Contents of Measure Theory, by D.H.Fremlin

Chapter 38: Automorphisms

381 Automorphisms of Boolean algebras
Assembling an automorphism; elements supporting an automorphism; periodic and aperiodic parts; full and countably full subgroups; recurrence; induced automorphisms of principal ideals; Stone spaces; cyclic automorphisms.

382 Factorization of automorphisms
Separators and transversals; Frolik's theorem; exchanging involutions; expressing an automorphism as the product of three involutions; subgroups of Aut(A) with many involutions; normal subgroups of full groups with many involutions; simple automorphism groups.

383 Automorphism groups of measure algebras
Measure-preserving automorphisms as products of involutions; normal subgroups of Aut(A) and Autμ(A); conjugacy in Aut(A) and Autμ(A).

384 Outer automorphisms
If G≤Aut(A), H≤Aut(B) have many involutions, any isomorphism between G and H arises from an isomorphism between A and B; if A is nowhere rigid, Aut(A) has no outer automorphisms; applications to localizable measure algebras.

385 Entropy
Entropy of a partition of unity in a probability algebra; conditional entropy; entropy of a measure-preserving homomorphism; calculation of entropy (Kolmogorov-Sinai theorem); Bernoulli shifts; isomorphic homomorphisms and conjugacy classes in Autμ(A); almost isomorphic inverse-measure-preserving functions.

386 More about entropy
The Halmos-Rokhlin-Kakutani lemma; the Shannon-McMillan-Breiman theorem; the Csiszár-Kullback inequality; various lemmas.

387 Ornstein's theorem
Bernoulli partitions; finding Bernoulli partitions with elements of given measure (Sinai's theorem); adjusting Bernoulli partitions; Ornstein's theorem (Bernoulli shifts of the same finite entropy are isomorphic); Ornstein's and Sinai's theorems in the case of infinite entropy.

388 Dye's theorem
Orbits of inverse-measure-preserving functions; von Neumann transformations; von Neumann transformations generating a given full subgroup; classification of full subgroups generated by a single automorphism.

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Revised 7.6.2016