Chapter 54: Real-valued-measurable cardinals
541
Saturated ideals
κ-saturated κ+-additive ideals; κ-saturated
κ-additive ideals; TrI(X;Y); normal ideals;
κ-saturated normal ideals; two-valued-measurable and weakly
compact cardinals; the Tarski-Solovay dichotomy;
covSh(2γ,κ,δ+,δ).
542
Quasi-measurable cardinals
Definition and basic properties; ω1-saturated
σ-ideals; and pcf theory; and cardinal arithmetic; cardinals
of quotient algebras; cofinality of [κ]<θ; cofinality
of product partial orders.
543
The Gitik-Shelah theorem
Real-valued-measurable and atomlessly-measurable cardinals; Ulam's
dichotomy; a Fubini inequality; Maharam types of witnessing
probabilities; compact measures, inverse-measure-preserving
functions and extensions of measures.
544
Measure theory with an atomlessly-measurable cardinal
Covering numbers of null ideals; repeated integrals;
measure-precalibers; functions from [κ]<ω to null
ideals; Sierpinski sets; uniformities of null ideals; weakly
Π11-indescribable cardinals; Cichon's diagram.
545
PMEA and NMA
The product measure extension axiom; the normal measure axiom;
Boolean algebras with many measurable subalgebras.
546
Power set σ-quotient algebras
Power set σ-quotient algebras; e-h families; cardinals from
Cichon's diagram; completed free products of measurable algebras
and Cohen algebras.
547
Cohen algebras and σ-measurable algebras
Harmless algebras and skew products of ideals; the Gitik-Shelah
theorem for Cohen algebras; σ-measurable algebras;
σ-measurable power set σ-quotient algebras.
548
Selectors and disjoint refinements
Equivalence relations with countable equivalence classes; disjoint
refinements of sequences of sets.
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