Corollary

Let $\mu$ be a measure on $X$. For a measure $\nu$ on $X$ to be in the vague closure of the set of measures $g \cdot \mu$, where $g$ runs over $\mathscr{K}(X; \mathbf{C})$, it is necessary and sufficient that $\operatorname{Supp}(\nu) \subset \operatorname{Supp}(\mu)$.

For, $\operatorname{Supp}(g \cdot \mu) \subset \operatorname{Supp}(\mu)$ by No. 3, Prop. 10; therefore the support of every vague limit of measures of the form $g \cdot \mu$ is also contained in $\operatorname{Supp}(\mu)$ (No. 2, Prop. 6). Conversely, if $\operatorname{Supp}(\nu) \subset \operatorname{Supp}(\mu)$ then $\nu$ is the vague limit of measures with finite support contained in $\operatorname{Supp}(\mu)$ (Th. 1), hence is in the vague closure of the set of measures $g \cdot \mu$ by Prop. 13.