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mt05I.tex
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\wheader{}{10}{4}{4}{100pt}
Introduction to Volume 5 \vtmpb{26.9.08}\pagereference{}{10}
\wheader{}{10}{4}{4}{100pt}
Chapter 51: Cardinal functions
%page-ends checked 8.1.15
\chapintrosection{5.9.03}{}{12}
\section{511}{Definitions}{10.10.13}{}{12}
{Cardinal functions of partially ordered sets, topological spaces,
Boolean algebras and measures; precalibers; ideals of sets.}
\section{512}{Galois-Tukey connections}{27.11.13}{}{20}
{Supported relations; Galois-Tukey connections;
covering numbers, additivity, saturation,
linking numbers; simple products;
sequential composition of supported relations.}
\section{513}{Partially ordered sets}{23.2.14}{}{25}
{Saturation and the Erd\H{o}s-Tarski theorem;
cofinalities of cardinal functions; Tukey functions;
Tukey equivalence of directed sets; $\sigma$-additivities;
*metrizably compactly based directed sets; *measurable Tukey functions.}
\section{514}{Boolean algebras}{16.5.14}{}{36}
{Stone spaces; cardinal
functions of Boolean algebras; order-preserving functions of Boolean
algebras; regular open algebras; regular open
algebras of partially ordered sets; finite-support products.}
\section{515}{The Balcar-Fran\v{e}k theorem}{29.8.14}{}{54}
{Boolean-independent sets; free subalgebras; refining systems; the
Balcar-Fran\v{e}k theorem; the Pierce-Koppelberg theorem.}
\section{516}{Precalibers}{9.10.14}{}{61}
{Precalibers of supported relations; and Galois-Tukey connections;
partially ordered sets, topological spaces and Boolean algebras;
saturation and linking numbers; saturation of product spaces.}
\section{517}{Martin numbers}{14.11.14}{}{68}
{Characterizations of $\frak{m}(P)$; regular open algebras, Stone spaces
and Nov\'ak numbers; precalibers, saturation and weak
distributivity; $\frak{m}$,
$\frakmctbl$, $\frak{p}$ and $\frak{m}_{\text{K}}$.}
\section{518}{Freese-Nation numbers}{24.12.14}{}{78}
{Freese-Nation numbers of partially ordered sets; Boolean algebras;
upper and lower bounds for $\FN(\frak{A})$ under special axioms; tight
filtrations and Geschke systems; large algebras are not tightly
filtered.}
\wheader{}{10}{4}{4}{100pt}
Chapter 52: Cardinal functions of measure theory
%page-ends checked 8.1.15
\chapintrosection{29.6.06}{}{90}
\section{521}{Basic theory}{3.3.14}{}{90}
{$\add\mu$ and $\add\Cal{N}(\mu)$; measure algebras and function spaces;
the topological density of a measure algebra;
shrinking numbers; $\pi(\mu)$;
subspace measures, direct sums, image measures,
products; perfect measures, compact measures;
complete locally determined measure spaces and strict localizability;
magnitudes; bounds on the Maharam type of a measure;
countably separated spaces; measurable additive
functionals on $\Cal{P}I$.}
\section{522}{Cicho\'n's diagram}{31.3.10}{}{104}
{The cardinals $\frak{b}$ and $\frak{d}$; inequalities linking them with
the additivity, cofinality, uniformity and covering numbers of measure
and category in the real line; the localization relation;
$\frakmctbl$ and other Martin numbers; $\FN(\Cal{P}\eightBbb{N})$;
cofinalities of the cardinals.}
\section{523}{The measure of $\{0,1\}^I$}{23.6.10}{}{121}
{The additivity, covering number, uniformity, shrinking number and
cofinality of the usual measure on $\{0,1\}^I$; Kraszewski's theorems;
what happens with GCH.}
\section{524}{Radon measures}{29.9.10}{}{128}
{The additivity, covering number, uniformity and cofinality of a Radon
measure; $\ell^1(\kappa)$ and localization; cardinal functions of
measurable algebras; countably compact and quasi-Radon
measures.}
\section{525}{Precalibers of measure algebras}{11.9.13}{}{145}
{Precalibers of measurable algebras; measure-precalibers of probability
algebras; (quasi-\nobreak)Radon measure spaces; under GCH; precaliber
triples $(\kappa,\kappa,k)$.}
\section{526}{Asymptotic density zero}{24.1.14}{}{154}
{$\Cal{Z}$ is metrizably compactly based;
$\NN\prT\Cal{Z}\prT\ell^1\prGT\NN\ltimes\Cal{Z}$; cardinal functions
of $\Cal{Z}$; meager sets and nowhere dense sets; sets with negligible
closures; $\CalNwd\not\prT\Cal{Z}$ and $\Cal{Z}\not\prT\CalNwd$.}
\section{527}{Skew products of ideals}{10.1.10}{}{167}
{$\Cal{N}\ltimes_{\Cal{B}}\Cal{N}$ and Fubini's theorem;
$\Cal{M}\ltimes_{\Cal{B}}\Cal{M}$ and the Kuratowski-Ulam theorem;
$\Cal{M}\ltimes_{\Cal{B}}\Cal{N}$;
$\Cal{N}\ltimes_{\Cal{B}}\Cal{M}$; harmless Boolean algebras.}
\section{528}{Amoeba algebras}{10.2.11}{}{178}
{Amoeba algebras; variable-measure amoeba algebras; isomorphic amoeba
algebras; regular embeddings of amoeba algebras; localization
posets; Martin numbers and other cardinal functions; algebras with
countable Maharam type.}
\section{529}{Further partially ordered sets of
analysis}{26.5.11}{}{200}
{$L^p$ and $L^0$; $L$-spaces; the localization poset and the regular
open algebra of $\{0,1\}^{\frakc}$; the Nov\'ak numbers $n(\{0,1\}^I)$;
the reaping numbers $\frak{r}(\omega_1,\lambda)$.}
\wheader{}{10}{4}{4}{100pt}
Chapter 53: Topologies and measures III
%page-ends checked 8.1.15
\chapintrosection{30.8.14}{}{208}
\section{531}{Maharam types of Radon measures}{19.2.11}{}{208}
{Topological and measure-theoretic cardinal functions; the set
$\MahR(X)$ of Maharam types of homogeneous Radon measures on $X$;
$\MahR(X)$, precalibers and continuous surjections onto
$[0,1]^{\kappa}$; $\MahR(X)$ and $\chi(X)$; a perfectly normal
hereditarily separable space
under CH; when $\frak{m}_{\text{K}}>\omega_1$.}
\section{532}{Completion regular measures on $\{0,1\}^I$}
{1.6.13}{}{226}
{The set $\MahcrR(X)$ of Maharam types of homogeneous completion regular
Radon measures on $X$; products of quasi-dyadic spaces; convexity of
the relation `$\lambda\in\MahcrR(\{0,1\}^{\kappa})$'; the measure algebra
of $\{0,1\}^{\lambda}$; $\frak{d}$, $\cov\Cal{N}$, $\FN(\Cal{P}\Bbb{N})$,
$\add\Cal{N}$ and
the case $\lambda=\omega$; $\square$, Chang's conjecture and the case
$\cf\lambda=\omega$.}
\section{533}{Special topics}{4.1.14}{}{236}
{$\add\Cal{N}$ and (quasi-)Radon measures of countable Maharam type;
uniformly regular measures; when $\eightBbb{R}^{\kappa}$ is
measure-compact.}
\section{534}{Hausdorff measures and strong measure zero}
{1.8.10}{}{243}
{Cardinal functions of Hausdorff measures;
strong measure zero in uniform spaces; Rothberger's property;
$\sigma$-compact groups;
$\non\CalSmz$, $\add\CalSmz$; $\CalSmz$-equivalence; uncountable sets
with strong measure zero.}
\section{535}{Liftings}{1.6.11}{}{257}
{Liftings of non-complete measure
spaces; Baire liftings for usual measures on $\{0,1\}^{\kappa}$;
tightly $\omega_1$-filtered measure algebras; Mokobodzki's theorems;
strong Borel liftings; Borel liftings for Radon measures on metrizable
spaces; linear liftings; problems.}
\section{536}{Alexandra Bellow's problem}{20.2.12}{}{269}
{The problem; consequences of a negative solution.}
\section{537}{Sierpi\'nski sets, shrinking numbers and strong Fubini
theorems}{12.8.13}{}{273}
{Sierpi\'nski and strongly Sierpi\'nski sets; entangled totally ordered
sets; non-ccc products; scalarly measurable functions; repeated
integrals of separately measurable functions; changing the order of
integration in multiply repeated integrals; $\shr^+$, $\cov$ and
repeated upper and lower integrals.}
\section{538}{Filters and limits}{18.2.14}{}{287}
{Filters on $\Bbb{N}$; the Rudin-Keisler ordering; products and iterated
products; Ramsey ultrafilters;
measure-centering ultrafilters; extending perfect measures with
measure-centering ultrafilters; Benedikt's theorem;
measure-converging filters; the Fatou property; medial
functionals and limits.}
\section{539}{Maharam submeasures}{24.5.14}{}{317}
{Maharam algebras; Maharam-algebra topology,
pre-ordered set of partitions of unity,
weak distributivity, $\pi$-weight, centering number, precalibers;
null ideals of Maharam submeasures; splitting reals; Quickert's ideal;
Todor\v{c}evi\'c's $p$-ideal dichotomy; a consistent
characterization of Maharam algebras; Souslin algebras; reflection
principles; exhaustivity rank, Maharam submeasure rank.}
\wheader{}{10}{4}{4}{20pt}
% Concordance to Part I \pagereference{}{329}
%Lulu 2015: 329 pages