Contents of Measure Theory, by D.H.Fremlin

Chapter 56: Choice and Determinacy

561 Analysis without choice
Elementary facts; Tychonoff's theorem; Baire's theorem; Stone's theorem; Haar measure; Kakutani's representation of L-spaces; Hilbert space.

562 Borel codes
Coding with trees in second-countable spaces; codable Borel sets in Polish spaces are analytic; resolvable sets are self-coding; codable families of codable sets; codable Borel functions, codable Borel equivalence; real-valued functions; codable families of codable functions; codable Baire sets and functions for general topological spaces.

563 Borel measures without choice
Borel-coded measures on second-countable spaces; construction of measures; analytic sets are universally measurable; Baire-coded measures on general topological spaces; measure algebras.

564 Integration without choice
Integration with respect to Baire-coded measures; convergence theorems for codable sequences of functions; Riesz representation theorem; when L1 is a Banach space; Radon-Nikodým theorem; conditional expectations; products of measures on second-countable spaces.

565 Lebesgue measure without choice
Construction of Lebesgue measure as a Borel-coded measure; Vitali's theorem; Fundamental Theorem of Calculus; Hausdorff measures as Borel-coded measures.

566 Countable choice
Basic measure theory survives; exhaustion; σ-finite spaces and algebras; atomless countably additive functionals; Vitali's theorem; bounded additive functionals; infinite products without DC; topological product measures; the Loomis-Sikorski theorem; the usual measure on {0,1}N and its measure algebra; weak compactness; automorphisms of measurable algebras; Baire σ-algebras; dependent choice.

567 Determinacy
Infinite games; closed games are determined; the axiom of determinacy; AC(R;ω); universal measurability and the Baire property; automatic continuity of group homomorphisms and linear operators; countable additivity of functionals; reflexivity of L-spaces; ω1 is two-valued-measurable; surjections from PN onto ordinals; two-valued-measurable cardinals and determinacy in ZFC; measurability of PCA sets.

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Revised 12.7.08