  Let $\Gamma$ be a countable group. A classical theorem of Thorisson states
that if $X$ is a standard Borel $\Gamma$-space and $\mu$ and $\nu$ are Borel
probability measures on $X$ which agree on every $\Gamma$-invariant subset,
then $\mu$ and $\nu$ are equidecomposable, i.e. there are Borel measures
$(\mu_\gamma)_{\gamma\in\Gamma}$ on $X$ such that $\mu = \sum_\gamma
\mu_\gamma$ and $\nu = \sum_\gamma \gamma\mu_\gamma$. We establish a
generalization of this result to cardinal algebras.
