Exercise 3, § 5

Let X be a locally compact space, $\Lambda : t \mapsto \lambda_t$ and $\Lambda' : t \mapsto \lambda'_t$ two $\mu$-adequate mappings of T into $\mathscr{M}_+(X)$, and let $\nu = \int \lambda_t d\mu(t)$, $\nu' = \int \lambda'_t d\mu(t)$. Show that if, locally almost everywhere for $\mu$, $\lambda'_t$ is a measure with base $\lambda_t$, then $\nu'$ is a measure with base $\nu$. (Consider a $\nu$-negligible compact subset K of X. It is $\lambda_t$-negligible locally almost everywhere, hence $\lambda'_t$-negligible locally almost everywhere. Apply Prop. 5 of §3.)