Exercise 13, § 6

For every number $p$ such that $0 < p < 1$, denote by $\mathscr{L}_F^p$ the set of measurable mappings $f$ of $X$ into a Banach space $F$ such that $N_p(f) < +\infty$.

a) Show that $\mathscr{L}_F^p$ is a vector space and that, if $B_a$ denotes the set of $f \in \mathscr{L}_F^p$ such that $N_p(f) \leqslant a$, the sets $B_a$ form, as $a$ runs over the set of numbers $> 0$, a fundamental system of neighborhoods of $0$ for a metrizable topology compatible with the vector space structure of $\mathscr{L}_F^p$.

b) Show that the mapping $f \mapsto |f|^{p-1} \cdot f$ is a uniformly continuous mapping of $\mathscr{L}_F^p$ onto $\mathscr{L}_F^1$ and that the inverse mapping is uniformly continuous on every bounded subset of $\mathscr{L}_F^1$ (cf. Exer. 10). Deduce from this that the space $\mathscr{L}_F^p$ is complete, and that $\mathscr{K}_F(X)$ is dense in $\mathscr{L}_F^p$.

c) If the measure $\mu$ is bounded, then $\mathscr{L}_F^1 \subset \mathscr{L}_F^p$ and the topology of convergence in mean is finer than the topology induced on $\mathscr{L}_F^1$ by that of $\mathscr{L}_F^p$.

d) Take $\mu$ to be Lebesgue measure on $X = [0, 1]$. Show that for every continuous function $f \geqslant 0$, there exists a decomposition $f = \frac{1}{2}(f_1 + f_2)$, where $f_1$ and $f_2$ are two functions $\geqslant 0$ of $\mathscr{L}^p$ such that $N_p(f_1) = N_p(f_2) = 2^{1-(1/p)} N_p(f)$. Deduce from this that, in $\mathscr{L}^p$, the closed convex envelope of every neighborhood $B_a$ is the entire space $\mathscr{L}^p$, consequently every continuous linear form on $\mathscr{L}^p$ is identically zero.