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Proceedings of the American Mathematical Society

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



Homeomorphic measures in metric spaces

Author: John C. Oxtoby
Journal: Proc. Amer. Math. Soc. 24 (1970), 419-423
MSC: Primary 28.13
MathSciNet review: 0260961
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Abstract: For any nonatomic, normalized Borel measure $ \mu $ in a complete separable metric space $ X$ there exists a homeomorphism $ h:\mathfrak{N} \to X$ such that $ \mu = \lambda {h^{ - 1}}$ on the domain of $ \mu $, where $ \mathfrak{N}$ is the set of irrational numbers in $ (0,1)$ and $ \lambda $ denotes Lebesgue-Borel measure in $ \mathfrak{N}$. A Borel measure in $ \mathfrak{N}$ is topologically equivalent to $ \lambda $ if and only if it is nonatomic, normalized, and positive for relatively open subsets.

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Keywords: Topologically equivalent Borel measures, homeomorphic measure spaces, measure-preserving mapping, complete separable metric space, space of irrational numbers, Cantor set
Article copyright: © Copyright 1970 American Mathematical Society