Proposition 1

For a mapping $f$ of $X$ into $F$ to be measurable, it is necessary and sufficient that, for every compact set $K \subset X$ and every number $\varepsilon > 0$, there exist a compact set $K_1 \subset K$ such that $|\mu|(K - K_1) \leq \varepsilon$ and the restriction of $f$ to $K_1$ is continuous.

If this condition is fulfilled, we may define recursively a sequence of pairwise disjoint compact sets $K_n \subset K$ such that $|\mu|(K - \bigcup_{i=1}^n K_i) \leq 1/n$ and such that the restriction of $f$ to each $K_n$ is continuous ($§ 4$, No. 6, Th. 4); the complement with respect to $K$ of the union of the $K_n$ is then negligible ($§ 4$, No. 5, Cor. of Prop. 7), thus $f$ is measurable. Conversely, suppose there exist a negligible set $N \subset K$ and a partition $(K_n)$ of $K - N$ formed of compact sets such that the restriction of $f$ to each $K_n$ is continuous; for every $\varepsilon > 0$ there exists an integer $n$ such that, if $H = \bigcup_{i=1}^n K_i$, then $|\mu|(K - H) \leq \varepsilon$ (§ 4, No. 5, Cor. of Prop. 7); the set $H$ is compact, the $K_i$ ($1 \leq i \leq n$) form a finite partition of $H$ into compact sets, and the restriction of $f$ to each $K_i$ is continuous; therefore the restriction of $f$ to $H$ is continuous.