Proposition 5

Suppose that the space X is compact, and let f be a continuous mapping of X into a Hausdorff locally convex space E. The closed convex envelope of f(X) in E'** (for $\sigma(E', E')$) is equal to the set of vectors $\int f\, d\mu$ for all of the positive measures $\mu$ on X of total mass 1.

Let C be the closed convex envelope of f(X) in E'; since f(X) is compact and E' is complete, C is compact. We already know (Prop. 4) that $\int f d\mu \in C$ for every measure $\mu$ belonging to the convex set $H$ of positive measures on $X$ of total mass equal to 1. On the other hand, $H$ is convex and compact for the vague topology (§ 1, No. 9, Cor. 3 of Prop. 15) and is the closure (for this topology) of the convex set $H_0$ of positive measures of mass 1 and finite support (§ 2, No. 4, Cor. 3 of Th. 1). Now, the image of $H_0$ under the mapping $\mu \mapsto \int f d\mu$ is the convex envelope $C_0$ of $f(X)$ in ${E'}^*$. On the other hand, this mapping is continuous for the vague topology on $\mathscr{M}(X; C)$ and the topology $\sigma({E'}^*, E')$ on ${E'}^*$ since $\langle \int f d\mu, z' \rangle = \int \langle f, z' \rangle d\mu$ by definition; thus the image of $H = \overline{H_0}$ is a compact convex set containing $C_0$ and contained in $C$; since $C = \overline{C_0}$, this image is equal to $C$.