Applications of a Theorem of Grothendieck to Vector Measures

Let R be a ring of subsets of a nonempty set V and S R. the Banach space of uniform limits of sequences of R-simple functions in V. Let X be a quasicom- plete locally convex Hausdorff space briefly, lcHs.. Given a bounded X-valued vector measure m on R, the concepts of m-integrability of functions in S R. and of representing measure of a continuous linear mapping u : S R.ªX are introduced. Based on these concepts and a theorem of Grothendieck on the range of the biadjoint uUU of ugL S R., X., it is shown that such a mapping u is weakly compact if and only if its representing measure is strongly additive. The result subsumes the range theorems of I. Tweddle Glasgow Math. J. 9, 1968, 123]127. and I. Kluv´anek Math. Systems Theory 7, 1973, 44]54.. Also the theorem on extension is deduced. The method of proof for all these results in vector measures is more natural than the known ones.