Chapter 52: Cardinal functions of measure theory
521
Basic theory
addμ and addN(μ); measure algebras and function
spaces; the topological density of a measure algebra; shrinking
numbers; π(μ); 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 P(I).
522
Cichon's diagram
The cardinals b and d; inequalities linking them
with the additivity, cofinality, uniformity and covering numbers of
measure and category in the real line; the localization relation;
mcountable and other Martin numbers; FN(PN);
cofinalities of the cardinals.
523
The measure of {0,1}I
The additivity, covering number, uniformity, shrinking number and
cofinality of the usual measure on {0,1}I; Kraszewski's
theorems; what happens with GCH.
524
Radon measures
The additivity, covering number, uniformity and cofinality of a Radon
measure; l1(κ) and localization; cardinal functions of
measurable algebras; countably compact and quasi-Radon measures.
525
Precalibers of measure algebras
Precalibers of measurable algebras; measure-precalibers of
probability algebras; (quasi-)Radon measure spaces; under
GCH; precaliber triples (κ,κ,k).
526
Asymptotic density zero
Z is metrizably compactly based;
NN≤TZ≤Tl1≤GTNN⧔Z; cardinal functions
of Z; meager sets and nowhere dense sets; sets with
negligible closures; not-Nwd≤TZ and
not-Z≤TNwd.
527
Skew products of ideals
N⧔BN and Fubini's theorem;
M⧔BM and the Kuratowski-Ulam theorem;
M⧔BN; N⧔BM;
harmless Boolean algebras.
528
Amoeba algebras
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.
529
Further partially ordered sets of analysis
Lp and L0; L-spaces; the localization poset and the regular
open algebra of {0,1}c; the Novák numbers
n({0,1}I); the reaping numbers r(ω1,λ).
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