A class of weakly compact sets in Lebesgue-Bochner spaces

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