Exercise 12, § 1

Let $\mu$ be Lebesgue measure on $T = [0,1]$, and let $F$ be a Hilbert space having an orthonormal basis $(e_t)_{t \in T}$ equipotent to $T$.

a) Let $f$ be the mapping $t \mapsto e_t$ of $T$ into $F$. Show that $f$ is scalarly $\mu$-negligible, but is not $\mu$-measurable when $F$ is equipped with the weak topology $\sigma(F,F')$ (nor, a fortiori, when $F$ is equipped with its original topology).

b) Let $A$ be a non-measurable set in $T$ (Ch. IV, §4, Exer. 8); show that the function $g = f \varphi_A$ is scalarly $\mu$-negligible, but that the numerical function $|g|$ is not $\mu$-measurable.