Exercise 15, § 1

Let $F$ be a metrizable locally convex space, $f$ a scalarly $\mu$-measurable mapping of $T$ into $F$ such that, for every compact subset $K$ of $T$, there exists a countable subset $H$ of $F$ such that $f(t) \in \overline{H}$ for almost every $t \in K$; show that under these conditions, $f$ is $\mu$-measurable for the original topology of $F$. (Embed $F$ in a countable product of normed spaces, after having reduced to the case that $F$ is separable.)