Exercise 12, § 1

a) Show that the bilinear mapping $(g, \mu) \mapsto g \cdot \mu$ of $\mathscr{C}(X; \mathbf{C}) \times \mathscr{M}(X; \mathbf{C})$ into $\mathscr{M}(X; \mathbf{C})$ is continuous when $\mathscr{C}(X; \mathbf{C})$ is equipped with the topology of compact convergence and $\mathscr{M}(X; \mathbf{C})$ with the quasi-strong topology (Exer. 8) or with the topology of strictly compact convergence.

b) Show that the bilinear mapping $(g, \mu) \mapsto g \cdot \mu$ of $\mathscr{C}(X; \mathbf{C}) \times \mathscr{M}(X; \mathbf{C})$ into $\mathscr{M}(X; \mathbf{C})$ is $(\mathscr{S}, \mathscr{T})$-hypocontinuous when $\mathscr{C}(X; \mathbf{C})$ is equipped with the topology of compact convergence and $\mathscr{M}(X; \mathbf{C})$ with the vague topology, where $\mathscr{S}$ denotes the set of finite subsets of $\mathscr{C}(X; \mathbf{C})$ and $\mathscr{T}$ the set of vaguely bounded subsets of $\mathscr{M}(X; \mathbf{C})$.

c) Show that the mapping $(g, \mu) \mapsto g \cdot \mu$ of $\mathscr{C}(X; \mathbf{C}) \times \mathscr{M}_+(X)$ into $\mathscr{M}(X; \mathbf{C})$ is continuous when $\mathscr{C}(X; \mathbf{C})$ is equipped with the topology of compact convergence, $\mathscr{M}(X; \mathbf{C})$ and $\mathscr{M}_+(X)$ with the vague topology.

d) Give an example of a compact space $X$ such that the bilinear mapping $(g, \mu) \mapsto g \cdot \mu$ of $\mathscr{C}(X; \mathbf{C}) \times \mathscr{M}(X; \mathbf{C})$ into $\mathscr{M}(X; \mathbf{C})$ is not continuous when $\mathscr{C}(X; \mathbf{C})$ is equipped with the topology of uniform convergence and $\mathscr{M}(X; \mathbf{C})$ with the vague topology.