Theorem 1

— Every measure $\mu$ on a locally compact space $X$ is in the vague closure of the vector space $V$ of measures whose support is finite and contained in $\mathrm{Supp}(\mu)$.

It suffices to prove that $\mu$ is orthogonal to the subspace $V^\circ$ of $\mathscr{K}(X; \mathbf{C})$ orthogonal to $V$ (TVS, II, §6, No. 3, Cor. 2 of Th. 1), that is, that the relations $\langle f, \varepsilon_a \rangle = 0$, where $a$ runs over the support of $\mu$, imply $\langle f, \mu \rangle = 0$; but this is just Prop. 8 of No. 3.