WEAK COMPACTNESS IN Ll(p,X)

Let (Q.,l,ß) be a probability space, X a Banach space, and L (fi, X) the Banach space of Bochner integrable functions f: Í2 —> X . Let W = {/ € Ll(p, X) : for a.e. w in Í2, ||/(w)|| < 1} . In this paper we characterize the rwc (relatively weakly compact) subsets of L (ß, X). The main results are as follows: Theorem A. A subset H of W is rwc iff given any sequence (/„) in H there exists a sequence (/„), with fn € Co(/n , fn+l, ...) such that, for a.e. a> in Í2 , the sequence (/„(«)) converges weakly in X. Theorem B. A subset A of L (ß, X) is rwc iff given any e > 0 there exist an integer N and a rwc subset H of NW suchthat A Ç H + eB{0), where B(0) is the unit ball of L [ß, X). Introduction Let (Q., X, p) be a probability space, X an arbitrary Banach space, and Lx(p, X) the Banach space of Bochner integrable functions /: Q —» X equipped with its usual norm [6, p. 50]. The problem of characterizing the rwc (relatively weakly compact) subsets of the space L (p, X) is a well-known longstanding open problem, see Chapter IV of [6] for a review of known results about this problem up to 1977 and [1, 2, 3, 7, 9, 12] for some more recent results. In this note we present a characterization of the rwc subsets of the space L (p, X). The characterization is obtained in two steps. In the first step we characterize the rwc subsets of the set W = {/ e Ll (p, X) : for a.e. coinQ, ||/(w)|| < 1}. This result is as follows: A subset H of W is rwc iff given any sequence (fn) in H there exists a sequence (fn) with fn £ Co(fn, fn+x, ■■■) such that, for a.e. co in £2, the sequence (fn(co)) converges weakly in X. In the second step we show that a subset A of L (p, X) is rwc iff it is a small perturbation of a rwc subset of NW for some integer N. More precisely, we prove the following result: A subset A of the space L (p, X) is rwc iff given any e > 0 there exist Received by the editors March 10, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 46E40; Secondary 46B20.