Exercise 10, § 3

Let T, X, Y be three locally compact spaces that are countable at infinity, $\Lambda : t \mapsto \lambda_t$ a diffusion of T into X, and H : $x \mapsto \eta_x$ a diffusion of X into Y; $\Lambda$ and H are said to be composable if the function $\Lambda(Hg)$ is locally bounded on T for every $g \in \mathscr{K}_+(Y)$. Show that if $\Lambda$ and H are composable, then $\lambda_t$ belongs to the domain of H for all $t \in T$, and the mapping $t \mapsto \lambda_t H$ is a diffusion of T into Y. Extend to this situation the formulas (15) (Prop. 13).