Exercise 25, § 2

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

(a) Let f be a scalarly $\mu$-measurable mapping from T into a Banach space F, such that $f(T)$ is relatively compact for $\sigma(F, F')$. Now prove that there exists a $\mu$-measurable mapping g from T into F, such that $f - g$ is scalarly locally negligible. (Consider the mapping $\tilde{h} \to \int f h d\mu$ from $L^1(\mu)$ into F, and apply Exercise 24 to it, using also Exercise 19 of Chapter IV, § 4.)

(b) Let f be a scalarly $\mu$-measurable mapping from T into a reflexive Fréchet space F. Now prove that there exists a $\mu$-measurable mapping g from T into F such that $f - g$ is scalarly locally negligible (cf. Exercise 15 b)). (Reduce to the case where T is compact, using Proposition 4 of Chapter V, § 1, No. 4. Then apply Exercise 23 c) of § 1 to reduce to the case where $f(T)$ is bounded in F. Then embed F in a countable product of Banach spaces, and use a).)

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