## Abstract

Given a σ-finite infinite measure space (Ω, µ), it is shown that any Dun-ford–Schwartz operator T: L^{1}(Ω) → L^{1}(Ω) can be uniquely extended to the space L^{1}(Ω) + L^{∞}(Ω). This allows one to find the largest subspace R_{µ} of L^{1}(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^{β}kT^{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^{β}kf(τ^{k}ω) converge for all ω ∈ Ω_{f} and any bounded Besicovitch sequence {β_{k}}. Applications to fully symmetric subspaces E ⊂ R_{µ} are outlined.

Original language | English (US) |
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Pages (from-to) | 219-238 |

Number of pages | 20 |

Journal | Colloquium Mathematicum |

Volume | 167 |

Issue number | 2 |

DOIs | |

State | Published - 2022 |

## All Science Journal Classification (ASJC) codes

- General Mathematics