Corollary 4

— Let f be a mapping of X into a topological space F. If the set N of points of discontinuity of f is locally negligible, then f is measurable.

For every compact set $K \subset X$, $K \cap N$ is negligible (Prop. 5), therefore, for every $\varepsilon > 0$, there exists a compact set $K_1 \subset K - (K \cap N)$ such that $|\mu|(K - K_1) \leq \varepsilon$ (§ 4, No. 6, Th. 4), and by hypothesis the restriction of f to $K_1$ is continuous, whence the conclusion.

If $P\{x\}$ is a property, the property « P\{x\} locally almost everywhere (with respect to $\mu$)» is by definition equivalent to the property «the set of x such that* (x $\in$ X and not $P\{x\}$) is locally negligible (for the measure $\mu$)». If F is any set, the relation «$f(x) = g(x)$ locally almost everywhere» is an equivalence relation in the set of mappings of X into F. In particular, if F is a vector space, a mapping $f$ of X into F such that f(x) = 0\text{ locally almost everywhere is said to be } \textit{locally negligible}. We leave to the reader the task of establishing for these concepts most of the properties corresponding to those that have been enumerated in §2, Nos. 4, 5 and 6 for functions equal almost everywhere. We shall limit ourselves to observing that if two \emph{continuous} mappings $f, g$ of X into a Hausdorff topological space F are equal \emph{locally almost everywhere}, then they are equal \emph{almost everywhere} by virtue of Cor. 2 of Prop. 5 (hence are equal at every point of the support of $\mu$ (§ 2, No. 4, Prop. 9)); however, we state explicitly the following proposition, which is an immediate consequence of the principle of localization: