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Proceedings of the American Mathematical Society

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

 

 

Extensions of isometrically invariant measures on Euclidean spaces


Author: Piotr Zakrzewski
Journal: Proc. Amer. Math. Soc. 110 (1990), 325-331
MSC: Primary 28C10; Secondary 03E05
DOI: https://doi.org/10.1090/S0002-9939-1990-1021216-3
MathSciNet review: 1021216
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Abstract: We consider countably additive, nonnegative, extended real-valued measures that vanish on singletons.

Given a group $ G$ of isometries of $ {{\mathbf{R}}^n}$ and a $ G$-invariant $ \sigma $-finite measure on $ {{\mathbf{R}}^n}$ we study the problem of determining whether it has a proper $ G$-invariant extension. We prove that it does, provided that the set of all points with uncountable $ G$-orbits is not of measure zero. We also characterize those groups $ G$ for which every $ G$-invariant $ \sigma $-finite measure on $ {{\mathbf{R}}^n}$ has a proper $ G$-invariant extension.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1990-1021216-3
Keywords: Invariant $ \sigma $-finite measure, isometries of $ {{\mathbf{R}}^n}$, real-valued measurable cardinal
Article copyright: © Copyright 1990 American Mathematical Society