The de la Vallée-Poussin theorem and Orlicz spaces associated to a vector measure
type: Item
instance of: scholarly article
title: The de la Vallée-Poussin theorem and Orlicz spaces associated to a vector measure
zbMATH Open document ID: 1423.46047
author: Ricardo del Campo
author: Antonio Fernández
author: Fernando Mayoral
author: Francisco Naranjo
published in: Journal of Mathematical Analysis and Applications
publication date: 2018-10-19T00:00:00Z
review text: The paper under review deals with the analysis of uniform integrability of bounded subsets of spaces of (scalar) integrable functions with respect to a vector measure. Let \((\Omega, \Sigma, \mu)\) be a finite measure space. Recall that a subset \(H\) of a Banach function space is uniformly integrable if \[ \lim_{c \to \infty} \sup_{f \in H} \int_{[ |f| >c]} |f|\, d\mu =0. \] The de la Vallée-Poussin theorem allows to relate this property with the following (equivalent) boundedness property in the Orlicz class \(L^\Phi(\mu).\) Let \(D\) be the class of non-decreasing functions \(\Phi:[0,\infty) \to [0,\infty)\) such that \(\lim_{x \to \infty} \Phi(x)/x =\infty.\) Then there exists \(\Phi \in D\) such that \[ \sup_{f \in H} \| \Phi(|f|)\|_{L^1(\mu)} < \infty. \] On the other hand, the Dunford-Pettis theorem allows to relate uniform integrability of subsets of \(L^1(\mu)\) also with the property of being relatively weakly (sequentially) compact. This paper works in the context of the spaces \(L^1(m)\) of scalar integrable functions with respect to a vector measure \(m\). Using elements of the Dunford-Pettis circle of ideas, the results presented here allow to locate each compact subset of \(L^1(m)\) in a smaller Orlicz space of a vector measure \(L^\Phi(m),\) also as a compact subset. This is presented mainly in Theorem 4.1 and the corresponding corollaries. For it, the authors obtain in the preceding Sections 2 and 3 some useful results on the Orlicz spaces of a vector measure, whose interest goes beyond its application in Section 4.
reviewed by: E. A. Sánchez-Pérez
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 46B50
zbMATH DE Number: 6958708
zbMATH Keywords: Orlicz spaces
zbMATH Keywords: vector measure
zbMATH Keywords: uniform integrability
zbMATH Keywords: compactness
MaRDI profile type: Publication
full work available at URL: j.jmaa.2018.10.001
OpenAlex ID: W2894728180
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DOI: 10.1016/J.JMAA.2018.10.001
description: scientific article; zbMATH DE number 6958708