A survey on recent advances on the Nikodým boundedness theorem and spaces of simple functions
type: Item
instance of: scholarly article
title: A survey on recent advances on the Nikodým boundedness theorem and spaces of simple functions
zbMATH Open document ID: 1061.46002
published in: Rocky Mountain Journal of Mathematics
publication date: 2004-09-27T00:00:00Z
review text: This clear and informative survey gives the main results on barrelledness properties of the space of simple functions, \(S(A)\), on an algebra \(A\) with ``strong enough'' properties, e.g., a \(\sigma\)-algebra. Also vector valued simple functions are covered. A fundamental observation, which may be taken as a starting point, is that of Schachermayer saying that \(S(A)\) is barrelled if and only if \(A\) is such that the Nikodým boundedness theorem holds in \(ba(A)\), the dual of \(S(A)\). A lot of such beautiful theorems exist giving connections between some ``bigness''-property of \(A\) and the barrelledness properties of \(S(A)\), in particular, attention is paid on the case when \(A\) is a \(\sigma\)-algebra. Being very central in the development of this theory, the authors present to the reader the main ideas and techniques used to prove these connections, thus also providing a very useful text for students entering the area.
reviewed by: Olav Nygaard
Mathematics Subject Classification ID: 46A08
Mathematics Subject Classification ID: 46E40
Mathematics Subject Classification ID: 46E27
Mathematics Subject Classification ID: 46G10
Mathematics Subject Classification ID: 28A33
Mathematics Subject Classification ID: 28B05
Mathematics Subject Classification ID: 46-02
zbMATH DE Number: 2103620
zbMATH Keywords: space of simple functions
zbMATH Keywords: barrelledness properties
zbMATH Keywords: Nikodým boundedness theorem
author: Juan Carlos Ferrando
author: L. M. Sánchez Ruiz
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181069896
OpenAlex ID: W2038197265
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A survey on recent advances on the Nikodým boundedness theorem and spaces of simple functions (DOI)
DOI: 10.1216/RMJM/1181069896