Invariant signed measures and the cancellation law
Miklós Laczkovich · Proceedings of the American Mathematical Society · 1991
Let X X be a set, and let the group G G act on X X . We show that, for every A , B ⊂ X A,B \subset X , the following are equivalent: (i) A A and B B are G G -equidecomposable; and (ii) ϑ ( A ) = ϑ ( B ) \vartheta (A) = \vartheta (B) for every G G -invariant finitely additive signed measure ϑ \vartheta . If the sets and the pieces of the decompositions are restricted to belong to a given G G -invariant field A \mathcal {A} , then ( i ) ⇔ ( ii ) ({\text {i}}) \Leftrightarrow ({\text {ii}}) if and only if the cancellation law ( n [ A ] = n [ B ] ⇒ [ A ] = [ B ] ) (n[A] = n[B] \Rightarrow [A] = [B]) holds in the space ( X , G , A ) (X,G,\mathcal {A}) . We show that the cancellation law may fail even if the transformation group G G is Abelian.