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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