Equidecomposition in cardinal algebras

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 is an element of Gamma) on X such that mu = Sigma(gamma)mu(gamma) and nu = Sigma(gamma)mu(gamma). We establish a generalization of this result to cardinal algebras.