Chapter 32: Measure algebras

321
Measure algebras

Measure algebras; elementary properties; the measure algebra of a
measure space; Stone spaces.

322
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.

323
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.

324
Homomorphisms

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.

325
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.

326
Additive functionals on Boolean algebras

Additive, countably additive and completely additive functionals;
Jordan decomposition; Hahn decomposition; Liapounoff's convexity
theorem; the region [[μ>ν]].

327
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.

*328
Reduced products and other constructions

Reduced products of probability algebras; inductive and projective
limits; converting homomorphisms into automorphisms.

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Revised 3.12.2017