Exercise 9, § 3

a) Let T and X be two locally compact spaces, $\mathscr{C}_+(T)$ the convex cone of all positive continuous functions on T, and V a mapping of $\mathscr{K}_+(T)$ into $\mathscr{C}_+(T)$ having the following properties:

Show that there exists a unique diffusion $\Lambda$ of T into X such that $\Lambda f = Vf$ for all $f \in \mathscr{K}_+(X)$. Show that this result is again true if the cone $\mathscr{C}_+(T)$ is replaced by the cone of lower semi-continuous functions on T that are positive and locally bounded (cf. Exer. 6).

b) Suppose that the topology of X admits a countable base. Show that the above result is again true if the cone $\mathscr{C}_+(T)$ is replaced by the cone of universally measurable functions on T that are positive and locally bounded.