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Proceedings of the American Mathematical Society

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

 

 

A note on the ergodic theorem


Author: Takeshi Yoshimoto
Journal: Proc. Amer. Math. Soc. 90 (1984), 401-404
MSC: Primary 47A35; Secondary 40C05
DOI: https://doi.org/10.1090/S0002-9939-1984-0728356-1
MathSciNet review: 728356
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Abstract: Let $ M$ be a positive regular shift-invariant method of summability and let $ T$ be a one-to-one transformation which maps $ X$ onto $ X$ and which is $ \mathcal{B}$-bimeasurable, i.e., $ A \in \mathcal{B}$ if and only if $ TA \in \mathcal{B}$, where $ (X,\mathcal{B})$ is a measurable space. Then it is proved that if for a finite measure $ \mu $ on $ \mathcal{B}$ the sequence $ \left\{ {\mu ({T^m}A)} \right\}$ is $ M$-summable for each $ A \in \mathcal{B}$, then for any real $ r \geqslant 1$ the sequence $ \left\{ {f \circ {T^m}} \right\}$ is $ (C,r)$-summable $ \mu $-almost everywhere for every bounded $ \mathcal{B}$-measurable function $ f$ defined on $ X$. The result includes the Blum-Hanson theorem.


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DOI: https://doi.org/10.1090/S0002-9939-1984-0728356-1
Keywords: Regular summation method, $ M$-summable, $ (C,r)$-summable, Blum-Hanson theorem
Article copyright: © Copyright 1984 American Mathematical Society