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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
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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