  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.
