A survey on recent advances on the Nikodým boundedness theorem and spaces of simple functions
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.
scientific article; zbMATH DE number 2103620
scientific article