Chapter 33: Maharam's theorem

331
Classification of homogeneous measure algebras

Relatively atomless algebras; one-step extension of
measure-preserving homomorphisms; Maharam type of a measure algebra;
Maharam-type-homogeneous probability algebras of the same Maharam type are isomorphic;
the measure algebra of {0,1}^{κ} is homogeneous.

332
Classification of localizable measure algebras

Any localizable measure algebra is isomorphic to a simple product of
homogeneous totally finite algebras; complete description of
isomorphism types; closed subalgebras.

333
Closed subalgebras

Relative Maharam types; extension of measure-preserving Boolean
homomorphisms; complete classification of closed subalgebras of
probability algebras as triples (**A**,μ,**C**);
fixed-point subalgebras.

334
Products

Maharam types of product measures; infinite powers of probability
spaces are Maharam-type-homogeneous.

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Revised 2.10.08