Corollary

*Let E be a Hausdorff locally convex space over $\mathbf{R}$, C a proper convex cone in E with vertex 0, that is complete and metrizable for the uniform structure induced by the weakened uniform structure of E. Let M be the union of the extremal generators of C. For every $\mathbf{x} \in C$ there exist a compact convex subset K of C and a measure $\lambda \geq 0$ on K of total mass 1, such that $K - (M \cap K)$ is $\lambda$-negligible and the barycenter of $\lambda$ is equal to $\mathbf{x}$.

For, $\mathbf{x}$ belongs to a cap K of C (TVS, II, §7, No. 2, Prop. 5), and $M \cap K$ contains the set of extremal points of K (loc. cit., Cor. 1 of Prop. 4). It then suffices to apply Th. 1.