Radon--Nikodým derivatives for vector measures belonging to Köthe function spaces
type: Item
instance of: scholarly article
title: Radon--Nikodým derivatives for vector measures belonging to Köthe function spaces
zbMATH Open document ID: 1160.46027
author: E. A. Sánchez-Pérez
author: J. M. Calabuig
author: Pablo Gregori
published in: Journal of Mathematical Analysis and Applications
publication date: 2008-10-06T00:00:00Z
review text: Let \(\mu\) and \(\nu\) be countably additive vector measures and let \(\int f\; d\nu\) denote the Bartle--Dunford--Schwartz integral of \(f\). A general problem can be formulated informally as follows. Is it possible to obtain a characterization of when the Radon--Nikodým derivative of \(\mu\) with respect to \(\nu\) exists in some given particular sublattice of \(L^1(\nu)\)? Some theorems of this type are known; as an important example, \textit{K.\,Musiał} [Atti Sem.\ Mat.\ Fis.\ Univ.\ Modena 41, No.\,1, 227--233 (1993; Zbl 0794.46036)] showed that the Radon--Nikodým derivative of \(\mu\) with respect to \(\nu\) exists and belongs to \(L^\infty(\nu)\) if and only if \(\mu\) is scalarly dominated by \(\nu\). The main theorem of the paper under review goes as follows. Let \((\Omega,\Sigma,\lambda)\) be a finite measure space and assume \(\mu\) and \(\nu\) are absolutely continuous with respect to \(\lambda\). Let \(Z(\lambda)\) be a \(\sigma\)-order continuous Köthe function space over \((\Omega,\Sigma,\lambda)\) and assume \(Z(\lambda)^\times\) is continuously included in \(L^1(\mu)\). Denote by \(RN(\mu,\lambda)\) the set \(\{\frac{dx'\mu}{d\lambda}\::x'\in X'\}\); this can be seen to be a subset of \((Z(\lambda)^\times)'\). Now assume \(RN(\mu,\lambda)\) actually sits in \(Z(\lambda)\). In this situation two necessary and sufficient conditions (of domination type) are found in order that the Radon--Nikodým derivative of \(\mu\) with respect to \(\nu\) exists in \(Z(\lambda)^\times\).
reviewed by: Olav Nygaard
Mathematics Subject Classification ID: 46G10
Mathematics Subject Classification ID: 28B05
zbMATH DE Number: 5349107
zbMATH Keywords: Radon-Nikodým derivative
zbMATH Keywords: Bartle-Dunford-Schwartz integral
zbMATH Keywords: vector measures
zbMATH Keywords: Köthe function spaces
MaRDI profile type: Publication
full work available at URL: j.jmaa.2008.07.024
OpenAlex ID: W2041390006
cites work: Vector-valued harmonic functions and cone absolutely summing operators
cites work: LORENTZ SPACES OF VECTOR-VALUED MEASURES
cites work: Q4475044
cites work: Q4148368
cites work: Representation theorems on Banach function spaces
cites work: Bounded Linear Operators on Banach Function Spaces of Vector-Valued Functions
cites work: On L(p,q) spaces
cites work: On Orlicz sequence spaces. II
cites work: Q3229647
cites work: Q5535810
cites work: Q5524733
cites work: Q4278311
cites work: Optimal domain and integral extension of operators, acting in function spaces
cites work: Q5760572
cites work: Orlicz spaces of finitely additive set functions, linear operators, and martingales
cites work: Orlicz spaces of finitely additive set functions
DOI: 10.1016/J.JMAA.2008.07.024
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description: scientific article; zbMATH DE number 5349107