Proposition 4 (Principle of localization)

— Let f be a mapping of X into a topological space F. Suppose that for every $x \in X$, there exist an integrable neighborhood $V_x$ of x and a measurable mapping $g_x$ of X into F such that $f(y) = g_x(y)$ almost everywhere in $V_x$. Then f is measurable.

Let K be a compact set; there exists a finite number of points $x_i \in K$ such that the $V_{x_i}$ form a covering of K. It follows at once (§4, No. 9, Lemma) that there exist a negligible set $N \subset K$ and a finite partition of $K - N$ formed of integrable sets $M_j$ such that each of the sets $K \cap V_{x_i}$ is the union of a subset of N and a certain number of the $M_j$, and such that on each of the $M_j$, $f$ is equal to one of the functions $g_{x_i}$. Now, for each $M_j$ there exist a negligible set $N_j \subset M_j$ and a partition of $M_j - N_j$ formed by a sequence of compact sets $K_{nj}\ (\mathbf{n} \in \mathbf{N})$; on the other hand, for each $K_{nj}$ there exist a negligible set $P_{nj} \subset K_{nj}$ and a partition of $K_{nj} - P_{nj}$ formed by a sequence of compact sets $K_{mnj}\ (\mathbf{m} \in \mathbf{N})$ such that the restriction of $f$ to each of the $K_{mnj}$ is continuous. Since the union of N, the $N_j$ and the $P_{nj}$ is negligible, $f$ is measurable.

The concept of measurable function is therefore a concept of local character.