INDIVIDUAL ERGODIC THEOREMS FOR INFINITE MEASURE

Given a σ-finite infinite measure space (Ω, µ), it is shown that any Dun-ford–Schwartz operator T: L¹(Ω) → L¹(Ω) can be uniquely extended to the space L¹(Ω) + L^∞(Ω). This allows one to find the largest subspace R_µ of L¹(Ω) + L^∞(Ω) such that the ergodic averages n^{−1 ∑}n−1 k=0^{T k} (f) converge almost uniformly (in Egorov’s sense) for every f ∈ R_µ and every Dunford–Schwartz operator T. Utilizing this result, almost uniform convergence of the averages n^{−1 ∑}n−1 k=0^βkT^k (f) for every f ∈ R_µ, any Dunford– Schwartz operator T and any bounded Besicovitch sequence {β_k} is established. Further, given a measure preserving transformation τ: Ω → Ω, Assani’s extension of Bourgain’s Return Times theorem to σ-finite measures is employed to show that for each f ∈ R_µ there exists a set Ω_f ⊂ Ω such that µ(Ω \ Ω_f) = 0 and the averages n^{−1 ∑}n−1 k=0^βkf(τ^kω) converge for all ω ∈ Ω_f and any bounded Besicovitch sequence {β_k}. Applications to fully symmetric subspaces E ⊂ R_µ are outlined.