The measure derivative has unbounded variation
type: Item
instance of: scholarly article
title: The measure derivative has unbounded variation
zbMATH Open document ID: 0828.46041
author: M. U. Khafizov
published in: Russian Mathematics
publication date: 1994-10-05T00:00:00Z
review text: Let \((X, \Sigma, \mu)\) be a measure space, where \(X\) is locally convex, \(\Sigma\) contains all the cylindrical sets in \(X\) and \(\mu\) is a finite measure. Let \(D(\mu)\) be the set of all vectors in whose direction \(\mu\) is differentiable. Let \(W\) be a linear subspace of \(D(\mu)\), normed by \(h\mapsto |d_n \mu|\). Define a \(W^*\)-valued vector measure \(\mu'\) by \(\langle \mu' (A), h\rangle= d_h \mu(A)\), \(A\in \Sigma\). The author shows that \(\mu'\) has bounded variation if and only if \(\dim W<\infty\). This result was known under the restriction that \(W= D(\mu)\) is a Hilbert space.
reviewed by: Luboš Pick
Mathematics Subject Classification ID: 46G05
Mathematics Subject Classification ID: 46E27
Mathematics Subject Classification ID: 46G12
zbMATH DE Number: 640208
zbMATH Keywords: measure derivative
zbMATH Keywords: bounded variation
zbMATH Keywords: cylindrical sets
zbMATH Keywords: \(W^*\)-valued vector measure
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 640208