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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Invariant signed measures and the cancellation law

Author: M. Laczkovich
Journal: Proc. Amer. Math. Soc. 111 (1991), 421-431
MSC: Primary 28D15; Secondary 20B99
MathSciNet review: 1036988
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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.

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