Proposition 9

Let $V$ be a linear subspace of $\mathscr{K}(X; \mathbf{R})$ having the following property: (P) For every compact subset $K$ of $X$, there exists a function $f \in V$ such that $f \geq 0$ and such that $f(x) > 0$ for all $x \in K$. Under these conditions, every positive linear form on $V$ for the ordering induced by that of $\mathscr{K}(X; \mathbf{R})$ (Ch. II, §2, No. 1, Def. 1) may be extended to a positive measure on $X$ (which is unique when $V$ is dense in $\mathscr{K}(X; \mathbf{R})$). For every function $f \in \mathscr{K}(X; \mathbf{R})$, with support $K$, there exists a function $g \in V$ such that $f \leq g$; for, there exists a function $h \geq 0$ in $V$ such that $h(x) > 0$ for all $x \in K$; setting $\alpha = \inf_{x \in K} h(x)$, we thus have $\alpha > 0$ and the function $g = (\alpha^{-1} \|f\|)h$ meets the requirements. It then suffices to apply Th. 1 of No. 5 and Prop. 1 of TVS, II, §3, No. 1.