Radial measures of pseudo-cones

We consider $C$-pseudo-cones, that is, closed convex sets $K \subset{\mathbb R}^n$ with $o\notin K\subset C$, for which $C$ is the recession cone. Here $C$ is a given closed convex cone in ${\mathbb R}^n$, pointed and with nonempty interior. We define a class of measures for such pseudo-cones and show how they can be interpreted as derivative measures. For a subclass of these measures, namely for dual curvature measures with negative indices, we solve a Minkowski type existence problem.