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 *L*^{1} 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; ω_{1} 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