Convex Komlo's sets in Banach function spaces

In 1967 Komlos proved that for any sequence {f(n)}(n) in L-1(mu), with parallel to f(n)parallel to <= M < infinity (where it is a probability measure), there exists a subsequence {g(n)}(n) of {f(n)}(n) and a function g is an element of L-1(mu) such that for any further subsequence {h(n)}(n) of {g(n)}(n). 1/n Sigma(n)(i=1)hi ->(n)g mu-a.e. Later. Lermard proved that every convex subset of L-1(mu) 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 Komlos' Theorem. We also extend Komlos' 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 mu is sigma-finite. Banach function spaces satisfying the hypotheses of both theorems include L-p(R) (1 <= p <= infinity, mu = Lebesgue measure), Lorentz, Orlicz and Orlicz-Lorentz spaces. (C) 2009 Elsevier Inc. All rights reserved.