Exercise 3, § 3

a) Let E be a quasi-complete Hausdorff locally convex space. Show that when $\mathscr{M}(X; \mathbf{C})$ is equipped with the quasi-strong topology ($§ 1$, Exer. 8) (resp. the topology of strictly compact convergence), the mapping $(f, \mu) \mapsto \int f\ d\mu$ is $(\mathscr{G}, \mathscr{T})$-hypocontinuous, $\mathscr{T}$ being the set of vaguely bounded subsets of $\mathscr{M}(X; \mathbf{C})$, $\mathscr{G}$ the set of bounded (resp. compact) subsets of $\mathscr{K}(X; E)$ that are contained in $\mathscr{K}(X, K; E)$ for a suitable compact set K.

b) For every compact subset K of X, the mapping $(f, \mu) \mapsto \int f\ d\mu$ is continuous on $\mathscr{K}(X, K; E) \times \mathscr{M}(X; \mathbf{C})$ when $\mathscr{M}(X; \mathbf{C})$ is equipped with the quasi-strong topology, and on $\mathscr{K}(X, K; E) \times \mathscr{M}_+(X)$ when $\mathscr{M}_+(X)$ is equipped with the vague topology.