Corollary 1

*Let $X$ be a locally compact space, $\mu$ a measure on $X$, $A$ a $\mu$-measurable subset of $X$, and $f$ a $\mu$-measurable mapping of $A$ into a topological space $F$. Let $\mathscr{K}$ be a set of compact subsets of $X$, $\mu$-dense in $X$. Then, there exists a partition of $A$ formed by a locally negligible set $N$ and a locally countable family $(K_\lambda)_{\lambda \in L}$ of sets $K_\lambda \in \mathscr{K}$, such that $f|K_\lambda$ is continuous for every $\lambda \in L$.

In view of No. 9, Prop. 14, it suffices to show that the set $\mathscr{H} \subset \mathscr{K}$ of subsets $K \in \mathscr{K}$ such that $K \subset A$ and $f|K$ is continuous, is $\mu$-dense in $A$. Now, it follows at once from Prop. 1 of No. 1 and condition d) of Prop. 15 that, for every compact subset $K_0$ of $A$ and every $\varepsilon > 0$, there exists a subset $K \subset K_0$ belonging to $\mathscr{K}$ such that $|\mu|(K_0 - K) \leq \varepsilon$ and $f|K$ is continuous; the conclusion therefore follows from Prop. 12 of No. 8.