Published online by Cambridge University Press: 04 December 2020
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.