Almost uniform and strong convergences in ergodic theorems for 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 Cesàro averages $M_n(T)(f)=\frac{1}{n}{\sum }_{k=0}^{n-1}{T}^k(f)$ 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 $X$.