Conical measures and closed vector measures

Abstract

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 \emph{closedness} of $m$, a concept introduced by I. Kluvánek in the early 1970's. He characterized the closedness of $m$ via the existence of a certain kind of localizable, $[0,\infty]$- valued measure $\iota$ such that every scalar measure $\langle m,x^*\rangle:E\mto \langle m(E),x^*\rangle$, for $x^*\in X^*$, satisfies $\langle m,x^*\rangle<< \iota$. The construction of $\ia$ 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 \emph{Maharam measures} and strengthening the requirement of absolute continuity to the condition that every $\langle m,x^*\rangle$, for $x^*\in X^*$, is \emph{truly continuous } with respect to $\iota$ (a notion investigated by D. Fremlin in connection with the Radon Nikodým Theorem), it is shown that an adequate characterization of the closedness of $m$ is indeed available.