminus A\in\Cal F$. 
      
If $\Cal F$ is an ultrafilter on $X$ and $f:D\to Y$ is a function, where 
$D\in\Cal F$, then 
$f[[\Cal F]]$ is an ultrafilter on $Y$\prooflet{ (because 
$f^{-1}[Y\setminus A]=D\setminus f^{-1}[A]$ for every $A\subseteq Y$)}. 
      
One type of ultrafilter can be described easily:  if $x$ is 
any point of a set $X$, then $\Cal F=\{F:x\in F\subseteq X\}$ is an 
ultrafilter on $X$.   (\prooflet{You need only read the 
definitions.  }Ultrafilters of this type are called {\bf principal ultrafilters}.) 
\cmmnt{But it is not obvious that there are any further ultrafilters, 
and 
indeed it is not possible to prove that there are any, without using a 
strong form of the axiom of choice, as follows.} 
      
\leader{2A1O}{The Ultrafilter Theorem}\cmmnt{ As an example of the use 
of Zorn's lemma which will be of great value in studying compact 
topological spaces (2A3N {\it et seq.}, and \S247), I give the following 
result. 
      
\medskip 
      
\noindent{\bf Theorem}} Let $X$ be any non-empty set, and $\Cal F$ a 
filter on $X$.   Then there is an ultrafilter $\Cal H$ on $X$ such that 
$\Cal F\subseteq\Cal H$. 
      
\proof{ (Cf.\ \Henle, 9.4;  \Roitman, 3.6.37.)   Let $\frak P$ be the 
set of all filters on $X$ 
including $\Cal F$, and order $\frak P$ by inclusion, so that, for 
$\Cal G_1$, $\Cal G_2\in\frak P$, $\Cal G_1\le \Cal G_2$ in $\frak P$ iff $\Cal G_1\subseteq\Cal G_2$.  It is easy to see that $\frak P$ is a 
partially ordered set, and it is non-empty because $\Cal F\in \frak P$.   If $\frak Q$ is any non-empty totally ordered 
subset of $\frak P$, then $\Cal H_{\frak Q}=\bigcup\frak Q\in\frak P$. 
\Prf\ Of course $\Cal H_{\frak Q}$ is a family of subsets of $X$. (i) 
Take any $\Cal G_0\in\frak Q$;  then 
$X\in\Cal G_0\subseteq\Cal H_{\frak Q}$.   If $\Cal G\in\frak Q$, then $\Cal G$ is a filter, so 
$\emptyset\notin\Cal G$;  accordingly $\emptyset\notin\Cal H_{\frak Q}$. 
(ii) If $E$, $F\in\Cal H_{\frak Q}$, then there are $\Cal G_1$, $\Cal 
G_2\in\frak Q$ such that $E\in\Cal G_1$ and $F\in\Cal G_2$.   Because 
$\frak Q$ is totally ordered, either $\Cal G_1\subseteq\Cal G_2$ or 
$\Cal G_2\subseteq\Cal G_1$.   In either case, $\Cal G=\Cal G_1\cup\Cal 
G_2\in\frak Q$.   Now $\Cal G$ is a filter containing both $E$ and $F$, 
so it contains $E\cap F$, and $E\cap F\in\Cal H_{\frak Q}$.   (iii) If 
$X\supseteq E\supseteq F\in\Cal H_{\frak Q}$, there is a $\Cal G\in\frak 
Q$ such that $F\in\Cal G$;   and $E\in\Cal G\subseteq\Cal H_{\frak Q}$. 
This shows that $\Cal H_{\frak Q}$ is a filter on $X$.   (iv)  Finally, 
$\Cal H_{\frak Q}\supseteq\Cal G_0\supseteq\Cal F$, so 
$\Cal H_{\frak Q}\in\frak P$.\ \QeD\ Now $\Cal H_{\frak Q}$ is evidently  an upper bound for $\frak Q$ in $\frak P$. 
      
We may therefore apply Zorn's Lemma to find a maximal element $\Cal H$ 
of $\frak P$.   This $\Cal H$ is surely a filter on $X$ including 
$\Cal F$. 
      
Now let $A\subseteq X$ be such that $A\notin \Cal H$.   Consider 
      
\Centerline{$\Cal H_1=\{E:E\subseteq X,\,E\cup A\in\Cal H\}$.} 
      
\noindent This is a filter on $X$.   \Prf\ Of course it is a family of 
subsets of $X$.   (i) $X\cup A=X\in\Cal H$, so $X\in\Cal H_1$. 
$\emptyset\cup A=A\notin\Cal H$ so $\emptyset\notin\Cal H_1$.   (ii) If 
$E$, $F\in\Cal H_1$ then 
      
\Centerline{$(E\cap F)\cup A=(E\cup A)\cap(F\cup A) 
\in\Cal 
H$,} 
      
\noindent so $E\cap F\in\Cal H_1$.   (iii) If 
$X\supseteq E\supseteq F\in\Cal H_1$ then 
$E\cup A\supseteq F\cup A\in\Cal H$, so $E\cup A\in\Cal H$ and 
$E\in\Cal H_1$.\ \QeD\  Also $\Cal H_1\supseteq\Cal H$, so 
$\Cal H_1\in\frak P$.   But $\Cal H$ is a maximal element of $\frak P$, so $\Cal H_1=\Cal H$.   Since $(X\setminus A)\cup A=X\in\Cal H$, 
$X\setminus A\in\Cal H_1$ and $X\setminus A\in\Cal H$. 
      
As $A$ is arbitrary, $\Cal H$ is an ultrafilter, as required. 
}%end of proof of 2A1O 
      
      
      
\leader{2A1P}{}\cmmnt{ I come now to a result from infinitary 
combinatorics for 
which I give a detailed proof, not because it cannot be found in many 
textbooks, but because it is usually given in enormously greater 
generality, to the point indeed that it may be harder to understand why 
the stated theorem covers the present result than to prove the latter 
from first principles. 
      
\medskip 
      
\noindent}{\bf Theorem} (a) Let 
$\langle K_{\alpha}\rangle_{\alpha\in A}$ be 
a family of countable sets, with $\#(A)$ strictly greater than 
$\frak c$, the cardinal of the continuum.   Then there are a set $M$, with cardinal at 
most $\frak c$, and a set $B\subseteq A$, with cardinal strictly greater 
than $\frak c$, such that $K_{\alpha}\cap K_{\beta}\subseteq M$ whenever 
$\alpha$, $\beta$ are distinct members of $B$. 
      
(b) Let $I$ be a set, and 
$\langle f_{\alpha}\rangle_{\alpha\in A}$ a family in $\{0,1\}^I$, the set of functions from $I$ to $\{0,1\}$, with $\#(A)>\frak c$.    If $\langle K_{\alpha}\rangle_{\alpha\in A}$ is any family of countable subsets of 
$I$, then there is a set $B\subseteq A$, with cardinal greater than 
$\frak c$, such that $f_{\alpha}$ and $f_{\beta}$ agree on 
$K_{\alpha}\cap K_{\beta}$ for all $\alpha$, $\beta\in B$. 
      
(c) In particular, under the conditions of (b), there are distinct 
$\alpha$, $\beta\in A$ such that $f_{\alpha}$ and $f_{\beta}$ agree on 
$K_{\alpha}\cap K_{\beta}$. 
      
\proof{{\bf (a)} Choose inductively a family 
$\langle M_{\xi}\rangle_{\xi<\omega_1}$ of sets by the rule 
      
\inset{if there is any set $N$ such that 
      
\inset{($*$) $N$ is disjoint from $\bigcup_{\eta<\xi}M_{\eta}$, 
$\#(N)\le\frak c$ and
%
%\qquad\qquad
$\#(\{\alpha:\alpha\in A,\,K_{\alpha}\cap N=\emptyset\}) 
  \le\frak c$,} 
      
\noindent choose such a set for $M_{\xi}$; 
      
otherwise set $M_{\xi}=\emptyset$.} 
      
\noindent When $M_{\xi}$ has been chosen for every $\xi<\omega_1$, set 
$M=\bigcup_{\xi<\omega_1}M_{\xi}$.   The rule ensures that $\ofamily{\xi}{\omega_1}{M_{\xi}}$ is disjoint and that 
$\#(M_{\xi})\le\frak c$ for every $\xi<\omega_1$, while 
$\omega_1\le\frak c$, so $\#(M)\le\frak c$. 
      
Let $\frak P$ be the family of sets $P\subseteq A$ such that 
$K_{\alpha}\cap K_{\beta}\subseteq M$ for all distinct $\alpha$, $\beta\in P$.   Order 
$\frak P$ by inclusion, so that it is a partially ordered set.   If 
$\frak Q\subseteq\frak P$ is totally ordered, then $\bigcup\frak Q\in\frak P$. 
\Prf\ If $\alpha$, $\beta$ are distinct members of $\bigcup\frak Q$, 
there are $Q_1$, $Q_2\in \frak Q$ such that $\alpha\in Q_1$, 
$\beta\in Q_2$; now 
$P=Q_1\cup Q_2$ is equal to one of $Q_1$, $Q_2$, and in either case 
belongs to $\frak P$ and contains both $\alpha$ and $\beta$, so $K_{\alpha}\cap K_{\beta}\subseteq M$.\ \QeD\  By Zorn's Lemma, 
$\frak P$ has a maximal 
element $B$, and we surely have $K_{\alpha}\cap K_{\beta}\subseteq M$ 
for all distinct $\alpha$, $\beta\in B$. 
      
\Quer\ Suppose, if possible, that $\#(B)\le\frak c$.   Set 
$N=\bigcup_{\alpha\in B}K_{\alpha}\setminus M$.   Then $N$ has cardinal 
at most $\frak c$, being included in a union of at most $\frak c$ 
countable sets.   For every $\gamma\in A\setminus B$, 
$B\cup\{\gamma\}\notin\frak P$, so there must be some $\alpha\in B$ such 
that $K_{\alpha}\cap K_{\gamma}\not\subseteq M$;  that is, 
$K_{\gamma}\cap N\ne\emptyset$.   Thus 
$\{\gamma:K_{\gamma}\cap N=\emptyset\}\subseteq B$ 
has cardinal at most $\frak c$.   But this means that in the rule for 
choosing $M_{\xi}$, there was always an $N$ satisfying the condition 
($*$), and therefore $M_{\xi}$ also did.   Thus 
$C_{\xi}=\{\alpha:K_{\alpha}\cap M_{\xi}=\emptyset\}$ has cardinal at most $\frak c$ for every 
$\xi<\omega_1$.   So $C=\bigcup_{\xi<\omega_1}C_{\xi}$ also has.   But 
the original hypothesis was that $\#(A)>\frak c$, so there is an $\alpha\in A\setminus C$.   In this case, 
$K_{\alpha}\cap M_{\xi}\ne\emptyset$ for 
every $\xi<\omega_1$.   But this means that we have a surjection 
$\phi:K_{\alpha}\cap M\to\omega_1$ given by setting 
      
\Centerline{$\phi(i)=\xi$ if $i\in K_{\alpha}\cap M_{\xi}$.} 
      
\noindent Since $\#(K_{\alpha})\le\omega<\omega_1$, this is 
impossible.\ \Bang 
      
Accordingly $\#(B)>\frak c$ and we have found a suitable pair $M$, $B$. 
      
\medskip 
      
{\bf (b)} By (a), we can find a set $M$, with cardinal at most $\frak c$, 
and a set $B_0\subseteq A$, with cardinal greater than $\frak c$, such that $K_{\alpha}\cap K_{\beta}\subseteq M$ for all distinct $\alpha$, 
$\beta\in B_0$.   Let $H$ be the set of functions from countable subsets of $M$ to $\{0,1\}$;  then 
$f'_{\alpha}=f_{\alpha}\restr(K_{\alpha}\cap M)\in H$ for 
each $\alpha\in B_0$.   Now 
$B_0=\bigcup_{h\in H}\{\alpha:\alpha\in B_0,\,f'_{\alpha}=h\}$ has cardinal greater than $\frak c$, while 
$\#(H)\le\frak c$ (2A1Le), so there must be some $h\in H$ such that 
$B=\{\alpha:\alpha\in B_0,\,f'_{\alpha}=h\}$ has cardinal greater than 
$\frak c$. 
      
If $\alpha$, $\beta$ are distinct members of $B$, then 
$K_{\alpha}\cap K_{\beta}\subseteq M$, because $\alpha$, $\beta\in B_0$;  but this means that 
      
\Centerline{$f_{\alpha}\restr K_{\alpha}\cap K_{\beta} 
=h\restr K_{\alpha}\cap K_{\beta} 
=f_{\beta}\restr K_{\alpha}\cap K_{\beta}$.} 
      
\noindent Thus $B$ has the required property.   (Of course $f_{\alpha}$ 
and $f_{\beta}$ agree on $K_{\alpha}\cap K_{\beta}$ if $\alpha=\beta$.) 
      
\medskip 
      
{\bf (c)} follows at once. 
}%end of proof of 2A1P 
%elementary submodel argument 
      
\cmmnt{\medskip 
      
\noindent{\bf Remark} The result we need in this volume (in 216E) is 
part (c) above.   There are other proofs of this,  
perhaps a little simpler; but the stronger result in part (b) will be 
useful in Volume 3. 
}%end of comment 
      
\discrpage 
      
\frfilename{mt2a2.tex} 
\versiondate{30.11.09} 
\copyrightdate{2000} 
 
\def\chaptername{Appendix} 
\def\sectionname{The topology of Euclidean space} 
 
\newsection{2A2} 
 
In the appendix to Volume 1\cmmnt{ (\S1A2)} I 
discussed open and closed sets in $\BbbR^r$;  the 
chief aim there was to support the idea of `Borel set', which is 
vital in the theory of Lebesgue measure, but of course they are also 
fundamental to the study of continuous functions, and indeed to all 
aspects of real analysis. 
I give here a very brief introduction to the further elementary facts 
about closed and compact sets and continuous functions which we need for 
this volume.   Much of this material can be derived from the 
generalizations in \S2A3, but nevertheless I sketch the proofs, since 
for the greater 
part of the volume (most of the exceptions are in Chapter 24) Euclidean 
space is sufficient for our needs. 
 
\leader{2A2A}{Closures:  Definition} For any $r\ge 1$ and any 
$A\subseteq\BbbR^r$, the {\bf closure} of $A$, $\overline{A}$, is the 
intersection of all the closed subsets of $\BbbR^r$ including $A$. 
This is\cmmnt{ itself closed (being the intersection of a non-empty 
family of closed sets, see 1A2Fd), so is} the smallest closed set 
including $A$.   \cmmnt{In particular, }$A$ is closed iff $\overline{A}=A$. 
 
\leader{2A2B}{Lemma} Let $A\subseteq\BbbR^r$ be any set.   Then for 
$x\in\BbbR^r$ the following are equiveridical: 
 
(i) $x\in\overline{A}$\cmmnt{, the closure of $A$}; 
 
(ii) $B(x,\delta)\cap A\ne\emptyset$ for every $\delta>0$, where 
$B(x,\delta)=\{y:\|y-x\|\le\delta\}$; 
 
(iii) there is a sequence $\sequencen{x_n}$ in $A$ such that 
$\lim_{n\to\infty}\|x_n-x\|=0$. 
 
\proof{{\bf (a)(i)$\Rightarrow$(ii)} Suppose that $x\in\overline{A}$ and 
$\delta>0$.   Then $U(x,\delta)=\{y:\|y-x\|<\delta\}$ is an open set 
(1A2D), so $F=\BbbR^r\setminus U(x,\delta)$ is closed, while  
$x\notin F$.   Now 
 
\Centerline{$x\in\overline{A}\setminus F 
\Longrightarrow\overline{A}\not\subseteq F 
\Longrightarrow A\not\subseteq F 
\Longrightarrow A\cap U(x,\delta)\ne\emptyset 
\Longrightarrow A\cap B(x,\delta)\ne\emptyset$.} 
 
\noindent As $\delta$ is arbitrary, (ii) is true. 
 
\medskip 
 
{\bf (b)(ii)$\Rightarrow$(iii)} If (ii) is true, then for each $n\in\Bbb 
N$ we can find an $x_n\in A$ such that $\|x_n-x\|\le 2^{-n}$, and now 
$\lim_{n\to\infty}\|x_n-x\|=0$. 
 
\medskip 
 
{\bf (c)(iii)$\Rightarrow$(i)} Assume (iii).   \Quer\ Suppose, if 
possible, that $x\notin\overline{A}$.   Then $x$ belongs to the open set 
$\BbbR^r\setminus\overline{A}$ and there is a $\delta>0$ such that 
$U(x,\delta)\subseteq\BbbR^r\setminus\overline{A}$.   But now there 
is an $n$ such that $\|x_n-x\|<\delta$, in which case $x_n\in 
U(x,\delta)\cap A\subseteq U(x,\delta)\cap\overline{A}$.   \Bang 
}%end of proof of 2A2B 
 
\leader{2A2C}{Continuous functions (a)} \cmmnt{I begin with a 
characterization of continuous functions 
in terms of open sets.}   If $r$, $s\ge 1$, $D\subseteq\BbbR^r$ and 
$\phi:D\to\BbbR^s$ is a function, we say that $\phi$ is {\bf 
continuous} if for every $x\in D$ and $\epsilon>0$ there is a $\delta>0$ 
such that $\|\phi(y)-\phi(x)\|\le\epsilon$ whenever $y\in D$ and 
$\|y-x\|\le\delta$.   \cmmnt{Now }$\phi$ is continuous 
iff for every open set 
$G\subseteq\BbbR^s$ there is an open set $H\subseteq\BbbR^r$ such that 
$\phi^{-1}[G]=D\cap H$. 
 
\prooflet{\Prf\ {\bf (i)} Suppose that $\phi$ is continuous and that 
$G\subseteq\BbbR^s$ is open.   Set 
 
\Centerline{$H=\bigcup\{U:U\subseteq\BbbR^r\text{ is open},\,\phi[U\cap 
D]\subseteq G\}$.} 
 
\noindent Then $H$ is a union of open sets, therefore open (1A2Bd), and 
$H\cap D\subseteq\phi^{-1}[G]$.   If $x\in\phi^{-1}[G]$, then 
$\phi(x)\in G$, so there is an $\epsilon>0$ such that 
$U(\phi(x),\epsilon)\subseteq G$;  now there is a $\delta>0$ such 
that $\|\phi(y)-\phi(x)\|\le\bover12\epsilon$ whenever $y\in D$ and 
$\|y-x\|\le\delta$, so that 
 
\Centerline{$\phi[U(x,\delta)\cap D]
\subseteq U(\phi(x),\epsilon)\subseteq G$} 
 
\noindent and 
 
\Centerline{$x\in U(x,\delta)\subseteq H$.} 
 
\noindent As $x$ is arbitrary, $\phi^{-1}[G]=H\cap D$.   As $G$ is 
arbitrary, $\phi$ satisfies the condition. 
 
\quad{\bf (ii)} Now suppose that $\phi$ satisfies the condition.   Take 
$x\in D$ and $\epsilon>0$.   Then $U(\phi(x),\epsilon)$ is open, 
so 
there is an open $H\subseteq\BbbR^r$ such that 
$H\cap D=\phi^{-1}[U(\phi(x),\epsilon)]$;  we see that $x\in H$, so 
there is a $\delta>0$ such that $U(x,\delta)\subseteq H$;  now if 
$y\in 
D$ and $\|y-x\|\le\bover12\delta$ then $y\in D\cap H$, $\phi(y)\in 
U(\phi(x),\epsilon)$ and $\|\phi(y)-\phi(x)\|\le\epsilon$.   As $x$ and 
$\epsilon$ are arbitrary, $\phi$ is continuous.   \Qed} 
 
\header{2A2Cb}{\bf (b)} Using the $\epsilon$-$\delta$ definition of 
continuity, it is 
easy to see that a function $\phi$ from a subset $D$ of $\BbbR^r$ to 
$\BbbR^s$ is continuous iff all its components $\phi_i$ are continuous, 
writing $\phi(x)=(\phi_1(x),\ldots,\phi_s(x))$ for $x\in D$. 
\prooflet{\Prf\ 
{\bf (i)} If $\phi$ is continuous, $i\le s$, $x\in D$ and $\epsilon>0$, 
then there is a $\delta>0$ such that 
 
\Centerline{$|\phi_i(y)-\phi_i(x)|\le\|\phi(y)-\phi(x)\|\le\epsilon$} 
 
\noindent whenever $y\in D$ and $\|y-x\|\le\delta$.   {\bf (ii)} If 
every $\phi_i$ is continuous, $x\in D$ and $\epsilon>0$, then there are 
$\delta_i>0$ such that $|\phi_i(y)-\phi_i(x)|\le\epsilon/\sqrt{s}$ 
whenever $y\in D$ and $\|y-x\|\le\delta_i$;  setting  
$\delta=\min_{1\le i\le r}\delta_i>0$, we have  
$\|\phi(y)-\phi(x)\|\le\epsilon$ whenever $y\in D$ and  
$\|y-x\|\le\delta$.  \Qed} 

\spheader 2A2Cc\dvAnew{2015}\cmmnt{ At one or two points, 
we shall encounter the 
following strengthening of the notion of `continuous function'.}    
If $r$, $s\ge 1$, $D\subseteq\BbbR^r$ and 
$\phi:D\to\BbbR^s$ is a function, we say that $\phi$ is 
{\bf uniformly continuous} if for every $\epsilon>0$ there is a $\delta>0$ 
such that $\|\phi(y)-\phi(x)\|\le\epsilon$ whenever $x$, $y\in D$ and 
$\|y-x\|\le\delta$.    A uniformly continuous function is\cmmnt{ of course}
continuous.

\leader{2A2D}{Compactness in $\BbbR^r$: Definition} A subset $F$ of 
$\BbbR^r$ is called {\bf compact} if whenever 
$\Cal G$ is a family of open sets covering $F$ then there is a finite 
subset $\Cal G_0$ of $\Cal G$ still covering $F$. 
 
\leader{2A2E}{Elementary properties of compact sets} Take any $r\ge 1$, and
subsets $D$, $F$, $G$ and $K$ of $\BbbR^r$. 
 
\header{2A2Ea}{\bf (a)} If $K$ is compact and 
$F$ is closed, then $K\cap F$ is 
compact.   \prooflet{\Prf\ Let $\Cal G$ be an open cover of $F\cap K$. 
Then $\Cal G\cup\{\BbbR^r\setminus F\}$ is an open cover of $K$, so has a finite subcover $\Cal G_0$ say.   Now  
$\Cal G_0\setminus\{\BbbR^r\setminus F\}$ is a finite subset of $\Cal G$ covering $K\cap F$.   As $\Cal G$ is arbitrary, $K\cap F$ is compact. 
\Qed} 
 
\header{2A2Eb}{\bf (b)} If $s\ge 1$, $\phi:D\to\BbbR^s$ is a 
continuous function, $K$ is compact and $K\subseteq D$, then $\phi[K]$ is 
compact. 
\prooflet{\Prf\ Let $\Cal V$ be an open cover of $\phi[K]$.   Let 
$\Cal H$ be 
 
\Centerline{$\{H:H\subseteq\BbbR^r\text{ is open},\,\exists\enskip 
V\in\Cal V,\,\phi^{-1}[V]=D\cap H\}$.} 
 
\noindent If $x\in K$, then $\phi(x)\in\phi[K]$ so there is a 
$V\in\Cal V$ such that $\phi(x)\in V$;  now there is an $H\in\Cal H$ such 
that $D\cap H\phi^{-1}[V]$ contains $x$ (2A2Ca);  as $x$ is arbitrary, 
$K\subseteq\bigcup\Cal H$.   Let $\Cal H_0$ be a finite subset of $H$ 
covering $K$.   For each 
$H\in\Cal H_0$, let $V_H\in\Cal V$ be such that 
$\phi^{-1}[V_H]=D\cap H$;  then $\{V_H:H\in\Cal H_0\}$ is a finite subset 
of $\Cal V$ covering 
$\phi[K]$.   As $\Cal V$ is arbitrary, $\phi[K]$ is compact.   \Qed} 
 
\spheader 2A2Ec If $K$ is compact, it is closed. 
\prooflet{\Prf\ Write $H=\BbbR^r\setminus K$.   Take any $x\in H$. 
Then $G_n=\BbbR^r\setminus B(x,2^{-n})$ is open for every $n\in\Bbb N$ 
(1A2G).   Also 
 
\Centerline{$\bigcup_{n\in\Bbb N}G_n=\{y:y\in\BbbR^r,\,\|y-x\|>0\}
=\Bbb R^r\setminus\{x\}\supseteq K$.} 
 
\noindent So there is some finite set 
$\Cal G_0\subseteq\{G_n:n\in\Bbb N\}$ which covers $K$.   
There must be an $n$ such that $\Cal G_0\subseteq\{G_i:i\le n\}$, so that 
 
\Centerline{$K\subseteq\bigcup\Cal G_0\subseteq\bigcup_{i\le n}G_i=G_n$,} 
 
\noindent and $B(x,2^{-n})\subseteq H$.   As $x$ is arbitrary, $H$ is 
open and $K$ is closed.  \Qed} 
 
\spheader 2A2Ed If $K$ is compact and $G$ is open and $K\subseteq G$, 
then there is a $\delta>0$ such that $K+B(\tbf{0},\delta)\subseteq G$. 
\prooflet{\Prf\ If $K=\emptyset$, this is trivial, as then 
 
\Centerline{$K+B(\tbf{0},1)=\{x+y:x\in K,\,y\in B(\tbf{0},1)\}=\emptyset$.} 
 
\noindent  Otherwise, set 
 
\Centerline{$\Cal G
=\{U(x,\delta):x\in \Bbb R^r,\,\delta>0,\,U(x,2\delta)\subseteq G\}$.} 
 
\noindent Then $\Cal G$ is a family of open sets and 
$\bigcup\Cal G=G$ (because $G$ is open), 
so $\Cal G$ is an open cover of $K$ and has 
a finite subcover $\Cal G_0$.   Express $\Cal G_0$ as 
$\{U(x_0,\delta_0),\ldots,U(x_n,\delta_n)\}$ where 
$U(x_i,2\delta_i)\subseteq G$ for each $i$.   Set 
$\delta=\min_{i\le n}\delta_i>0$.   If $x\in K$ and 
$y\in B(\tbf{0},\delta)$, then there 
is an $i\le n$ such that $x\in U(x_i,\delta_i)$;  now 
 
\Centerline{$\|(x+y)-x_i\|\le\|x-x_i\|+\|y\|<\delta_i+\delta
\le 2\delta_i$,} 
 
\noindent so $x+y\in U(x_i,2\delta_i)\subseteq G$.   As $x$ and $y$ 
are arbitrary, $K+B(\tbf{0},\delta)\subseteq G$.   \Qed} 
 
\leader{2A2F}{}\cmmnt{ The value of the concept of `compactness' is 
greatly 
increased by the fact that there is an effective characterization of the 
compact subsets of $\BbbR^r$. 
 
\medskip 
 
\noindent}{\bf Theorem} For any $r\ge 1$, a subset $K$ of $\BbbR^r$ is 
compact iff it is closed and bounded. 
 
\proof{{\bf (a)} Suppose that $K$ is compact.   By 2A2Ec, it is closed. 
To see that it is bounded, consider $\Cal G=\{U(\tbf{0},n):n\in\Bbb 
N\}$.   $\Cal 
G$ consists entirely of open sets, and $\bigcup\Cal G=\Bbb 
R^r\supseteq K$, so there is a finite $\Cal G_0\subseteq\Cal G$ covering 
$K$. There must be an $n$ such that $\Cal G_0\subseteq\{G_i:i\le n\}$, 
so that 
 
\Centerline{$K\subseteq\bigcup\Cal G_0\subseteq\bigcup_{i\le 
n}U(\tbf{0},i)=U(\tbf{0},n)$,} 
 
\noindent and $K$ is bounded. 
 
\medskip 
 
{\bf (b)} Thus we are left with the converse;  I have to show that a 
closed bounded set is compact.   The main part of the argument is a 
proof by induction on $r$ that the closed interval $[-\tbf{n},\tbf{n}]$ 
is compact for all $n\in\Bbb N$, writing $\tbf{n}=(n,\ldots,n)\in\Bbb 
R^r$. 
 
\medskip 
 
\quad{\bf (i)} If $r=1$ and $n\in\Bbb N$ and $\Cal G$ is a family of 
open sets in $\Bbb R$ covering $[-n,n]$, set 
 
\Centerline{$A=\{x:x\in[-n,n],\text{ there is a finite }\Cal 
G_0\subseteq\Cal G\text{ such that }[-n,x]\subseteq\bigcup\Cal G_0\}$.} 
 
\noindent Then 
 $-n\in A$, because if $-n\in G\in\Cal G$ then 
$[-n,-n]\subseteq\bigcup\{G\}$, and $A$ is bounded above by $n$, so 
$c=\sup A$ exists and belongs to $[-n,n]$. 
 
Next, $c\in[-n,n]\subseteq\bigcup\Cal G$, so there is a $G\in\Cal G$ 
containing $c$.   Let $\delta>0$ be such that $U(c,\delta)\subseteq 
G$. 
There is an $x\in A$ such that $x\ge c-\delta$.   Let $\Cal G_0$ be a 
finite subset of $\Cal G$ covering $[-n,x]$.  Then $\Cal G_1=\Cal 
G_0\cup\{G\}$ is a finite subset of $\Cal G$ covering $[-n,c+{1\over 
2}\delta]$. 
But $c+{1\over2}\delta\notin A$ so $c+{1\over2}\delta>n$ and $\Cal G_1$ 
is a finite subset of $\Cal G$ covering $[-n,n]$.   As $\Cal G$ is 
arbitrary, $[-n,n]$ is compact and the induction starts. 
 
\medskip 
 
\quad{\bf (ii)} For the inductive step to $r+1$, regard the closed 
interval $F=[-\tbf{n},\tbf{n}]$, taken in $\BbbR^{r+1}$, as the product 
of the closed interval $E=[-\tbf{n},\tbf{n}]$, taken in $\BbbR^r$, with 
the closed interval $[-n,n]\subseteq\Bbb R$;  by the inductive 
hypothesis, both $E$ and $[-n,n]$ are compact.   Let $\Cal G$ be a 
family of open subsets of $\BbbR^{r+1}$ covering $F$.   Write $\Cal H$ 
for the family of open subsets $H$ of $\BbbR^r$ such that 
$H\times[-n,n]$ is covered by a finite subfamily of $\Cal G$.   Then 
$E\subseteq\bigcup\Cal H$.   \Prf\ Take $x\in E$.   Set 
 
\Centerline{$\Cal U_x=\{U:U\subseteq\Bbb R\text{ is 
open},\,\exists\enskip G\in\Cal G,\text{ open }H\subseteq\BbbR^r,\, 
x\in H\text{ and }H\times U\subseteq G\}$.} 
 
\noindent Then $\Cal U_x$ is a family of open subsets of $\Bbb R$.   If 
$\xi\in[-n,n]$, there is a $G\in\Cal G$ containing $(x,\xi)$;  there is 
a $\delta>0$ such that $U((x,\xi),\delta)\subseteq G$;  now 
$U(x,\bover12\delta)$ and $U(\xi,\bover12\delta)$ are open sets in 
$\BbbR^r$, $\Bbb R$ respectively and 
 
\Centerline{$U(x,\bover12\delta)\times U(\xi,\bover12\delta)
\subseteq U((x,\xi),\delta)\subseteq G$,} 
 
\noindent so $U(\xi,\bover12\delta)\in\Cal U_x$.   As $\xi$ is 
arbitrary, $\Cal U_x$ is an open cover of $[-n,n]$ in $\Bbb R$.   By 
(i), it has a finite subcover $U_0,\ldots,U_k$ say.   For each $j\le k$ 
we can find $H_j$, $G_j$ such that $H_j$ is an open subset of $\BbbR^r$ 
containing $x$ and $H_j\times U_j\subseteq G_j\in\Cal G$.   Now set 
$H=\bigcap_{j\le k}H_j$.  This is an open subset of $\BbbR^r$ 
containing $x$, and $H\times[-n,n]\subseteq\bigcup_{j\le n}G_j$ is 
covered by a finite subfamily of $\Cal G$.   So $x\in H\in\Cal H$.   As 
$x$ is arbitrary, $\Cal H$ covers $E$.   \Qed 
 
\medskip 
 
\quad{\bf (iii)} Now the inductive hypothesis tells us that $E$ is 
compact, so there is a finite subfamily $\Cal H_0$ of $\Cal H$ covering 
$E$.   For each $H\in\Cal H_0$ let $\Cal G_H$ be a finite subfamily of 
$\Cal G$ covering $H\times[-n,n]$.   Then $\bigcup_{H\in\Cal H_0}\Cal 
G_H$ is a finite subfamily of $\Cal G$ covering $E\times[-n,n]=F$.   As 
$\Cal G$ is arbitrary, $F$ is compact and the induction proceeds. 
 
\medskip 
 
\quad{\bf (iv)} Thus the interval $[-\tbf{n},\tbf{n}]$ is compact in 
$\BbbR^r$ for every $r$, $n$.   Now suppose that $K$ is a closed 
bounded set in $\BbbR^r$.   Then there is an $n\in\Bbb N$ such that 
$K\subseteq[-\tbf{n},\tbf{n}]$, that is, $K=K\cap[-\tbf{n},\tbf{n}]$. 
As $K$ is closed and $[-\tbf{n},\tbf{n}]$ is compact, $K$ is compact, by 
2A2Ea. 
 
This completes the proof. 
}%end of proof of 2A2F 
 
\leader{2A2G}{Corollary} If $\phi:D\to\Bbb R$ is continuous, where 
$D\subseteq\BbbR^r$, and $K\subseteq D$ is a non-empty compact set, 
then $\phi$ is bounded and attains its bounds on $K$. 
 
\proof{ By 2A2Eb, $\phi[K]$ is compact; by 2A2F it is closed and 
bounded.   To say that $\phi[K]$ is bounded is just to say that $\phi$ 
is bounded on $K$.   Because $\phi[K]$ is a non-empty bounded set, it 
has an infimum $a$ and a supremum $b$;  now both belong to 
$\overline{\phi[K]}$ (by the criterion 2A2B(ii), or otherwise);  because 
$\phi[K]$ is closed, both belong to $\phi[K]$, that is, $\phi$ attains 
its bounds. 
}%end of proof of 2A2G 
 
\leader{2A2H}{Lim sup and lim inf revisited} In \S1A3 I briefly 
discussed $\limsup_{n\to\infty}a_n$, $\liminf_{n\to\infty}a_n$ for real 
sequences $\sequencen{a_n}$.   In this volume we need the notion of 
$\limsup_{\delta\downarrow 0}f(\delta)$,  
$\liminf_{\delta\downarrow 0}f(\delta)$ for real functions $f$.   I say 
that $\limsup_{\delta\downarrow 0}f(\delta)=u\in[-\infty,\infty]$ if 
(i) for every 
$v>u$ there is an $\eta>0$ such that $f(\delta)$ is defined and less than 
or equal to $v$ for every $\delta\in\ocint{0,\eta}$ 
(ii) for every $v<u$ and $\eta>0$ there is a $\delta\in\ocint{0,\eta}$ 
such that $f(\delta)$ is 
defined and greater than or equal to $v$.   Similarly, 
$\liminf_{\delta\downarrow 0}f(\delta)=u\in[-\infty,\infty]$ if (i) for 
every $v<u$ there is an $\eta>0$ such that $f(\delta)$ is defined and  
greater than 
or equal to $v$ for every $\delta\in\ocint{0,\eta}$ (ii) for every 
$v>u$ and $\eta>0$ there is an $\delta\in\ocint{0,\eta}$ such that  
$f(\delta)$ is defined and less than or equal to $v$. 
 
\leader{2A2I}{}\cmmnt{ In the one-dimensional case, we have a particularly simple description of the open sets. 
 
\medskip 
 
\noindent}{\bf Proposition} If $G\subseteq\Bbb R$ is any open set, it is 
expressible as the union of a countable disjoint family of open 
intervals. 
 
\proof{ For $x$, $y\in G$ write $x\sim y$ if either $x\le y$ and 
$[x,y]\subseteq G$ or $y\le x$ and $[y,x]\subseteq G$.   It is easy to 
check that $\sim$ is an equivalence relation on $G$.   Let $\Cal C$ be 
the set of equivalence classes under $\sim$.   Then $\Cal C$ is a 
partition of $G$.   Now every $C\in\Cal C$ is an open interval. 
\Prf\ Set $a=\inf C$, $b=\sup C$ (allowing $a=-\infty$ and/or $b=\infty$ 
if $C$ is unbounded).   If $a<x<b$, there are $y$, $z\in C$ such that 
$y\le x\le z$, so that $[y,x]\subseteq[y,z]\subseteq G$ and $y\sim x$ 
and $x\in C$;  thus $\ooint{a,b}\subseteq C$.   If $x\in C$, there is an 
open interval $I$ containing $x$ and included in $G$;  since $x\sim y$ 
for every $y\in I$, $I\subseteq C$;  so 
 
\Centerline{$a\le\inf I<x<\sup I\le b$} 
 
\noindent and $x\in\ooint{a,b}$.   Thus $C=\ooint{a,b}$ is an open 
interval.\ \Qed 
 
To see that $\Cal C$ is countable, observe that every member of $\Cal C$ 
contains a member of $\Bbb Q$, so that we have a surjective function 
from a subset of $\Bbb Q$ onto $\Cal C$, and $\Cal C$ is countable (1A1E). 
}%end of proof of 2A2I 
 
\discrpage 
 
\frfilename{mt2a3.tex}
\versiondate{25.7.07}
\copyrightdate{1995}

\def\varBbb#1{\mathchoice{\hbox{$\Bbb#1$\hskip0.02em}}
  {\hbox{$\Bbb#1$\hskip0.02em}}
  {\hbox{$\scriptstyle\Bbb#1$\hskip0.04em}}
  {\hbox{$\scriptscriptstyle\Bbb#1$\hskip0.04em}}}

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

\newsection{2A3}

At various points -- principally \S\S245-247, %\S245 \S246 \S247
but also for certain
ideas in Chapter 27 -- we need to know something about non-metrizable
topologies.   I must say that you should probably
take the time to look at some book on elementary functional analysis
which has the phrases `weak compactness' or `weakly compact' in the
index.   But I can
list here the concepts actually used in this volume, in a good deal less
space than any orthodox, complete treatment would employ.

\leader{2A3A}{Topologies} \cmmnt{First we need to know what a {\lq
topology'}
is.}   If $X$ is
any set, a {\bf topology} on $X$ is a family $\frak T$ of subsets of $X$
such that (i) $\emptyset$, $X\in\frak T$ (ii) if $G$, $H\in\frak T$ then
$G\cap H\in\frak T$ (iii) if $\Cal G\subseteq\frak T$ then
$\bigcup\Cal G\in\frak T$\cmmnt{ (cf.\ 1A2B)}.   \cmmnt{The pair} $(X,\frak T)$ is now a
{\bf topological space}.   In this context, members of $\frak T$ are
called {\bf open} and their complements\cmmnt{ (in $X$)} are called
{\bf closed}\cmmnt{ (cf.\ 1A2E-1A2F)}.

\leader{2A3B}{Continuous functions (a)} If $(X,\frak T)$ and $(Y,\frak
S)$ are topological spaces, a
function $\phi:X\to Y$ is {\bf continuous} if $\phi^{-1}[G]\in\frak T$
for every $G\in\frak S$.   \cmmnt{(By 2A2Ca above, this is consistent
with the
$\epsilon$-$\delta$ definition of continuity for functions from one
Euclidean space to another.   See also 2A3H below.)}

\header{2A3Bb}{\bf (b)} If $(X,\frak T)$, $(Y,\frak S)$ and
$(Z,\frak U)$ are topological spaces
and $\phi:X\to Y$ and $\psi:Y\to Z$ are continuous, then
$\psi\phi:X\to Z$
is continuous.   \prooflet{\Prf\ If $G\in\frak U$ then
$\psi^{-1}[G]\in\frak S$ so
$(\psi\phi)^{-1}[G]=\phi^{-1}[\psi^{-1}[G]]\in\frak T$.\ \Qed}

\header{2A3Bc}{\bf (c)} If $(X,\frak T)$ is a topological space, a
function $f:X\to\Bbb R$ is continuous iff $\{x:a<f(x)<b\}$ is open
whenever $a<b$ in $\Bbb R$.  \prooflet{\Prf\ {\bf (i)} Every interval
$\ooint{a,b}$ is open in $\Bbb R$, so if $f$ is continuous its inverse
image $\{x:a<f(x)<b\}$ must be open.   {\bf (ii)} Suppose that
$f^{-1}[\,\ooint{a,b}\,]$ is open whenever $a<b$, and let
$H\subseteq\Bbb R$ be any open set.   By the definition of `open' set
in $\Bbb R$\cmmnt{ (1A2A)},

$$\eqalign{H&
=\bigcup\{\ooint{y-\delta,y+\delta}:y\in\Bbb R,\,\delta>0,\,
\ooint{y-\delta,y+\delta}\subseteq H\},\cr}$$

\noindent so

\Centerline{$f^{-1}[H]=
\bigcup\{f^{-1}[\,\ooint{y-\delta,y+\delta}\,]:y\in\Bbb R,\,\delta>0,\,
\ooint{y-\delta,y+\delta}\subseteq H\}$}

\noindent is a union of open sets in $X$, therefore open.\ \Qed}

\header{2A3Bd}{\bf (d)} If $r\ge 1$, $(X,\frak T)$ is a topological
space, and $\phi:X\to\BbbR^r$ is a function, then $\phi$ is continuous
iff $\phi_i:X\to\Bbb R$ is continuous for each $i\le r$, where
$\phi(x)=(\phi_1(x),\ldots,\phi_r(x))$ for each $x\in X$.
\prooflet{\Prf\ {\bf (i)} Suppose that $\phi$ is continuous.   For $i\le
r$, $y=(\eta_1,\ldots,\eta_r)\in\BbbR^r$, set $\pi_i(y)=\eta_i$.   Then
$|\pi_i(y)-\pi_i(z)|\le\|y-z\|$ for all $y$, $z\in\BbbR^r$ so
$\pi_i:\BbbR^r\to\Bbb R$
is continuous.   Consequently $\phi_i=\pi_i\phi$ is continuous, by
(b) above.   {\bf (ii)} Suppose that every $\phi_i$ is continuous, and that
$H\subseteq\BbbR^r$ is open.   Set

\Centerline{$\Cal G
=\{G:G\subseteq X$ is open, $G\subseteq\phi^{-1}[H]\}$.}

\noindent Then $G_0=\bigcup\Cal G$ is open, and
$G_0\subseteq \phi^{-1}[H]$.   But suppose that $x_0$ is any point of
$\phi^{-1}[H]$.   Then there is a $\delta>0$ such that
$U(\phi(x_0),\delta)\subseteq H$, because $H$ is open and contains
$\phi(x_0)$.   For $1\le i\le r$ set
$V_i=\{x:\phi_i(x_0)-\bover{\delta}{\sqrt{r}}
<\phi_i(x)<\phi_i(x_0)+\bover{\delta}{\sqrt{r}}\}$;
then $V_i$ is the inverse image of an open set under the continuous map
$\phi_i$, so is open.   Set $G=\bigcap_{i\le r}V_i$.    Then $G$ is open
(using (ii) of the definition 2A3A), $x_0\in G$, and
$\|\phi(x)-\phi(x_0\|<\delta$ for every $x\in G$, so
$G\subseteq\phi^{-1}[H]$, $G\in\Cal G$ and $x_0\in G_0$.   This shows
that $\phi^{-1}[H]=G_0$ is open.   As $H$ is arbitrary, $\phi$ is
continuous.\ \Qed}

\spheader 2A3Be If $(X,\frak T)$ is a topological space,
$f_1,\ldots,f_r$ are continuous functions from $X$ to $\Bbb R$, and
$h:\BbbR^r\to\Bbb R$ is continuous, then $h(f_1,\ldots,f_r):X\to\Bbb R$
is continuous.   \prooflet{\Prf\ Set
$\phi(x)=(f_1(x),\ldots,f_r(x))\in\BbbR^r$ for $x\in X$.   By (d),
$\phi$ is continuous, so by 2A3Bb $h(f_1,\ldots,f_r)=h\phi$ is
continuous.\ \QeD}   In particular, $f+g$, $f\times g$ and $f-g$ are
continuous for all continuous functions $f$, $g:X\to\Bbb R$.

\spheader 2A3Bf If $(X,\frak T)$ and $(Y,\frak S)$ are topological
spaces and $\phi:X\to Y$ is a continuous function, then $\phi^{-1}[F]$
is closed in $X$ for every closed set $F\subseteq Y$.   \prooflet{(For
$X\setminus\phi^{-1}[F]=\phi^{-1}[Y\setminus F]$ is open.)}

\leader{2A3C}{Subspace topologies}
If $(X,\frak T)$ is a topological space and $D\subseteq X$,
then $\frak T_D=\{G\cap D:G\in\frak T\}$ is a topology on $D$.
\prooflet{\Prf\
(i) $\emptyset=\emptyset\cap D$ and $D=X\cap D$ belong to $\frak T_D$.
(ii) If $G$, $H\in\frak T_D$ there are $G'$, $H'\in\frak T$ such that
$G=G'\cap D$, $H=H'\cap D$;  now $G\cap H=G'\cap H'\cap D\in\frak T_D$.
(iii) If $\Cal G\subseteq\frak T_D$ set $\Cal H=\{H:H\in\frak T,\,H\cap
D\in\Cal G\}$;  then
$\bigcup\Cal G=(\bigcup\Cal H)\cap D\in\frak T_D$.\ \Qed}

$\frak T_D$ is called the {\bf subspace topology} on $D$, or the
topology on $D$ {\bf induced} by $\frak T$.   If $(Y,\frak S)$ is
another topological space, and $\phi:X\to Y$ is
$(\frak T,\frak S)$-continuous, then $\phi\restr D:D\to Y$ is
$(\frak T_D,\frak S)$-continuous.   \prooflet{(For if $H\in\frak S$ then

\Centerline{$(\phi\restr D)^{-1}[H]=D\cap\phi^{-1}[H]\in\frak T_D$.)}}

\leader{2A3D}{Closures and interiors (a)} \cmmnt{In the proof of 2A3Bd
I have already used the following idea.}   Let $(X,\frak T)$ be any
topological space and $A$ any subset of $X$.   Write

\Centerline{$\interior A=\bigcup\{G:G\in\frak T,\,G\subseteq A\}$.}

\noindent Then $\interior A$ is\cmmnt{ an open set, being a union of
open sets,
and is of course included in $A$;  it must be} the largest open set
included in $A$, and is called the {\bf interior} of $A$.


\header{2A3Db}{\bf (b)} Because a set is closed iff its complement is
open, we have a complementary notion:

\dvro{$$\eqalign{\overline{A}
&=\bigcap\{F:F\text{ is closed},\,A\subseteq F\}
=X\setminus\interior(X\setminus A).\cr}$$}
{$$\eqalign{\overline{A}
&=\bigcap\{F:F\text{ is closed},\,A\subseteq F\}\cr
&=X\setminus\bigcup\{X\setminus F:F\text{ is closed},\,A\subseteq F\}\cr
&=X\setminus\bigcup\{G:G\text{ is open},\,A\cap G=\emptyset\}\cr
&=X\setminus\bigcup\{G:G\text{ is open},\,G\subseteq X\setminus A\}
=X\setminus\interior(X\setminus A).\cr}$$}

\noindent$\overline{A}$ is\cmmnt{ closed (being the complement of an
open set) and is} the smallest closed set including $A$;  it is called
the {\bf closure} of $A$.   \cmmnt{(Compare 2A2A.)}   \cmmnt{Because
the union of two closed sets is closed (cf.\ 1A2Fc),}
$\overline{A\cup B}=\overline{A}\cup\overline{B}$ for all $A$,
$B\subseteq X$.
%4{}A2G

\header{2A3Dc}{\bf (c)}\cmmnt{ There are innumerable ways of looking
at these concepts;  a useful description of the closure of a set is}

\dvro{$$\eqalign{x\in\overline A
&\iff\,\text{every open set containing }x\text{ meets }A.\cr}$$}
{$$\eqalign{x\in\overline A
&\iff\,x\notin\interior(X\setminus A)\cr
&\iff\,\text{there is no open set containing }x
\text{ and included in }X\setminus A\cr
&\iff\,\text{every open set containing }x\text{ meets }A.\cr}$$}

\leader{2A3E}{Hausdorff topologies (a)}\dvro{ A}{ The concept
of `topological space' is so widely drawn, and
so widely applicable, that a vast number of different types of
topological space have been studied.   For this volume we shall not need
much of the (very extensive) vocabulary which has been developed to
describe this variety.   But one useful word (and one of the most
important concepts) is that of `Hausdorff space';  a} topological
space $X$ is {\bf Hausdorff} if for all distinct $x$, $y\in X$ there are
disjoint open sets $G$, $H\subseteq X$ such that $x\in G$ and $y\in H$.

\spheader 2A3Eb In a Hausdorff space $X$, finite sets are closed.
\prooflet{\Prf\ If $z\in X$, then for any $x\in X\setminus\{z\}$ there
is an open set containing $x$ but not $z$, so $X\setminus\{z\}$ is open
and
$\{z\}$ is closed.   So a finite set is a finite union of closed sets
and is therefore closed.\ \Qed}

\leader{2A3F}{Pseudometrics}\cmmnt{ Many important topologies (not
all!) can be
defined by families of pseudometrics;  it will be useful to have a
certain amount of technical skill with these.

\medskip

}{\bf (a)} Let $X$ be a set.   A {\bf pseudometric} on $X$
is a function $\rho:X\times X\to\coint{0,\infty}$ such that

\inset{$\rho(x,z)\le\rho(x,y)+\rho(y,z)$ for all $x$, $y$, $z\in X$}

\cmmnt{\noindent (the `triangle inequality';)}

\inset{$\rho(x,y)=\rho(y,x)$ for all $x$, $y\in X$;

$\rho(x,x)=0$ for all $x\in X$.}

\noindent A {\bf metric} is a pseudometric $\rho$ satisfying the further
condition

\inset{if $\rho(x,y)=0$ then $x=y$.}

\header{2A3Fb}{\bf (b) Examples (i)} For $x$, $y\in\Bbb R$, set
$\rho(x,y)=|x-y|$\cmmnt{;  then $\rho$ is a metric on $\Bbb R$} (the
`usual metric' on $\Bbb R$).

\medskip

\quad{\bf (ii)} For $x$, $y\in\BbbR^r$, where $r\ge 1$, set
$\rho(x,y)=\|x-y\|$, defining $\|z\|=\sqrt{\sum_{i=1}^r\zeta_i^2}$,
as usual.   Then $\rho$ is\cmmnt{ a metric,} the {\bf Euclidean
metric} on $\Bbb
R^r$.   \cmmnt{(The triangle inequality for $\rho$ comes from Cauchy's
inequality in 1A2C:  if $x$, $y$, $z\in\BbbR^r$, then

\Centerline{$\rho(x,z)=\|x-z\|=\|(x-y)+(y-z)\|
\le\|x-y\|+\|y-z\|=\rho(x,y)+\rho(y,z)$.}

\noindent The other required properties of $\rho$ are elementary.
Compare 2A4Bb below.)}

\cmmnt{\medskip

\quad{\bf (iii)} For an example of a pseudometric which is not a metric,
take $r\ge 2$ and define $\rho:\BbbR^r\times\Bbb
R^r\to\coint{0,\infty}$ by setting $\rho(x,y)=|\xi_1-\eta_1|$ whenever
$x=(\xi_1,\ldots,\xi_r)$, $y=(\eta_1,\ldots,\eta_r)\in\BbbR^r$.
}%end of comment

\header{2A3Fc}{\bf (c)} Now let $X$ be a set and $\Rho$ a non-empty
family of
pseudometrics on $X$.   Let $\frak T$ be the family of those subsets $G$
of $X$ such that for every $x\in G$ there are
$\rho_0,\ldots,\rho_n\in\Rho$ and $\delta>0$ such that

\Centerline{$U(x;\rho_0,\ldots,\rho_n;\delta)
=\{y:y\in X,\,\max_{i\le n}\rho_i(y,x)<\delta\}\subseteq G$.}

\noindent Then $\frak T$ is a topology on $X$.

\prooflet{\Prf\ (Compare 1A2B.)  (i)
$\emptyset\in\frak T$ because the condition is vacuously satisfied.
$X\in\frak T$ because $U(x;\rho;1)\subseteq X$ for any $x\in X$,
$\rho\in\Rho$.   (ii) If $G$, $H\in\frak T$ and $x\in G\cap H$, take
$\rho_0,\ldots,\rho_m,\rho'_0,\ldots,\rho'_n\in\Rho$, $\delta$,
$\delta'>0$ such that $U(x;\rho_0,\ldots,\rho_m;\delta)\subseteq G$,
$U(x;\rho'_0,\ldots,\rho'_n;\delta')
\ifdim\pagewidth=390pt\penalty-1000\fi
\subseteq G$;   then


\Centerline{
$U(x;\rho_0,\ldots,\rho_m,\rho'_0,\ldots,\rho'_n;\min(\delta,\delta'))
\subseteq G\cap H$.}

\noindent   As $x$ is arbitrary, $G\cap H\in\frak T$.   (iii)
If $\Cal G\subseteq\frak T$ and $x\in\bigcup\Cal G$, there is a
$G\in\Cal G$ such that $x\in G$;  now there are
$\rho_0,\ldots,\rho_n\in\Rho$ and $\delta>0$ such that


\Centerline{$U(x;\rho_0,\ldots,\rho_n;\delta)\subseteq
G\subseteq\bigcup\Cal G$.}

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

$\frak T$ is the {\bf topology defined by}
$\Rho$.

\header{2A3Fd}{\bf (d)} You may wish to have a convention to deal with
the case in which $\Rho$
is the empty set;   in this case the topology on $X$ defined by $\Rho$
is $\{\emptyset,X\}$.

\cmmnt{\spheader 2A3Fe In many important cases,
$\Rho$ is upwards-directed in the sense that for any $\rho_1$,
$\rho_2\in\Rho$ there is a
$\rho\in\Rho$ such that $\rho_i(x,y)\le\rho(x,y)$ for all $x$, $y\in X$
and both $i$.   In this case, of course, any set
$U(x;\rho_0,\ldots,\rho_n;\delta)$, where $\rho_0,\ldots,\rho_n\in\Rho$,
includes some set of the form $U(x;\rho;\delta)$, where $\rho\in\Rho$.
Consequently, for instance, a set $G\subseteq X$ is open iff for every
$x\in G$ there are $\rho\in\Rho$, $\delta>0$ such that
$U(x;\rho;\delta)\subseteq G$.
}

\spheader 2A3Ff A topology $\frak T$ is {\bf metrizable} if it
is the topology
defined
by a family $\Rho$ consisting of a single metric.   Thus the {\bf
Euclidean topology} on $\BbbR^r$ is the metrizable topology defined by
$\{\rho\}$, where $\rho$ is the metric of (b-ii) above.

\leader{2A3G}{Proposition} Let $X$ be a set with a topology defined by  a non-empty set $\Rho$
of pseudometrics on $X$.   Then $U(x;\rho_0,\ldots,\rho_n;\epsilon)$ is
open for all $x\in X$, $\rho_0,\ldots,\rho_n\in\Rho$ and $\epsilon>0$.

\proof{ (Compare 1A2D.)  Take $y\in U(x;\rho_0,\ldots,\rho_n;\epsilon)$.   Set

\Centerline{$\eta=\max_{i\le n}\rho_i(y,x)$,
\quad $\delta=\epsilon-\eta>0$.}

\noindent    If $z\in U(y;\rho_0,\ldots,\rho_n;\delta)$ then

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

\noindent for each $i\le n$, so $U(y;\rho_0,\ldots,\rho_n;\delta)
\subseteq U(x;\rho_0,\ldots,\rho_n;\epsilon)$.   As $y$ is arbitrary,
$U(x;\rho_0,\ldots,\rho_n;\epsilon)$ is open.
}%end of proof of 2A3G

\leader{2A3H}{}\cmmnt{ Now we have a result corresponding to 2A2Ca,
describing continuous functions between topological spaces defined by
families of pseudometrics.

\medskip

\noindent}{\bf Proposition} Let $X$ and $Y$ be sets;  let $\Rho$ be a
non-empty family of
pseudometrics on $X$, and $\Theta$ a non-empty family of pseudometrics
on $Y$;  let $\frak T$ and $\frak S$ be the corresponding topologies.
Then a function $\phi:X\to Y$ is continuous iff whenever $x\in X$,
$\theta\in\Theta$ and $\epsilon>0$, there are
$\rho_0,\ldots,\rho_n\in\Rho$ and $\delta>0$ such that
$\theta(\phi(y),\phi(x))\le\epsilon$ whenever $y\in X$ and $\max_{i\le
n}\rho_i(y,x)\le\delta$.

\proof{{\bf (a)} Suppose that $\phi$ is continuous;  take $x\in X$,
$\theta\in\Theta$ and $\epsilon>0$.   By 2A3G,
$U(\phi(x);\theta;\epsilon)\in\frak S$.
So $G=\phi^{-1}[U(\phi(x);\theta;\epsilon)]\in\frak T$.   Now $x\in G$,
so there are $\rho_0,\ldots,\rho_n\in\Rho$ and $\delta>0$ such that
$U(x;\rho_0,\ldots,\rho_n;\delta)\subseteq G$.   In this case
$\theta(\phi(y),\phi(x))\le\epsilon$ whenever $y\in X$ and $\max_{i\le
n}\rho_i(y,x)\le\bover12\delta$.   As $x$, $\theta$ and $\epsilon$ are
arbitrary, $\phi$ satisfies the condition.

\medskip

{\bf (b)} Suppose $\phi$ satisfies the condition.   Take
$H\in\frak S$ and consider $G=\phi^{-1}[H]$.   If $x\in G$, then
$\phi(x)\in H$, so there are $\theta_0,\ldots,\theta_n\in\Theta$ and
$\epsilon>0$ such that
$U(\phi(x);\theta_0,\ldots,\theta_n;\epsilon)\subseteq H$.   For each
$i\le n$ there are $\rho_{i0},\ldots,\rho_{i,m_i}\in\Rho$ and
$\delta_i>0$ such that $\theta(\phi(y),\phi(x))\le\bover12\epsilon$
whenever $y\in X$ and $\max_{j\le m_i}\rho_{ij}(y,x)\le\delta_i$.   Set
$\delta=\min_{i\le n}\delta_i>0$;  then

\Centerline{$U(x;\rho_{00},\ldots,\rho_{0,m_0},\ldots,\rho_{n0},\ldots,
\rho_{n,m_n};\delta)\subseteq G$.}

\noindent As $x$ is arbitrary, $G\in\frak T$.   As $H$ is arbitrary,
$\phi$ is continuous.
}%end of proof of 2A3H

\cmmnt{
\leader{2A3I}{Remarks (a)} If $\Rho$ is upwards-directed, the condition
simplifies to:  for every $x\in X$,
$\theta\in\Theta$ and $\epsilon>0$, there are
$\rho\in\Rho$ and $\delta>0$ such that
$\theta(\phi(y),\phi(x))\le\epsilon$ whenever $y\in X$ and
$\rho(y,x)\le\delta$.




\spheader 2A3Ib Suppose we have a set $X$ and two non-empty
families $\Rho$,
$\Theta$ of pseudometrics on $X$, generating topologies $\frak T$ and
$\frak S$ on $X$.   Then $\frak S\subseteq\frak T$ iff the identity map
$\phi$ from $X$ to itself is a continuous function when regarded as a
map from $(X,\frak T)$ to $(X,\frak S)$, because this will mean that
$G=\phi^{-1}[G]$ belongs to $\frak T$ whenever $G\in\frak S$.   Applying
the proposition above to $\phi$, we see that this happens iff for every
$\theta\in\Theta$,
$x\in X$ and $\epsilon>0$ there are $\rho_0,\ldots,\rho_n\in\Rho$ and
$\delta>0$ such that $\theta(y,x)\le\epsilon$ whenever $y\in X$ and
$\max_{i\le n}\rho_i(y,x)\le\delta$.   Similarly, reversing the roles
of $\Rho$ and $\Theta$, we get a criterion for when $\frak
T\subseteq\frak S$, and putting the two together we obtain a criterion
to determine when $\frak T=\frak S$.
}%end of comment

\leader{2A3J}{Subspaces:  Proposition} If $X$ is a set, $\Rho$ a
non-empty family of pseudometrics on
$X$ defining a topology $\frak T$ on $X$, and $D\subseteq X$, then

(a) for every $\rho\in\Rho$, the restriction $\rho^{(D)}$ of $\rho$
to $D\times D$ is a pseudometric on $D$;

(b) the topology defined by $\Rho_D=\{\rho^{(D)}:\rho\in\Rho\}$ on
$D$ is precisely the subspace topology $\frak T_D$ described in 2A3C.

\proof{(a) is just a matter of reading through the
definition in 2A3Fa.   For (b), we have to think for a moment.

\medskip

{\bf (i)} Suppose that $G$ belongs to the topology defined by $\Rho_D$.
Set

\Centerline{$\Cal H=\{H:H\in\frak T,\,H\cap D\subseteq G\}$,}

\Centerline{$H^*=\bigcup\Cal
H\in\frak T$,\quad $G^*=H^*\cap D\in\frak T_D$;}

\noindent  then $G^*\subseteq G$.   On
the other hand, if $x\in G$, then there are
$\rho_0,\ldots,\rho_n\in\Rho$ and $\delta>0$ such that

\Centerline{$U(x;\rho_0^{(D)},\ldots,\rho_n^{(D)};\delta)
=\{y:y\in D,\,\max_{i\le n}\rho_i^{(D)}(y,x)<\delta\}\subseteq G$.}

\noindent Consider

\Centerline{$H=U(x;\rho_0,\ldots,\rho_n;\delta)
=\{y:y\in X,\,\max_{i\le n}\rho_i(y,x)<\delta\}\subseteq X$.}

\noindent Evidently

\Centerline{$H\cap D
=U(x;\rho_0^{(D)},\ldots,\rho_n^{(D)};\delta)\subseteq G$.}

\noindent   Also $H\in\frak T$.   So $H\in\Cal H$ and

\Centerline{$x\in H\cap D\subseteq H^*\cap D=G^*$.}

\noindent Thus $G=G^*\in\frak T_D$.

\medskip

{\bf (ii)} Now suppose that $G\in\frak T_D$.   Then there is an
$H\in\frak T$ such that $G=H\cap D$.
Consider the identity map
$\phi:D\to
X$, defined by saying that $\phi(x)=x$ for every $x\in D$.   $\phi$
obviously satisfies the criterion of 2A3H, if we endow $D$ with $\Rho_D$
and $X$ with $\Rho$, because $\rho(\phi(x),\phi(y))=\rho^{(D)}(x,y)$
whenever $x$, $y\in D$ and $\rho\in\Rho$;  so $\phi$ must be continuous
for the associated topologies, and $\phi^{-1}[H]$ must belong to the
topology defined by $\Rho_D$.   But $\phi^{-1}[H]=G$.   Thus every set
in $\frak T_D$ belongs to the topology defined by $\Rho_D$, and the two
topologies are the same, as claimed.
}%end of proof of 2A3J

\leader{2A3K}{Closures and interiors} Let $X$ be a set, $\Rho$ a
non-empty family of pseudometrics on $X$ and $\frak T$ the topology
defined by $\Rho$.

\header{2A3Ka}{\bf (a)} For any $A\subseteq X$ and $x\in X$,

\dvro{$$\eqalign{x\in\interior A
&\iff\,\text{there are }\rho_0,\ldots,\rho_n\in\Rho,\,\delta>0
\text{ such that }U(x;\rho_0,\ldots,\rho_n;\delta)\subseteq A.\cr}$$}
{$$\eqalign{x\in\interior A
&\iff\,\text{there is an open set included in }A
\text{ containing }x\cr
&\iff\,\text{there are }\rho_0,\ldots,\rho_n\in\Rho,\,\delta>0
\text{ such that }U(x;\rho_0,\ldots,\rho_n;\delta)\subseteq A.\cr}$$}

\header{2A3Kb}{\bf (b)} For any $A\subseteq X$ and $x\in X$,
$x\in\overline A$ iff
$U(x;\rho_0,\ldots,\rho_n;\delta)\cap A\ne\emptyset$ for every
$\rho_0,\ldots,\rho_n\in\Rho$ and $\delta>0$.
\cmmnt{(Compare 2A2B(ii), 2A3Dc.)}

\leader{2A3L}{Hausdorff topologies}\cmmnt{ Recall that a topology
$\frak T$ is Hausdorff if any two
points can be separated by open sets (2A3E).}   Now a topology defined
on a set $X$ by a non-empty family $\Rho$ of pseudometrics is Hausdorff
iff for any
two different points $x$, $y$ of $X$ there is a $\rho\in\Rho$ such that
$\rho(x,y)>0$.   \prooflet{\Prf\ {\bf (i)} Suppose that the topology is
Hausdorff and that $x$, $y$ are distinct points in $X$.   Then there is
an open set $G$ containing $x$ but not containing $y$.   Now there are
$\rho_0,\ldots,\rho_n\in\Rho$ and $\delta>0$ such that
$U(x;\rho_0),\ldots,\rho_n;\delta)\subseteq G$, in which case
$\rho_i(y,x)\ge\delta>0$ for some $i\le n$.   {\bf (ii)} If $\Rho$
satisfies the condition, and $x$, $y$ are distinct points of $X$, take
$\rho\in\Rho$ such that $\rho(x,y)>0$, and set
$\delta=\bover12\rho(x,y)$.   Then $U(x;\rho;\delta)$ and
$U(y;\rho;\delta)$ are disjoint (because if $z\in X$, then

\Centerline{$\rho(z,x)+\rho(z,y)\ge\rho(x,y)=2\delta$,}

\noindent so at least one of $\rho(z,x)$, $\rho(z,y)$ is greater than or
equal to $\delta$), and they are open sets containing $x$, $y$
respectively.   As $x$ and $y$ are arbitrary, the topology is Hausdorff.
\Qed}

In particular, metrizable topologies are Hausdorff.

\leader{2A3M}{Convergence of sequences (a)} If $(X,\frak T)$ is any
topological space, and $\sequencen{x_n}$ is a sequence in $X$, we say
that $\sequencen{x_n}$ {\bf converges} to $x\in X$, or that $x$ is a
{\bf limit} of $\sequencen{x_n}$, or $\sequencen{x_n}\to x$, if for
every open set $G$ containing $x$ there is an $n_0\in\Bbb N$ such that
$x_n\in G$ for every $n\ge n_0$.

\header{2A3Mb}{\bf (b) Warning} In general topological spaces, it is
possible for a sequence to have more than one limit\cmmnt{, and we cannot
safely write $x=\lim_{n\to\infty}x_n$}.   But in Hausdorff spaces, this
does not occur.   \prooflet{\Prf\ If $\frak T$ is Hausdorff, and $x$,
$y$ are distinct points of $X$, there are disjoint open sets $G$, $H$
such that $x\in G$ and $y\in H$.   If now $\sequencen{x_n}$ converges to
$x$, there is an $n_0$ such that $x_n\in G$ for every $n\ge n_0$, so
$x_n\notin H$ for every $n\ge n_0$, and $\sequencen{x_n}$ cannot
converge to $y$.\ \Qed}   In particular, a sequence in a metrizable space
can have at most one limit.

\header{2A3Mc}{\bf (c)} Let $X$ be a set, and $\Rho$ a non-empty family
of pseudometrics on $X$, generating a topology $\frak T$;  let
$\sequencen{x_n}$ be a sequence in $X$ and $x\in X$.   Then
$\sequencen{x_n}$ converges to $x$ iff $\lim_{n\to\infty}\rho(x_n,x)=0$
for every $\rho\in\Rho$.   \prooflet{\Prf\ {\bf (i)} Suppose that
$\sequencen{x_n}\to x$ and that $\rho\in\Rho$.   Then for any
$\epsilon>0$ the set $G=U(x;\rho;\epsilon)$ is an open set containing
$x$, so there is an $n_0$ such that $x_n\in G$ for every $n\ge n_0$,
that is, $\rho(x_n,x)<\epsilon$ for every $n\ge n_0$.   As $\epsilon$ is
arbitrary, $\lim_{n\to\infty}\rho(x_n,x)=0$.    {\bf (ii)} If the
condition is satisfied, take any open set $G$ containing $X$.   Then
there are $\rho_0,\ldots,\rho_k\in\Rho$ and $\delta>0$ such that
$U(x;\rho_0,\ldots,\rho_k;\delta)\subseteq G$.   For each $i\le k$ there
is an $n_i\in\Bbb N$ such that $\rho_i(x_n,x)<\delta$ for every $n\ge
n_i$.   Set $n^*=\max(n_0,\ldots,n_k)$;  then $x_n\in
U(x;\rho_0,\ldots,\rho_k;\delta)\subseteq G$ for every $n\ge n^*$.   As
$G$ is arbitrary, $\sequencen{x_n}\to x$.\ \Qed}

\spheader 2A3Md Let $(X,\rho)$ be a metric space, $A$ a subset of $X$
and $x\in X$.   Then $x\in\overline{A}$ iff there is a sequence in $A$
converging to $x$.   \prooflet{\Prf {\bf (i)} If $x\in\overline{A}$,
then for every $n\in\Bbb N$ we can choose a point
$x_n\in A\cap U(x;\rho;2^{-n})$
(2A3Kb);  now $\sequencen{x_n}\to x$.   {\bf (ii)} If $\sequencen{x_n}$
is a sequence in $A$ converging to $x$, then for every open set $G$
containing $x$ there is an $n$ such that $x_n\in G$, so that $A\cap
G\ne\emptyset$;  by 2A3Dc, $x\in\overline{A}$.\ \Qed}

\leader{2A3N}{Compactness} \cmmnt{ The next concept we need is the
idea of `compactness' in general topological spaces.

\header{2A3Na}}{\bf (a)} If $(X,\frak T)$ is any
topological space, a subset $K$ of $X$ is {\bf compact} if whenever
$\Cal G$ is a family in $\frak T$ covering $K$, then there is a finite
$\Cal G_0\subseteq\Cal G$ covering $K$.   \cmmnt{(Cf.\ 2A2D.  A {\bf
warning}:  many authors reserve the term `compact' for Hausdorff
spaces.)}   A set $A\subseteq X$ is {\bf relatively compact} in $X$ if
there is a compact subset of $X$ including $A$.   \cmmnt{({\bf Warning!}
in non-Hausdorff spaces, this is not the same thing as saying that
$\overline{A}$ is compact.)}

\header{2A3Nb}{\bf (b)}\cmmnt{ Just as in 2A2E-2A2G %2A2E 2A2F 2A2G
(and the proofs
are the same in the general case), we have the following results.

\medskip

\quad}{\bf (i)} If $K$ is compact and $E$ is closed, then $K\cap E$ is
compact.

\medskip

\quad {\bf (ii)} If $K\subseteq X$ is compact and $\phi:K\to Y$ is
continuous,
where $(Y,\frak S)$ is another topological space, then $\phi[K]$ is a
compact subset of $Y$.

\medskip

\quad{\bf (iii)} If $K\subseteq X$ is compact and $\phi:K\to\Bbb R$ is
continuous, then $\phi$ is bounded and attains its bounds.

\leader{2A3O}{Cluster points (a)}  If
$(X,\frak T)$ is a topological space, and $\sequencen{x_n}$ is a
sequence
in $X$, then a {\bf cluster point} of $\sequencen{x_n}$ is an $x\in X$
such that whenever $G$ is an open set containing $x$ and $n\in\Bbb N$
then there is a $k\ge n$ such that $x_k\in G$.

\header{2A3Ob}{\bf (b)} Now if $(X,\frak T)$ is
a topological space and $A\subseteq X$ is relatively compact, every
sequence $\sequencen{x_n}$ in $A$ has a cluster point in $X$.
\prooflet{\Prf\
Let $K$ be a compact subset of $X$ including $A$.    Set

\Centerline{$\Cal G=\{G:G\in\frak T,\,\{n:x_n\in G\}\text{ is
finite}\}$.}

\noindent\Quer\ If $\Cal G$ covers $K$, then there is a finite
$\Cal G_0\subseteq\Cal G$ covering $K$.   Now

\Centerline{$\Bbb N=\{n:x_n\in A\}=\{n:x_n\in\bigcup\Cal G_0\}
=\bigcup_{G\in\Cal G_0}\{n:x_n\in G\}$}

\noindent is a finite union of finite sets, which is absurd.   \Bang\
Thus $\Cal G$ does not cover $K$.
Take any $x\in K\setminus\bigcup\Cal G$.   If $G\in\frak T$ and $x\in G$
and $n\in\Bbb N$, then $G\notin\Cal G$ so $\{k:x_k\in G\}$ is infinite
and there is a $k\ge n$ such that $x_k\in G$.   Thus $x$ is a cluster
point of $\sequencen{x_n}$, as required.\ \Qed}

\cmmnt{
\leader{2A3P}{Filters} In $\BbbR^r$, and more generally in all
metrizable spaces, topological ideas can be effectively
discussed in terms of convergent sequences.   (To be sure, this
occasionally necessitates the use of a weak form of the axiom of choice,
in order to
choose a sequence;  but as measure theory without such choices is
changed utterly -- see Chapter 56 in Volume 5 -- there
is no point in fussing about them here.)   For
topological spaces in general, however, sequences are quite inadequate,
for very interesting reasons which I shall not enlarge upon.   Instead
we need to use `nets' or `filters'.   The latter take a moment's
more effort at the beginning, but are then (in my view) much easier to
work with, so I describe this method now.
}%end of comment

\leader{2A3Q}{Convergent filters (a)} Let $(X,\frak T)$ be a topological
space, $\Cal F$ a filter on $X$\cmmnt{ (see 2A1I)} and $x$ a point of
$X$.   We say that $\Cal F$ is
{\bf convergent} to $x$, or that $x$ is a {\bf limit} of $\Cal F$, and
write $\Cal F\to x$, if
every open set containing $x$ belongs to $\Cal F$.

\header{2A3Qb}{\bf (b)} Let $(X,\frak T)$ and $(Y,\frak S)$ be
topological spaces,
$\phi:X\to Y$ a continuous function, $x\in X$ and $\Cal F$ a filter on
$X$ converging to $x$.   Then $\phi[[\Cal F]]$\cmmnt{ (as defined in
2A1Ib)} converges to $\phi(x)$\prooflet{ (because $\phi^{-1}[G]$ is an
open set containing $x$ whenever $G$ is an
open set containing $\phi(x)$)}.

\leader{2A3R}{}\cmmnt{ Now we have the following characterization of
compactness.

\medskip

\noindent}{\bf Theorem} Let $X$ be a topological space, and $K$ a subset
of $X$.   Then $K$ is compact iff every ultrafilter on $X$  containing
$K$ has a limit in $K$.

\proof{{\bf (a)} Suppose that $K$ is compact and that $\Cal F$
is an ultrafilter on $X$ containing $K$.   Set

\Centerline{$\Cal G=\{G:G\subseteq X$ is open,
$X\setminus G\in\Cal F\}$.}

\noindent Then the union of any two members of $\Cal G$ belongs to $\Cal
G$, so the union of any finite number of members of $\Cal G$ belongs to
$\Cal G$;  also no member of $\Cal G$ can include $K$, because
$X\setminus K\notin\Cal F$.   Because $K$ is compact, it follows that
$\Cal G$ cannot cover $K$.   Let $x$ be any point of
$K\setminus\bigcup\Cal G$.   If $G$ is any open set containing $x$, then
$G\notin\Cal G$ so $X\setminus G\notin\Cal F$;  but this means that $G$
must belong to $\Cal F$, because $\Cal F$ is an ultrafilter.   As $G$ is
arbitrary, $\Cal F\to x$.
Thus every ultrafilter on $X$ containing $K$ has a limit in $K$.

\medskip

{\bf (b)} Now suppose that every ultrafilter on $X$ containing $K$ has a
limit in $K$.   Let $\Cal G$ be a cover of $K$ by open sets in $X$.
\Quer\ Suppose, if possible, that $\Cal G$ has no finite subcover.   Set

\Centerline{$\Cal F=\{F:$ there is a finite $\Cal G_0\subseteq
\Cal G,\,F\cup\bigcup\Cal G_0\supseteq K\}$.}

\noindent Then $\Cal F$ is a filter on $X$.   \Prf\ (i)
$X\cup\bigcup\emptyset\supseteq K$ so $X\in\Cal F$.


\Centerline{$\emptyset\cup\bigcup\Cal G_0=\bigcup\Cal G_0\not\supseteq
K$}

\noindent for any
finite $\Cal G_0\subseteq\Cal G$, by hypothesis, so
$\emptyset\notin\Cal F$.
(ii) If $E$, $F\in\Cal F$ there are finite sets $\Cal G_1$,
$\Cal G_2\subseteq\Cal G$ such that $E\cup\bigcup\Cal G_1$ and
$F\cup\bigcup\Cal G_2$ both include $K$;  now
$(E\cap F)\cup\bigcup(\Cal G_1\cup\Cal G_2)\supseteq K$ so
$E\cap F\in\Cal F$.   (iii) If
$X\supseteq E\supseteq F\in\Cal F$ then there is a finite
$\Cal G_0\subseteq\Cal G$ such that $F\cup\Cal G_0\supseteq K$;  now
$E\cup\bigcup\Cal G_0\supseteq K$ and $E\in\Cal F$.\ \Qed

By the Ultrafilter Theorem (2A1O), there is an ultrafilter $\Cal F^*$ on
$X$ including $\Cal F$.   Of course $K$ itself belongs to $\Cal F$, so
$K\in\Cal F^*$.   By hypothesis, $\Cal F^*$ has a limit $x\in K$.   But
now there is a set $G\in\Cal G$ containing $x$, and
$(X\setminus G)\cup G\supseteq K$, so
$X\setminus G\in\Cal F\subseteq\Cal F^*$;  which means
that $G$ cannot belong to $\Cal F^*$, and $x$ cannot be a limit of
$\Cal F^*$.   \Bang

So $\Cal G$ has a finite subcover.   As $\Cal G$ is arbitrary, $K$ must
be compact.
}%end of proof of 2A3R

\cmmnt{
\medskip

\noindent{\bf Remark} Note that part (b) of the proof of
this theorem depends vitally on the
Ultrafilter Theorem and therefore on the axiom of choice.
}

\leader{2A3S}{Further calculations with filters (a)}
I\cmmnt{n general, i}t is
possible for a filter to have more than one limit;  but in Hausdorff
spaces this does not occur.  \prooflet{\Prf\ (Compare 2A3Mb.)  If
$(X,\frak T)$ is Hausdorff, and $x$,
$y$ are distinct points of $X$, there are disjoint open sets $G$, $H$
such that $x\in G$ and $y\in H$.   If now a filter $\Cal F$ on $X$
converges to
$x$, $G\in\Cal F$ so $H\notin\Cal F$ and $\Cal F$ does not converge to
$y$.\ \Qed}

Accordingly we can safely write $x=\lim\Cal F$ when $\Cal F\to x$ in a
Hausdorff space.

\spheader 2A3Sb Now suppose that $X$ is a set, $\Cal F$ is a filter on
$X$, $(Y,\frak S)$ is a Hausdorff space, $D\in\Cal F$ and $\phi:D\to Y$
is a function.   Then we write $\lim_{x\to\Cal F}\phi(x)$ for
$\lim\phi[[\Cal F]]$ if this is defined in $Y$;  that is,
$\lim_{x\to\Cal F}\phi(x)=y$ iff $\phi^{-1}[H]\in\Cal F$ for every open
set $H$ containing $y$.

\dvAnew{2014}If $Z$ is another set, $\Cal G$ is a filter on $Z$, and
$\psi:Z\to X$ is such that $\Cal F=\psi[[\Cal G]]$, then 
the composition $\phi\psi$ is defined on $\psi^{-1}[D]\in\Cal G$, and
if one of the limits $\lim_{x\to\Cal F}\phi(x)$, 
$\lim_{z\to\Cal G}\phi\psi(z)$ is defined in $Y$ so is the other, and they
are then equal.   \prooflet{\Prf\ Suppose that $y\in Y$ and let 
$\Cal U$ be the family of open subsets of $Y$ containing $y$.   Then

$$\eqalign{\lim_{x\to\Cal F}\phi(x)=y
&\iff\phi^{-1}[G]\in\Cal F\text{ for every }G\in\Cal U
\cr&\iff\psi^{-1}[\phi^{-1}[G]]\in\Cal G\text{ for every }G\in\Cal U
\cr&\iff(\phi\psi)^{-1}[G]\in\Cal G\text{ for every }G\in\Cal U
\iff\lim_{z\to\Cal G}\phi\psi(z)=y. \text{ \Qed}\cr}$$}

In the special case $Y=\Bbb R$, $\lim_{x\to\Cal F}\phi(x)=a$ iff
$\{x:|\phi(x)-a|\le\epsilon\}\in\Cal F$ for every
$\epsilon>0$\prooflet{ (because every open set containing $a$ includes a
set of the form $[a-\epsilon,a+\epsilon]$, which in turn includes the
open set $\ooint{a-\epsilon,a+\epsilon}$)}.

\spheader 2A3Sc Suppose that $X$ and $Y$ are sets, $\Cal F$ is a filter
on $X$, $\Theta$ is a non-empty family
of pseudometrics on $Y$ defining a topology $\frak S$ on
$Y$, and
$\phi:X\to Y$ is a function.   Then the image filter $\phi[[\Cal F]]$
converges to $y\in Y$ iff $\lim_{x\to\Cal F}\theta(\phi(x),y)=0$ in
$\Bbb R$
for every $\theta\in\Theta$.   \prooflet{\Prf\ {\bf (i)} Suppose that
$\phi[[\Cal F]]\to
y$.   For every $\theta\in\Theta$ and $\epsilon>0$,
$U(y;\theta;\epsilon)=\{z:\theta(z,y)<\epsilon\}$ is an open set
containing $y$ (2A3G), so belongs to $\phi[[\Cal F]]$, and its inverse
image $\{x:0\le\theta(\phi(x),y)<\epsilon\}$ belongs to $\Cal F$.
As $\epsilon$ is arbitrary, $\lim_{x\to\Cal F}\theta(\phi(x),y)=0$.
As $\theta$ is arbitrary, $\phi$ satisfies the condition.   {\bf (ii)}
Now
suppose that $\lim_{x\to\Cal F}\theta(\phi(x),y)=0$ for every
$\theta\in\Theta$.   Let $G$ be any open set in $Y$ containing $y$.
Then there are $\theta_0,\ldots,\theta_n\in\Theta$ and $\epsilon>0$ such
that

\Centerline{$U(y;\theta_0,\ldots,\theta_n;\epsilon)
=\bigcap_{i\le n}U(y;\theta_i;\epsilon)\subseteq G$.}

\noindent For each $i\le n$,

\Centerline{$\phi^{-1}[U(y;\theta_i;\epsilon)]
=\{x:\theta(\phi(x),y)<\epsilon\}$}

\noindent belongs to $\Cal F$;  because $\Cal F$ is closed under finite
intersections, so do
$\phi^{-1}[U(y;\theta_0,\ldots,\theta_n;\epsilon)]$ and its superset
$\phi^{-1}[G]$.   Thus $G\in\phi[[\Cal F]]$.   As $G$ is arbitrary,
$\phi[[\Cal F]]\to y$.\ \Qed}

%2A3S

\spheader 2A3Sd In particular,\cmmnt{ taking $X=Y$ and $\phi$ the
identity map,} if
$X$ has a topology $\frak T$ defined by a non-empty family $\Rho$ of
pseudometrics, then a filter $\Cal F$ on $X$ converges to $x\in X$ iff
$\lim_{y\to\Cal F}\rho(y,x)=0$ for every $\rho\in\Rho$.

\spheader 2A3Se{\bf (i)} If $X$ is any set, $\Cal F$ is an ultrafilter
on $X$, $(Y,\frak S)$ is a Hausdorff space, and
$h:X\to Y$ is a function such that $h[F]$ is relatively compact in $Y$
for some $F\in\Cal F$, then $\lim_{x\to\Cal F}h(x)$ is defined in $Y$.
\prooflet{\Prf\ Let $K\subseteq Y$ be a compact set including $h[F]$.
Then $K\in h[[\Cal F]]$, which is an ultrafilter (2A1N), so
$h[[\Cal F]]$ has a limit in $Y$ (2A3R), which is
$\lim_{x\to\Cal F}h(x)$.\ \Qed}

\medskip

\quad{\bf (ii)} If $X$ is any set, $\Cal F$ is an ultrafilter on $X$,
and $h:X\to\Bbb R$ is a function such that $h[F]$ is bounded in $\Bbb R$
for some set $F\in\Cal F$, then $\lim_{x\to\Cal F}h(x)$ exists in
$\Bbb R$.
\prooflet{\Prf\ $\overline{h[F]}$ is closed and bounded, therefore
compact (2A2F), so $h[F]$ is relatively compact and we can use
(i).\ \Qed}

\spheader 2A3Sf \cmmnt{ The concepts of $\limsup$, $\liminf$ can be
applied to filters.}  Suppose that $\Cal F$ is a filter on a set $X$,
and that $f:X\to[-\infty,\infty]$ is any function.   Then

\dvro{$$\eqalign{\limsup_{x\to\Cal F}f(x)
&=\inf_{F\in\Cal F}\sup_{x\in F}f(x)
\in[-\infty,\infty],\cr}$$}
{$$\eqalign{\limsup_{x\to\Cal F}f(x)
&=\inf\{u:u\in[-\infty,\infty],\,\{x:f(x)\le u\}\in\Cal F\}\cr
&=\inf_{F\in\Cal F}\sup_{x\in F}f(x)
\in[-\infty,\infty],\cr}$$}

\dvro{$$\eqalign{\liminf_{x\to\Cal F}f(x)
&=\sup_{F\in\Cal F}\inf_{x\in F}f(x).\cr}$$}
{$$\eqalign{\liminf_{x\to\Cal F}f(x)
&=\sup\{u:u\in[-\infty,\infty],\,\{x:f(x)\ge u\}\in\Cal F\}\cr
&=\sup_{F\in\Cal F}\inf_{x\in F}f(x).\cr}$$}

\noindent\dvro{For}{It is easy to see that, for} any two functions $f$,
$g:X\to\Bbb R$,

\Centerline{$\lim_{x\to\Cal F}f(x)=a$\quad iff\quad
$a=\limsup_{x\to\Cal F}f(x)=\liminf_{x\to\Cal F}f(x)$,}

\noindent and

\Centerline{$\limsup_{x\to\Cal F}f(x)+g(x)
\le\limsup_{x\to\Cal F}f(x)+\limsup_{x\to\Cal F}g(x)$,}

\Centerline{$\liminf_{x\to\Cal F}f(x)+g(x)
\ge\liminf_{x\to\Cal F}f(x)+\liminf_{x\to\Cal F}g(x)$,}

\Centerline{$\liminf_{x\to\Cal F}(-f(x))=-\limsup_{x\to\Cal F}f(x)$,
\quad$\limsup_{x\to\Cal F}(-f(x))=-\liminf_{x\to\Cal F}f(x)$,}

\Centerline{$\liminf_{x\to\Cal F}cf(x)=c\liminf_{x\to\Cal F}f(x)$,
\quad$\limsup_{x\to\Cal F}cf(x)=c\limsup_{x\to\Cal F}f(x)$}

\noindent whenever the right-hand-sides are defined in
$[-\infty,\infty]$ and $c\ge 0$.   So if $a=\lim_{x\to\Cal F}f(x)$ and
$b=\lim_{x\to\Cal F}(x)$ exist in $\Bbb R$, $\lim_{x\to\Cal F}f(x)+g(x)$
exists and is equal to $a+b$ and $\lim_{x\to\Cal F}cf(x)$ exists and is
equal to $c\lim_{x\to\Cal F}f(x)$ for every $c\in\Bbb R$.

\dvro{If}{We also see that if} $f:X\to\Bbb R$ is such that

\Centerline{for every $\epsilon>0$ there is an $F\in\Cal F$ such that
$\sup_{x\in F}f(x)\le\epsilon+\inf_{x\in F}f(x)$,}

\noindent then
\cmmnt{$\limsup_{x\to\Cal F}f(x)\le\epsilon+\liminf_{x\to\Cal F}f(x)$
for every $\epsilon>0$, so that} $\lim_{x\to\Cal F}f(x)$ is defined in
$[-\infty,\infty]$.

\spheader 2A3Sg \cmmnt{Note that the standard limits of real analysis
can be represented in the form described here.   For instance,}
$\lim_{n\to\infty}$, $\limsup_{n\to\infty}$, $\liminf_{n\to\infty}$
correspond to $\lim_{n\to\CalFr}$, $\limsup_{n\to\CalFr}$,
$\liminf_{n\to\CalFr}$ where $\CalFr$ is the {\bf Fr\'echet filter}
on $\Bbb N$, the filter $\{\Bbb N\setminus A:A\subseteq\Bbb N$ is
finite$\}$ of cofinite subsets of $\Bbb N$.   Similarly,
$\lim_{\delta\downarrow a}$, $\limsup_{\delta\downarrow a}$,
$\liminf_{\delta\downarrow a}$ correspond to $\lim_{\delta\to\Cal F}$,
$\limsup_{\delta\to\Cal F}$, $\liminf_{\delta\to\Cal F}$ where

\Centerline{$\Cal F=\{A:A\subseteq\Bbb R,\,\exists\,h>0$ such that
$\ocint{a,a+h}\subseteq A\}$.}
%2A3S

\leader{2A3T}{Product topologies}\cmmnt{ We need some brief remarks
concerning topologies on product spaces.

\medskip

} {\bf (a)} Let $(X,\frak T)$ and $(Y,\frak S)$ be
topological spaces.
Let $\frak U$ be the set of subsets $U$ of $X\times Y$ such that for
every $(x,y)\in U$ there are $G\in\frak T$, $H\in\frak S$ such that
$(x,y)\in G\times H\subseteq U$.   Then $\frak U$ is a topology on
$X\times Y$.   \prooflet{\Prf\ (i) $\emptyset\in\frak U$ because the
condition for
membership of $\frak U$ is vacuously satisfied.   $X\times Y\in\frak U$
because $X\in\frak T$, $Y\in\frak S$ and $(x,y)\in X\times Y\subseteq
X\times Y$ for every $(x,y)\in X\times Y$.   (ii) If $U$, $V\in\frak U$
and $(x,y)\in U\cap V$, then there are $G$, $G'\in\frak T$, $H$,
$H'\in\frak S$ such that

\Centerline{$(x,y)\in G\times H\subseteq U$,
\quad $(x,y)\in G'\times H'\subseteq V$;}

\noindent now $G\cap G'\in \frak T$, $H\cap H'\in\frak S$ and

\Centerline{$(x,y)\in(G\cap G')\times(H\cap H')\subseteq U\cap V$.}

\noindent As $(x,y)$ is arbitrary, $U\cap V\in\frak U$.   (iii) If
$\Cal U\subseteq\frak U$ and $(x,y)\in\bigcup\Cal U$, then there is a
$U\in\Cal U$ such that $(x,y)\in U$;  now there are $G\in\frak T$,
$H\in\frak S$ such that $(x,y)\in G\times H\subseteq
U\subseteq\bigcup\Cal U$.   As $(x,y)$ is arbitrary, $\bigcup\Cal
U\in\frak U$.\ \Qed}

$\frak U$ is called the {\bf product topology} on $X\times Y$.


\spheader 2A3Tb Suppose, in (a), that $\frak T$ and $\frak S$ are
defined by
non-empty families $\Rho$, $\Theta$ of pseudometrics\cmmnt{ in the manner of 2A3F}.   Then $\frak U$ is defined by the family
$\Upsilon=\{\tilde\rho:\rho\in\Rho\}\cup\{\bar\theta:\theta\in\Theta\}$
of pseudometrics on $X\times Y$, where

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

\noindent whenever $x$, $x'\in X$, $y$, $y'\in Y$, $\rho\in\Rho$ and
$\theta\in\Theta$.

\prooflet{\Prf\ {\bf (i)} Of course you should check that every
$\tilde\rho$, $\bar\theta$ is a pseudometric on $X\times Y$.

\quad{\bf (ii)} If $U\in\frak U$ and $(x,y)\in U$, then there are
$G\in\frak T$, $H\in\frak S$ such that $(x,y)\in G\times H\subseteq U$.
There are $\rho_0,\ldots,\rho_m\in\Rho$,
$\theta_0,\ldots,\theta_n\in\Theta$, $\delta$, $\delta'>0$ such that (in
the language of 2A3Fc) $U(x;\rho_0,\ldots,\rho_m;\delta)\subseteq G$,
$U(x;\theta_0,\ldots,\theta_n;\delta)\subseteq H$.   Now

\Centerline{$U((x,y);\tilde\rho_0,\ldots,\tilde\rho_m,
\bar\theta_0,\ldots,\bar\theta_n;\min(\delta,\delta'))\subseteq U$.}

\noindent As $(x,y)$ is arbitrary, $U$ is open for the topology
generated by $\Upsilon$.

\quad{\bf (iii)} If $U\subseteq X\times Y$ is open for the topology
defined by $\Upsilon$, take any $(x,y)\in U$.   Then there are
$\upsilon_0,\ldots,\upsilon_k\in\Upsilon$ and $\delta>0$ such that
$U((x,y);\upsilon_0,\ldots,\upsilon_k;\delta)\subseteq U$.   Take
$\rho_0,\ldots,\rho_m\in\Rho$ and $\theta_0,\ldots,\theta_n\in\Theta$
such that $\{\upsilon_0,\ldots,\upsilon_k\}
\subseteq\{\tilde\rho_0,\ldots,\tilde\rho_m,
\bar\theta_0,\ldots,\bar\theta_n\}$;    then
$G=U(x;\rho_0,\ldots,\rho_m;\delta)\in\frak T$ (2A3G),
$H=U(y;\theta_0,\ldots,\theta_n;\delta)\in\frak S$, and

\Centerline{$(x,y)\in G\times H
=U((x,y);\tilde\rho_0,\ldots,\rho_m,
\bar\theta_0,\ldots,\bar\theta_n;\delta)
\subseteq U((x,y);\upsilon_0,\ldots,\upsilon_k;\delta)\subseteq U$.}

\noindent As $(x,y)$ is arbitrary, $U\in\frak U$.   This completes the
proof that $\frak U$ is the topology defined by $\Upsilon$.\ \Qed}

\spheader 2A3Tc In particular, the product topology on
$\BbbR^r\times\BbbR^s$ is the Euclidean topology if we identify
$\BbbR^r\times\BbbR^s$ with $\BbbR^{r+s}$.  \prooflet{\Prf\ The product
topology is defined by the two pseudometrics $\upsilon_1$, $\upsilon_2$,
where for $x$, $x'\in \BbbR^r$ and $y$, $y'\in\BbbR^s$ I write

\Centerline{$\upsilon_1((x,y),(x',y'))=\|x-x'\|$,
\quad$\upsilon_2((x,y),(x',y'))=\|y-y'\|$}

\noindent (2A3F(b-ii)).   Similarly, the Euclidean topology on
$\BbbR^r\times\BbbR^s\cong\BbbR^{r+s}$ is defined by the metric $\rho$,
where

\Centerline{$\rho((x,y),(x',y'))=\|(x-y)-(x',y')\|
=\sqrt{\|x-x'\|^2+\|y-y'\|^2}$.}

\noindent Now if $(x,y)\in\BbbR^r\times\BbbR^s$ and $\epsilon>0$, then

\Centerline{$U((x,y);\rho;\epsilon)
\subseteq U((x,y);\upsilon_j;\epsilon)$}

\noindent for both $j$, while

\Centerline{$U((x,y);\upsilon_1,\upsilon_2;\Bover{\epsilon}{\sqrt2})
\subseteq U((x,y);\rho;\epsilon)$.}

\noindent Thus, as remarked in 2A3Ib, each topology is included in the
other, and they are the same.\ \Qed}

\leader{2A3U}{Dense sets (a)} If $X$ is a topological space, a set
$D\subseteq X$ is {\bf dense} in $X$ if $\overline{D}=X$\cmmnt{, that
is, if every non-empty open set meets $D$}.   More
generally, if $D\subseteq A\subseteq X$, then $D$ is dense in $A$ if it
is dense for the subspace topology of $A$\cmmnt{ (2A3C), that is, if
$A\subseteq\overline{D}$}.   %for 253

\spheader 2A3Ub If $\frak T$ is defined by a non-empty family $\Rho$ of
pseudometrics on $X$, then $D\subseteq X$ is dense iff
\discrcenter{468pt}
{$U(x;\rho_0,\ldots,\rho_n;\delta)\cap D\ne\emptyset$ }whenever
$x\in X$, $\rho_0,\ldots,\rho_n\in\Rho$ and $\delta>0$.

\spheader 2A3Uc If $(X,\frak T)$, $(Y,\frak S)$ are topological spaces,
of which $Y$ is Hausdorff (in particular, if $(X,\rho)$
and $(Y,\theta)$ are metric spaces),
and $f$, $g:X\to Y$ are continuous functions
which agree on some dense subset $D$ of $X$, then $f=g$.
\prooflet{\Prf\Quer\ Suppose, if possible, that there is an $x\in X$
such that $f(x)\ne g(x)$.   Then there are open sets $G$, $H\subseteq Y$
such that $f(x)\in G$, $g(x)\in H$ and $G\cap H=\emptyset$.   Now
$f^{-1}[G]\cap g^{-1}[H]$ is an open set, containing $x$ and therefore
not empty, but it cannot meet $D$, so $x\notin\overline{D}$ and $D$ is
not dense.\ \Bang\QeD}

\spheader 2A3Ud A topological space is called {\bf separable} if it has
a countable dense subset.   \cmmnt{For instance, $\BbbR^r$ is
separable for every $r\ge 1$, since $\varBbb{Q}^r$ is dense.}

\discrpage

\frfilename{mt2a4.tex} 
\versiondate{4.3.14} 
\copyrightdate{1996} 
 
\def\chaptername{Appendix} 
\def\sectionname{Normed spaces} 
 
\newsection{2A4} 
 
In Chapter 24 I discuss the spaces $L^p$, for 
$1\le p\le\infty$, and describe their most basic properties.   These 
spaces form a cluster of fundamental examples for the general theory of 
`normed spaces', the basis of functional analysis.   This is not the 
book from which you should learn that theory, but once again it may save 
you trouble if I briefly outline those parts of the general theory which 
are essential if you are to make sense of the ideas here. 
 
\leader{2A4A}{The real and complex fields} While the most important 
parts of the theory, from the point 
of view of measure theory, are most effectively dealt with in terms of 
{\it real} linear spaces, there are many applications in which {\it 
complex} 
linear spaces are essential.   I will therefore use the phrase 
 
\Centerline{`$U$ is a linear space over $\RoverC$'} 
 
\noindent to mean that $U$ is either a linear space over the field 
$\Bbb R$ or a linear space over the field $\Bbb C$;   it being 
understood that 
in any particular context all linear spaces considered will be 
over the same field.   In the same way, I will write 
`$\alpha\in\RoverC$' to mean that $\alpha$ belongs to whichever is the 
current underlying field. 
 
\leader{2A4B}{Definitions  (a)} A {\bf normed space} is a linear space 
$U$ over $\RoverC$ together with a {\bf norm}, that is, a functional 
$\|\,\|:U\to\coint{0,\infty}$ such that 
 
\qquad $\|u+v\|\le\|u\|+\|v\|$ for all $u$, $v\in U$, 
 
\qquad $\|\alpha u\|=|\alpha|\|u\|$ for $u\in U$, $\alpha\in\RoverC$, 
 
\qquad $\|u\|=0$ only when $u=0$, the zero vector of $U$. 
 
\cmmnt{\noindent (Observe that if $u=0$ (the zero vector) then $0u=u$ 
(where this $0$ is the zero scalar) so that $\|u\|=|0|\|u\|=0$.) 
} 
 
\header{2A4Bb}{\bf (b)} If $U$ is a normed space, then we have a metric 
$\rho$ on $U$ defined by saying that $\rho(u,v)=\|u-v\|$ for $u$, 
$v\in U$.   \prooflet{\Prf\ $\rho(u,v)\in\coint{0,\infty}$ for all $u$, 
$v$ because $\|u\|\in\coint{0,\infty}$ for every $u$. 
$\rho(u,v)=\rho(v,u)$ for all $u$, $v$ because 
$\|v-u\|=|-1|\|u-v\|=\|u-v\|$ for all $u$, $v$. 
If $u$, $v$, $w\in U$ then 
 
\Centerline{$\rho(u,w)=\|u-w\|=\|(u-v)+(v-w)\|\le\|u-v\|+\|v-w\| 
=\rho(u,v)+\rho(v,w)$.} 
 
\noindent If $\rho(u,v)=0$ then $\|u-v\|=0$ so $u-v=0$ and $u=v$.\ \Qed} 
 
\cmmnt{We therefore have a corresponding topology, with open and 
closed sets, closures, convergent sequences and so on.} 
 
\spheader 2A4Bc If $U$ is a normed space, a set $A\subseteq U$ is {\bf 
bounded}\cmmnt{ (for the norm)} if $\{\|u\|:u\in A\}$ is bounded in 
$\Bbb R$\cmmnt{;  that is, there is some $M\ge 0$ such that 
$\|u\|\le M$ for every $u\in A$}. 
 
\leader{2A4C}{Linear subspaces (a)} If $U$ is any normed space and $V$ 
is a linear subspace of $U$, then $V$ is also a normed space, if we take 
the norm of $V$ to be just the restriction to $V$ of the norm of 
$U$\cmmnt{;  the verification is trivial}. 
 
\spheader 2A4Cb If $V$ is a linear subspace of $U$, so is its
closure\cmmnt{ $\overline{V}$}.   
\prooflet{\Prf\ Take $u$, $u'\in\overline{V}$ and 
$\alpha\in\RoverC$.   If $\epsilon>0$, set 
$\delta=\epsilon/(2+|\alpha|)>0$;  then there are $v$, $v'\in V$ such 
that $\|u-v\|\le\delta$ and $\|u'-v'\|\le\delta$.   Now $v+v'$, 
$\alpha v\in V$ and 
 
\Centerline{$\|(u+u')-(v+v')\|\le\|u-v\|+\|u'-v'\|\le\epsilon$, 
\quad$\|\alpha u-\alpha v\|\le|\alpha|\|u-v\|\le\epsilon$.} 
 
\noindent As $\epsilon$ is arbitrary, $u+u'$ and $\alpha u$ belong to 
$\overline{V}$;  as $u$, $u'$ and $\alpha$ are arbitrary, and $0$ surely 
belongs to $V\subseteq\overline{V}$, $\overline{V}$ is a linear subspace 
of $U$.\ \Qed} 
 
\leader{2A4D}{Banach spaces (a)} If $U$ is a normed space, a sequence 
$\sequencen{u_n}$ in $U$ is {\bf Cauchy} if $\|u_m-u_n\|\to 0$ as $m$, 
$n\to\infty$\cmmnt{, that is, for every 
$\epsilon>0$ there is an $n_0\in\Bbb N$ such that 
$\|u_m-u_n\|\le\epsilon$ for all $m$, $n\ge n_0$}. 
 
\spheader 2A4Db A normed space $U$ is {\bf complete} if every 
Cauchy sequence has a limit;  a 
complete normed space is called a {\bf Banach space}. 
 
\leader{2A4E}{}\cmmnt{ It is helpful to know the following result. 
 
\medskip 
 
\noindent}{\bf Lemma}  Let $U$ be a normed 
space such that $\sequencen{u_n}$ is convergent\cmmnt{ (that is, has a 
limit)} in $U$ whenever $\sequencen{u_n}$ is a sequence in $U$ such that 
$\|u_{n+1}-u_n\|\le 4^{-n}$ for every $n\in\Bbb N$.   Then $U$ is 
complete. 
 
\proof{ Let $\sequencen{u_n}$ be any Cauchy sequence in $U$.   For 
each $k\in\Bbb N$, let $n_k\in\Bbb N$ be such that 
$\|u_{m}-u_n\|\le 4^{-k}$ whenever $m$, $n\ge n_k$.   Set $v_k=u_{n_k}$ 
for each $k$. 
Then $\|v_{k+1}-v_k\|\le 4^{-k}$ (whether $n_k\le n_{k+1}$ or 
$n_{k+1}\le n_k$).   So $\sequence{k}{v_k}$ has a limit $v\in U$.   I 
seek to show that $v$ is the required limit of $\sequencen{u_n}$. 
Given $\epsilon>0$, let $l\in\Bbb N$ be such that $\|v_k-v\|\le\epsilon$ 
for every $k\ge l$;  let $k\ge l$ be such that $4^{-k}\le\epsilon$; 
then if $n\ge n_k$, 
 
\Centerline{$\|u_n-v\|=\|(u_n-v_k)+(v_k-v)\| 
\le\|u_n-v_k\|+\|v_k-v\| 
\le\|u_n-u_{n_k}\|+\epsilon 
\le 2\epsilon$.} 
 
\noindent As $\epsilon$ is arbitrary, $v$ is a limit of 
$\sequencen{u_n}$.   As $\sequencen{u_n}$ is arbitrary, $U$ is complete. 
}%end of proof of 2A4E 
 
\leader{2A4F}{Bounded linear operators (a)} Let $U$, $V$ be two normed 
spaces.   A linear operator $T:U\to V$ 
is {\bf bounded} if $\{\|Tu\|:u\in U,\,\|u\|\le 1\}$ is bounded. 
\cmmnt{({\bf Warning!} in this context, we do not ask for the whole 
set of values $T[U]$ to be bounded;  a `bounded linear operator' need 
not be what we ordinarily call a `bounded function'.)} 
Write $\eurm B(U;V)$ for the space of all bounded linear operators from 
$U$ to $V$, and for $T\in \eurm B(U;V)$ write 
$\|T\|= \sup\{\|Tu\|:u\in U,\,\|u\|\le 1\}$. 
 
\header{2A4Fb}{\bf (b)} 
\cmmnt{A useful fact:}  $\|Tu\|\le\|T\|\|u\|$ whenever
$T\in\eurm B(U;V)$ and $u\in U$.   \prooflet{\Prf\ If $|\alpha|>\|u\|$ then 
 
\Centerline{$\|\Bover1{\alpha}u\|=\Bover1{|\alpha|}\|u\|\le 1$,} 
 
\noindent so 
 
\Centerline{$\|Tu\|=\|\alpha T(\Bover1{\alpha}u)\| 
=|\alpha|\|T(\Bover1{\alpha}u)\|\le|\alpha|\|T\|$;} 
 
\noindent as $\alpha$ is arbitrary, $\|Tu\|\le\|T\|\|u\|$.  \Qed} 
 
\spheader 2A4Fc A linear operator $T:U\to V$ is bounded iff it is 
continuous for the norm topologies on $U$ and $V$.   \prooflet{\Prf\ 
(i) If $T$ is bounded, $u_0\in U$ and $\epsilon>0$, then 
 
\Centerline{$\|Tu-Tu_0\|=\|T(u-u_0)\|\le\|T\|\|u-u_0\|\le\epsilon$} 
 
\noindent whenever $\|u-u_0\|\le\Bover{\epsilon}{1+\|T\|}$;  by 2A3H, 
$T$ is continuous.   (ii) If $T$ is continuous, then there is some 
$\delta>0$ such that $\|Tu\|=\|Tu-T0\|\le 1$ whenever 
$\|u\|=\|u-0\|\le\delta$.   If now $\|u\|\le 1$, 
 
\Centerline{$\|Tu\|=\Bover1{\delta}\|T(\delta u)\|\le\Bover1{\delta}$,} 
 
\noindent so $T$ is a bounded operator.\ \Qed} 
 
\spheader 2A4Fd\dvAnew{2013} If $U$, $V$ and $W$ are normed spaces,  
$S\in\eurm B(U;V)$ and $T\in\eurm B(V;W)$ then $TS\in\eurm B(U;W)$ and 
$\|TS\|\le\|T\|\|S\|$. 
\prooflet{\Prf\ I am rather supposing that you are aware, but 
in any case you will find it easy to check, that $TS:U\to W$ is a linear 
operator.   Now if $u\in U$ and $\|u\|\le 1$, 
 
\Centerline{$\|TSu\|=\|T(Su)\|\le\|T\|\|Su\|\le\|T\|\|S\|$} 
 
\noindent (using (b) for the middle inequality), 
so $TS$ is bounded and $\|TS\|\le\|T\|\|S\|$.\ \Qed} 
 
\leader{2A4G}{Theorem} $\eurm B(U;V)$ is a linear space over $\RoverC$, 
and $\|\,\|$ is a norm on $\eurm B(U;V)$. 
 
\proof{ As in 2A4Fd, it is easy to check, that if $S:U\to V$ and 
$T:U\to V$ are linear operators, and $\alpha\in\RoverC$, then we have 
linear operators $S+T$ and $\alpha T$ from $U$ to $V$ defined by the 
formulae 
 
\Centerline{$(S+T)(u)=Su+Tu$,\quad$(\alpha T)(u)=\alpha(Tu)$} 
 
\noindent for every $u\in U$;  moreover, that under these definitions of 
addition and scalar multiplication the space of all linear operators 
from $U$ to $V$ is a linear space.   Now we see that whenever $S$, $T\in 
\eurm B(U;V)$, $\alpha\in\RoverC$, $u\in U$ and $\|u\|\le 1$, 
 
\Centerline{$\|(S+T)(u)\|=\|Su+Tu\|\le\|Su\|+\|Tu\|\le\|S\|+\|T\|$,} 
 
\Centerline{$\|(\alpha 
T)u\|=\|\alpha(Tu)\|=|\alpha|\|Tu\|\le|\alpha|\|T\|$;} 
 
\noindent so that $S+T$ and $\alpha T$ belong to $\eurm B(U;V)$, with 
$\|S+T\|\le\|S\|+\|T\|$ and $\|\alpha T\|\le|\alpha|\|T\|$.   This shows 
that $\eurm B(U;V)$ is a linear subspace of the space of all linear 
operators and is therefore a linear space over $\RoverC$ in its own 
right.   To 
check that the given formula for $\|T\|$ defines a norm, most of the 
work has just been done;   I suppose I should remark, for the sake of 
form, that $\|T\|\in\coint{0,\infty}$ for every $T$;  if $\alpha=0$, 
then of course $\|\alpha T\|=0=|\alpha|\|T\|$;  for other $\alpha$, 
 
\Centerline{$|\alpha|\|T\|=|\alpha|\|\alpha^{-1}\alpha T\| 
\le|\alpha||\alpha^{-1}|\|\alpha T\|=\|\alpha T\|\le|\alpha|\|T\|$,} 
 
\noindent so $\|\alpha T\|=|\alpha|\|T\|$.    Finally, if $\|T\|=0$ then 
$\|Tu\|\le\|T\|\|u\|=0$ for every $u\in U$, so $Tu=0$ for every $u$ and 
$T$ is the zero operator (in the space of all linear operators, and 
therefore in its subspace $\eurm B(U;V)$). 
}%end of proof of 2A4G 
 
\leader{2A4H}{Dual spaces} The most important case of $\eurm B(U;V)$ is 
when $V$ is the scalar 
field $\RoverC$ itself\cmmnt{ (of course we can think of $\RoverC$ as 
a normed space over itself, writing $\|\alpha\|=|\alpha|$ for each 
scalar $\alpha$)}.   In this case we call $\eurm B(U;\RoverC)$ the 
{\bf dual} of $U$; it is commonly denoted $U'$ or $U^*$;   I use the 
latter. 
 
\leader{2A4I}{Extensions of bounded operators:  Theorem} Let $U$ be a 
normed space and $V\subseteq U$ a dense linear subspace.   Let $W$ be a 
Banach space and $T_0:V\to 
W$ a bounded linear operator;  then there is a unique bounded linear 
operator $T:U\to W$ extending $T_0$, and $\|T\|=\|T_0\|$. 
 
\proof{{\bf (a)} For any $u\in U$, there is a sequence $\sequencen{v_n}$ 
in $V$ converging to $u$.   Now 
 
\Centerline{$\|T_0v_m-T_0v_n\|=\|T_0(v_m-v_n)\| 
\le\|T_0\|\|v_m-v_n\|\le\|T_0\|(\|v_m-u\|+\|u-v_n\|) 
\to 0$} 
 
\noindent as $m$, $n\to\infty$, so $\sequencen{T_0v_n}$ is Cauchy and 
$w=\lim_{n\to\infty}T_0v_n$ is defined in $W$.   If $\sequencen{v'_n}$ 
is another sequence in $V$ converging to $u$, then 
 
$$\eqalign{\|w-T_0v'_n\| 
&\le\|w-T_0v_n\|+\|T_0(v_n-v'_n)\|\cr 
&\le\|w-T_0v_n\|+\|T_0\|(\|v_n-u\|+\|u-v'_n\|)\to 0\cr}$$ 
 
\noindent as $n\to\infty$, so $w$ is also the limit of 
$\sequencen{T_0v'_n}$. 
 
\medskip 
 
{\bf (b)} We may therefore define $T:U\to W$ by setting 
$Tu=\lim_{n\to\infty}T_0v_n$ whenever $\sequencen{v_n}$ is a sequence in 
$V$ converging to $u$.   If $v\in V$, then we can set $v_n=v$ for every 
$n$ to see that $Tv=T_0v$;  thus $T$ extends $T_0$.   If $u$, $u'\in U$ 
and $\alpha\in\RoverC$, take sequences $\sequencen{v_n}$, 
$\sequencen{v'_n}$ in $V$ converging to $u$, $u'$ respectively;  in this 
case 
 
\Centerline{$\|(u+u')-(v_n+v'_n)\|\le\|u-v_n\|+\|u'-v'_n\|\to 0$, 
\quad$\|\alpha u-\alpha u_n\|=|\alpha|\|u-u_n\|\to 0$} 
 
\noindent as $n\to\infty$, so that 
$T(u+u')=\lim_{n\to\infty}T_0(v_n+v'_n)$, $T(\alpha 
u)=\lim_{n\to\infty}T_0(\alpha v_n)$, and 
 
$$\eqalign{\|T(u+u')-Tu-Tu'\| 
&\le\|T(u+u')-T_0(v_n+v'_n)\|+\|T_0v_n-Tu\|+\|T_0v'_n-Tu'\|\cr 
&\to 0,\cr}$$ 
 
\Centerline{$\|T(\alpha u)-\alpha Tu\| 
\le\|T(\alpha u)-T_0(\alpha v_n)\|+|\alpha|\|T_0v_n-Tu\|\to 0$} 
 
\noindent as $n\to\infty$.   This means that $\|T(u+u')-Tu-Tu'\|=0$, 
$\|T(\alpha u)-\alpha Tu\|=0$ so $T(u+u')=Tu+Tu'$, $T(\alpha u)=\alpha 
Tu$;  as $u$, $u'$ and $\alpha$ are arbitrary, $T$ is linear. 
 
\medskip 
 
{\bf (c)} For any $u\in U$, let $\sequencen{v_n}$ be a sequence in $V$ 
converging to $u$.   Then 
 
$$\eqalign{\|Tu\| 
&\le\|T_0v_n\|+\|Tu-T_0v_n\| 
\le\|T_0\|\|v_n\|+\|Tu-T_0v_n\|\cr 
&\le\|T_0\|(\|u\|+\|v_n-u\|)+\|Tu-T_0v_n\| 
\to\|T_0\|\|u\|\cr}$$ 
 
\noindent as $n\to\infty$, so $\|Tu\|\le\|T_0\|\|u\|$.   As $u$ is 
arbitrary, $T$ is bounded and $\|T\|\le\|T_0\|$.  Of course 
$\|T\|\ge\|T_0\|$ just because $T$ extends $T_0$. 
 
\medskip 
 
{\bf (d)} Finally, let $\tilde T$ be any other bounded linear operator 
from $U$ to $W$ extending $T$.   If $u\in U$, there is a sequence 
$\sequencen{v_n}$ in $V$ converging to $u$;  now 
 
\Centerline{$\|\tilde Tu-Tu\|\le\|\tilde T(u-v_n)\|+\|T(v_n-u)\| 
\le(\|\tilde T\|+\|T\|)\|u-v_n\|\to 0$} 
 
\noindent as $n\to\infty$, so $\|\tilde Tu-Tu\|=0$ and $\tilde Tu=Tu$. 
As $u$ is arbitrary, $\tilde T=T$.   Thus $T$ is unique. 
}%end of proof of 2A4I 
 
\leader{2A4J}{Normed algebras (a)} A {\bf normed algebra} is a normed 
space $(U,\|\,\|)$ together with a multiplication, a binary operator 
$\times$ on $U$, such that 
 
\Centerline{$u\times(v\times w)=(u\times v)\times w$,} 
 
\Centerline{$u\times(v+w)=(u\times v)+(u\times w)$, 
\quad$(u+v)\times w=(u\times w)+(v\times w)$,} 
 
\Centerline{$(\alpha u)\times v=u\times(\alpha v)=\alpha(u\times v)$,} 
 
\Centerline{$\|u\times v\|\le\|u\|\|v\|$} 
 
\noindent for all $u$, $v$, $w\in U$ and $\alpha\in\RoverC$. 
 
\spheader 2A4Jb A {\bf Banach algebra} is a normed algebra which is a 
Banach space.   A normed algebra\cmmnt{ $U$} is {\bf commutative} if its multiplication is commutative\cmmnt{, that is,  
$u\times v=v\times u$ for all $u$, $v\in U$}. 
 
\leader{*2A4K}{Definition} A  
normed space $U$ is {\bf uniformly convex} if 
for every $\epsilon>0$ there is a $\delta>0$ such that  
$\|u+v\|\le 2-\delta$ whenever $u$, $v\in U$, $\|u\|=\|v\|=1$ and  
$\|u-v\|\ge\epsilon$. 
 
\discrpage 
 
\frfilename{mt2a5.tex}
\versiondate{13.11.07}
\copyrightdate{1996}

\def\chaptername{Appendix}
\def\sectionname{Linear topological spaces}

\newsection{2A5}

The principal objective of \S2A3 is in fact the study of certain
topologies on the linear spaces of Chapter 24.   I give some fragments
of the general theory.

\leader{2A5A}{Linear space \dvrocolon{topologies}}\cmmnt{ Something
which is not covered in
detail by every introduction to functional analysis is the general
concept of `linear topological space'.   The ideas needed for the
work of \S245 are reasonably briefly expressed.

\medskip

\noindent}{\bf Definition} A {\bf linear topological space} or
{\bf topological vector space} over $\RoverC$\cmmnt{ (see 2A4A)} is a linear
space $U$ over $\RoverC$ together with a topology $\frak T$ such that
the maps

\Centerline{$(u,v)\mapsto u+v:U\times U\to U$,}

\Centerline{$(\alpha,u)\mapsto \alpha u:\RoverC\times U\to U$}

\noindent are both continuous, where the product spaces $U\times U$ and
$\RoverC\times U$ are given their product topologies\cmmnt{ (2A3T)}.
Given a linear space $U$, a topology on $U$ satisfying the conditions
above is a {\bf linear space topology}.   Note that

\Centerline{$(u,v)\mapsto u-v=u+(-1)v:U\times U\to U$}

\noindent will also be continuous.

\leader{2A5B}{}\cmmnt{ All the linear topological spaces we need
turn out to be readily presentable in the following terms.

\medskip

\noindent}{\bf Proposition} Suppose that $U$ is a linear space over
$\RoverC$, and $\Tau$
is a family of functionals $\tau:U\to\coint{0,\infty}$ such that

\quad (i) $\tau(u+v)\le\tau(u)+\tau(v)$ for all $u$, $v\in U$ and
$\tau\in\Tau$;

\quad (ii) $\tau(\alpha u)\le\tau(u)$ if $u\in U$, $|\alpha|\le 1$ and
$\tau\in\Tau$;

\quad (iii) $\lim_{\alpha\to 0}\tau(\alpha u)=0$ whenever $u\in U$ and
$\tau\in \Tau$.

\noindent For $\tau\in\Tau$, define
$\rho_{\tau}:U\times U\to\coint{0,\infty}$ by setting
$\rho_{\tau}(u,v)=\tau(u-v)$ for all $u$, $v\in U$.   Then each
$\rho_{\tau}$
is a pseudometric on $U$, and the topology defined by
$\Rho=\{\rho_{\tau}:\tau\in\Tau\}$ renders $U$ a linear topological
space.

\proof{{\bf (a)} It is worth noting immediately that

\Centerline{$\tau(0)=\lim_{\alpha\to 0}\tau(\alpha 0)=0$}

\noindent for every $\tau\in\Tau$.

\medskip

{\bf (b)} To see that every $\rho_{\tau}$ is a pseudometric, argue as
follows.

\medskip

\quad{\bf (i)} $\rho_{\tau}$ takes values
in $\coint{0,\infty}$ because $\tau$ does.

\medskip

\quad{\bf (ii)} If $u$, $v$, $w\in U$ then

$$\eqalign{\rho_{\tau}(u,w)
&=\tau(u-w)
=\tau((u-v)+(v-w))\cr
&\le\tau(u-v)+\tau(v-w)
=\rho_{\tau}(u,v)+\rho_{\tau}(v,w).\cr}$$

\medskip

\quad{\bf (iii)} If $u$, $v\in U$, then

\Centerline{$\rho(v,u)=\tau(v-u)
=\tau(-1(u-v))\le\tau(u,v)=\rho_{\tau}(u,v)$,}

\noindent and similarly $\rho_{\tau}(u,v)\le\rho_{\tau}(v,u)$, so the
two are equal.

\medskip

\quad{\bf (iv)} If $u\in U$ then $\rho_{\tau}(u,u)=\tau(0)=0$.

\medskip

{\bf (c)} Let $\frak T$ be the topology on $U$ defined by
$\{\rho_{\tau}:\tau\in\Tau\}$ (2A3F).

\medskip

\quad{\bf (i)} Addition is continuous because, given $\tau\in\Tau$, we
have

$$\eqalign{\rho_{\tau}(u'+v',u+v)
&=\tau((u'+v')-(u+v))\cr
&\le\tau(u'-u)+\tau(v'-v)
\le\rho_{\tau}(u',u)+\rho_{\tau}(v',v)\cr}$$

\noindent for all $u$, $v$, $u'$, $v'\in U$.   This means that, given
$\epsilon>0$ and $(u,v)\in U\times U$, we shall have

\Centerline{$\rho_{\tau}(u'+v',u+v)\le\epsilon$ whenever
$(u',v')\in
U((u,v);\tilde\rho_{\tau},\bar\rho_{\tau};\Bover{\epsilon}2)$,}

\noindent using the language of 2A3Tb.   Because $\tilde\rho_{\tau}$,
$\bar\rho_{\tau}$ are two of the pseudometrics defining the product
topology of $U\times U$ (2A3Tb), $(u,v)\mapsto u+v$ is continuous, by
the criterion of 2A3H.

\medskip

\quad{\bf (ii)} Scalar multiplication is continuous because if $u\in U$
and $n\in\Bbb N$
then $\tau(nu)\le n\tau(u)$ for every $\tau\in\Tau$ (induce on $n$).
Consequently, if $\tau\in\Tau$,

\Centerline{$\tau(\alpha u)\le n\tau(\Bover{\alpha}{n}u)\le n\tau(u)$}

\noindent whenever $|\alpha|<n\in\Bbb N$ and $\tau\in\Tau$.  Now, given
$(\alpha,u)\in \RoverC\times U$ and $\epsilon>0$, take $n>|\alpha|$ and
$\delta>0$ such that $\delta\le\min(n-|\alpha|,\bover{\epsilon}{2n})$
and $\tau(\gamma u)\le\bover{\epsilon}2$ whenever $|\gamma|\le\delta$;
then

$$\eqalign{\rho_{\tau}(\alpha'u',\alpha u)
&=\tau(\alpha'u'-\alpha u)
\le\tau(\alpha'(u'-u))+\tau((\alpha'-\alpha)u)\cr
&\le n\tau(u'-u)+\tau((\alpha'-\alpha)u)\cr}$$

\noindent whenever  $u'\in U$ and  $\alpha'\in\RoverC$
and $|\alpha'|<n\in\Bbb N$.   Accordingly, setting
$\theta(\alpha',\alpha)=|\alpha'-\alpha|$ for $\alpha'$,
$\alpha\in\RoverC$,

\Centerline{$\rho_{\tau}(\alpha'u',\alpha u)
\le n\delta+\Bover{\epsilon}2\le\epsilon$}

\noindent whenever

\Centerline{$(\alpha',u')\in
U((\alpha,u);\tilde\theta,\bar\rho_{\tau};\delta)$.}

\noindent Because $\tilde\theta$ and $\bar\rho_{\tau}$ are among the
pseudometrics defining the topology of $\RoverC\times U$, the map
$(\alpha,u)\mapsto\alpha u$ satisfies the criterion of 2A3H and is
continuous.

Thus $\frak T$ is a linear space topology on $U$.
}%end of proof of  2A5B

\medskip

\noindent{\bf Remark}\dvAnew{2013} 
Functionals satisfying the conditions
(i)-(iii) above are called {\bf F-seminorms};  an F-seminorm $\tau$ such
that $\tau(u)\ne 0$ for every non-zero $u$ is an {\bf F-norm}.
 
\leader{*2A5C}{}\cmmnt{ We do not need it for Chapter 24, but the
following is worth knowing.

\medskip

\noindent}{\bf Theorem} Let $U$ be a linear space and $\frak T$ a linear
space topology on $U$.

(a) There is a family $\Tau$ of F-seminorms defining $\frak T$ as in
2A5B.

(b) If $\frak T$ is metrizable, we can take $\Tau$ to consist of a
single functional.

\proof{{\bf (a)} {\smc Kelley \& Namioka 76}, p.\ 50.

\medskip

{\bf (b)} {\smc K\"othe 69}, \S15.11.
}%end of proof of 2A5C

\vleader{72pt}{2A5D}{Definition} Let $U$ be a linear space over $\RoverC$.
Then a {\bf seminorm} on $U$ is a functional $\tau:U\to\coint{0,\infty}$
such that

\quad (i) $\tau(u+v)\le\tau(u)+\tau(v)$ for all $u$, $v\in U$;

\quad (ii) $\tau(\alpha u)=|\alpha|\tau(u)$ if $u\in U$,
$\alpha\in\RoverC$.

\cmmnt{\noindent Observe that a norm is always a seminorm, and that a
seminorm is always an F-seminorm.   In
particular, the association of a metric with a norm (2A4Bb) is a special
case of 2A5B.}

\leader{2A5E}{Convex sets (a)} Let $U$ be a linear space over $\RoverC$.
A subset $C$ of $U$ is {\bf convex} if
$\alpha u+(1-\alpha)v\in C$ whenever $u$, $v\in C$ and $\alpha\in[0,1]$.
The intersection of any family of convex sets is convex, so for every
set $A\subseteq U$ there is a smallest convex set including $A$;
this is\cmmnt{ just} the set of vectors expressible as
$\sum_{i=0}^n\alpha_iu_i$
where $u_0,\ldots,u_n\in A$, $\alpha_0,\ldots,\alpha_n\in[0,1]$ and
$\sum_{i=0}^n\alpha_i=1$\prooflet{ ({\smc Bourbaki 87}, II.2.3)};  it
is the {\bf convex hull} of $A$.   If $C$, $C'\subseteq U$ 
are convex, and $\alpha\in\RoverC$, 
then $\alpha C$ and $C+C'$ are convex\dvAnew{2012}.   If $C\subseteq U$ is
convex, $V$ is another linear space over $\RoverC$, and $T:U\to V$ is a linear
operator, then $T[C]\subseteq V$ is convex\dvAnew{2021}.

\spheader 2A5Eb If $U$ is a linear topological space, the closure of
any convex set is convex\prooflet{ ({\smc Bourbaki 87}, II.2.6)}.   It
follows that, for any $A\subseteq U$, the closure of the convex hull of
$A$ is the smallest closed convex set including $A$;  this is the
{\bf closed convex hull} of $A$.

\spheader 2A5Ec I note for future reference that in a linear topological
space, the closure of any linear subspace is a linear subspace.
\prooflet{({\smc Bourbaki 87}, I.1.3; {\smc K\"othe 69}, \S15.2.   Compare
2A4Cb.)}

\leader{2A5F}{Completeness in linear topological
\dvrocolon{spaces}}\cmmnt{ In normed spaces,
completeness can be described in terms of Cauchy sequences (2A4D).   In
general linear topological spaces this is inadequate.   The true theory
of `completeness' demands the concept of `uniform space' (see \S3A4 in
the next volume, or
{\smc Kelley 55}, chap.\ 6;  {\smc Engelking 89}, \S8.1:
{\smc Bourbaki 66}, chap.\ II);  I shall not
describe this here, but will give a version adapted
to linear spaces.   I mention this only because you will I hope some day
come to the general theory (in Volume 3 of this treatise, if not
before), and you should be aware that the special
case described here gives a misleading emphasis at some points.

\medskip

\noindent}{\bf Definitions} Let $U$ be a linear space over $\RoverC$,
and $\frak T$ a
linear space topology on $U$.   A filter $\Cal F$ on $U$ is {\bf Cauchy}
if for every open set $G$ in $U$ containing $0$ there is an $F\in\Cal F$
such that $F-F=\{u-v:u,\,v\in F\}$ is included in $G$.
$U$ is {\bf complete} if every Cauchy filter on $U$ is convergent.

\leader{2A5G}{}\cmmnt{ Cauchy filters have a simple description when a
linear space topology is defined by the method of 2A5B.

\medskip

\noindent}{\bf Lemma} Let $U$ be a linear space over $\RoverC$, and let
$\Tau$ be a family of F-seminorms defining a linear space topology on
$U$, as in 2A5B.   Then a filter $\Cal F$ on $U$ is
Cauchy iff for every $\tau\in\Tau$ and $\epsilon>0$ there is an
$F\in\Cal F$ such that $\tau(u-v)\le\epsilon$ for all $u$, $v\in F$.

\proof{{\bf (a)} Suppose that $\Cal F$ is Cauchy, $\tau\in\Tau$ and
$\epsilon>0$.   Then
$G=U(0;\rho_{\tau};\epsilon)$ is open (using the language of 2A3F-2A3G),
so there is an $F\in\Cal F$ such that $F-F\subseteq G$;  but this just
means that $\tau(u-v)<\epsilon$ for all $u$, $v\in F$.

\medskip

{\bf (b)} Suppose that
$\Cal F$ satisfies the criterion, and that $G$ is an open set containing
$0$.   Then there are $\tau_0,\ldots,\tau_n\in\Tau$ and $\epsilon>0$
such that $U(0;\rho_{\tau_0},\ldots,\rho_{\tau_n};\epsilon)\subseteq G$.
For each $i\le n$ there is an $F_i\in\Cal F$ such that
$\tau_i(u,v)<\bover{\epsilon}2$ for all $u$, $v\in F_i$;  now
$F=\bigcap_{i\le n}F_i\in\Cal F$ and $u-v\in G$ for all $u$, $v\in F$.
}%end of proof of 2A5G

\leader{2A5H}{Normed spaces and sequential
\dvrocolon{completeness}}\cmmnt{ I had
better point out that for normed spaces the definition of 2A5F agrees
with that of 2A4D.

\medskip

\noindent}{\bf Proposition} Let $(U,\|\,\|)$ be a normed space over
$\RoverC$, and let $\frak T$ be the linear space topology on $U$ defined
by the method of 2A5B from the set $\Tau=\{\|\,\|\}$.   Then $U$ is
complete in the sense of 2A5F iff it is complete in the sense of 2A4D.

\proof{{\bf (a)} Suppose first that $U$ is complete in the sense of
2A5F.   Let $\sequencen{u_n}$ be a sequence in $U$ which is Cauchy in
the sense of 2A4Da.   Set

\Centerline{$\Cal F
=\{F:F\subseteq U,\,\{n:u_n\notin F\}\text{ is finite}\}$.}

\noindent Then it is easy to check that $\Cal F$ is a filter on $U$, the
image of the Fr\'echet filter under the map $n\mapsto u_n:\Bbb N\to U$.
If $\epsilon>0$, take $m\in\Bbb N$ such that $\|u_j-u_k\|\le\epsilon$
whenever $j$, $k\ge m$;  then $F=\{u_j:j\ge m\}$ belongs to $\Cal F$,
and $\|u-v\|\le\epsilon$ for all $u$, $v\in F$.   So $\Cal F$ is Cauchy
in the sense of 2A5F, and has a
limit $u$ say.   Now, for any $\epsilon>0$, the set
$\{v:\|v-u\|<\epsilon\}=U(u;\rho_{\|\,\|};\epsilon)$ is an open set
containing $u$, so belongs to $\Cal F$, and $\{n:\|u_n-u\|\ge\epsilon\}$
is finite, that is, there is an $m\in\Bbb N$ such that
$\|u_m-u\|<\epsilon$ whenever $n\ge m$.   As $\epsilon$ is arbitrary,
$u=\lim_{n\to\infty}u_n$ in the sense of 2A3M.   As $\sequencen{u_n}$
is arbitrary, $U$ is complete in the sense of 2A4D.

\medskip

{\bf (b)} Now suppose that $U$ is complete in the sense of 2A4D.
Let $\Cal F$ be a Cauchy filter on $U$.   For each $n\in\Bbb N$, choose
a set $F_n\in\Cal F$ such that $\|u-v\|\le 2^{-n}$ for all $u$,
$v\in F_n$.   For each $n\in\Bbb N$, $F'_n=\bigcap_{i\le n}F_i$ belongs
to $\Cal F$, so
is not empty;  choose $u_n\in F'_n$.   If $m\in\Bbb N$ and $j$,
$k\ge m$, then both $u_j$ and $u_k$ belong to $F_m$, so
$\|u_j-u_k\|\le 2^{-m}$;  thus $\sequencen{u_n}$ is a Cauchy sequence in
the sense of 2A4Da, and has a limit $u$ say.   Now take
any $\epsilon>0$ and $m\in\Bbb N$ such that $2^{-m+1}\le\epsilon$.
There is surely a $k\ge m$ such that $\|u_k-u\|\le 2^{-m}$;  now
$u_k\in F_m$, so

\Centerline{$F_m\subseteq\{v:\|v-u_k\|\le 2^{-m}\}
\subseteq\{v:\|v-u\|\le 2^{-m+1}\}
\subseteq\{v:\rho_{\|\,\|}(v,u)\le\epsilon\}$,}

\noindent and $\{v:\rho_{\|\,\|}(v,u)\le\epsilon\}\in\Cal F$.
As $\epsilon$ is arbitrary, $\Cal F$ converges to $u$, by 2A3Sd.   As
$\Cal F$ is arbitrary, $U$ is complete.

\medskip

{\bf (c)}
Thus the two definitions coincide, provided at least that we allow the
countably many simultaneous choices of the $u_n$ in part (b) of the
proof.
}%end of proof of 2A5H

\leader{2A5I}{Weak topologies}\cmmnt{ I come now to brief notes on
`weak topologies' on normed spaces;
from the point of view of this volume, these
are in fact the primary examples of linear space topologies.}
Let $U$ be a normed linear space over $\RoverC$.

\spheader 2A5Ia  Write $U^*$ for its dual
$\eurm B(U;\RoverC)$\cmmnt{ (2A4H)}.   \cmmnt{If $h\in U^*$, then
$|h|:U\to\coint{0,\infty}$ is a seminorm, so}
$\Tau=\{|h|:h\in U^*\}$ defines a linear space topology on $U$\cmmnt{, 
by 2A5B};  
this is\cmmnt{ called} the {\bf weak topology} of $U$.

\spheader 2A5Ib A filter $\Cal F$ on $U$ converges to $u\in U$ for the
weak topology of $U$ iff\cmmnt{ $\lim_{v\to\Cal F}\rho_{|h|}(v,u)=0$
for every
$h\in U^*$ (2A3Sd), that is, iff $\lim_{v\to\Cal F}|h(v-u)|=0$ for every
$h\in U^*$, that is,
iff} $\lim_{v\to\Cal F}h(v)=h(u)$ for every $h\in U^*$.

\spheader 2A5Ic A set $C\subseteq U$ is called {\bf weakly compact} if
it is compact for the weak topology of $U$.   So\cmmnt{ (subject to
the axiom of choice)} a set $C\subseteq U$ is weakly compact iff for
every ultrafilter
$\Cal F$ on $U$ containing $C$ there is a $u\in C$ such that
$\lim_{v\to\Cal F}h(v)=h(u)$ for every $h\in U^*$\cmmnt{ (put 2A3R
together with (b) above)}.

\spheader 2A5Id A subset $A$ of $U$ is called {\bf relatively weakly
compact} if it is a subset of some weakly compact subset of $U$.

\spheader 2A5Ie If $h\in U^*$, then $h:U\to\RoverC$ is continuous for
the weak
topology on $U$ and the usual topology of $\RoverC$\cmmnt{;  this is
obvious if we apply the criterion of 2A3H}.   So if
$A\subseteq U$ is relatively weakly compact, $h[A]$ must be bounded in
$\RoverC$.   \prooflet{\Prf\ Let $C\supseteq A$ be a weakly compact set.
Then $h[C]$ is compact in $\RoverC$, by 2A3Nb, so is bounded, by 2A2F
(noting that if the underlying field is $\Bbb C$, then it can be
identified, as metric space, with $\BbbR^2$).
Accordingly $h[A]$ also is bounded.\ \Qed}

\spheader 2A5If If $V$ is another normed space and $T:U\to V$ is a
bounded linear operator, then $T$ is continuous for the respective weak
topologies.   \prooflet{\Prf\ If $h\in V^*$ then the composition $hT$
belongs to $U^*$.   Now, for any $u$, $v\in U$,

\Centerline{$\rho_{|h|}(Tu,Tv)=|h(Tu-Tv)|=|hT(u-v)|=\rho_{|hT|}(u,v)$,}

\noindent taking $\rho_{|h|}$, $\rho_{|hT|}$ to be the pseudometrics on
$V$, $U$ respectively defined by the formula of 2A5B.   By 2A3H, $T$ is
continuous.\ \Qed}

\spheader 2A5Ig Corresponding to the weak topology on a normed space
$U$, we have the {\bf weak*} or {\bf w*-}topology on its dual
$U^*$, defined by the set $\Tau=\{|\hat u|:u\in U\}$, where I write
$\hat u(f)=f(u)$ for every $f\in U^*$, $u\in U$.   As in (a), this is
a linear space topology on $U^*$.   \cmmnt{(It is essential to
distinguish between the `weak*' topology and the `weak' topology on
$U^*$.   The former depends only on the action of $U$ on $U^*$, the
latter on the action of $U^{**}=(U^*)^*$.   You will have no difficulty
in checking that $\hat u\in U^{**}$ for every $u\in U$, but the point is
that there may be members of $U^{**}$ not representable in this way,
leading to open sets for the weak topology which are not open for the
weak* topology.)}

\leader{*2A5J}{Angelic spaces}\cmmnt{ I do not rely on the
following ideas, but they may throw light on some results in
\S\S246-247.}   First, a topological space $X$ is {\bf regular} if
whenever $G\subseteq X$ is open and $x\in G$ then there is an open set
$H$ such that $x\in H\subseteq\overline{H}\subseteq G$.   Next, a
regular Hausdorff space $X$ is {\bf angelic} if whenever $A\subseteq X$
is such that every sequence in $A$ has a cluster point in $X$, then
$\overline{A}$ is compact and every point of $\overline{A}$ is the limit
of a sequence in $A$.\cmmnt{  What this means is that compactness in
$X$, and
the topologies of compact subsets of $X$, can be effectively described
in terms of sequences.}   Now\cmmnt{ the theorem (due to Eberlein and
\v Smulian) is that} any normed space is
angelic in its weak topology.\cmmnt{   (462D in Volume 4;
{\smc K\"othe 69},
\S24;  {\smc Dunford \& Schwartz 57}, V.6.1.)   In
particular, this is true of $L^1$ spaces, which makes it less surprising
that there should be criteria for weak compactness in $L^1$ spaces which
deal only with sequences.
}%end of comment

\discrpage

\frfilename{mt2a6.tex} 
\versiondate{10.11.14} 
\copyrightdate{1994} 
 
\def\chaptername{Appendix} 
\def\sectionname{Factorization of matrices} 
 
\newsection{2A6} 
 
I spend a couple of pages on the linear algebra of $\BbbR^r$ required for 
Chapter 26.  I give 
only one proof, because this is material which can be found in any textbook 
of elementary linear algebra;  but I think it may be helpful to run 
through the basic ideas in the language which I use for this treatise. 
 
\leader{2A6A}{Determinants}\cmmnt{ We need to know the 
following things about determinants.} 
 
\inset{(i) Every $r\times r$ real matrix $T$ has a real determinant 
$\det T$.} 
 
\inset{(ii) For any $r\times r$ matrices $S$ and $T$,  
$\det ST=\det S\det T$.} 
 
\inset{(iii) If $T$ is a diagonal matrix, its determinant is just the 
product of its diagonal entries.} 
 
\inset{(iv) For any $r\times r$ matrix $T$, $\det T\trs=\det T$,  
where $T\trs$ is the transpose of $T$.} 
 
\inset{(v) $\det T$ is a continuous function of the coefficients of 
$T$.} 
 
\cmmnt{\noindent There are so many routes through this topic that I avoid
even a definition of `determinant';  I invite you to check your memory, or 
your favourite text, to confirm that you are indeed happy with the facts 
above. 
}%end of comment 
 
\leader{2A6B}{Orthonormal families} For $x=(\xi_1,\ldots,\xi_r)$, 
$y=(\eta_1,\ldots,\eta_r)\in\BbbR^r$, write $x\dotproduct 
y=\sum_{i=1}^r\xi_i\eta_i$; \cmmnt{of course} $\|x\|$\cmmnt{, as 
defined in 1A2A,} is $\sqrt{x\dotproduct x}$. 
\cmmnt{Recall that} $x_1,\ldots,x_k$ are {\bf orthonormal} if 
$x_i\dotproduct 
x_j=0$ for $i\ne j$, $1$ for $i=j$.   \cmmnt{The results we need here 
are:} 
 
\inset{(i) If $x_1,\ldots,x_k$ are orthonormal vectors in $\BbbR^r$, 
where $k<r$, then there are vectors $x_{k+1},\ldots,x_r$ in $\BbbR^r$ such 
that $x_1,\ldots,x_r$ are orthonormal.} 
 
\inset{(ii) An $r\times r$ matrix $P$ is {\bf orthogonal} if $P\trs P$ is 
the identity matrix;  equivalently, if the columns of $P$ are 
orthonormal.} 
 
\inset{(iii) For an orthogonal matrix $P$, $\det P$ must be 
$\pm 1$\cmmnt{ (put (ii)-(iv) of 2A6A together)}.} 
\inset{(iv) If $P$ is orthogonal, then $Px\dotproduct 
Py\cmmnt{\mskip5mu =P\trs Px\dotproduct 
y}=x\dotproduct y$ for all $x$, $y\in\BbbR^r$.} 
 
\inset{(v) If $P$ is orthogonal, so is $P\trs=P^{-1}$.} 
 
\inset{(vi) If $P$ and $Q$ are orthogonal, so is $PQ$.} 
 
\leader{2A6C}{}\cmmnt{ I now give a proposition which is not always 
included in 
elementary presentations.   Of course there are many approaches to this; 
I offer a direct one. 
 
\medskip 
 
\noindent}{\bf Proposition} Let $T$ be any real $r\times r$ matrix. 
Then $T$ is expressible as $PDQ$ where $P$ and $Q$ are orthogonal 
matrices and $D$ is a diagonal matrix with non-negative coefficients. 
 
\proof{ I induce on $r$. 
 
\medskip 
 
{\bf (a)} If $r=1$, then $T=(\tau_{11})$.   Set $D=(|\tau_{11}|)$, 
$P=(1)$ and $Q=(1)$ if $\tau_{11}\ge 0$, $(-1)$ otherwise. 
 
\medskip 
 
{\bf (b)(i)} For the inductive step to $r+1\ge 2$, consider the unit 
ball $B=\{x:x\in\BbbR^{r+1},\,\|x\|\le 1\}$.   This is a closed bounded 
set in $\BbbR^{r+1}$, so is compact (2A2F).   The maps  
$x\mapsto Tx:\BbbR^{r+1}\to\BbbR^{r+1}$ and  
$x\mapsto\|x\|:\BbbR^{r+1}\to\Bbb R$ are continuous, so the function  
$x\mapsto\|Tx\|:B\to\Bbb R$ is 
bounded and attains its bounds (2A2G), and there is a $u\in B$ such 
that $\|Tu\|\ge\|Tx\|$ for every $x\in B$.   Observe that $\|Tu\|$ must 
be the norm $\|T\|$ of $T$ as defined in 262H.   Set 
$\delta=\|T\|=\|Tu\|$.   If $\delta=0$, then $T$ must be the zero 
matrix, and the result is trivial;  so let us suppose that $\delta>0$.    
In this case $\|u\|$ must be exactly $1$, since otherwise we should have 
$u=\|u\|u'$ where $\|u'\|=1$ and $\|Tu'\|>\|Tu\|$. 
 
\medskip 
 
\quad{\bf (ii)} If $x\in\BbbR^{r+1}$ and $x\dotproduct u=0$, then 
$Tx\dotproduct Tu=0$.   \Prf\Quer\ If not, set  
$\gamma=Tx\dotproduct Tu\ne 0$.   Consider $y=u+\eta\gamma x$ for small 
$\eta>0$.   We have 
 
\Centerline{$\|y\|^2=y\dotproduct y 
=u\dotproduct u+2\eta\gamma u\dotproduct x+\eta^2\gamma^2 x\dotproduct x 
=\|u\|^2+\eta^2\gamma^2\|x\|^2 
=1+\eta^2\gamma^2\|x\|^2$,} 
 
\noindent while 
 
\Centerline{$\|Ty\|^2=Ty\dotproduct Ty 
=Tu\dotproduct Tu+2\eta\gamma Tu\dotproduct Tx 
  +\eta^2\gamma^2 Tx\dotproduct Tx 
=\delta^2+2\eta\gamma^2+\eta^2\gamma^2\|Tx\|^2$.} 
 
\noindent But also 
$\|Ty\|^2\le\delta^2\|y\|^2$ (2A4Fb), so 
 
\Centerline{$\delta^2+2\eta\gamma^2+\eta^2\gamma^2\|Tx\|^2 
\le\delta^2(1+\eta^2\gamma^2\|x\|^2)$} 
 
\noindent and 
 
\Centerline{$2\eta\gamma^2 
\le\delta^2\eta^2\gamma^2\|x\|^2-\eta^2\gamma^2\|Tx\|^2$,} 
 
\noindent that is, 
 
\Centerline{$2\le\eta(\delta^2\|x\|^2-\|Tx\|^2)$.} 
 
\noindent But this surely cannot be true for all $\eta>0$, so we have a 
contradiction.\ \Bang\Qed 
 
\medskip 
 
\quad{\bf (iii)} Set $v=\delta^{-1}Tu$, so that $\|v\|=1$.   Let 
$u_1,\ldots,u_{r+1}$ be orthonormal vectors such that $u_{r+1}=u$, and 
let $Q_0$ be the orthogonal $(r+1)\times(r+1)$ matrix with columns 
$u_1,\ldots,u_{r+1}$;  then, writing $e_1,\ldots,e_{r+1}$ for the 
standard orthonormal basis of $\BbbR^{r+1}$, we have $Q_0e_i=u_i$ for 
each $i$, and $Q_0e_{r+1}=u$.   Similarly, there is an orthogonal matrix 
$P_0$ such that $P_0e_{r+1}=v$. 
 
Set $T_1=P_0^{-1}TQ_0$.   Then 
 
\Centerline{$T_1e_{r+1}=P_0^{-1}Tu=\delta P_0^{-1}v=\delta e_{r+1}$,} 
 
\noindent while if $x\dotproduct e_{r+1}=0$ then  
$Q_0x\dotproduct u=0$ (2A6B(iv)), so that 
 
\Centerline{$T_1x\dotproduct e_{r+1}=P_0T_1x\dotproduct P_0e_{r+1} 
=TQ_0x\dotproduct v=0$,} 
 
\noindent by (ii).   This means that $T_1$ must be of the form 
 
$$\Matrix{S&0\\0&\delta},$$ 
 
\noindent where $S$ is an $r\times r$ matrix. 
 
\medskip 
 
\quad{\bf (iv)} By the inductive hypothesis, $S$ is expressible as 
$\tilde P\tilde D\tilde Q$, where $\tilde P$ and $\tilde Q$ are 
orthogonal $r\times r$ matrices and $\tilde D$ is a diagonal $r\times r$ 
matrix with non-negative coefficients.   Set 
 
$$P_1=\Matrix{\tilde P&0\\0&1},\quad Q_1=\Matrix{\tilde Q&0\\0&1},\quad 
D=\Matrix{\tilde D&0\\0&\delta}.$$ 
 
\noindent Then $P_1$ and $Q_1$ are orthogonal and $D$ is diagonal, with 
non-negative coefficients, and $P_1DQ_1=T_1$.   Now set 
 
\Centerline{$P=P_0P_1$,\quad $Q=Q_1Q_0^{-1}$,} 
\noindent so that $P$ and $Q$ are orthogonal (2A6B(v)-(vi)) and 
 
\Centerline{$PDQ=P_0P_1DQ_1Q_0^{-1}=P_0T_1Q_0^{-1}=T$.} 
 
\noindent Thus the induction proceeds. 
}%end of proof of 2A6C 
 
\frnewpage 
 
\references{
\frfilename{mt2r.tex}
\versiondate{17.12.12}
\copyrightdate{2000}

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

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

\centerline{\bf References for Volume 2}

\medskip

%A

Alexits G.\ [78] (ed.) {\it Fourier Analysis and Approximation Theory.}
North-Holland, 1978 (Colloq.\ Math.\ Soc.\ Janos Bolyai 19).

Antonov N.Yu.\ [96] `Convergence of Fourier series', East J.\ Approx.\ 7 (1996) 187-196.
\cmmnt{[\S286 {\it notes\/}.]}

Arias de Reyna J.\ [02] {\it Pointwise Convergence of Fourier Series.} Springer, 2002 (Lecture Notes in Mathematics 1785).
\cmmnt{[\S286 {\it notes\/}.]}

\medskip%B

Baker R.\ [04] `{}``Lebesgue measure'' on $\Bbb R^{\infty}$, II',
Proc.\ Amer.\ Math.\ Soc.\ 132 (2004) 2577-2591.
\cmmnt{[254Yb.]}

Bergelson V., March P.\ \& Rosenblatt J.\ [96] (eds.) {\it Convergence
in Ergodic Theory and Probability.}  de Gruyter, 1996.

\indexheader{Bogachev}
Bogachev V.I.\ [07] {\it Measure theory.}  Springer, 2007.

du Bois-Reymond P.\ [1876] `Untersuchungen \"uber die Convergenz und
Divergenz der Fouriersche Darstellungformeln', Abh.\ Akad.\ M\"unchen 12
(1876) 1-103.   \cmmnt{[\S282 {\it notes}.]}

Bourbaki N.\ [66] {\it General Topology.}  Hermann/Addison-Wesley, 1968.
\cmmnt{[2A5F.]}

Bourbaki N.\ [87] {\it Topological Vector Spaces.}  Springer, 1987.
\cmmnt{[2A5E.]}

\medskip%C

Carleson L.\ [66] `On convergence and growth of partial sums of
Fourier series', Acta Math.\ 116 (1966) 135-157.  \cmmnt{[\S282 {\it
notes}, \S286 {\it intro.}, \S286 {\it notes}.]}

Clarkson J.A.\ [1936] `Uniformly convex spaces', Trans.\ Amer.\ Math.\ Soc.\
40 (1936) 396-414.
\cmmnt{[244O.]}

\medskip%D

Defant A.\ \& Floret K.\ [93] {\it Tensor Norms and Operator Ideals},
North-Holland, 1993.   \cmmnt{[\S253 {\it notes}.]}

Doob J.L.\ [53] {\it Stochastic Processes.}  Wiley, 1953.
%used for 275Ye

Dudley R.M.\ [89] {\it Real Analysis and Probability.} Wadsworth \&
Brooks/Cole, 1989.   \cmmnt{[\S282 {\it notes}.]}

Dunford N.\ \& Schwartz J.T.\ [57] {\it Linear Operators I.}  Wiley,
1957 (reprinted 1988).  \cmmnt{[\S244 {\it notes}, 2A5J.]}

\medskip%E

Enderton H.B.\ [77] {\it Elements of Set Theory.}   Academic, 1977.
\cmmnt{[\S2A1.]}

Engelking R.\ [89] {\it General Topology.} Heldermann, 1989 (Sigma
Series in Pure Mathematics 6).   \cmmnt{[2A5F.]}

Etemadi N.\ [96] `On convergence of partial sums of independent random
variables', pp.\ 137-144 in {\smc Bergelson March \& Rosenblatt 96}.
\cmmnt{[272V.]}

Evans L.C.\ \& Gariepy R.F.\ [92] {\it Measure Theory and Fine
Properties of Functions.}   CRC Press, 1992.
\cmmnt{[263Ec, \S265 {\it notes}.]}
%Kirszbraun's theorem mentioned but not proved

\medskip%F

Federer H.\ [69] {\it Geometric Measure Theory.}  Springer, 1969
(reprinted 1996).
\cmmnt{[262C, 263Ec, \S264 {\it notes}, \S265 {\it notes},
\S266 {\it notes}.]}

Feller W.\ [66] {\it An Introduction to Probability Theory and its
Applications,} vol.\ II.   Wiley, 1966.  \cmmnt{[Chap.\ 27 {\it
intro.}, 274H, 275Xc, 285N.]}

Fremlin D.H.\ [74] {\it Topological Riesz Spaces and
Measure Theory.}   Cambridge U.P., 1974.
\cmmnt{[\S232 {\it notes}, 241F, \S244 {\it notes}, \S245 {\it notes},
\S247 {\it notes}.]}

Fremlin D.H.\ [93] `Real-valued-measurable cardinals', pp.\ 151-304 in
{\smc Judah 93}.  \cmmnt{[232H.]}

\medskip%G\medskip%H

Haimo D.T.\ [67] (ed.) {\it Orthogonal Expansions and their Continuous
Analogues.}   Southern Illinois University Press, 1967.

Hall P.\ [82] {\it Rates of Convergence in the Central Limit Theorem.}
Pitman, 1982.  \cmmnt{[274H.]}

Halmos P.R.\ [50] {\it Measure Theory.}  Van Nostrand, 1950.
\cmmnt{[\S251 {\it notes}, \S252 {\it notes}, 255Yn.]}

Halmos P.R.\ [60] {\it Naive Set Theory.} Van Nostrand, 1960.
\cmmnt{[\S2A1.]}

Hanner O.\ [56] `On the uniform convexity of $L^p$ and $l^p$',
Arkiv f\"or Matematik 3 (1956) 239-244.
\cmmnt{[244O.]}

Henle J.M.\ [86] {\it An Outline of Set Theory.}  Springer, 1986.
\cmmnt{[\S2A1.]}

Hoeffding W.\ [63]
`Probability inequalities for sums of bounded random variables',
J.\ Amer.\ Statistical Association 58 (1963) 13-30.
\cmmnt{[272W.]}

Hunt R.A.\ [67] `On the convergence of Fourier series', pp.\ 235-255
in {\smc Haimo 67}.  \cmmnt{[\S286 {\it notes}.]}

\medskip%I\medskip%J

Jorsb{\o}e O.G.\ \& Mejlbro L.\ [82] {\it The Carleson-Hunt Theorem on
Fourier Series.}  Springer, 1982 (Lecture Notes in Mathematics 911).
\cmmnt{[\S286 {\it notes}.]}

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

\medskip%K

\indexheader{Kelley}
Kelley J.L.\ [55] {\it General Topology.} Van Nostrand, 1955.
\cmmnt{[2A5F.]}

Kelley J.L.\ \& Namioka I.\ [76] {\it Linear Topological Spaces.}
Springer, 1976.   \cmmnt{[2A5C.]}

\indexheader{Kirszbraun}
Kirszbraun M.D.\ [1934] `\"Uber die zusammenziehenden und Lipschitzian
Transformationen', Fund.\ Math.\ 22 (1934) 77-108.  \cmmnt{[262C.]}

Kolmogorov A.N.\ [1926] `Une s\'erie de Fourier-Lebesgue divergente
partout', C.\ R.\ Acad.\ Sci.\ Paris 183 (1926) 1327-1328.
\cmmnt{[\S282 {\it notes}.]}

Koml\'os J.\ [67] `A generalization of a problem of Steinhaus', Acta
Math.\ Acad.\ Sci.\ Hung.\ 18 (1967) 217-229.  \cmmnt{[276H.]}

K\"orner T.W.\ [88] {\it Fourier Analysis.}  Cambridge U.P., 1988.
\cmmnt{[\S282 {\it notes}.]}

K\"othe G.\ [69] {\it Topological Vector Spaces I.}   Springer, 1969.
\cmmnt{[2A5C, 2A5E, 2A5J.]}

Krivine J.-L.\ [71] {\it Introduction to Axiomatic Set Theory.}
D.~Reidel, 1971.  \cmmnt{[\S2A1.]}

\medskip%L

Lacey M.T.\ [05] {\it Carleson's Theorem: Proof, Complements, Variations.}
{\tt http://arxiv.org/pdf/math{\bsp}0307008v4{\penalty-100}.pdf}.
%expanded from 
% Publ.\ Mat.\ 48 (2004) 251-307
\cmmnt{[\S286 {\it notes}.]}

Lacey M.T.\ \& Thiele C.M.\ [00] `A proof of boundedness of the Carleson
operator', Math.\ Research Letters 7 (2000) 1-10.
\cmmnt{[\S286 {\it intro.}, 286H.]}

Lebesgue H.\ [72] {\it Oeuvres Scientifiques.}   L'Enseignement
Math\'ematique, Institut de Math\'ematiques, Univ.\ de Gen\`eve, 1972.
\cmmnt{[Chap.\ 27 {\it intro.}]}

Liapounoff A.\ [1901] `Nouvelle forme du th\'eor\`eme sur la limite
de probabilit\'e', M\'em.\ Acad.\ Imp.\ Sci.\ St-P\'etersbourg 12(5)
(1901) 1-24. \cmmnt{[274Xh.]}

Lighthill M.J.\ [59] {\it Introduction to Fourier Analysis and
Generalised Functions.}  Cambridge U.P., 1959.
\cmmnt{[\S284 {\it notes}.]}

Lindeberg J.W.\ [1922] `Eine neue Herleitung des Exponentialgesetzes in
der Wahrscheinlichkeitsrechnung', Math.\ Zeitschrift 15 (1922) 211-225.
\cmmnt{[274H, \S274 {\it notes}.]}

Lipschutz S.\ [64] {\it Set Theory and Related Topics.}  McGraw-Hill,
1964 (Schaum's Outline Series).  \cmmnt{[\S2A1.]}

Lo\`eve M.\ [77] {\it Probability Theory I.} Springer, 1977.
\cmmnt{[Chap.\ 27 {\it intro.}, 274H.]}

Luxemburg W.A.J.\ \& Zaanen A.C.\ [71] {\it Riesz Spaces
I.} North-Holland, 1971.
\cmmnt{[241F.]}


\medskip%M

Mozzochi C.J.\ [71] {\it On the Pointwise Convergence of Fourier
Series.}   Springer, 1971 (Lecture Notes in Mathematics 199).
\cmmnt{[\S286 {\it notes}.]}

\medskip%N

Naor A.\ [04] `Proof of the uniform convexity lemma',
{\tt http://www.cims.nyu.edu/\penalty-100$^{\sim}$naor/homepage files/\ifnum\stylenumber=12\break\fi{\vthsp}inequality\discretionary{}{}{}.pdf},
26.2.04.
\cmmnt{[244O.]}
%anon04.PDF

\medskip%O\medskip%P\medskip%Q\medskip%R

R\'enyi A.\ [70] {\it Probability Theory.}  North-Holland, 1970.
\cmmnt{[274H.]}

Roitman J.\ [90] {\it  An Introduction to Set Theory.}  Wiley, 1990.
\cmmnt{[\S2A1.]}

Roselli P.\ \& Willem M.\ [02] `A convexity inequality',
Amer.\ Math.\ Monthly 109 (2002) 64-70.
\cmmnt{[244Ym.]}

\medskip%S

%Saks S.\ [1924] 'Sur les nombres d\'eriv\'es des fonctions', 
%  Fundamenta Math.\ 5 (1924) 98-104.
% D-Y-S theorem
% books/saks.24.pdf   http://matwbn.icm.edu.pl/ksiazki/fm/fm5/fm5112.pdf

Saks S.\ [1924] 'Sur les nombres d\'eriv\'es des fonctions', Fund

Schipp F.\ [78] `On Carleson's method', pp.\ 679-695 in
{\smc Alexits 78}.   \cmmnt{[\S286 {\it notes}.]}

Semadeni Z.\ [71] {\it Banach spaces of continuous functions I.}
Polish Scientific Publishers, 1971.  \cmmnt{[\S253 {\it notes}.]}

Shiryayev A.N.\ [84] {\it Probability.} Springer, 1984. \cmmnt{[285N.]}

Steele J.M.\ [86] `An Efron-Stein inequality of nonsymmetric statistics',
Annals of Statistics 14 (1986) 753-758.
\cmmnt{[274Ya.]}

\medskip%T\medskip%U\medskip%V\medskip%W\medskip%Y\medskip%Z

Zaanen A.C.\ [83] {\it Riesz Spaces II.} North-Holland, 1983.
\cmmnt{[241F.]}

Zygmund A. [59] {\it Trigonometric Series.}   Cambridge U.P, 1959.
\cmmnt{[\S244 {\it notes}, \S282 {\it notes}, 284Xk.]}

\frnewpage

}%end of references

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

\end



