Almost uniform and strong convergences in ergodic theorems for symmetric spaces

V. Chilin, S. Litvinov

Research output: Contribution to journalArticle

Abstract

Let (Ω , μ) be a σ-finite measure space, and let X⊂ L 1 (Ω) + L (Ω) be a fully symmetric space of measurable functions on (Ω , μ). If μ(Ω) = ∞, necessary and sufficient conditions are given for almost uniform convergence in X (in Egorov’s sense) of Cesàro averages Mn(T)(f)=1n∑k=0n-1Tk(f) for all Dunford–Schwartz operators T in L 1 (Ω) + L (Ω) and any f∈ X. If (Ω , μ) is quasi-non-atomic, it is proved that the averages M n (T) converge strongly in X for each Dunford–Schwartz operator T in L 1 (Ω) + L (Ω) if and only if X has order continuous norm and L 1 (Ω) is not contained in X.

Original languageEnglish (US)
Pages (from-to)229-253
Number of pages25
JournalActa Mathematica Hungarica
Volume157
Issue number1
DOIs
StatePublished - Feb 1 2019

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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