Corollary 3

Every bounded positive measure $\mu$ on $X$ is in the vague closure of the convex set of positive measures whose support is finite and contained in that of $\mu$ and whose norm is equal to $\| \mu \|$.

The same reasoning as in Cor. 2 shows that we can limit ourselves to proving that $\mu$ is in the vague closure of the convex set $B$ formed by the positive measures with finite support contained in $\mathrm{Supp}(\mu)$ and with norm $\leq \| \mu \|$. Again, it suffices to establish that $\mu$ belongs to the polar set of $B^\circ$, the polar set of $B$ in the space $\mathscr{K}(X; \mathbf{R})$ (TVS, II, §6, No. 3, Th. 1); but this means that for $f \in \mathscr{K}(X; \mathbf{R})$ the relations $\langle f, \varepsilon_a \rangle \geq -1 / \| \mu \|$ for all $a \in \mathrm{Supp}(\mu)$ imply $\langle f, \mu \rangle \geq -1$, which is a consequence of No. 3, Cor. 2 of Prop. 8.