Proposition 13

— Let $T, X, Y$ be three locally compact spaces, $\Lambda : t \mapsto \lambda_t$ a bounded diffusion of $T$ in $X$, and $H : x \mapsto \eta_x$ a bounded diffusion of $X$ in $Y$. The mapping $t \mapsto \lambda_t H$ is then a bounded diffusion of $T$ in $Y$, which is denoted by $\Lambda H$, and

Let $f$ be a universally measurable function $\geq 0$ defined on $Y$, and $\mu$ a measure on $T$. Suppose that $\mu$ belongs to the domain of $\Lambda$, and that μΛ belongs to the domain of H; then μ belongs to the domain of ΛH, and

(15)

Set $\gamma_t = \lambda_t H$; we shall denote by $\Gamma$ the mapping $\Lambda H$ of T into $\mathscr{M}_+(Y)$, and by $\Gamma f$ the function $t \mapsto \langle \gamma_t, f \rangle$ (an abuse of notation, since we do not yet know whether $\Gamma$ is a diffusion). Then $\langle \gamma_t, f \rangle = \langle \lambda_t H, f \rangle = \langle \lambda_t, Hf \rangle$ by (13); since the function $Hf$ is positive and universally measurable on X (Cor. of Prop. 12), it follows first of all that $\Gamma f = \Lambda(Hf)$, and then that $\Gamma f$ is universally measurable on T (same reference). It is clear that all of the measures $\gamma_t$ have total mass at most equal to $\| \Lambda \| \| H \|$. Consequently $\Gamma g$ is universally measurable and bounded for every function $g \in \mathscr{K}_+(Y)$; $\Gamma$ is therefore scalarly essentially integrable for every bounded measure on T, and in particular for every measure with compact support. More generally, if $\mu$ is a measure in the domain of $\Lambda$, such that $\mu \Lambda$ belongs to the domain of H, then, for $g \in \mathscr{K}_+(Y)$,

Since the last quantity is finite, we see that $\Gamma$ is scalarly essentially $\mu$-integrable. Let us denote by $\mu \Gamma$ the integral $\int \gamma_t d\mu(t)$ (an abuse of notation, since we do not yet know whether $\Gamma$ is a diffusion). The preceding relations may then be written

or also $\mu \Gamma = (\mu \Lambda)H$ since $g$ is arbitrary in $\mathscr{K}_+(Y)$.

Consider anew the universally measurable function $f \geqslant 0$. We have

When $f$ is lower semi-continuous and $\mu$ runs over the set of positive measures with compact support, these relations express that $\Gamma$ is a diffusion of T in Y. The assertion then does no more than make explicit the relations obtained in the course of the above proof.