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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
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.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1982-0660613-8
PII: S 0002-9939(1982)0660613-8
Keywords: Irreducibility, random walks, ergodicity
Article copyright: © Copyright 1982 American Mathematical Society