Corollary 1

With hypotheses as in Lemma 4, suppose in addition that there exists a compact subset $K_0$ of $T$ containing the supports of all of the measures $\mu \in M$. Then $\mathfrak{F}$ also converges to $\varepsilon_a$ in $\mathscr{C}'(T)$ equipped with the topology of compact convergence in $\mathscr{C}(T)$.

For, the restriction mapping of $\mathscr{C}(T)$ into $\mathscr{C}(K_0)$ is continuous; therefore, if $H$ is a compact subset of $\mathscr{C}(T)$, then the restrictions to $K_0$ of the functions in $H$ form a compact subset of $\mathscr{C}(K_0)$. It then suffices to apply Lemma 4 on replacing $T$ by $K_0$.