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Individual ergodic theorems for infinite measure

Authors :
Chilin, Vladimir
Comez, Dogan
Litvinov, Semyon
Publication Year :
2019

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 $\frac1n\sum\limits_{k=0}^{n-1}T^k(f)$ 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 $\frac1n\sum\limits_{k=0}^{n-1}\beta_kT^k(f)$ 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 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 $\frac1n\sum\limits_{k=0}^{n-1}\beta_kf(\tau^k\omega)$ 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 given.<br />Comment: arXiv admin note: text overlap with arXiv:1612.05802

Subjects

Subjects :
Mathematics - Functional Analysis

Details

Database :
arXiv
Publication Type :
Report
Accession number :
edsarx.1907.04678
Document Type :
Working Paper