Exercise 7, § 5

Let $X$ be the interval $[0, 1]$ in $\mathbf{R}$, and $F$ the Hilbert space having an orthonormal basis $(e_t)_{0 \leqslant t \leqslant 1}$ equipotent to $X$.

a) Show that the mapping $f$ of $X$ into $F$, such that $f(t) = e_t$ for $0 \leqslant t \leqslant 1$, is not measurable for Lebesgue measure, but that the inverse image under $f$ of every closed ball in $F$ is measurable and, for every continuous linear form $a'$ on $F$, $\langle f, a' \rangle$ is negligible.

b) Let $H$ be a non-measurable set in $X$ (§ 4, Exer. 8); show that if $g = f \varphi_H$ then the function $\langle g, a' \rangle$ is negligible for every continuous linear form $a'$ on $F$, but the numerical function $|g|$ is not measurable.