Exercise 1, § 5

Let T and X be two locally compact spaces, $\mu$ a measure $\geq 0$ on T, $\Lambda : t \mapsto \lambda_t$ a $\mu$-adequate mapping of T into $\mathscr{M}_+(X)$, and $g$ a function locally integrable for the measure $\nu = \int \lambda_t d\mu(t)$. Give an example showing that if X is not countable at infinity, $g$ may not be locally $\lambda_t$-integrable for any value of $t$ (cf. §3, Exer. 4).