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mt03I.tex
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\wheader{}{10}{4}{4}{100pt}
Introduction to Volume 3 \vtmpb{31.12.01}\pagereference{11}{}
%wheader parameters: #1 new paragraph name
% #2#3#4 \vskip parameters if page break not forced (no. of points)
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\wheader{}{10}{4}{4}{100pt}
Chapter 31: Boolean algebras
\chapintrosection{31.12.01}{13}{}
\section{311}{Boolean algebras}{15.10.08}{13}{}
{Boolean rings and algebras; ideals and ring homomorphisms to
$\Bbb{Z}_2$; Stone's theorem; the operations $\Bcup$, $\Bcap$,
$\Bsymmdiff$, $\Bsetminus$ and the relation $\Bsubseteq$; partitions of
unity; topology of
the Stone space; Boolean algebras as complemented distributive
lattices.}
\section{312}{Homomorphisms}{29.5.07}{21}{}
{Subalgebras; ideals; Boolean homomorphisms; the ordering
determines the ring structure; quotient algebras; extension of
homomorphisms; homomorphisms and Stone spaces.}
\section{313}{Order-continuity}{8.6.11}{29}{}
{General distributive laws; order-closed sets; order-closures;
Monotone Class Theorem; order-preserving functions; order-continuity;
order-dense sets; order-continuous Boolean homomorphisms; and Stone
spaces; regularly embedded subalgebras; upper envelopes.}
\section{314}{Order-completeness}{26.7.07}{39}{}
{Dedekind completeness and $\sigma$-completeness; quotients,
subalgebras, principal ideals; order-continuous homomorphisms;
extension of homomorphisms; Loomis-Sikorski representation of a
$\sigma$-complete algebra as a quotient of a $\sigma$-algebra of sets;
regular open algebras; Stone spaces; Dedekind completion of a Boolean
algebra.}
\section{315}{Products and free products}{13.11.12}{49}{}
{Simple product of Boolean algebras; free product of Boolean
algebras; algebras of sets and their quotients; projective and inductive
limits.}
\section{316}{Further topics}{26.1.09}{59}{}
{The countable chain condition; weak
$(\sigma,\infty)$-distributivity; Stone spaces; atomic and atomless
Boolean algebras; homogeneous Boolean algebras.}
\wheader{}{10}{4}{4}{100pt}
Chapter 32: Measure algebras
\chapintrosection{6.1.02}{68}{}
\section{321}{Measure algebras}{3.1.11}{68}{}
{Measure algebras; elementary properties; the measure algebra of
a measure space; Stone spaces.}
\section{322}{Taxonomy of measure algebras}{24.4.06}{71}{}
{Totally finite, $\sigma$-finite, semi-finite and localizable
measure algebras; relation to corresponding types of measure space;
completions and c.l.d.\ versions of measures; semi-finite measure
algebras are \wsid; subspace measures and indefinite-integral measures;
simple products of measure
algebras; Stone spaces of localizable measure algebras; localizations
of semi-finite measure algebras.}
\section{323}{The topology of a measure algebra}{20.7.06}{81}{}
{Defining a topology and uniformity on a measure algebra;
continuity of algebraic operations; order-closed sets; Hausdorff and
metrizable topologies, complete uniformities; closed subalgebras;
products.}
\section{324}{Homomorphisms}{6.2.02}{87}{}
{Homomorphisms induced by measurable functions; order-continuous
and continuous homomorphisms; the topology of a semi-finite measure
algebra is determined by the algebraic structure; measure-preserving
homomorphisms.}
\section{325}{Free products and product measures}{30.8.06}{93}{}
{The measure algebra of a product measure; the
localizable measure algebra free product of two semi-finite measure
algebras; the measure algebra of a product of probability measures;
the probability algebra free product of probability algebras;
factorizing through subproducts.}
\section{326}{Additive functionals on Boolean algebras}{21.5.11}{102}{}
{Additive, countably additive and completely additive functionals;
Jordan decomposition; Hahn decomposition; Liapounoff's convexity
theorem; the region $\Bvalue{\mu>\nu}$.}
\section{327}{Additive functionals on measure algebras}{13.7.11}{114}{}
{Absolutely continuous and continuous additive functionals;
Radon-Nikod\'ym theorem; the standard extension of a continuous
additive functional on a closed subalgebra.}
\section{*328}{Reduced products and other constructions}{2.6.09}{120}{}
{Reduced products of probability algebras; inductive and projective
limits; converting homomorphisms into automorphisms.}
\wheader{}{10}{4}{4}{100pt}
Chapter 33: Maharam's theorem
\chapintrosection{13.1.02}{127}{}
\section{331}{Maharam types and homogeneous measure
algebras}{1.2.05}{127}{}
{Relatively atomless algebras; one-step extension of
measure-preserving homomorphisms; Maharam type of a measure algebra;
\Mth\ probability algebras of the same Maharam type are
isomorphic; the measure algebra of $\{0,1\}^{\kappa}$ is homogeneous.}
\section{332}{Classification of localizable measure
algebras}{19.3.05}{135}{}
{Any localizable measure algebra is isomorphic to a simple product
of homogeneous totally finite algebras; complete description of
isomorphism types; closed subalgebras.}
\section{333}{Closed subalgebras}{27.6.08}{145}{}
{Relative Maharam types; extension of measure-preserving Boolean
homomorphisms; complete classification of closed subalgebras of
probability algebras as triples $(\frak{A},\bar\mu,\frak{C})$; fixed-point
subalgebras.}
\section{334}{Products}{26.9.08}{160}{}
{Maharam types of product measures; infinite powers of probability spaces
are Maharam-type-homogeneous.}
\wheader{}{10}{4}{4}{100pt}
Chapter 34: Liftings
\chapintrosection{19.1.02}{163}{}
\section{341}{The lifting theorem}{9.4.10}{163}{}
{Liftings and lower densities; strictly localizable spaces have
lower densities; construction of a lifting from a density; complete
strictly localizable spaces have liftings; liftings and Stone spaces.}
\section{342}{Compact measure spaces}{9.7.10}{174}{}
{Inner regular measures; compact classes; compact and locally
compact measures; perfect measures.}
\section{343}{Realization of homomorphisms}{17.11.10}{181}{}
{Representing homomorphisms between measure algebras by functions;
possible when target measure space is locally compact; countably
separated measures and uniqueness of representing functions; the split
interval; perfect measures.}
\section{344}{Realization of automorphisms}{22.3.06}{189}{}
{Simultaneously representing groups of automorphisms of
measure algebras by functions -- Stone spaces,
countably separated measure spaces,
measures on $\{0,1\}^I$; characterization of Lebesgue measure as a
measure space; strong homogeneity of usual measure on $\{0,1\}^I$.}
\section{345}{Translation-invariant liftings}{27.6.06}{198}{}
{Translation-invariant liftings on $\Bbb{R}^r$ and $\{0,1\}^I$;
there is no t.-i.\ Borel lifting on $\Bbb{R}$.}
\section{346}{Consistent liftings}{17.12.10}{206}{}
{Liftings of product measures which respect the product structure;
translation-invariant liftings on $\{0,1\}^I$; products of \Mth\
probability spaces; lower densities respecting product
structures; consistent liftings; the Stone space of Lebesgue measure.}
\wheader{}{10}{4}{4}{0pt}
% Concordance to part I \pagereference{216}{}
%213 pp