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 $s = 1 - r$.References
- J. C. Oxtoby and S. M. Ulam, Measure-preserving homeomorphisms and metrical transitivity, Ann. of Math. (2) 42 (1941), 874–920. MR 5803, DOI 10.2307/1968772
- John C. Oxtoby, Homeomorphic measures in metric spaces, Proc. Amer. Math. Soc. 24 (1970), 419–423. MR 260961, DOI 10.1090/S0002-9939-1970-0260961-1
- John C. Oxtoby and Vidhu S. Prasad, Homeomorphic measures in the Hilbert cube, Pacific J. Math. 77 (1978), no. 2, 483–497. MR 510936, DOI 10.2140/pjm.1978.77.483 F. J. Navarro-Bermúdez, Topologically equivalent measures in the Cantor space, Ph. D. Thesis, Bryn Mawr College, 1977.
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