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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Strongly ergodic sequences of integers and the individual ergodic theorem


Authors: J. R. Blum and J. I. Reich
Journal: Proc. Amer. Math. Soc. 86 (1982), 591-595
MSC: Primary 28D05; Secondary 47A35, 60F15
DOI: https://doi.org/10.1090/S0002-9939-1982-0674086-2
MathSciNet review: 674086
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Abstract: Let $ S = \{ {k_1},{k_2}, \ldots \} $ be an increasing sequence of positive integers. We call $ S$ strongly ergodic if for every measure preserving transformation $ T$ on a probability space $ (\Omega ,\mathcal{F},P)$ and every $ f \in {L_1}(\Omega )$ we have $ {\lim _{n \to \infty }}(1/n)\sum\nolimits_{j = 1}^n {f({T^{kj}}\omega ) = Pf(\omega )} $ a.e. where $ Pf$ is the appropriate limit guaranteed by the individual ergodic theorem. We give sufficient conditions for a sequence $ S$ to be strongly ergodic and provide examples.


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DOI: https://doi.org/10.1090/S0002-9939-1982-0674086-2
Keywords: Individual ergodic theorems, subsequences, strongly ergodic sequences
Article copyright: © Copyright 1982 American Mathematical Society