The range of the integration map of a vector measure
type: Item
instance of: scholarly article
title: The range of the integration map of a vector measure
zbMATH Open document ID: 0832.28014
DOI: 10.1007/BF01195133
published in: Archiv der Mathematik
publication date: 1995-07-03T00:00:00Z
review text: Let be \(X\) a locally convex Hausdorff space, \(m\) a countably additive \(X\)-valued measure, \(X [m]\) the sequential closure of the linear span of the range \(m (\Sigma)\), \(L^1 (m)\) the space of the \(\mathbb{C}\)-valued \(m\)- integrable functions and \(I_m : L^1 (m) \to X\) the integration map. Several criteria are stated which ensure that \(I_m (L^1(m))\) is a sequentially closed subspace of \(X\) (in which case \(I_m (L^1(m)) = X[m])\). Also it is proved that if \(X\) is Fréchet, \(I_m\) is weakly compact and \(I_m (L^1(m))\) is infinite dimensional then \(I_m (L^1(m))\) is not closed.
Mathematics Subject Classification ID: 28B05
zbMATH DE Number: 769538
zbMATH Keywords: vector measure
zbMATH Keywords: integrable function
zbMATH Keywords: Fréchet space
zbMATH Keywords: locally convex Hausdorff space
zbMATH Keywords: sequentially closed
zbMATH Keywords: weakly compact
author: S. Okada
author: Werner J. Ricker
reviewed by: P. Jiménez Guerra
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 769538