Exercise 14, § 1

Let $\mu$ be Lebesgue measure on $T = [0, 1]$; denote by $F$ the vector space over $\mathbf{R}$ of $\mu$-measurable finite numerical functions on $T$, equipped with the topology of pointwise convergence, which makes it a Hausdorff locally convex space.

a) Show that there exists in $F$ a countable dense subset (consider a dense sequence in the Banach space $C(T)$ of continuous numerical functions on $T$).

b) For every $t \in T$, let $f(t)$ be the element of the dual $F'$ of $F$ defined by $\langle f(t), z \rangle = z(t)$ for all $t \in T$. Show that, when $F'$ is equipped with the topology $\sigma(F', F)$, $f$ is scalarly $\mu$-measurable but is not $\mu$-measurable (cf. Prop. 13).