Corollary

Let E be a Hausdorff locally convex space, C a proper convex cone with vertex 0 in E, admitting a compact sole M, and let G be the union of the extremal generators of C. Let $x \in M$. If C is lattice-ordered, then there exists at most one positive measure $\lambda$ of mass 1 on M, such that $\lambda^*(M - (G \cap M)) = 0$, and admitting x as barycenter.

Replacing the topology of E by the weakened topology (which does not change the topology of M), one can suppose E to be a weak space. Let $\lambda$ and $\lambda'$ be two measures on M having the stated properties, and let h be a continuous linear form on E such that M is the intersection of C and the hyperplane with equation $h(x) = 1$. Let $\mathscr{S}$ be the subset of $\mathscr{C}(M)$ consisting of the restrictions to M of the positively homogeneous and continuous convex functions $\geq 0$ on C. The cone C is weakly complete (TVS, II, §7, No. 3). By Th. 3, $\lambda(f) = \lambda'(f)$ for every $f \in \mathscr{S}$.

If $f_1, f_2, f_3, f_4$ belong to $\mathscr{S}$, then

Since $h|M \in \mathscr{S}$, $\mathscr{S} - \mathscr{S}$ contains the constant functions. If x and y are two distinct points of M, there exists a continuous linear form on E that separates x and y, and this form is the difference of two continuous linear forms that are positive on C (TVS, II, §6, No. 8, Lemma 1). It follows from the foregoing that for $\alpha, \beta$ real, there exists $f \in \mathscr{S} - \mathscr{S}$ such that $f(x) = \alpha, f(y) = \beta$. Then $\mathscr{S} - \mathscr{S}$ is dense in $\mathscr{C}(M)$ for the topology of uniform convergence (GT, X, §4, No. 1, Cor. of Prop. 2). Since $\lambda$ and $\lambda'$ coincide on $\mathscr{S} - \mathscr{S}$, we have $\lambda = \lambda'$.

Exercises