Exercise 11, § 6

Let $g$ be a numerical function $\geqslant 0$ belonging to $\mathscr{L}^p$ ($1 \leqslant p < +\infty$). Denote by $I_g$ the set of functions $f \in \mathscr{L}_F^p$ such that $|f| \leqslant g$.

a) Show that on $I_g$, the topology of convergence in mean of order $p$ is identical to the topology of convergence in measure.

b) If $p \leqslant q < r$ (resp. $q < r \leqslant p$), show that on the set $I_g \cap \mathscr{L}_F^q \cap \mathscr{L}_F^r$ the topology of convergence in mean of order $q$ is coarser (resp. finer) than the topology of convergence in mean of order $r$ (for two functions $f, f_0$ belonging to this set, write $|f - f_0|^q \leqslant |f - f_0|^s (2g)^{q-s}$ and use Hölder’s inequality, suitably choosing $s$ and the pair of conjugate exponents). Show by means of examples that these topologies can be distinct (cf. Exer. 8).

c) Let $\mu$ be Lebesgue measure on $X = ]0, +\infty[$; the function

belongs to $\mathscr{L}^p$, but to no $\mathscr{L}^q$ for $q \neq p$. Show that if $q \neq p$, the topology of convergence in mean of order $q$ on the set $I_g \cap \mathscr{L}^q$ is distinct from the topology of convergence in measure.