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.

中文翻译：

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可数Borel等价关系的分解与测度

我们证明了可数 Borel 等价关系的统一测度理论遍历分解$(X, E)$可以实现为可数群的连续动作的拓扑遍历分解$\Gamma \curvearrowright X$生成乙. 然后我们将其应用于基数代数的研究$\数学{K}(E)$Borel 集关于可压缩可数 Borel 等价关系的等分解类型$(X, E)$. 我们还对拓扑遍历分解上的商拓扑进行了一些一般性观察，并将其应用于保测动作的弱等价性。