Completeness of \(L_p\)-spaces in the finitely additive setting and related stories
type: Item
instance of: scholarly article
title: Completeness of \(L_p\)-spaces in the finitely additive setting and related stories
zbMATH Open document ID: 0976.46017
author: Achille Basile
author: K. P. S. Bhaskara Rao
published in: Journal of Mathematical Analysis and Applications
publication date: 2001-02-28T00:00:00Z
review text: There are obtained necessary and sufficient conditions for completeness of spaces \(L_p(\Omega,{\mathcal F},\mu)\), \(0\leq p<\infty\), where \(\mu\) is a charge (i.e., finitely additive measure over the field \({\mathcal F}\) of subsets of \(\Omega\)), and convergence almost everywhere is replaces by convergence \(f_n\to f\) hazily, defined by the condition \(\mu^*(\{x:|f_n(x)- f(x)|> \varepsilon\})\to 0\) as \(n\to\infty\) for every \(\varepsilon> 0\), \(\mu^*\) being the ``outer measure'' defined by means of \(\mu\). Among else it is proved that completeness of \(L_0(\Omega,{\mathcal F},\mu)\) is equivalent to completeness of \(L_p(\Omega,{\mathcal F},\mu)\) for any \(p\in [0,\infty)\); there are given several necessary and sufficient conditions for this completeness, also by means of the Stone space of \({\mathcal F}/\mu\). Moreover, there are obtained necessary and sufficient conditions in order that the space \({\mathcal F}/\mu\) with Radon-Nikodým metric \(d_\mu\) be complete. Further, investigations are derived to completeness of \(L_1(\Omega,{\mathcal F},\mu)\).
reviewed by: Julian Musielak
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 28A33
Mathematics Subject Classification ID: 46A30
Mathematics Subject Classification ID: 46G10
zbMATH DE Number: 1523557
zbMATH Keywords: convergence hazily
zbMATH Keywords: Hahn decomposition
zbMATH Keywords: Radon-Nikodým derivative
zbMATH Keywords: completeness
zbMATH Keywords: charge
zbMATH Keywords: finitely additive measure
zbMATH Keywords: Stone space
zbMATH Keywords: Radon-Nikodým metric
MaRDI profile type: Publication
full work available at URL: jmaa.2000.6946
OpenAlex ID: W2070541788
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DOI: 10.1006/JMAA.2000.6946
description: scientific article; zbMATH DE number 1523557