Exercise 1, § 1

Let $\Gamma$ be a proper$^1$ closed convex cone in $\mathbf{R}^n$. Show that the mapping $(x, y) \mapsto x + y$ of $\Gamma \times \Gamma$ into $\Gamma$ is proper. Deduce from this that any two measures on $\Gamma$ are convolvable for the mapping $(x, y) \mapsto x + y$.