Radon--Nikodým derivatives for vector measures belonging to Köthe function spaces
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 \(Z(\lambda)^\times\).
scientific article; zbMATH DE number 5349107
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