Decompositions and measures on countable Borel equivalence relations

TL;DR

This paper connects measure-theoretic and topological ergodic decompositions for countable Borel equivalence relations, and explores their implications for the algebra of equidecomposition types and quotient topologies.

Contribution

It demonstrates that measure-theoretic ergodic decompositions can be realized as topological decompositions via continuous group actions, and applies this to the study of equidecomposition algebra and quotient topologies.

Findings

Measure-theoretic ergodic decomposition can be realized as topological ergodic decomposition.

Analysis of the algebra of equidecomposition types for Borel sets.

Observations on quotient topologies and their relation to measure-preserving actions.

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.