A Komlós Theorem for abstract Banach lattices of measurable functions

Abstract

Consider a Banach function space X ( μ ) of (classes of) locally integrable functions over a σ-finite measure space ( Ω , Σ , μ ) with the weak σ-Fatou property. Day and Lennard (2010) [9] proved that the theorem of Komlós on convergence of Cesàro sums in L 1 [ 0 , 1 ] holds also in these spaces; i.e. for every bounded sequence ( f n ) n in X ( μ ) , there exists a subsequence ( f n k ) k and a function f ∈ X ( μ ) such that for any further subsequence ( h j ) j of ( f n k ) k , the series 1 n ∑ j = 1 n h j converges μ-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 ( ν ) of integrable functions with respect to a vector measure ν on a δ-ring — and explore to which point the Fatou property and the Komlós property are equivalent. In particular we prove that this always holds for ideals of spaces L 1 ( ν ) with the weak σ-Fatou property, and provide an example of a Banach lattice of measurable functions that is Fatou but do not satisfy the Komlós Theorem.