Almost uniform and strong convergences in ergodic theorems for symmetric spaces

TL;DR

This paper establishes necessary and sufficient conditions for almost uniform and strong convergence of Cesàro averages of Dunford-Schwartz operators in symmetric function spaces over infinite measure spaces.

Contribution

It provides a complete characterization of convergence behaviors of Cesàro averages in symmetric spaces, extending ergodic theorems to a broad class of function spaces.

Findings

Necessary and sufficient conditions for almost uniform convergence in symmetric spaces.

Characterization of strong convergence of averages based on space properties.

Identification of conditions under which Cesàro averages converge in symmetric spaces.

Abstract

Let $(\Omega,\mu)$ be a $\sigma$-finite measure space, and let $X\subset L^1(\Omega)+L^\infty(\Omega)$ be a fully symmetric space of measurable functions on $(\Omega,\mu)$. If $\mu(\Omega)=\infty$, necessary and sufficient conditions are given for almost uniform convergence in $X$ (in Egorov's sense) of Ces\`aro averages $M_n(T)(f)=\frac1n\sum_{k = 0}^{n-1}T^k(f)$ for all Dunford-Schwartz operators $T$ in $L^1(\Omega)+ L^\infty(\Omega)$ and any $f\in X$. Besides, it is proved that the averages $M_n(T)$ converge strongly in $X$ for each Dunford-Schwartz operator $T$ in $L^1(\Omega)+L^\infty(\Omega)$ if and only if $X$ has order continuous norm and $L^1(\Omega)$ is not contained in $X$.