TY - JOUR
T1 - INDIVIDUAL ERGODIC THEOREMS FOR INFINITE MEASURE
AU - Chilin, Vladimir
AU - Çömez, Doğan
AU - Litvinov, Semyon
N1 - Publisher Copyright:
© Instytut Matematyczny PAN, 2022.
PY - 2022
Y1 - 2022
N2 - 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 Rµ of L1(Ω) + L∞(Ω) such that the ergodic averages n−1 ∑n−1 k=0T k (f) converge almost uniformly (in Egorov’s sense) for every f ∈ Rµ and every Dunford–Schwartz operator T. Utilizing this result, almost uniform convergence of the averages n−1 ∑n−1 k=0βkTk (f) for every f ∈ Rµ, any Dunford– Schwartz operator T and any bounded Besicovitch sequence {βk} 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 ∈ Rµ there exists a set Ωf ⊂ Ω such that µ(Ω \ Ωf) = 0 and the averages n−1 ∑n−1 k=0βkf(τkω) converge for all ω ∈ Ωf and any bounded Besicovitch sequence {βk}. Applications to fully symmetric subspaces E ⊂ Rµ are outlined.
AB - 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 Rµ of L1(Ω) + L∞(Ω) such that the ergodic averages n−1 ∑n−1 k=0T k (f) converge almost uniformly (in Egorov’s sense) for every f ∈ Rµ and every Dunford–Schwartz operator T. Utilizing this result, almost uniform convergence of the averages n−1 ∑n−1 k=0βkTk (f) for every f ∈ Rµ, any Dunford– Schwartz operator T and any bounded Besicovitch sequence {βk} 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 ∈ Rµ there exists a set Ωf ⊂ Ω such that µ(Ω \ Ωf) = 0 and the averages n−1 ∑n−1 k=0βkf(τkω) converge for all ω ∈ Ωf and any bounded Besicovitch sequence {βk}. Applications to fully symmetric subspaces E ⊂ Rµ are outlined.
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U2 - 10.4064/cm8271-2-2021
DO - 10.4064/cm8271-2-2021
M3 - Article
AN - SCOPUS:85133713528
SN - 0010-1354
VL - 167
SP - 219
EP - 238
JO - Colloquium Mathematicum
JF - Colloquium Mathematicum
IS - 2
ER -