CONICAL MEASURES AND CLOSED VECTOR MEASURES

Let X be a locally convex Hausdorff space with topological dual X* and m be a (sigma-additive) X-valued vector measure defined on a sigma-algebra. The completeness of the associated L-1-space of m is determined by the closedness of m, a concept introduced by I. Kluvanek in the early 1970's. He characterized the closedness of m via the existence of a certain kind of localizable, [0, infinity]- valued measure iota such that every scalar measure < m, x*> : E bar right arrow < m(E),x*>, for x* is an element of X*, satisfies < m, x*> << iota. The construction iota relies on the theory of conical measures. Unfortunately, in this generality the characterization is invalid; a counterexample is exhibited. However, by restricting iota to the class of Maharam measures and strengthening the requirement of absolute continuity to the condition that every < m, x*>, for x* is an element of X*, is truly continuous with respect iota (a notion investigated by D. Fremlin in connection with the Radon Nikodym Theorem), it is shown that an adequate characterization of the closedness of m is indeed available.