Measurable equidecompositions for group actions with an expansion property

Given an action of a group $Gamma$ on a measure space $Omega$, we provide a sufficient criterion under which two sets $A, Bsubseteq Omega$ are measurably equidecomposable, i.e., $A$ can be partitioned into finitely many measurable pieces which can be rearranged using the elements of $Gamma$ to form a partition of $B$. In particular, we prove that every bounded measurable subset of $R^n$, $nge 3$, with non-empty interior is measurably equidecomposable to a ball via isometries. The analogous result also holds for some other spaces, such as the sphere or the hyperbolic space of dimension $nge 2$.