Exercise 17, § 1

Let $\mu$ be Lebesgue measure on $T = [0, 1]$, $F$ the Banach space $L^1(\mathbf{N})$, $F' = L^\infty(\mathbf{N})$ its dual. For every $t \in [0, 1[$, let $t = \sum_{n=0}^\infty \xi_n 2^{-n}$ be the dyadic expansion of $t$; denote by $f(t)$ the sequence $(\xi_n) \in F'$, and set $f(1) = 0$.

a) Show that, for $1 \leq p \leq +\infty$, the function $f$ belongs to the space $\Lambda_{F'}^p$ (cf. Exer. 16).

b) Show that $f$ is measurable for the topology $\sigma(F', F)$ and that, for this topology, there exists a sequence of step functions that converges almost everywhere to $f$.

c) Show that $f$ is not measurable for the strong topology on $F'$ (observe that $|f(t) - f(t')| = 1$ if $0 \leq t < t' < 1$).

d) Show that, in the space $\Lambda_{F'}^p$ ($1 \leq p < +\infty$), $f$ is not the limit of a sequence of step functions. (Reduce to considering only step functions that are linear combinations (with coefficients in $F'$) of characteristic functions of intervals whose endpoints are of the form $k/2^n$; if $g$ is such a step function, show that, in the notations of Exer. 16 b), $M_1'(f - g) \geq 1/4$.)

e) Show that there does not exist any mapping $h$ of $T$ into $F'$, measurable for the strong topology, such that $f - h$ is scalarly negligible. (Observe that, for such a mapping, one would necessarily have $N_\infty(h) \leq 1$; then obtain a contradiction with the result of d).)