Exercise 12, § 6

Let $h$ be a numerical function $\geqslant 0$ such that $h$ and $h^2$ are integrable; let $I_h$ be the set of measurable numerical functions $f$ such that $|f| \leqslant h$. Show that the mapping $(f, g) \mapsto fg$ of $I_h \times I_h$ into $\mathscr{L}^1$ is continuous for the topology of convergence in mean (on $I_h$ and on $\mathscr{L}^1$).