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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Topologically equivalent measures in the Cantor space
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by Francisco J. Navarro-Bermúdez PDF
Proc. Amer. Math. Soc. 77 (1979), 229-236 Request permission

Abstract:

The Cantor space is realized as a countable product X of two-element sets. The measures $\mu$ and $\nu$ in X are topologically equivalent if there is a homeomorphism h of X onto itself such that $\mu = \nu h$. Let $\mathcal {F}$ be the family of product measures in X which are shift invariant. The members $\mu (r)$ of $\mathcal {F}$ are in one-to-one correspondence with the real numbers r in the unit interval. The relation of topological equivalence partitions the family $\mathcal {F}$ into classes with at most countably many measures each. A class contains only the measures $\mu (r)$ and $\mu (1 - r)$ when r is a rational or a transcendental number. Equivalently, if r is rational or transcendental and $\mu (s)$ is topologically equivalent to $\mu (r)$ then $s = r$ or $s = 1 - r$.
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Additional Information
  • © Copyright 1979 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 77 (1979), 229-236
  • MSC: Primary 28D05
  • DOI: https://doi.org/10.1090/S0002-9939-1979-0542090-0
  • MathSciNet review: 542090