Proposition 7

Let $\mu$ be a positive measure on T, and let $f$ be a $\mu$-measurable mapping of T into a topological space F. Then, there exists a universally measurable mapping $f'$ of T into F such that $f = f'$ locally $\mu$-almost everywhere.

Let $\mathscr{K}$ be the set of compact sets in T such that the restriction of $f$ to K is continuous; since $\mathscr{K}$ is $\mu$-dense (Ch. IV, §5, No. 10, Prop. 15), there exists a locally countable family $(K_i)_{i \in I}$ of pairwise disjoint elements of $\mathscr{K}$ such that the set $N = T - \bigcup_{i \in I} K_i$ is locally $\mu$-negligible (Ch. IV, §5, No. 9, Prop. 14). Let $x$ be an element of F; set

The functions $f$ and $f'$ are equal locally $\mu$-almost everywhere. On the other hand, $N \cap K$ is a Borel subset of K for every compact set K in T, since the family $(K_i)$ is locally countable. It follows that N is a universally measurable set, and that $f'$ is a universally measurable function (Ch. IV, §5, No. 10, Prop. 16).