The bounded vector measure associated to a conical measure and Pettis differentiability (Q2726647)

The authors consider finitely additive bounded vector measures with values in a locally convex space \(X\). It is proved that every conical measure \(u\) on \(X\), whose associated zonoform \(K_u\) is contained in \(X\), is associated to a bounded additive vector measure \(\sigma (u)\) defined on \(X\), and satisfying \(\sigma (u)(H)\in H\), for every finite intersection \(H\) of closed half-spaces. When \(X\) is a complete weak space, \(\sigma (u)\) is countably additive. When \(X\) is a Banach space, it is also proved that \(\sigma (u)\) is countably additive if and only if \(u\) is the connical measure associated to a Pettis differentiable vector measure.