Proposition 9

— Let $\Lambda : t \mapsto \lambda_t$ be a mapping of T into $\mathscr{M}_+(X)$, such that the function $t \mapsto \lambda_t(g)$ is universally measurable and locally bounded in T for every $g \in \mathscr{K}_+(X)$. One can affirm that $\Lambda$ is a diffusion in each of the following cases:

a) the topology of X admits a countable base; b) Λ is universally measurable for the vague topology.

For, let $\mu$ be a positive measure on $T$ with compact support; the mapping $\Lambda$ is scalarly essentially $\mu$-integrable, hence $\mu$-adequate if either a) or b) is satisfied (No. 1, Prop. 2).

For the rest of this section, we shall adopt the following notations: we will denote by $⟨\eta, h⟩$ the upper essential integral, for a positive measure $\eta$, of a positive $\eta$-measurable function $h$. The mapping $\Lambda : t \mapsto \lambda_t$ will be a diffusion of $T$ in $X$. If $f$ is a positive universally measurable function defined on $X$, we shall denote by $\Lambda f$ the mapping $t \mapsto \lambda_t^*(f)$. If $\mu$ is a positive measure on $T$ such that $\Lambda$ is scalarly essentially $\mu$-integrable, we shall denote by $\mu\Lambda$ the measure $\int \lambda_t d\mu(t)$. The definition of the integral then takes the form

We shall say that a positive measure $\mu$ on $T$ belongs to the domain of $\Lambda$ if $\Lambda$ is $\mu$-adequate: this amounts to saying (in view of Prop. 8) that $\Lambda$ is scalarly essentially $\mu$-integrable and $⟨\mu'\Lambda, f⟩ = ⟨\mu', \Lambda f⟩$ for every positive measure $\mu' \leq \mu$ and every lower semi-continuous positive function $f$.