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 $(μ_γ)_{γ\inΓ}$ on $X$ such that $μ= \sum_γμ_γ$ and $ν= \sum_γγμ_γ$. We establish a generalization of this result to cardinal algebras.

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