Exercise 3, § 6

Let T be a noncompact locally compact space, $\mu$ a positive measure on T, and $\mathscr{G}$ the group of homeomorphisms of T. Give an example showing that the mapping $\pi \mapsto \pi(\mu)$ of $\mathscr{G}$ into $\mathscr{M}(T)$ is not necessarily continuous when $\mathscr{G}$ is equipped with the topology of compact convergence and $\mathscr{M}(T)$ with the vague topology (take for T the subspace of $\mathbf{R}$ formed by 0 and the points n and $1/n$ ($n$ an integer $\geqslant 1$) and for $\mu$ the measure defined by $\mu(\{0\}) = 0, \mu(\{1/n\}) = 1/n^2, \mu(\{n\}) = 1$).