Exercise 20, § 6

Let T, X, Y be three locally compact spaces, $\mu$ a positive measure on T, $t \mapsto \lambda_t$ a $\mu$-adequate mapping of T into $\mathscr{M}_+(X)$, $\nu = \int \lambda_t d\mu(t)$, and $\pi$ a $\nu$-proper mapping of X into Y. Assume one of the following hypotheses: a) Y is countable at infinity and $\nu$ is moderated; b) Y is countable at infinity and $\lambda_t$ is bounded locally almost everywhere in T. Then $\pi$ is $\lambda_t$-proper locally almost everywhere in T, the mapping $t \mapsto \pi(\lambda_t)$ is $\mu$-adequate, and its integral is equal to $\pi(\mu)$.