Convex Komlós sets in Banach function spaces

Abstract: In 1967 Komlós proved that for any sequence in , with (where μ is a probability measure), there exists a subsequence of and a function such that for any further subsequence of , Later, Lennard proved that every convex subset of satisfying the conclusion of the previous theorem is norm bounded. In this paper, we isolate a very general class of Banach function spaces (those satisfying the Fatou property), to which we generalize Lennard''s converse to Komlós'' Theorem. We also extend Komlós'' Theorem itself to a broad class of Banach function spaces: those that satisfy the Fatou property and are finitely integrable (or even weakly finitely integrable), when the measure μ is σ-finite. Banach function spaces satisfying the hypotheses of both theorems include (, ), Lorentz, Orlicz and Orlicz–Lorentz spaces. [Copyright &y& Elsevier]