Convex Komlós sets in Banach function spaces

In 1967 Komlós proved that for any sequence {fn}n in L1(μ), with ‖fn‖⩽M<∞ (where μ is a probability measure), there exists a subsequence {gn}n of {fn}n and a function g∈L1(μ) such that for any further subsequence {hn}n of {gn}n,1n∑i=1nhi→ngμ-a.e. Later, Lennard proved that every convex subset of L1(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 Lp(R) (1⩽p⩽∞, μ=Lebesgue measure), Lorentz, Orlicz and Orlicz-Lorentz spaces.