A class of weakly compact sets in Lebesgue-Bochner spaces

TL;DR

This paper investigates a specific class of weakly compact sets in Lebesgue-Bochner spaces, introduces the property ($ ext{}oldsymbol{ ext{ extdelta}} ext{ extS}_oldsymbol{ extmu} ext{}$), and explores its implications and examples, including its relation to SWCG spaces.

Contribution

The paper defines the property ($ ext{ extdelta} ext{ extS}_ ext{ extmu}$) for Banach spaces, provides criteria for this property, and presents new examples and counterexamples involving SWCG spaces.

Findings

Testing on uniformly bounded sets suffices to verify the property.

Identified new spaces with the property ($ ext{ extdelta} ext{ extS}_ ext{ extmu}$).

Showed a separable Schur space failing the property with Lebesgue measure.

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