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Introduction to Volume 1 \vtmpb{30.9.02}\pagereference{7}{7}

\medskip

Chapter 11:  Measure Spaces

\chapintrosection{10.9.96}{9}{9}

\section{111}{$\sigma$-algebras}{26.1.05}{10}{10}
{Definition of $\sigma$-algebra;  countable sets;
$\sigma$-algebra generated by a family of sets;  Borel
$\sigma$-algebras.}

\section{112}{Measure spaces}{20.2.05}{14}{11}
{Definition of measure space;  the use of $\infty$;  elementary
properties;  negligible sets;  point-supported
measures;  image measures.}

\section{113}{Outer measures and \Caratheodory's
construction}{6.4.05}{19}{13}
{Outer measures;  \Caratheodory's construction of a measure from
an outer measure.}

\section{114}{Lebesgue measure on $\Bbb{R}$}{14.6.05}{23}{13}
{Half-open intervals;  Lebesgue outer measure;  Lebesgue measure;
Borel sets are measurable.}

\section{115}{Lebesgue measure on $\BbbR^r$}{21.7.05}{28}{14}
{Half-open intervals;  Lebesgue outer measure;  Lebesgue measure;
Borel sets are measurable.}

\medskip

 Chapter 12:  Integration

\chapintrosection{7.4.05}{35}{16}

\section{121}{Measurable functions}{21.12.03}{35}{16}
{Subspace $\sigma$-algebras;  measurable real-valued functions;
partially defined functions;  Borel measurable functions;  operations on
measurable functions;  generating Borel sets from half-spaces.}

\section{122}{Definition of the integral}{4.1.04}{43}{18}
{Simple functions;  non-negative integrable functions;  integrable
real-valued functions;  virtually measurable functions;  linearity of
the integral.}

\section{123}{The convergence theorems}{18.11.04}{52}{20}
{B.Levi's theorem;  Fatou's lemma;  Lebesgue's Dominated
Convergence Theorem;  differentiating through an integral.}

\medskip

 Chapter 13:  Complements

\chapintrosection{16.6.01}{56}{21}

\section{131}{Measurable subspaces}{18.3.05}{56}{21}
{Subspace measures on measurable subsets;  integration over
measurable subsets.}

\section{132}{Outer measures from measures}{6.4.05}{58}{22}
{The outer measure associated with a measure;  Lebesgue outer
measure again;  measurable envelopes.}

\section{133}{Wider concepts of integration}{29.3.10}{61}{23}
{$\infty$ as a value of an integral;  complex-valued functions;
upper and lower integrals.}

\section{134}{More on Lebesgue measure}{7.1.04}{68}{25}
{Translation-invariance;  non-measurable sets;  inner and outer
regularity;  the Cantor set and function;  *the Riemann integral.}

\section{135}{The extended real line}{14.9.04}{79}{27}
{The algebra of $\pm\infty$;  Borel sets and convergent sequences
in $[-\infty,\infty]$;   measurable and integrable
$[-\infty,\infty]$-valued functions.}

\section{*136}{The Monotone Class Theorem}{22.6.05}{84}{30}
{The $\sigma$-algebra generated by a family $\Cal{I}$;
algebras of sets.}

\medskip

 Appendix to Volume 1

\chapintrosection{10.10.96}{89}{32}

\section{1A1}{Set theory}{5.11.03}{89}{32}
{Notation;  countable and uncountable sets.}

\section{1A2}{Open and closed sets in $\BbbR^r$}{21.11.03}{92}{33}
{Definitions;  basic properties of open and closed sets;  Cauchy's
inequality;  open balls.}

\section{1A3}{Lim sups and lim infs}{18.12.03}{94}{34}
{$\limsup_{n\to\infty}a_n$, $\liminf_{n\to\infty}a_n$ in
$[-\infty,\infty]$.}

\wheader{}{6}{2}{2}{100pt}
%\medskip

% Concordance \pagereference{97}{}

\medskip

References for Volume 1 \vtmpb{4.10.96}\pagereference{97}{35}

% \medskip

% Index to Volume 1

% \qquad Principal topics and results \pagereference{98}{}

% \qquad General index \pagereference{99}{}

%102 pages