Corollary 1

Let F be a Banach space. In order that a mapping f of X into F be measurable, it is necessary and sufficient that it satisfy the following two conditions:

a) for every $a' \in F'$, the scalar function $x \mapsto \langle f(x), a' \rangle$ is measurable;

b) for every compact set $K \subset X$, there exists a countable subset H of F such that $f(x) \in \overline{H}$ for almost every $x \in K$.

The necessity of the conditions follows from No. 3, Th. 1, and Th. 4. To prove that the conditions are sufficient, it again suffices to verify condition a) of Th. 4. With notations as in the proof of Prop. 10, we may (on account of b)) suppose, after modifying $f$ if necessary on a negligible set, that $f(K) \subset \overline{H}$, where $H$ is a countable subset of $F$. If $V$ is the closed linear subspace of $F$ generated by the set $H \cup \{a\}$, then $V$ is a separable Banach space and every continuous linear form on $V$ is the restriction of a form $a' \in F'$; the same reasoning as in Prop. 10 then shows that $K \cap f^{-1}(B)$ is measurable.