Proposition 8

Let $(F_n)$ be a countable family (finite or infinite) of sets. For every index n, let $f_n, g_n$ be two equivalent mappings of X into $F_n$; then, there exists a negligible set H such that, for every $x \notin H$, $f_n(x) = g_n(x)$ for all n.

For, the set $H_n$ of $x \in X$ such that $f_n(x) \neq g_n(x)$ is negligible, therefore so is their union H (No. 2, Prop. 4), and this set meets the requirements.