Bounded common extensions of vector measures

Abstract

Let \(\mathcal{A}\) and \(\mathcal{B}\) be fields of subsets of a set \({\Omega}\), let \({\bf X}\) be a normed space with the Hahn-Banach extension property and let \({\mu: {\mathcal A}\rightarrow {\bf X}}\) and \({\nu: {\mathcal B} \rightarrow {\bf X}}\) be consistent, bounded, vector measures. We give necessary and sufficient conditions for \({\mu}\) and \({\nu}\) to have a bounded common extension to \({{\mathcal A} \vee {\mathcal B}}\), generalizing already known results for real valued charges.