Proposition 17

— Let $(A_\lambda)_{\lambda \in L}$ be a locally countable family of $\mu$-measurable subsets of $A$, pairwise disjoint and such that $A - \bigcup_{\lambda \in L} A_\lambda$ is locally $\mu$-negligible. If, for every class $f \in S(A, \mu; F)$ and every $\lambda \in L$, $f_\lambda$ denotes the class of the restriction to $A_\lambda$ of any of the functions in the class $f$, then the mapping $\psi : f \mapsto (f_\lambda)_{\lambda \in L}$ is an isomorphism of the uniform space $S(A, \mu; F)$ onto the product uniform space $\prod_{\lambda \in L} S(A_\lambda, \mu; F)$.

It follows from No. 10, Prop. 16 that $\psi$ is bijective. Consider an entourage $T$ of $S(A, \mu; F)$ that is the canonical image of a $W(V, B, \delta)$, where $B$ is a compact subset of $A$; we know that the set $J$ of $\lambda \in L$ such that $B \cap A_\lambda \neq \varnothing$ is countable (No. 9), and $|\mu|(B) = \sum_{\lambda \in J} |\mu|(B \cap A_\lambda)$; therefore, there exists a finite subset $H$ of $J$ such that

The image of $T$ under $\psi \times \psi$ is then contained in the canonical image of the product $\prod_{\lambda \in H} W(V, B \cap A_\lambda, \delta)$. On the other hand, if $m$ is the number of elements of $H$, then the image of $T$ under $\psi \times \psi$ contains the canonical image of the entourage $\prod_{\lambda \in H} W(V, A_\lambda, \delta/2m)$, which proves the proposition.