Chapter 54: Real-valued-measurable cardinals
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(2g,κ,δ+,δ).
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.
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.
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.
PMEA and NMA
The product measure extension axiom; the normal measure axiom; Boolean algebras with many measurable subalgebras.
Power set σ-quotient algebras
Power set σ-quotient algebras; completed free products of measurable algebras and Cohen algebras.
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.
Selectors and disjoint refinements
Equivalence relations with countable equivalence classes; disjoint refinements of sequences of sets.
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