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Proceedings of the American Mathematical Society

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

 

 

Paradoxical decompositions and invariant measures


Author: Piotr Zakrzewski
Journal: Proc. Amer. Math. Soc. 111 (1991), 533-539
MSC: Primary 04A20; Secondary 03E05, 28A99
DOI: https://doi.org/10.1090/S0002-9939-1991-1039268-4
MathSciNet review: 1039268
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Abstract: Suppose $ G$ is a certain group of bijections of a given set $ X$. A subset $ E$ of $ X$ is countably $ G$-paradoxical if it contains disjoint subsets $ A,B$, each of which can be taken apart into countably many pieces that may be rearranged via $ G$ to form a partition of $ E$. We prove that the existence of a countably additive measure on $ P(X)$ that normalizes $ X$ and vanishes on all countably $ G$-paradoxical sets implies the existence of a countably additive, $ G$-invariant measure on $ P(X)$ normalizing $ X$.


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DOI: https://doi.org/10.1090/S0002-9939-1991-1039268-4
Keywords: Invariant measure, paradoxical set, real-valued measurable cardinal
Article copyright: © Copyright 1991 American Mathematical Society