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Abstract

Let $(\Omega,\Sigma, \mu)$ be a probability space, $X$ a Banach space, and $L_1(\mu,X)$ the Banach space of Bochner integrable functions $f:\Omega\to X$. Let $W=\{f\in L_1(\mu,X): \text{for a.e.}$ $\omega \in \Omega, \|f(\omega)\|\le 1\}$. In this paper we characterize the weakly precompact subsets of $L_1(\mu,X)$. We prove that a bounded subset $A$ of $L_1(\mu,X)$ is weakly precompact if and only if $A$ is uniformly integrable and for any sequence $(f_n)$ in $A$, there exists a sequence $(g_n)$ with $g_n\in {\rm co}\{f_i: i\ge n\}$ for each $n$ such that for a.e. $\omega \in \Omega$, the sequence $(g_n(\omega))$ is weakly Cauchy in $X$. We also prove that if $A$ is a bounded subset of $L_1(\mu,X)$, then $A$ is weakly precompact if and only if for every $\epsilon >0$, there exist a positive integer $N$ and a weakly precompact subset $H$ of $ NW$ such that $A\subseteq H+\epsilon B(0)$, where $B(0)$ is the unit ball of $L_1(\mu,X)$.