Exercise 5, § 3

Give an example of a $\mu$-adequate family $t \mapsto \lambda_t$ and a numerical function $f$ defined on X, such that $f$ is $\lambda_t$-measurable for all $t \in T$ but is not measurable for the measure $\nu = \int \lambda_t d\mu(t)$ (take $X = T$ and $\lambda_t = \varepsilon_t$ for all $t \in T$).