Weak compactness in $L^ 1(\mu ,X)$

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 $W = \{ f \in {L^1}(\mu ,X):{\text { for a}}{\text {.e}}{\text {. }}\omega {\text { in }}\Omega ,||f(\omega )|| \leq 1\}$. In this paper we characterize the rwc (relatively weakly compact) subsets of ${L^1}(\mu ,X)$. The main results are as follows: Theorem A. A subset $H$ of $W$ is rwc iff given any sequence $({f_n})$ in $H$ there exists a sequence $({\tilde f_n})$, with ${\tilde f_n} \in \operatorname {Co}({f_n},{f_{n + 1}}, \ldots )$ such that, for a.e. $\omega$ in $\Omega$, the sequence $({\tilde f_n}(\omega ))$ converges weakly in $X$. Theorem B. A subset $A$ of ${L^1}(\mu ,X)$ is rwc iff given any $\varepsilon > 0$ there exist an integer $N$ and a rwc subset $H$ of NW such that $A \subseteq H + \varepsilon B(0)$, where $B(0)$ is the unit ball of ${L^1}(\mu ,X)$.