A class of weakly compact sets in Lebesgue-Bochner spaces

Let $X$ be a Banach space and $μ$ a probability measure. A set $K \subseteq L^1(μ,X)$ is said to be a $δ\mathcal{S}$-set if it is uniformly integrable and for every $δ>0$ there is a weakly compact set $W \subseteq X$ such that $μ(f^{-1}(W)) \geq 1-δ$ for every $f\in K$. This is a sufficient, but in general non necessary, condition for relative weak compactness in $L^1(μ,X)$. We say that $X$ has property ($δ\mathcal{S}_μ$) if every relatively weakly compact subset of $L^1(μ,X)$ is a $δ\mathcal{S}$-set. In this paper we study $δ\mathcal{S}$-sets and Banach spaces having property ($δ\mathcal{S}_μ$). We show that testing on uniformly bounded sets is enough to check this property. New examples of spaces having property ($δ\mathcal{S}_μ$) 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 ($δ\mathcal{S}_μ$) when $μ$ is the Lebesgue measure on $[0,1]$.

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