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 sets with trees; codable Borel sets; in a Polish space, a set
is analytic and coanalytic iff it is a codable Borel set; 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; inner and outer regularity; 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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