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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 \({f\in X}\). 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.

Almost uniform and strong convergences in ergodic theorems for symmetric spaces