A class of weakly compact sets in Lebesgue–Bochner spaces

Abstract

Let X be a Banach space and μ a probability measure. A set K⊆L1(μ,X) is said to be a δS-set if it is uniformly integrable and for every δ>0 there is a weakly compact set W⊆X such that μ(f−1(W))≥1−δ for every f∈K. This is a sufficient, but in general non-necessary, condition for relative weak compactness in L1(μ,X). We say that X has property (δSμ) if every relatively weakly compact subset of L1(μ,X) is a δS-set. In this paper we study δS-sets and Banach spaces having property (δSμ). We show that testing on uniformly bounded sets is enough to check this property. New examples of spaces having property (δSμ) are provided. Special attention is paid to the relationship with strongly weakly compactly generated (SWCG) spaces. In particular, we show an example of a SWCG (in fact, separable Schur) space failing property (δSμ) when μ is the Lebesgue measure on [0,1].