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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
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