Proposition 8

— For a mapping $\Lambda : t \mapsto \lambda_t$ of T into $\mathscr{M}_+(X)$ to be a diffusion, it is necessary and sufficient that the following conditions be satisfied: 1) For every lower semi-continuous function $f \geqslant 0$ defined on X, the function $t \mapsto \lambda_t^\bullet(f)$ is universally measurable on T. 2) For every function $g \in \mathscr{K}_+(X)$, the function $t \mapsto \lambda_t(g)$ is locally bounded in T. 3) For every lower semi-continuous function $f \geqslant 0$ defined on X and for every positive measure $\mu$ on T with compact support, the following relation holds, where $\nu$ denotes $\int \lambda_t d\mu(t)$:

Suppose that $\Lambda$ is a diffusion. The condition 1) is then satisfied by the definition of adequate mappings (No. 1, Def. 1) and Prop. 6; the condition 3) is satisfied by formula (4), since $\Lambda$ is $\mu$-adequate. Let $g \in \mathscr{K}_+(X)$ and let $u$ be the function $t \mapsto \lambda_t(g)$ (universally measurable, by 1)); suppose that $u$ is not locally bounded. There would then exist a compact set K such that $u$ is not bounded on K, hence there would exist a sequence $(t_n)$ of elements of K such that $u(t_n) \geqslant n^2$ for all $n \geqslant 1$; then $u$ is not integrable for the measure $\mu = \sum_{n \geqslant 1} \frac{1}{n^2} \varepsilon_{t_n}$ with compact support, contrary to the hypothesis on $\Lambda$, which implies that $t \mapsto \lambda_t(g)$ is integrable for every positive measure with compact support. The above three conditions are thus necessary. Conversely, the conditions 1) and 2) imply that $\Lambda$ is scalarly essentially $\mu$-integrable for every measure $\mu$ with compact support. Since every measure $\mu' \geqslant 0$ that is bounded above by a measure $\mu$ with compact support also has compact support, the conditions 1) and 3) express that $\Lambda$ is $\mu$-adequate for every positive measure with compact support, which is indeed the sought-for result.