A note on the ergodic theorem

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.