Almost uniform and strong convergences in ergodic theorems for symmetric spaces

Let (Ω , μ) be a σ-finite measure space, and let X⊂ L ¹ (Ω) + L ^∞ (Ω) be a fully symmetric space of measurable functions on (Ω , μ). If μ(Ω) = ∞, necessary and sufficient conditions are given for almost uniform convergence in X (in Egorov’s sense) of Cesàro averages Mn(T)(f)=1n∑k=0n-1Tk(f) for all Dunford–Schwartz operators T in L ¹ (Ω) + L ^∞ (Ω) and any f∈ X. If (Ω , μ) 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 (Ω) + L ^∞ (Ω) if and only if X has order continuous norm and L ¹ (Ω) is not contained in X.