Equidecomposition in cardinal algebras

Let Γ be a countable group. A classical theorem of Thorisson states that if X is a standard Borel Γ-space and μ and ν are Borel probability measures on X which agree on every Γ-invariant subset, then μ and ν are equidecomposable, i.e. there are Borel measures (μ_γ)_(γ∈Γ) on X such that μ=∑_γμ_γ and ν = ∑_(γ γμγ). We establish a generalization of this result to cardinal algebras.