Proposition 16

Let $\mathfrak{A}$ be a locally countable set of $\mu$-measurable subsets of $X$ and let $B = \bigcup_{A \in \mathfrak{A}} A$. For a mapping $f$ of $B$ into a topological space $F$ to be $\mu$-measurable, it is necessary and sufficient that the restriction of $f$ to every $A \in \mathfrak{A}$ be $\mu$-measurable.

We have already observed (No. 9) that $B$ is $\mu$-measurable. The condition being obviously necessary, let us prove that it is sufficient. Thus, let $K$ be a compact subset of $B$. By hypothesis, there exists a sequence $(A_n)$ of sets belonging to $\mathfrak{A}$ such that the $K \cap A_n$ form a covering of $K$. Set $C_0 = K \cap A_0$ and

for $n > 0$, so that the nonempty $C_n$ form a partition of $K$ into $\mu$-integrable sets. Since the restriction of $f$ to $C_n$ is $\mu$-measurable, there exists a partition of $C_n$ formed by a $\mu$-negligible set $N_n$ and a sequence $(L_{mn})_{m \geq 0}$ of compact sets such that $f|L_{mn}$ is continuous. Since $N = \bigcup_n N_n$ is $\mu$-negligible, we see that condition a) of Prop. 15 is satisfied, whence the proposition.

Property d) of Prop. 15 makes it possible to immediately generalize the properties of measurable functions defined on all of $X$, observed in Nos. 2 to 5, to measurable functions defined on a measurable subset $A$ of $X$; these generalizations are left to the reader. We only point out explicitly that the principle of localization (No. 2, Prop. 4) remains valid when it is assumed that each of the functions $g_x$ is only defined in $V_x$ (or almost everywhere in $V_x$) and is measurable.