Counter-examples to the Dunford-Schwartz pointwise ergodic theorem on\n L¹+L^\∞

2018/03/29 by Dávid Kunszenti-Kovács, Kunszenti-Kovács, Dávid
Economics, Econometrics and Finance · Mathematics · #47B38 #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #Advanced Topology and Set Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Functional Analysis (math.FA) #Stochastic processes and financial applications #primary: 47A35 #secondary: 37A30

paper · pdf · doi:10.48550/arxiv.1803.11040

openalex publication_date 2018/03/29 · openalex created_date 2022/09/05 · openalex updated_date 2026/07/28

Abstract

Extending a result by Chilin and Litvinov, we show by construction that given\nany \σ-finite infinite measure space (\Ω,\A, \μ) and a\nfunction f\∈ L¹(\Ω)+L^\∞(\Ω) with\n\μ( |f|>\ε )=\∞ for some \ε>0, there exists a\nDunford-Schwartz operator T over (\Ω,\A, \μ) such that\n\(1)/(N)\∑_n=1^N (Tⁿf)(x) fails to converge for almost every\nx\∈\Ω. In addition, for each operator we construct, the set of functions\nfor which pointwise convergence fails almost everywhere is residual in\nL¹(\Ω)+L^\∞(\Ω).\n

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