Exercise 4, § 6

For every locally compact space T, let $\mathscr{M}^c(T)$ be the subspace of $\mathscr{M}(T)$ formed by the real measures with compact support. If T and X are two locally compact spaces, and $\pi$ is a continuous mapping of T into X, then $\pi$ is $\lambda$-proper and $\pi(\lambda) \in \mathscr{M}^c(X)$ for every measure $\lambda \in \mathscr{M}^c(T)$. Give an example showing that the mapping $\lambda \mapsto \pi(\lambda)$ of $\mathscr{M}^c(T)$ into $\mathscr{M}^c(X)$ is not necessarily continuous when $\mathscr{M}^c(T)$ and $\mathscr{M}^c(X)$ are equipped with the vague topology (or with the quasi-strong topology (Ch. III, §1, Exer. 8)) and that $\pi$ is not a proper mapping (cf. Ch. III, §2, Exer. 1).