Convex Komlós sets in Banach function spaces

Abstract

In 1967 Komlós proved that for any sequence { f n } n in L 1 ( μ ) , with ‖ f n ‖ ⩽ M < ∞ (where μ is a probability measure), there exists a subsequence { g n } n of { f n } n and a function g ∈ L 1 ( μ ) such that for any further subsequence { h n } n of { g n } n , 1 n ∑ i = 1 n h i → n g μ -a.e. Later, Lennard proved that every convex subset of L 1 ( μ ) 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 L p ( R ) ( 1 ⩽ p ⩽ ∞ , μ = Lebesgue measure ), Lorentz, Orlicz and Orlicz–Lorentz spaces.