Proposition 14

In the space $\mathscr{M}(X; \mathbf{C})$ of measures on a locally compact space X, the cone $\mathscr{M}_+(X)$ of positive measures is complete for the uniform structure deduced from the vague topology (hence is vaguely closed in $\mathscr{M}(X; \mathbf{C})$).

For, consider a Cauchy filter $\Phi$ for the vague uniform structure on $\mathscr{M}_+(X)$; by definition, $\mu_0(f) = \lim_{\mu, \Phi} \mu(f)$ exists for every function $f \in \mathscr{K}(X; \mathbf{C})$ and, by the principle of extension of inequalities, $\mu_0(f) \geq 0$ for every function $f \in \mathscr{K}_+(X)$; it follows that $\mu_0$ is a positive measure on X (No. 5, Th. 1).

It should be noted that the space $\mathscr{M}(X; \mathbf{C})$ (or $\mathscr{M}(X; \mathbf{R})$) itself is not necessarily complete for the vague uniform structure (TVS, II, §6, No. 7).