A class of weakly compact sets in Lebesgue-Bochner spaces

Abstract

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 $\mu (f^{-1}(W))\ge 1-\delta$ 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 $[0,1]$.

References in corpus (1)

• On a certain class of $\K$ Banach spaces