Exercise 27, § 5

Let X be a locally compact space, $\Lambda : t \mapsto \lambda_t$ a scalarly essentially $\mu$-integrable mapping of T into $\mathscr{M}_+(X)$. Suppose that $\Lambda$ is pre-adequate for every measure $\varphi_K \cdot \mu$, where K is compact.

a) Show that $\Lambda$ is pre-adequate for every measure $\varphi_A \cdot \mu$, where A is a $\mu$-measurable subset of T (consider the compact subsets of A).

b) Show that $\Lambda$ is pre-adequate for every measure $f \cdot \mu$, where $f$ is positive, $\mu$-measurable, and takes on only finitely many values.

c) Show that $\Lambda$ is pre-adequate for every measure $f \cdot \mu$, where $f$ is $\mu$-measurable, between 0 and 1. From this, deduce that $\Lambda$ is $\mu$-adequate.