Almost uniform and strong convergences in ergodic theorems for symmetric\n spaces

2018/02/19 by Vladimir Chilin, Chilin, Vladimir, Semyon Litvinov +1
Economics, Econometrics and Finance · Mathematics · #37A30 #46E30 #47A35 #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #FOS: Mathematics #Functional Analysis (math.FA) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.1802.06932

openalex publication_date 2018/02/19 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28

Abstract

Let (\Ω,\μ) be a \σ-finite measure space, and let X\⊂\nL¹(\Ω)+L^\∞(\Ω) be a fully symmetric space of measurable\nfunctions on (\Ω,\μ). If \μ(\Ω)=\∞, necessary and sufficient\nconditions are given for almost uniform convergence in X (in Egorov's sense)\nof Ces `aro averages M_n(T)(f)= frac1n\∑_{k = 0}^n-1T^k(f) for all\nDunford-Schwartz operators T in L¹(\Ω)+ L^\∞(\Ω) and any f\∈\nX. Besides, it is proved that the averages M_n(T) converge strongly in X\nfor each Dunford-Schwartz operator T in L¹(\Ω)+L^\∞(\Ω) if and\nonly if X has order continuous norm and L¹(\Ω) is not contained in\nX.\n

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