Equidecomposition in cardinal algebras

Abstract

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.