Chapter 32: Measure algebras
Measure algebras; elementary properties; the measure algebra of a measure space; Stone spaces.
Taxonomy of measure algebras
Totally finite, σ-finite, semi-finite and localizable measure algebras; relation to corresponding types of measure space; completions and c.l.d. versions of measures; semi-finite measure algebras are weakly (σ,∞)-distributive; subspace measures and indefinite-integral measures; simple products of measure algebras; Stone spaces of localizable measure algebras; localizations of semi-finite measure algebras.
The topology of a measure algebra
Defining a topology and uniformity on a measure algebra; continuity of algebraic operations; order-closed sets; Hausdorff and metrizable topologies, complete uniformities; closed subalgebras; products.
Homomorphisms induced by measurable functions; order-continuous and continuous homomorphisms; the topology of a semi-finite measure algebra is determined by the algebraic structure; measure-preserving homomorphisms.
Free products and product measures
The measure algebra of a product measure; the localizable measure algebra free product of two semi-finite measure algebras; the measure algebra of a product of probability measures; the probability algebra free product of probability algebras; factorizing through subproducts.
Additive functionals on Boolean algebras
Additive, countably additive and completely additive functionals; Jordan decomposition; Hahn decomposition; Liapounoff's convexity theorem; the region [[μ>ν]].
Additive functionals on measure algebras
Absolutely continuous and continuous additive functionals; Radon-Nikodým theorem; the standard extension of a continuous additive functional on a closed subalgebra.
Reduced products and other constructions
Reduced products of probability algebras; inductive and projective limits; converting homomorphisms into automorphisms.
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