Quasi-measures with finitely or countably many extreme extensions
type: Item
instance of: scholarly article
title: Quasi-measures with finitely or countably many extreme extensions
zbMATH Open document ID: 0918.28003
published in: Manuscripta Mathematica
publication date: 1999-08-03T00:00:00Z
review text: Let \(E(\mu)\) denote the convex set consisting of all quasi-measure extensions of a quasi-measure \(\mu\), i.e., a positive additive function on an algebra \({\mathcal M}\) of subsets of a set \(\Omega\), to a larger algebra \({\mathcal R}\) of subsets of \(\Omega\). It is shown that the set \(\text{extr }E(\mu)\) consisting of the extreme points of \(E(\mu)\) is finite if and only if \(E(\mu)\) is finite-dimensional and that \(E(\mu)\) is finite-dimensional if and only if \(E(\mu)\) is affinely isomorphic to a finite product of finite-dimensional simplices. Moreover, it is proved in the case where \({\mathcal M}\) and \({\mathcal R}\) are \(\sigma\)-algebras and \(\mu\) is a measure that \(\text{extr }E(\mu)\) is either finite or has cardinality not smaller than the cardinality of \(\mathbb{R}\). In particular, \(E(\mu)\) consists only of measures in the case where \(\text{extr }E(\mu)\) is finite and the property of \(\mu\) that \(E(\mu)\) consists only of measures together with the property of \(\mu\) to have finite range implies that \(\text{extr }E(\mu)\) is finite.
Mathematics Subject Classification ID: 28A12
Mathematics Subject Classification ID: 46A55
Mathematics Subject Classification ID: 52A20
zbMATH DE Number: 1239591
zbMATH Keywords: quasi-measure extension
zbMATH Keywords: extreme quasi-measure extension
author: Zbigniew Lipecki
MaRDI profile type: Publication
full work available at URL: s002290050115
OpenAlex ID: W1987918555
DOI: 10.1007/S002290050115
description: scientific article; zbMATH DE number 1239591