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Proceedings of the American Mathematical Society

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

 
 

 

Topologically equivalent measures in the Cantor space


Author: Francisco J. Navarro-Bermúdez
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
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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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Article copyright: © Copyright 1979 American Mathematical Society