On weak compactness in the space of Pettis integrable functions
It is first stated that an unnecessary condition can be dropped in a weak compactness criterion for subsets K in the space \(L^ 1_ E(\mu)\) of all strongly measurable and Pettis integrable functions with values in a Banach space E, as given by the same authors in an earlier paper [Theorem 1, Contemp. Math. 2, 161-187 (1980)]. For the proof of this improvement the authors refer to a forthcoming paper of the second author. The improved version states that K is conditionally weakly compact if and only if the following two conditions are satisfied: (1) For each measurable subset A of finite measure the set of all \(\int_{A}fd\mu\) for f in K is conditionally weakly compact in E; (2) For each countable subset \(K_ 0\) of K there exists a sequence of finite partitions such that for f in \(K_ 0\) the conditional expectations of f with respect to these partitions converge weakly in \(L^ 1_ E(\mu)\) towards f. This result is applied for a new demonstration of Lewis' earlier characterization of conditionally weakly compact subsets of \(L^ 1_ E(\mu)\). The second half of the paper deals with the correction of a proof in an earlier paper [Theorem 8.8 in J. Math. Anal. Appl. 54, 348- 389 (1976; Zbl 0339.46029].
space of strongly measurable and Pettis integrable functions with values in a Banach space
scientific article; zbMATH DE number 3844394
scientific article