Corollary

— Let A be a $\nu$-measurable subset of $T \times T'$, and let M be the set of $t \in T$ such that the section $A(t)$ of A at t is not $\mu'$-measurable.

a) If A is $\nu$-moderated, then M is $\mu$-negligible.

b) If the projection of A on $T'$ is $\mu'$-moderated, then M is locally $\mu$-negligible.

The assertion a) follows immediately from Prop. 2. To establish b), denote by B a set, the union of a sequence of $\mu'$-integrable open sets in $T'$, that contains the projection of A on $T'$, and denote by $\mu'_1$ the moderated measure $\varphi_B \cdot \mu'$; since A is measurable with respect to $\mu \otimes \mu'_1 \leq \mu \otimes \mu'$, Prop. 2 implies that $A(t)$ is $\mu'_1$-measurable, except for $t$ forming a locally $\mu$-negligible set. But since $A(t) \subset B$, to say that $A(t)$ is $\mu'_1$-measurable is equivalent to saying that $A(t)$ is $\mu'$-measurable (§ 5, No. 3, Cor. of Prop. 4).