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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Extensions of measures invariant under countable groups of transformations


Authors: Adam Krawczyk and Piotr Zakrzewski
Journal: Trans. Amer. Math. Soc. 326 (1991), 211-226
MSC: Primary 28C10; Secondary 03E05, 03E55
DOI: https://doi.org/10.1090/S0002-9947-1991-0998127-8
MathSciNet review: 998127
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Abstract: We consider countably additive, nonnegative, extended real-valued measures vanishing on singletons. Given a group $G$ of bijections of a set $X$ and a $G$-invariant measure $m$ on $X$ we ask whether there exists a proper $G$-invariant extension of $m$. We prove, among others, that if $\mathbb {Q}$ is the group of rational translations of the reals, then there is no maximal $\mathbb {Q}$-invariant extension of the Lebesgue measure on $\mathbb {R}$. On the other hand, if ${2^\omega }$ is real-valued measurable, then there exists a maximal $\sigma$-finite $\mathbb {Q}$-invariant measure defined on a proper $\sigma$-algebra of subsets of $\mathbb {R}$.


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Article copyright: © Copyright 1991 American Mathematical Society