Applications of a Theorem of Grothendieck to Vector Measures

Abst rac t Let R be a ring of subsets of a nonempty set R and C(R) the Banach space of uniform limits of sequences of R-simple functions in 0. Let X be a quasicomplete locally convex Hausdorff space (briefly, IcHs). Given a bounded X-valued vector measure m on R, the concepts of m-integrability of functions in C(R) and of representing measure of a continuous linear mapping u: C(R) + X are introduced. Based on these concepts and a theorem of Grothendieck on the range of the biadjoint u " of u E C(C(R), X) , it is shown that such a mapping u is weakly compact if and only if its representing measure is strongly additive (which is the quasicomplete IcHs version of Theorem VI.l. l of [3]). The result subsumes the range theorems of Tweddle [12] and Kluvinek [9]. Also is deduced the theorem on extension in [lo]. The methods of proof for all these results in vector measures is more natural than the known ones. Dedicated to the memory of Professor I. Kluv6nek Let X be a quasicomplete locally convex Hausdorff space (briefly, a quasicomplete 1cHs). Using,lames ' criterion for weak compactness of a set, Tweddle [12] showed that the closed convex hull of the range of a a-additive X-valued vector measure defined on a a-ring of sets is weakly compact. His proof is first given for the case of a a-algebra, and then, by appealing t o the Eberlein theore