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Proceedings of the American Mathematical Society

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

 

 

When do equidecomposable sets have equal measures?


Author: Piotr Zakrzewski
Journal: Proc. Amer. Math. Soc. 113 (1991), 831-837
MSC: Primary 28C10; Secondary 03E05
DOI: https://doi.org/10.1090/S0002-9939-1991-1086587-1
MathSciNet review: 1086587
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Abstract: Suppose that $ G$ is a group of bijections of a set $ X$. Two subsets of $ X$ are called countably $ G$-equidecomposable if they can be partitioned into countably many respectively $ G$-congruent pieces.

We present a simple combinatorial approach to problems concerning countable equidecomposability. As an application, we prove that if $ G$ is a discrete group of isometries of $ {\mathbb{R}^n}$, then every two Lebesgue measurable, countably $ G$-equidecomposable subsets of $ {\mathbb{R}^n}$ have equal measures.


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DOI: https://doi.org/10.1090/S0002-9939-1991-1086587-1
Keywords: Countably equidecomposable sets, selector of orbits, invariant extensions of Lebesgue measure
Article copyright: © Copyright 1991 American Mathematical Society