Exercise 29, § 5

Let X be a locally compact space. Let $\Lambda : t \mapsto \lambda_t$ be a $\mu$-adequate mapping of T into $\mathscr{M}_+(X)$. Let g be a function on X, positive, universally measurable, and locally bounded. Show that the family $t \mapsto g \cdot \lambda_t$ is $\mu$-adequate, and that

under each of the following conditions: a) g is moderated for the measure $\int \lambda_t d\mu(t)$; b) the measures $\lambda_t$ are bounded.