Remarks on \(\mu''\)-measurable sets: regularity, \(\sigma\)-smoothness, and measurability
type: Item
instance of: scholarly article
title: Remarks on \(\mu''\)-measurable sets: regularity, \(\sigma\)-smoothness, and measurability
zbMATH Open document ID: 0936.28008
DOI: 10.1155/S0161171299223915
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2000-05-25T00:00:00Z
full work available at URL: https://eudml.org/doc/48399
review text: Summary: Let \({\mathbf X}\) be an arbitrary nonempty set and \({\mathcal L}\) a lattice of subsets of \({\mathbf X}\) such that \(\phi, {\mathbf X}\in{\mathcal L}\). \({\mathcal A}({\mathcal L})\) is the algebra generated by \({\mathcal L}\) and \({\mathcal M}({\mathcal L})\) denotes those nonnegative, finite, finitely additive measures \(\mu\) on \({\mathcal A}({\mathcal L})\). \(I({\mathcal L})\) denotes the subset of \({\mathcal M}({\mathcal L})\) of nontrivial zero-one valued measures. Associated with \(\mu\in I({\mathcal L})\) (or \(I_\sigma({\mathcal L})\)) are the outer measures \(\mu'\) and \(\mu''\) considered in detail. In addition, measurability conditions and regularity conditions are investigated and specific characteristics are given for \({\mathcal S}_{\mu''}\), the set of \(\mu''\)-measurable sets. Notions of strongly \(\sigma\)-smooth and vaguely regular measures are also discussed. Relationships between regularity, \(\sigma\)-smoothness and measurability are investigated for zero-one valued measures and certain results are extended to the case of a pair of lattices \({\mathcal L}_1\), \({\mathcal L}_2\) where \({\mathcal L}_1\subset{\mathcal L}_2\).
Mathematics Subject Classification ID: 28C15
Mathematics Subject Classification ID: 28A12
zbMATH DE Number: 1334840
zbMATH Keywords: outer measure
zbMATH Keywords: strongly \(\sigma\)-smooth and vaguely regular measures
zbMATH Keywords: zero-one valued measures
author: Carmen Vlad
MaRDI profile type: Publication
OpenAlex ID: W2095440625
description: scientific article; zbMATH DE number 1334840