Weak sequential convergence in \(L^1(\mu,X)\) and an exact version of Fatou's lemma
type: Item
instance of: scholarly article
title: Weak sequential convergence in \(L^1(\mu,X)\) and an exact version of Fatou's lemma
zbMATH Open document ID: 1326.46037
author: M. Ali Khan
author: Nobusumi Sagara
published in: Journal of Mathematical Analysis and Applications
publication date: 2015-03-27T00:00:00Z
review text: The central result of the paper is Theorem 3.7 which roughly states that when \((\Omega, \mathcal{F},\mu)\) is a nonatomic finite measure space and \(X\) is an infinite-dimensional Banach space, then the following are equivalent: \newline (a) \((\Omega, \mathcal{F},\mu)\) has the saturation property of Maharam;\newline (b) a resonable generalization of Fatou's lemma holds. A corresponding result for multifunctions is proved in Theorem 4.9. The paper under review is also a survey of concepts involved in looking for generalization(s) of Fatou's lemma to the vector-valued case; a generalization that is important for applications. As the authors aptly describe their paper, ``[t]he results are developed through a considered overview of the relevant literature on the exact and approximate Fatou lemma phrased in terms of Bochner integration.''
reviewed by: Olav Nygaard
Mathematics Subject Classification ID: 46G10
Mathematics Subject Classification ID: 28B05
Mathematics Subject Classification ID: 49J21
Mathematics Subject Classification ID: 46-02
zbMATH DE Number: 6420080
zbMATH Keywords: Fatou's lemma
zbMATH Keywords: Bochner integral
zbMATH Keywords: Maharam's saturation property
zbMATH Keywords: well dominated sequence
MaRDI profile type: Publication
full work available at URL: j.jmaa.2013.10.082
OpenAlex ID: W2062531601
Wikidata QID: Q125018979
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DOI: 10.1016/J.JMAA.2013.10.082
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description: scientific article; zbMATH DE number 6420080