Exercise 7, § 3

a) Let $\Lambda : t \mapsto \lambda_t$ be a scalarly essentially integrable mapping of T into $\mathscr{M}_+(X)$, and let $\nu = \int \lambda_t d\mu(t)$. Show that for every lower semi-continuous function $f \geq 0$ defined on X,

b) Suppose that $\Lambda$ is $\mu$-pre-adequate, denote by $\mu'$ a positive measure $\leq \mu$, and write $\mu = \mu' + \mu''$, $\nu' = \int \lambda_t d\mu'(t)$. Deduce from a) that, for every lower semi-continuous function $f \geq 0$ that is $\nu$-integrable,

Extend this result to a lower semi-continuous function $f \geq 0$ that is $\nu$-moderated. From this, deduce that if $\nu$ is a moderated measure (in particular, if X is countable at infinity), then $\Lambda$ is $\mu$-adequate.