Exercise 3, § 1

Let S, T, Y be three locally compact spaces, countable at infinity, with Y having a countable base. Let $\rho$ (resp. $\sigma$) be a positive measure on S (resp. T), $\nu = \rho \otimes \sigma$, and let $(\lambda_{s,t})_{(s,t) \in S \times T}$ be a $\nu$-adequate$^1$ family of positive measures on Y. Then, there exists a $\rho$-negligible set N such that, for every $s \notin N$, the family $(\lambda_{s,t})_{t \in T}$ is $\sigma$-adequate; the family $s \mapsto \int \lambda_{s,t} d\sigma(t)$ (defined almost everywhere) is $\rho$-adequate, and

(Apply Exer. 2 to the space $X = S \times T$, on noting that $\nu = \int (\varepsilon_s \otimes \sigma) d\rho(s).$)