The de la Vallée-Poussin theorem and Orlicz spaces associated to a vector measure
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.
scientific article; zbMATH DE number 6958708
scientific article