Quasi-Radon measures
type: Item
instance of: scholarly article
title: Quasi-Radon measures
zbMATH Open document ID: 0772.28004
DOI: 10.1007/BF00971179
published in: Siberian Mathematical Journal
publication date: 1992-09-27T00:00:00Z
review text: Let \(X\) be a complete regular topological space, \(\mathcal K\) (resp. \(\Sigma\)) the family of closed (resp. compact) subsets of \(X\), \(M(X)\) (resp. \(C_ b(X)\)) the space of Borel (resp. bounded continuous) real valued functions on \(X\), \(Y\) a lattice-normed space by means of the Riesz space \(F\) [ cf. \textit{L. V. Kantorovich}, \textit{B. Z. Vulich} and \textit{A. G. Pinsker}, ``Functional analysis in partially ordered spaces'' (1950; Zbl 0037.072)] with generalized norm \(| y|\in F\), whenever \(y\in Y\). By \(F-ba({\mathcal A},Y)\) (resp. \(F-bca({\mathcal A},Y))\) it is denoted the set of all finitely additive (resp. \(\sigma\)-additive) measures \(\mu: {\mathcal A}\to Y\) with finite \(F\)-variation \(|\mu|\) on a given algebra \(\mathcal A\) of subsets of \(X\). A finitely additive measure \(\mu: {\mathcal A}\to Y\) is called Radon (resp. quasi-Radon) measure on \(\mathcal A\) if and only if \(\mu(E)=o-\lim\{\mu(K): K\in\Sigma_ E\}\), for every \(E\in{\mathcal A}\) (resp. \(E\in\Sigma\cap\mathcal A)\) with \(\Sigma_ E:=\{K: K\in\Sigma\cap {\mathcal A}, K\subseteq E\}\). Further \(\mu\) is called regular (resp. quasi-regular) measure if \(\mu(E)=o-\lim\{\mu(F): F\in K_ E\}\) for every \(E\in{\mathcal A}\) (resp. \(E\in {\mathcal K}\cap{\mathcal A})\) with \(K_ E:=\{F: F\in{\mathcal K}\cap {\mathcal A}, F\subseteq E\}\). The author showed that if \(\mu\in F-ba({\mathcal A},Y)\) and either (i) \({\mathcal A}=\alpha({\mathcal K}\cap{\mathcal A})\) (the algebra generated by the class \({\mathcal K}\cap{\mathcal A})\) or (ii) \({\mathcal A}=\sigma({\mathcal K}\cap{\mathcal A})\) (the \(\sigma\)-algebra generated by the class \({\mathcal K}\cap{\mathcal A})\) and \(\mu\in F-bca({\mathcal A},Y)\), then \(\mu\) is a quasi-regular (resp. quasi-Radon) measure if and only if \(|\mu|\) is a quasi-regular (resp. quasi-Radon) measure. Moreover, the bounded quasi-Radon measure \(\mu\) on the algebra \(\alpha({\mathcal K}\cap{\mathcal A})\) is \(\sigma\)-additive …
Mathematics Subject Classification ID: 28B15
zbMATH DE Number: 56736
zbMATH Keywords: quasi-Radon measures
zbMATH Keywords: quasi-regular measures
zbMATH Keywords: integral representation theorem
zbMATH Keywords: extension
zbMATH Keywords: moment problem
zbMATH Keywords: integrals
zbMATH Keywords: partially ordered vector spaces
author: Sergey Artem'evich Malyugin
reviewed by: Panaiotis K. Pavlakos
MaRDI profile type: Publication
cites work: Q4052400
cites work: Stone-Algebra-Valued Measures and Integrals
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cites work: Q3712524
cites work: Lattice-Valued Borel Measures. II
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cites work: Q4725223
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description: scientific article; zbMATH DE number 56736