Geometry and integration for operator valued measures
type: Item
instance of: scholarly article
title: Geometry and integration for operator valued measures
zbMATH Open document ID: 0755.28004
DOI: 10.1016/0022-247X(92)90068-O
author: M. Ballvé
published in: Journal of Mathematical Analysis and Applications
publication date: 1992-09-27T00:00:00Z
review text: The average range of a measure, valued in a Banach space \(E\) with respect to an operator-valued measure \(m\) with values in \(L(E,E)\), the \(m\)-convex hull of a subset of \(E\) and the core with respect to \(m\) of an \(E\)-valued function are defined and studied assuming that \(m\) verifies the Price's axiom (some of the results hold without this assumption). Several interesting relations are studied between the average range of a measure defined by an \(m\)-integrable \(E\)-valued function \(f\), the core and the essential range of \(f\), and also between their closed \(m\)-convex hulls. As an application, a mean-value theorem and a criterion of measurability are stated for the Dobrakov integral. Also a Radon-Nikodým theorem about derivation of \(E\)-valued measures with respect to measures valued in \(L(E,E)\) is proved for the Dobrakov integral by means of the generalized concept of the average range of a measure (instead of the usual techniques of localization and control).
Mathematics Subject Classification ID: 28B05
Mathematics Subject Classification ID: 46G10
zbMATH DE Number: 62063
zbMATH Keywords: Radon-Nikodým derivative
zbMATH Keywords: Banach space
zbMATH Keywords: operator-valued measure
zbMATH Keywords: core
zbMATH Keywords: average range
zbMATH Keywords: essential range
zbMATH Keywords: mean-value theorem
zbMATH Keywords: Dobrakov integral
zbMATH Keywords: Radon-Nikodým theorem
reviewed by: P. Jiménez Guerra
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 62063