Proposition 2

— Let $f$ be a $\nu$-measurable function defined on $T \times T'$, with values in a topological space G, and let M be the set of $t \in T$ such that the mapping $t' \mapsto f(t, t')$ is not $\mu'$-measurable.

a) If $f$ is constant on the complement of a $\nu$-moderated subset of $T \times T'$, then M is $\mu$-negligible.

b) If $\mu'$ is moderated, then M is $\mu$-negligible.

The assertion a) follows from Prop. 4b) of §3, No. 2 and the remarks of No. 1. To treat b), note that $\mu'$ is the sum of a sequence $\mu'_n$ of bounded measures (§2, No. 3, Prop. 4); $f$ is measurable with respect to $\mu \otimes \mu'_n \leq \nu$, and the set $M$ is the union of the sets $M_n$ associated with the measures $\mu'_n$ (§ 2, No. 2, Prop. 2). One is thus reduced to the case that $\mu'$ is bounded, which follows from Prop. 4c) of § 3, No. 2.

This statement extends immediately to complex measures (Ch. III, § 4, No. 2, Prop. 3).