The ergodic decomposition of conservative Baire measures

Sturgeon, Michael L.

doi:10.1090/s0002-9939-1974-0331517-0

Proc. Amer. Math. Soc.

1974

DOI: 10.1090/s0002-9939-1974-0331517-0

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The ergodic decomposition of conservative Baire measures

Michael L. Sturgeon¹

Abstract: Certain topological conditions on a Markov transition function are shown sufficient for an integral representation of conservative invariant Baire measures. The analysis incorporates the Choquet-Bishop-de Leeuw extension of the Krein-Milman theorem.

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2000

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Ergodic decomposition of quasi-invariant probability measures

Greschonig¹,

Schmidt²

2000

Colloq. Math.

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Abstract. The purpose of this note is to prove various versions of the ergodic decomposition theorem for probability measures on standard Borel spaces which are quasiinvariant under a Borel action of a locally compact second countable group or a discrete nonsingular equivalence relation. In the process we obtain a simultaneous ergodic decomposition of all quasi-invariant probability measures with a prescribed Radon-Nikodym derivative, analogous to classical results about decomposition of invariant probability measures.1. Introduction. Throughout this note we assume that (X, S) is a standard Borel space, i.e. a measurable space which is isomorphic to the unit interval with its usual Borel structure. A Borel action T of a locally compact second countable group G on X is a group homomorphism g → T g from G into the group Aut(X, S) of Borel automorphisms of (X, S) such that the map (g, x) → T g x from G × X to X is Borel. Let T be a Borel action of a locally compact second countable group G on X, and let µ be a probability measure on S. The measure µ is quasi-invariant under T if µ(T g B) = 0 for every g ∈ G and every B ∈ S with µ(B) = 0, and µ is ergodic under T if µ(B) ∈ {0, 1} for every B ∈ S with µ(B T g B) = 0 for every g ∈ G.The following theorem is part of the mathematical folklore about group actions on measure spaces. Theorem 1.1 (Ergodic decomposition theorem). Let T be a Borel action of a locally compact second countable group G on a standard Borel space (X, S), and let µ be a probability measure on S which is quasi-invariant under T . Then there exist a standard Borel space (Y, T), a probability measure 2000 Mathematics Subject Classification: 28D05, 28D15, 28D99, 47A35.

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Ergodic decomposition of quasi-invariant probability measures

Greschonig¹,

Schmidt²

2000

Colloq. Math.

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The ergodic decomposition of conservative Baire measures

Abstract: Certain topological conditions on a Markov transition function are shown sufficient for an integral representation of conservative invariant Baire measures. The analysis incorporates the Choquet-Bishop-de Leeuw extension of the Krein-Milman theorem.

Cited by 1 publication

References 6 publications

Ergodic decomposition of quasi-invariant probability measures

Ergodic decomposition of quasi-invariant probability measures

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