A note on the ergodic theorem
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- by Takeshi Yoshimoto PDF
- Proc. Amer. Math. Soc. 90 (1984), 401-404 Request permission
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.References
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Additional Information
- © Copyright 1984 American Mathematical Society
- 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