Corollary 1

— Let F, F' and G be three topological spaces, and let u be a continuous mapping of $F \times F'$ into G. Let f (resp. $f'$) be a function defined on T (resp. $T'$) with values in F (resp. $F'$) and measurable for $\mu$ (resp. $\mu'$). Then the function $(t, t') \mapsto u(f(t), f'(t'))$ is measurable for $\mu \otimes \mu'$.

The mappings $(t, t') \mapsto f(t)$, $(t, t') \mapsto f'(t')$ being $\nu$-measurable by Prop. 3, this follows from Th. 1 of Ch. IV, § 5, No. 3.