Proposition 10

— Let F be a separable Banach space, F' its dual, and $(\mathbf{a}'_n)$ a weakly dense sequence in the unit ball of F' (TVS, III, § 3, No. 4, Cor. 2 of Prop. 6).¹ For a mapping f of X into F to be measurable, it is necessary and sufficient that for every n the scalar function $x \mapsto \langle \mathbf{f}(x), \mathbf{a}'_n \rangle$ be measurable.

The condition being obviously necessary (No. 3, Th. 1), let us prove that it is sufficient; it suffices to verify condition a) of Th. 4 and, to this end,

¹ See the footnote at the end of § 2, No. 4.

it will suffice by the principle of localization to prove that, for every compact subset K of X and every closed ball $B \subset F$, with center $a$ and radius $r$, the set $A = K \cap f^{-1}(B)$ is measurable; now, for every $z \in F$,

$A$ is thus the intersection of $K$ and the sets defined by

as these sets are measurable by hypothesis, $A$ is measurable.