Corollary 2

— Every measure with compact support (resp. every positive measure with compact support) $\mu$ is in the closure in $\mathscr{C}'(X; \mathbf{C})$, for the topology $\mathscr{T}$ of compact convergence, of the set of measures (resp.

(*) In fact, the mapping $x \mapsto \varepsilon_x$ is a homeomorphism of X into $\mathscr{C}'(X; \mathbf{C})$ (Ch. VI, §1, No. 6, Remark 1).

positive measures) whose support is finite and contained in Supp(\mu) and whose norm is equal to $\| \mu \|$.

For, on the set B of measures $\nu$ such that $\mathrm{Supp}(\nu) \subset \mathrm{Supp}(\mu)$ and $\| \nu \| \leq \| \mu \|$, the topology induced by the vague topology is identical to the topology induced by $\mathscr{T}$, and the corollary therefore follows from Ch. III, §2, No. 4, Cors. 2 and 3 of Th. 1.