Advertisement

Moore’s Ergodicity Theorem

  • Robert J. Zimmer
Part of the Monographs in Mathematics book series (MMA, volume 81)

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 0SS 0S′ and a measure class preserving Borel isomorphism (ϕ:S 0S 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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer Science+Business Media New York 1984

Authors and Affiliations

  • Robert J. Zimmer

There are no affiliations available

Personalised recommendations