Ergodic Theory and Semisimple Groups pp 8-31 | Cite as

# Moore’s Ergodicity Theorem

## Abstract

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*.

## Keywords

Unitary Representation Closed Subgroup Invariant Vector Measure Class Semisimple Group## Preview

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