Invariant signed measures and the cancellation law

Abstract:

Let $X$ be a set, and let the group $G$ act on $X$. We show that, for every $A,B \subset X$, the following are equivalent: (i) $A$ and $B$ are $G$-equidecomposable; and (ii) $\vartheta (A) = \vartheta (B)$ for every $G$-invariant finitely additive signed measure $\vartheta$. If the sets and the pieces of the decompositions are restricted to belong to a given $G$-invariant field $\mathcal {A}$, then $({\text {i}}) \Leftrightarrow ({\text {ii}})$ if and only if the cancellation law $(n[A] = n[B] \Rightarrow [A] = [B])$ holds in the space $(X,G,\mathcal {A})$. We show that the cancellation law may fail even if the transformation group $G$ is Abelian.