Individual ergodic theorems for infinite measure
Volume 167 / 2022
Abstract
Given a $\sigma $-finite infinite measure space $(\Omega ,\mu )$, it is shown that any Dun\-ford–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 one to find the largest subspace $\mathcal {R}_\mu $ of $\mathcal {L}^1(\Omega )+\mathcal {L}^\infty (\Omega )$ such that the ergodic averages $n^{{-1}}\sum _{k=0}^{n-1}T^k(f)$ 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 $n^{-1}\sum _{k=0}^{n-1}\beta _kT^k(f)$ 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 measures 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 $n^{-1}\sum _{k=0}^{n-1}\beta _kf(\tau ^k\omega )$ 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 outlined.
