Remark

— Th. 1 of Ch. V, §3, No. 4 shows (taking into account the fact that T and B have countable bases) that for every function $f$ defined on T, with values in $\overline{\mathbf{R}}$ or in a Banach space F and $\mu$-integrable, the set of $b \in B$ such that $f$ is not $\lambda_b$-integrable is $\nu$-negligible, the function $b \mapsto \int f(t) d\lambda_b(t)$, defined almost everywhere, is $\nu$-integrable, and

An analogous result holds for scalarly $\mu$-integrable functions, on applying Prop. 3 of §1, No. 1.