Moore’s Ergodicity Theorem
Let G be a locally compact second countable group. We shall consider actions of G on a Borel space S so that the action map S × G → S, (s, g) → sg is Borel. We shall assume that S is a standard Borel space, i.e., isomorphic as a Borel space to a Borel subset of a complete separable metric space. This includes, of course, many spaces arising naturally in analysis and geometry. If µ is a σ-finite measure on S, µ is called quasi-invariant under the action of G if for all A ⊂ S and g ⊂ G, µ(Ag) = 0 if and only if µ(A) = 0. The measure µ is called invariant if µ(Ag) = µ(A) for all A, g. Two measures are said to be in the same measure class if they have the same null sets. Any σ-finite measure is in the same class as a probability measure. An action with quasi-invariant measure can also be thought of as an action with an invariant measure class. If (S, µ), (S′, µ′) are two such G-spaces, they are called equivalent (or “isomorphic”, or “conjugate”) if there are conull G-invariant Borel sets S 0 ⊂ S′ S 0 ⊂ S′ and a measure class preserving Borel isomorphism (ϕ:S 0 →S′ 0 such that ϕ(sg) = ϕ(s)g for all s∈S 0, g∈G.
KeywordsUnitary Representation Closed Subgroup Invariant Vector Measure Class Semisimple Group
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