Exercise 9, § 5

Let $\mu$ be Lebesgue measure on $X = [0, 1]$; denote by $F$ the vector space over $\mathbf{R}$ of $\mu$-measurable finite numerical functions on $X$, 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 $\mathscr{C}(X; \mathbf{R})$ of continuous numerical functions on $X$).

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