Individual ergodic theorems for infinite measure

Abstract

Given a $\sigma$-finite infinite measure space $(\Omega ,\mu )$, it is shown that any Dunford-Schwartz operator $T:{\mathcal{L}}^1(\Omega )\to {\mathcal{L}}^1(\Omega )$ can be uniquely extended to the space ${\mathcal{L}}^1(\Omega )+{\mathcal{L}}^{\infty }(\Omega )$. This allows to find the largest subspace ${\mathcal{R}}_{\mu }$ of ${\mathcal{L}}^1(\Omega )+{\mathcal{L}}^{\infty }(\Omega )$ such that the ergodic averages $\frac{1}{n}\sum _{k=0}^{n-1}{T}^k(f)$ converge almost uniformly (in Egorov's sense) for every $f\in {\mathcal{R}}_{\mu }$ and every Dunford-Schwartz operator $T$. Utilizing this result, almost uniform convergence of the averages $\frac{1}{n}\sum _{k=0}^{n-1}{\beta }_k{T}^k(f)$ for every $f\in {\mathcal{R}}_{\mu }$, any Dunford-Schwartz operator $T$ and any bounded Besicovitch sequence $\{{\beta }_k\}$ is established. Further, given a measure preserving transformation $\tau :\Omega \to \Omega$, Assani's extension of Bourgain's Return Times theorem to $\sigma$-finite measure is employed to show that for each $f\in {\mathcal{R}}_{\mu }$ there exists a set ${\Omega }_f\subset \Omega$ such that $\mu (\Omega \setminus {\Omega }_f)=0$ and the averages $\frac{1}{n}\sum _{k=0}^{n-1}{\beta }_{k}f({\tau }^k\omega )$ converge for all $\omega \in {\Omega }_f$ and any bounded Besicovitch sequence $\{{\beta }_k\}$. Applications to fully symmetric subspaces $E\subset {\mathcal{R}}_{\mu }$ are given.

arXiv admin note: text overlap with arXiv:1612.05802