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 Cesaro 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 $${f\in X}$$ . If $${(\Omega,\mu)}$$ is quasi-non-atomic, 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.