Decompositions and measures on countable Borel equivalence relations

Abstract

We show that the uniform measure-theoretic ergodic decomposition of a countable Borel equivalence relation $(X,E)$ may be realized as the topological ergodic decomposition of a continuous action of a countable group $\Gamma \curvearrowright X$ generating $E$. We then apply this to the study of the cardinal algebra $\mathcal{K}(E)$ of equidecomposition types of Borel sets with respect to a compressible countable Borel equivalence relation $(X,E)$. We also make some general observations regarding quotient topologies on topological ergodic decompositions, with an application to weak equivalence of measure-preserving actions.

31 pages; revisions from refereeing

References in corpus (1)

• Notes on quasi-Polish spaces

Cited by in corpus (3)

• Borel and analytic sets in locales
• Equitable Colorings of Borel Graphs
• Structural, point-free, non-Hausdorff topological realization of Borel groupoid actions