Exercise 24, § 1

Let F be a complete Hausdorff locally convex space, containing a countable dense subset. Let f be a mapping of T into F, scalarly $\mu$-measurable and scalarly essentially bounded; show that for every function $g \in \mathscr{L}^1$, $gf$ is scalarly integrable and $\int gf\, d\mu \in F$. (Make use of Cor. 1 of Th. 2 of TVS, III, §3, No. 6, on observing that the equicontinuous subsets of $F'$ are metrizable for $\sigma(F', F)$, and applying Lebesgue’s theorem.)