Corollary 2

— Every point of C that belongs to a cap of C, also belongs to the convex closed envelope of the union of the extremal generators of C.

This follows immediately from cor. 1 and the Krein-Milman theorem (II, p. 55, th. 1).

\* Example. — Let X be a locally compact space that is σ-compact. Let C be a closed convex cone of vertex 0 in $\mathscr{M}_+(X)$ with the vague topology. We shall show that C is the union of its caps. Let $(X_n)$ be an increasing sequence of open, relatively compact sets of X whose union is X. Let μ be an element $\neq 0$ of C. There exist $\alpha_n > 0$ such that $\sum_n \alpha_n \mu(X_n) = 1$.

For every measure $v \in C$, put $p(v) = \sum_n \alpha_n v(X_n) \in [0, +\infty]$. The function p on C satisfies conditions (i) and (ii) of prop. 4. It is lower semi-continuous for the vague topology (INT, IV, 2nd ed., § 1, No. 1, prop. 4). The set A of the $\gamma \in C$ such that $p(\gamma) \leq 1$ is therefore closed and non-empty. On the other hand, every compact set of X is contained in one of the $X_n$, thus A being vaguely bounded is also vaguely compact (INT, III, 2nd ed., § 1, No. 9, prop. 15). The set A is therefore a cap of C containing μ. \*