The dual space of \({\mathcal L}^ 1(\mu)\) for a vector measure \(\mu\)
type: Item
instance of: scholarly article
title: The dual space of \({\mathcal L}^ 1(\mu)\) for a vector measure \(\mu\)
zbMATH Open document ID: 0804.46049
author: S. Okada
published in: Journal of Mathematical Analysis and Applications
publication date: 1994-01-10T00:00:00Z
review text: Let \(X\) be a Banach space and let \(\mu\) be an \(X\)-valued measure defined on a \(\sigma\)-algebra \(\Sigma\). The space \({\mathcal L}^ 1(\mu)\) of scalar- valued, \(\mu\)-integrable functions and its topological dual \({\mathcal L}^ 1(\mu)'\) are investigated. The author corrects an earlier characterization of \({\mathcal L}^ 1(\mu)'\) due to \textit{L. Egghe} [Rev. Roum. Math. Pures Appl. 29, 467-471 (1984; Zbl 0555.46015)]. \({\mathcal L}^ 1(\mu)'\) is now represented as the space of scalar measures \(\nu\) that are absolutely continuous with respect to \(\mu\) and satisfy the inclusion \({\mathcal L}^ 1(\mu)\subseteq {\mathcal L}^ 1(\nu)\). A concrete representation of each member of \({\mathcal L}^ 1(\mu)'\) is also given. Basically scattered vector measures with a bounded basis constant are characterized as those measues for which \({\mathcal L}^ 1(\mu)'\) consists of \(\{x'\mu: x'\in X'\}\). The weak convergence in \({\mathcal L}^ 1(\mu)\) is also investigated.
Mathematics Subject Classification ID: 46G10
Mathematics Subject Classification ID: 28B05
zbMATH DE Number: 480463
zbMATH Keywords: vector measure
zbMATH Keywords: Bartle-Dunford-Schwartz integral
zbMATH Keywords: \(X\)-valued measure
zbMATH Keywords: topological dual
zbMATH Keywords: representation
zbMATH Keywords: scattered vector measures
reviewed by: Kazimierz Musial
MaRDI profile type: Publication
full work available at URL: jmaa.1993.1279
OpenAlex ID: W1969718447
DOI: 10.1006/JMAA.1993.1279
description: scientific article; zbMATH DE number 480463