Exercise 25, § 2

Let $\mu$ be a positive measure on T.

a) Let f be a scalarly $\mu$-measurable mapping of T into a Banach space F, such that $f(T)$ is relatively compact for $\sigma(F, F')$. Show that there exists a $\mu$-measurable mapping g of T into F such that $f - g$ is scalarly locally negligible. (Consider the mapping $h \mapsto \int fh\, d\mu$ of $L^1(\mu)$ into F, and apply Exer. 24 to it, also using Exer. 10 of Ch. IV, §7.)

b) Let f be a scalarly $\mu$-measurable mapping of T into a reflexive Fréchet space F. Show that there exists a $\mu$-measurable mapping g of T into F such that $f - g$ is scalarly locally negligible (cf. Exer. 15 b)). (Reduce to the case that T is compact, using Prop. 14 of Ch. IV, §5, No. 9. Next, apply Exer. 23 c) of §1 to reduce to the case that $f(T)$ is bounded in F. Then embed F in a countable product of Banach spaces, and use a).)

c) Let f be a mapping of T into a Fréchet space F, $\mu$-measurable for the topology $\sigma(F, F')$. Show that f is $\mu$-measurable for the original topology of F. (Reduce to the case that T is compact and f is continuous for the topology $\sigma(F, F')$; conclude as in b).)