Individual ergodic theorems for infinite measure

2019/07/09 by Chilin, Vladimir, Comez, Dogan, Litvinov, Semyon
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1907.04678

Abstract

Given a σ-finite infinite measure space (Ω,μ), it is shown that any Dunford-Schwartz operator T: \mathcal L¹(Ω)→\mathcal L¹(Ω) can be uniquely extended to the space \mathcal L¹(Ω)+\mathcal L^∞(Ω). This allows to find the largest subspace \mathcal R_μ of \mathcal L¹(Ω)+\mathcal L^∞(Ω) such that the ergodic averages \frac1n∑_k=0^n-1T^k(f) converge almost uniformly (in Egorov's sense) for every f∈\mathcal R_μ and every Dunford-Schwartz operator T. Utilizing this result, almost uniform convergence of the averages \frac1n∑_k=0^n-1β_kT^k(f) for every f∈\mathcal 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 measure is employed to show that for each f∈\mathcal R_μ there exists a set Ω_f⊂Ω such that μ(Ω∖Ω_f)=0 and the averages \frac1n∑_k=0^n-1β_kf(τ^kω) converge for all ω∈Ω_f and any bounded Besicovitch sequence \β_k\. Applications to fully symmetric subspaces E⊂\mathcal R_μ are given.

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