Convex Komlós sets in Banach function spaces
Komlós's theorem establishes that, for any bounded sequence in \(L^1[0,1]\), there is a subsequence and a function \(g \in L^1[0,1]\) such that, for any further sequence, its Cesàro means converge to \(g\) (see [\textit{J.\,Komlós}, Acta Math.\ Acad.\ Sci.\ Hung.\ 18, 217--229 (1967; Zbl 0228.60012)]). A sort of converse of this result that states that every convex subset of \(L^1(\mu)\) satisfying the conclusion of Komlós Theorem is norm bounded was proved by \textit{C.\,Lennard} [Pac.\ J.\ Math.\ 159, No.\,1, 75--85 (1993; Zbl 0739.46017)]. In this paper, these results -- Komlós's theorem and the Lennard converse -- are extended to the class of Banach function spaces over \(\sigma\)-finite measure spaces that are finitely integrable and satisfy the Fatou property.
scientific article; zbMATH DE number 5687686