On compactness and extreme points of some sets of quasi-measures and measures. II
type: Item
instance of: scholarly article
title: On compactness and extreme points of some sets of quasi-measures and measures. II
zbMATH Open document ID: 0847.28001
DOI: 10.1007/BF02567525
published in: Manuscripta Mathematica
publication date: 1996-09-29T00:00:00Z
full work available at URL: https://eudml.org/doc/156170
review text: Miscellaneous properties of the extreme points of the convex set \(S(\mu)\) consisting of all quasi-measure extensions of a quasi-measure \(\mu\), i.e., a non-negative, additive set function on an algebra of subsets of a set to a larger algebra of subsets of the same underlying set, like compactness, denseness in \(S(\mu)\), local compactness, discreteness, zero-dimensionality, connectedness, pathwise connectedness with respect to the weak\(^*\), weak and strong topologies of \(S(\mu)\) are studied. Analogous results are derived for the extreme points of the convex set \(S_\sigma(\mu)\) consisting of all measure extensions of a measure \(\mu\), i.e., a non-negative, \(\sigma\)-additive set function on a \(\sigma\)-algebra of subsets of a set, to a larger \(\sigma\)-algebra of subsets of the same underlying set. In both cases, some results established by the author in Part I of the paper [Manuscr. Math. 86, No. 3, 349-365 (1995; Zbl 0826.28004)] which have the form of an implication are now improved so that they take the form of an equivalence. This is achieved with the help of the antimonogenic component of \(\mu\).
Mathematics Subject Classification ID: 28A12
Mathematics Subject Classification ID: 28A10
Mathematics Subject Classification ID: 28A33
zbMATH DE Number: 871042
zbMATH Keywords: extreme points
zbMATH Keywords: quasi-measure extensions
zbMATH Keywords: compactness
zbMATH Keywords: denseness
zbMATH Keywords: discreteness
zbMATH Keywords: zero-dimensionality
zbMATH Keywords: connectedness
author: Zbigniew Lipecki
MaRDI profile type: Publication
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cites work: Sigma-additivity of quasi-measure extensions of a measure
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cites work: On compactness and extreme points of some sets of quasi-measures and measures
cites work: Quasi-measures with finitely or countably many extreme extensions
cites work: On Monogenic Operators and Measures
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cites work: Der Wertebereich atomloser Inhalte
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OpenAlex ID: W4237778426
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description: scientific article; zbMATH DE number 871042