Corollary 4

In the space $\mathscr{M}(X; \mathbf{C})$, the set of point measures is total for the topology of strictly compact convergence (§1, No. 10).

On the cone $\mathscr{M}_+(X)$, the topology of strictly compact convergence is identical to the vague topology (§1, No. 10, Prop. 18), and every measure on $X$ may be written $\mu_1 - \mu_2 + i \mu_3 - i \mu_4$, where the $\mu_j$ ($1 \leq j \leq 4$) are positive measures; the conclusion therefore follows from Th. 1.