Chapter 54: Real-valued-measurable cardinals

541
Saturated ideals

κ-saturated κ^{+}-additive ideals; κ-saturated
κ-additive ideals; Tr_{I}(*X*;*Y*); normal ideals;
κ-saturated normal ideals; two-valued-measurable and weakly
compact cardinals; the Tarski-Solovay dichotomy;
cov_{Sh}(2^{g},κ,δ^{+},δ).

542
Quasi-measurable cardinals

Definition and basic properties; ω_{1}-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
Π_{1}^{1}-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; 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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Revised 2.9.2018