On measures integrating all functions of a given vector lattice
type: Item
instance of: scholarly article
title: On measures integrating all functions of a given vector lattice
zbMATH Open document ID: 0623.28006
DOI: 10.1007/BF01364337
author: Wolfgang Adamski
published in: Monatshefte für Mathematik
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/178323
review text: Let E be a vector lattice of real-valued functions defined on a set X, and let \({\mathcal M}(E)\) be the family of all (non-negative, countably additive) measures \(\mu\) on \(\sigma\) (E) such that \(E\subset {\mathcal L}_ 1(X,\sigma (E),\mu).\) Here \(\sigma\) (E) denotes the smallest \(\sigma\)- algebra in X making all functions in E measurable. This paper is concerned with the statement that every \(\mu \in {\mathcal M}(E)\) is \(\tau\)- smooth with respect to the lattice \({\mathcal H}(E):=\{\{f\geq 1\}: f\in E\},\) i.e. \(\lim_{\alpha}\mu (H_{\alpha})=0\) for each net \((H_{\alpha})\) in \({\mathcal H}(E)\) with \(H_{\alpha}\downarrow \emptyset.\) Under some additional assumptions on E, this statement can be described by smoothness properties of the nonnegative linear functionals on E as well as by closure properties of \({\mathcal H}(E)\)-ultrafilters. An application of this result to various topological situations leads to some new measure- theoretic characterizations of realcompact, Borel-complete, and N-compact spaces, respectively.
Mathematics Subject Classification ID: 28C05
Mathematics Subject Classification ID: 46E05
Mathematics Subject Classification ID: 54D60
zbMATH DE Number: 4010837
zbMATH Keywords: realcompact space
zbMATH Keywords: Borel-complete space
zbMATH Keywords: N-compact space
zbMATH Keywords: vector lattice of real-valued functions
zbMATH Keywords: \(\tau \)-smooth
zbMATH Keywords: ultrafilters
zbMATH Keywords: measure-theoretic characterizations
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4010837