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A note on the irreducibility of Lebesgue measure with applications to random walks on the unit circle


Author: Tzuu Shuh Chiang
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
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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 [Enhancements On Off] (What's this?)

  • Robert B. Ash and Melvin F. Gardner, Topics in stochastic processes, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Probability and Mathematical Statistics, Vol. 27. MR 0448463

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Keywords: Irreducibility, random walks, ergodicity
Article copyright: © Copyright 1982 American Mathematical Society