Exercise 1, § 4

Let X, Y be two locally compact spaces.

a) Show that the mapping $(\mu, \nu) \mapsto \mu \otimes \nu$ of $\mathscr{M}(X; \mathbf{C}) \times \mathscr{M}(Y; \mathbf{C})$ into $\mathscr{M}(X \times Y; \mathbf{C})$ is continuous when $\mathscr{M}(X; \mathbf{C}),\ \mathscr{M}(Y; \mathbf{C})$ and $\mathscr{M}(X \times Y; \mathbf{C})$ are equipped with the quasi-strong topology ($§ 1$, Exer. 8).

b) Show that the mapping $(\mu, \nu) \mapsto \mu \otimes \nu$ of $\mathscr{M}_+(X) \times \mathscr{M}_+(Y)$ into $\mathscr{M}_+(X \times Y)$ is continuous when $\mathscr{M}_+(X),\ \mathscr{M}_+(Y)$ and $\mathscr{M}_+(X \times Y)$ are equipped with the vague topology.