Statistical Ergodic Theorem in Symmetric Spaces for Infinite Measures

Let (Ω, μ) be a measurable space with a σ-finite continuous measure, μ(Ω) = ∞. A linear operator T : L₁(Ω) + L_∞(Ω) → L₁(Ω) + L_∞(Ω) is called a Dunford–Schwartz operator if ∥T(f)∥₁ ≤ ∥f∥₁ (respectively, ∥T(f)∥_∞ ≤ ∥f∥_∞) for all f ∈ L₁(Ω) (respectively, f ∈ L_∞(Ω)). If {T_t}_t≥0 is a strongly continuous in L₁(Ω) semigroup of Dunford–Schwartz operators, then each operator A_t(f) = \(\frac{1}{t}\underset{0}{\overset{t}{\int }}{T}_{s}\left(f\right)ds\in {L}_{1}\left(\Omega \right)\) $\frac{1}{t}\underset{0}{\stackrel{t}{\int }}{T}_s\left(f\right)ds\in L_1\left(\mathrm{\Omega }\right)$ , f ∈ L₁(Ω) has a unique extension to a Dunford–Schwartz operator, which is also denoted by A_t, t > 0. It is proved that in a completely symmetric space E(Ω) ⊈ L₁ 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.