A Komlós Theorem for abstract Banach lattices of measurable functions

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• 作者：E. Jim&#xe9 ; nez Fern&#xe1 ; ndez¹ ; ^a ; ^{edjimfer@mat.upv.es"" rel=""nofollow} ; M.A. Juan² ; ^a ; ^{majuabl1@mat.upv.es"" rel=""nofollow} ; E.A. S&#xe1 ; nchez P&#xe9 ; rez ; ¹ ; ^a ; ^{easancpe@mat.upv.es"" rel=""nofollow}
• 关键词：Komló ; s Theorem ; Ces&#xe0 ; ro convergence ; Fatou property ; Banach function space ; Vector measure
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2011
• 出版时间：1 November 2011
• 年：2011
• 卷：383
• 期：1
• 页码：130-136
• 全文大小：179 K

文摘

Consider a Banach function space X(μ) of (classes of) locally integrable functions over a σ-finite measure space (Ω,Σ,μ) with the weak σ-Fatou property. Day and Lennard (2010) [9] proved that the theorem of Komlós on convergence of Cesàro sums in L¹[0,1] holds also in these spaces; i.e. for every bounded sequence (f_n)_n in X(μ), there exists a subsequence (f_nk)_k and a function fX(μ) such that for any further subsequence (h_j)_j of (f_nk)_k, the series converges μ-a.e. to f. In this paper we generalize this result to a more general class of Banach spaces of classes of measurable functions — spaces L¹(;d;) of integrable functions with respect to a vector measure ;d; on a δ-ring — and explore to which point the Fatou property and the Komlós property are equivalent. In particular we prove that this always holds for ideals of spaces L¹(;d;) with the weak σ-Fatou property, and provide an example of a Banach lattice of measurable functions that is Fatou but do not satisfy the Komlós Theorem.