On weak compactness in the space of Pettis integrable functions
type: Item
instance of: scholarly article
title: On weak compactness in the space of Pettis integrable functions
zbMATH Open document ID: 0532.46024
DOI: 10.1016/S0001-8708(82)80005-4
author: James K. Brooks
author: Nicolae Dinculeanu
published in: Advances in Mathematics
publication date: 1982-01-01T00:00:00Z
review text: 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].
Mathematics Subject Classification ID: 46G10
Mathematics Subject Classification ID: 46A50
Mathematics Subject Classification ID: 46E40
Mathematics Subject Classification ID: 28B05
zbMATH DE Number: 3844394
zbMATH Keywords: space of strongly measurable and Pettis integrable functions with values in a Banach space
zbMATH Keywords: weak compactness criterion
reviewed by: Werner Strauß
MaRDI profile type: Publication
cites work: Q3660614
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description: scientific article; zbMATH DE number 3844394