INVARIANT SIGNED MEASURES AND THE CANCELLATION LAW

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