A Komlós Theorem for abstract Banach latticesof measurable functions

Abstract

Consider a Banach function space X(mu) of (classes of) locally integrable functions over a sigma-finite measure space (Omega, Sigma, mu) with the weak sigma-Fatou property. Day and Lennard (2010) [9] proved that the theorem of Komlos on convergence of Cesaro sums in L(1) [0, 1] holds also in these spaces; i.e. for every bounded sequence (f(n))(n) in X(mu), there exists a subsequence (f(nk))(k) and a function f is an element of X(mu) such that for any further subsequence (h(j))(j) of (f(nk))(k), the series 1/n Sigma(n)(j=1) h(j) converges mu-a.e. to f. In this paper we generalize this result to a more general class of Banach spaces of classes of measurable functions - spaces L(1) (nu) of integrable functions with respect to a vector measure nu on a delta-ring - and explore to which point the Fatou property and the Komlos property are equivalent. In particular we prove that this always holds for ideals of spaces L(1)(nu) with the weak sigma-Fatou property, and provide an example of a Banach lattice of measurable functions that is Fatou but do not satisfy the Komlos Theorem. (C) 2011 Elsevier Inc.