Proof: It is easy to prove that $[\omega]$ induces a continuous linear functional on $C_{c}^{\infty}(W)$ where this locally convex vector space is endowed with the subspace topology of $C_{c}(W)$. By 5.8, this subspace is dense and hence the Riesz representation theorem proves the first claim. If $\omega$ has compact support, then ${\rm supp}(\mu)$ is also compact and the last claim follows. $\square$