Definition 3

Let X be a locally compact space, and let $\Lambda : t \mapsto \lambda_t$ be a mapping of T into $\mathscr{M}_+(X)$. The mapping $\Lambda$ is said to be a diffusion of T in X if Λ is adequate for every positive measure on T with compact support. The diffusion Λ is said to be bounded if all of the measures λ_t are bounded and $\sup_{t \in T} \| \lambda_t \| < +\infty$; this quantity is then called the norm of Λ and is denoted $\| \Lambda \|$.

The following proposition merely translates the definition: