Submeasures and locally solid topologies on Riesz spaces
type: Item
instance of: scholarly article
title: Submeasures and locally solid topologies on Riesz spaces
zbMATH Open document ID: 0601.46006
DOI: 10.1007/BF01166456
author: Iwo Labuda
published in: Mathematische Zeitschrift
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/183685
review text: The main tool in this paper is a general representation theorem, presented in Section 2, which gives necessary and sufficient conditions in order that an Archimedean Riesz space L admits a suitable space \(L^ 0\) of measurable functions as its universal completion \(L^ u\). As no assumptions concerning the existence of order-continuous functionals on L are imposed, direct sums of (finite normal) submeasure spaces appear. Our main objective is to study, by applying the representation result, minimal Hausdorff locally solid topologies on Riesz spaces. This is done in the second part of the paper, improving upon earlier results by Fremlin, Aliprantis and Burkinshaw. In particular, it is shown that a Riesz space L admits a minimal Hausdorff locally solid topology \(\lambda\) if and only if L admits the \(L^ 0\) obtained in the representation theorem as its universal completion and (modulo the embedding) \(\lambda\) is the restriction of the canonical topology of ''convergence in submeasure'' on this \(L^ 0\).
Mathematics Subject Classification ID: 46A40
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 06F20
zbMATH DE Number: 3969356
zbMATH Keywords: representation theorem
zbMATH Keywords: Archimedean Riesz space
zbMATH Keywords: universal completion
zbMATH Keywords: submeasure spaces
zbMATH Keywords: minimal Hausdorff locally solid topologies on Riesz spaces
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3969356