Exercise 11, § 1

a) Show that when $\mathscr{M}(X; \mathbf{C})$ is equipped with the quasi-strong topology (resp. the topology of strictly compact convergence, resp. the vague topology), the bilinear form $(f, \mu) \mapsto \langle f, \mu \rangle$ is $(\mathscr{S}, \mathscr{T})$-hypocontinuous, where $\mathscr{T}$ is the set of vaguely bounded subsets of $\mathscr{M}(X; \mathbf{C})$, and $\mathscr{S}$ is the set of bounded subsets of $\mathscr{H}(X; \mathbf{C})$ that are contained in $\mathscr{H}(X, K; \mathbf{C})$ for a suitable compact set $K$ (resp. the set of strictly compact subsets of $\mathscr{H}(X; \mathbf{C})$, resp. the set of finite subsets of $\mathscr{H}(X; \mathbf{C})$) (cf. TVS, III, §5, Exer. 12).

b) For every compact subset $K$ of $X$, show that the bilinear form $(f, \mu) \mapsto \langle f, \mu \rangle$ is continuous on $\mathscr{H}(X, K; \mathbf{C}) \times \mathscr{M}(X; \mathbf{C})$ when $\mathscr{M}(X; \mathbf{C})$ is equipped with the quasi-strong topology; it is continuous on $\mathscr{H}(X, K; \mathbf{C}) \times \mathscr{M}_+(X)$ when $\mathscr{M}_+(X)$ is equipped with the vague topology (make use of Exer. 10).

c) Assume that $X$ is paracompact and noncompact. Show that the bilinear form $(f, \mu) \mapsto \langle f, \mu \rangle$ is not continuous on $\mathscr{H}(X; \mathbf{C}) \times \mathscr{M}_+(X)$ when $\mathscr{M}_+(X)$ is equipped with the strong topology (identical in this case to the quasi-strong topology).

d) Give an example where $X$ is compact and the bilinear form $(f, \mu) \mapsto \langle f, \mu \rangle$ is not continuous on $\mathscr{C}(X; \mathbf{C}) \times \mathscr{M}(X; \mathbf{C})$ when $\mathscr{M}(X; \mathbf{C})$ is equipped with the topology of compact convergence (TVS, IV, §1, Exer. 11).