Statistical Ergodic Theorem in Symmetric Spaces for Infinite Measures

Let (Ω,μ) be a measurable space with σ -finite continuous measure, μ(Ω)=∞. A linear operator T:L₁(Ω)+L_∞(Ω)→L₁(Ω)+L_∞(Ω) is called the Dunford-Schwartz operator if ||T(f)||₁<||f||₁ (respectively, ||T(f)||_∞<||f||_∞) for all f∈L₁(Ω) (respectively, f∈L_∞(Ω)).  {T_t}_t>0 is a strongly continuous in L₁(Ω) semigroup of Dunford-Schwartz operators, then each operator ${{{A_t(f)} ={\frac{1}{t}} {\int_0^t} {T_s(f)} ds \in L_1(\Omega)}}$ has a unique extension to the Dunford-Schwartz operator, which is also denoted by A_t, t>0. It is proved that in the completely symmetric space of measurable functions on (Ω,μ) the means A_t converge strongly as t→+∞ for each strongly continuous in L₁(Ω) semigroup {T_t}_t>0 of Dunford-Schwartz operators if and only if the norm ||.||_E(Ω) is order continuous.