Exercise 28, § 5

Let X be a locally compact space, $\Lambda : t \mapsto \lambda_t$ a $\mu$-adequate mapping of T into $\mathscr{M}_+(X)$. Let $\mu' = g \cdot \mu$ be a positive measure with base $\mu$, such that $\Lambda$ is scalarly essentially $\mu'$-integrable.

a) Show that $\Lambda$ is $\mu'$-integrable (use Exer. 27 b), or the relation $\mu' = \sup_n (\inf(\mu', n\mu))$ and Exer. 9 of §3).

b) Show that $\nu' = \int \lambda_t \, d\mu'(t)$ is a measure with base $\nu = \int \lambda_t \, d\mu(t)$.

c) Show that the mapping $t \mapsto g(t)\lambda_t$ is $\mu$-adequate, and that its integral is $\nu'$.