Proposition 12

If F is a separable, metrizable locally convex space, then every scalarly $\mu$-measurable mapping f of T into F is $\mu$-measurable.

For, F may be regarded as a subspace of a countable product $\prod_n E_n$ of Banach spaces (TVS, II, §1, No. 3, Prop. 3), and one can suppose that $\mathrm{pr}_n(F)$ is dense in $E_n$, which is therefore separable. For every n, the mapping $\mathrm{pr}_n \circ f$ is scalarly $\mu$-measurable, hence $\mu$-measurable (Ch. IV, §5, No. 5, Cor. 2 of Prop. 10), consequently f is $\mu$-measurable (Ch. IV, §5, No. 3, Th. 1).