Application of a theorem of Grothendieck to vector measures
Let \(X\) be a quasi-complete locally convex space, \({\mathfrak R}\) a ring of subsets of a non-empty set and \(\Sigma({\mathfrak R})\) the Banach space of all bounded complex functions which are uniform limits of sequences of \({\mathfrak R}\)-simple functions. Mainly, a direct proof of the following result is given, without using the range theorem of I. Tweddle (1968) and an extension theorem of I. Kluvánek (1972): ``A continuous linear mapping \(u:\Sigma({\mathfrak R})\to X\) is weakly compact if and only if its representing measure is strongly additive''. This result subsumes the range theorems of Tweddle (1968) and Kluvánek (1972), and it also extends to quasi-complete locally convex spaces, Theorem VI.1.1 in ``Vector measures'' (1977; Zbl 0369.46039) by \textit{J. Diestel} and \textit{J. J. Uhl jun.}. Several others extensions (in different directions) are given. For instance, a generalization of the second part of Corollary VI.1.2 in the above-mentioned Diestel-Uhl's book, is also included.
scientific article; zbMATH DE number 1115503