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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles 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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A note on the irreducibility of Lebesgue measure with applications to random walks on the unit circle
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by Tzuu Shuh Chiang PDF
Proc. Amer. Math. Soc. 85 (1982), 603-605 Request permission

Abstract:

Let $\mu$ be a probability measure on $R$. We say that a $\sigma$-finite measure $\lambda$ is irreducible with respect to $\mu$ if there does not exist a Borel set $A$ with $\mu (A)$, $\mu ({A^c}) > 0$ such that $\int _A {\mu ({A^c} - x)} \lambda (dx) = 0$. It is well known that the Lebesgue measure $m(dx)$ is irreducible with respect to any discrete measure whose support is $R$. We prove that every absolutely continuous measure is irreducible with respect to any probability measure whose support is $R$ and give an application of this fact to random walks on the unit circle.
References
  • Robert B. Ash and Melvin F. Gardner, Topics in stochastic processes, Probability and Mathematical Statistics, Vol. 27, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0448463
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Additional Information
  • © Copyright 1982 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 85 (1982), 603-605
  • MSC: Primary 60J15; Secondary 28A12, 46G99
  • DOI: https://doi.org/10.1090/S0002-9939-1982-0660613-8
  • MathSciNet review: 660613