Weakly precompact subsets of $L_1(\mu,X)$

Ioana Ghenciu¹

¹ Mathematics Department University of Wisconsin-River Falls River Falls, WI 54022, U.S.A.

Colloquium Mathematicum, Tome 129 (2012) no. 1, pp. 133-143

Résumé

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 $L_1(\mu,X)$.

Détail

DOI : 10.4064/cm129-1-10

Keywords: omega sigma probability space banach space banach space bochner integrable functions omega text omega omega omega paper characterize weakly precompact subsets prove bounded subset weakly precompact only uniformly integrable sequence there exists sequence each omega omega sequence omega weakly cauchy prove bounded subset weakly precompact only every epsilon there exist positive integer weakly precompact subset subseteq epsilon where unit ball