Exercise 2, § 6

Let T and X be two locally compact spaces, $\mathscr{P}$ the set of proper continuous mappings of T into X, equipped with the topology of compact convergence. Let H be a subset of $\mathscr{P}$ such that, for every compact subset K of X, the union of the sets $\pi^{-1}(K)$, where $\pi$ runs over H, is relatively compact in T.

a) Show that the mapping $(\pi, \lambda) \mapsto \pi(\lambda)$ of $H \times \mathscr{M}(T)$ into $\mathscr{M}(X)$ is not necessarily continuous when each of the spaces $\mathscr{M}(T)$ and $\mathscr{M}(X)$ is equipped with the quasi-strong topology (Ch. III, §1, Exer. 8).

b) If B is a bounded subset of $\mathscr{M}(T)$, show that the restriction to $H \times B$ of the mapping $(\pi, \lambda) \mapsto \pi(\lambda)$ is continuous when each of the spaces $\mathscr{M}(T), \mathscr{M}(X)$ is equipped with the vague topology (make use of Prop. 15 of Ch. III, §1).

c) Show that the mapping $(\pi, \lambda) \mapsto \pi(\lambda)$ of $H \times \mathscr{M}_+(T)$ into $\mathscr{M}_+(X)$ is continuous when $\mathscr{M}(T)$ and $\mathscr{M}(X)$ are equipped with the vague topology (cf. Ch. III, §1, Exer. 10).

d) Give an example where $T = X$ is compact and the mapping $(\pi, \lambda) \mapsto \pi(\lambda)$ of $\mathscr{P} \times \mathscr{M}(T)$ into $\mathscr{M}(X) = \mathscr{M}(T)$ is not continuous, when $\mathscr{M}(T)$ is equipped with the vague topology (take for T the torus T and use Exer. 2 b) of Ch. III, §4).