Upper envelopes of inner premeasures
type: Item
instance of: scholarly article
title: Upper envelopes of inner premeasures
zbMATH Open document ID: 0962.28001
publication date: 2000-05-17T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/75423
review text: Let \({\mathfrak S}\) and \({\mathfrak T}\) be lattices of subsets of a nonempty set \(X\) both containing \(\emptyset\). Let, further, \(\beta:{\mathfrak S}\to [0, \infty[\) vanish at \(\emptyset\), be isotone and submodular (= strongly subadditive), and have the following regularity property: given \(A\in{\mathfrak S}\) and \(\varepsilon> 0\) there exists \(U\in{\mathfrak T}\) such that \(A\subset U\) and \(\beta(K)< \beta(A)+ \varepsilon\) whenever \(K\in{\mathfrak S}\) and \(K\subset U\). According to a classical theorem, in the special case where \({\mathfrak S}\) and \({\mathfrak T}\) are the families of compact and open subsets of a Hausdorff topological space \(X\), respectively, for every \(B\in{\mathfrak S}\) there exists a Radon premeasure \(\varphi:{\mathfrak S}\to [0,\infty[\) with \(\varphi\leq \beta\) and \(\varphi(B)= \beta(B)\). If, in addition, \(\sup\beta< \infty\), then there exists a Radon premeasure \(\psi:{\mathfrak S}\to [0,\infty[\) with \(\psi\leq\beta\) and \(\sup\psi= \sup\beta\). This theorem, originally formulated by G. Choquet (1953-54), has been studied by many authors, with the first complete proof given by F. Topsøe (1978). An abstract, non-topological generalization of the theorem was then found by W. Adamski (1977), who was also able to replace the single set \(B\) above by some systems of subsets of \(X\), e.g., increasing sequences of elements of \({\mathfrak S}\). Adamski's list of postulates on \({\mathfrak S}\), \({\mathfrak T}\) and \(\beta\) is, however, quite long and includes the stability of \({\mathfrak S}\) under countable intersections and the \(\sigma\)-continuity of \(\beta\) at \(\emptyset\). The present paper offers a simpler and (essentially) more general setting for the theorem under discussion. Namely, it merely assumes that each pair of disjoint elements of \({\mathfrak S}\) can be separated by a pair of disjoint elements of \({\mathfrak T}\) and for all \(U\in{\mathfrak T}\) and \(K\in{\mathfrak S}\) we have …
Mathematics Subject Classification ID: 28A12
Mathematics Subject Classification ID: 28C05
Mathematics Subject Classification ID: 28C15
zbMATH DE Number: 1448493
zbMATH Keywords: submodular isotone set functions
zbMATH Keywords: inner premeasures
zbMATH Keywords: supportive properties
zbMATH Keywords: lattice of sets
zbMATH Keywords: supermodular
zbMATH Keywords: upper envelope
zbMATH Keywords: tight
zbMATH Keywords: Radon premeasure
author: Heinz König
reviewed by: Zbigniew Lipecki
MaRDI profile type: Publication
OpenAlex ID: W4248510623
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DOI: 10.5802/AIF.1759
published in: Annales de l’institut Fourier
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description: scientific article; zbMATH DE number 1448493