Proposition 11

— If a function $f$ belongs to $\mathscr{L}_F^p$, then the function $|f|$ belongs to $\mathscr{L}^p$, and the mapping $f \mapsto |f|$ of $\mathscr{L}_F^p$ into $\mathscr{L}^p$ is uniformly continuous (for the topology of convergence in mean of order $p$).

For every $\varepsilon > 0$ there exists a continuous function $g$ with compact support, such that $N_p(f - g) \leq \varepsilon$; since $||f| - |g|| \leq |f - g|$, we have $N_p(|f| - |g|) \leq \varepsilon$, which proves that $|f| \in \mathscr{L}^p$. On the other hand, if $f_1, f_2$ are two functions in $\mathscr{L}_F^p$ then $N_p(|f_1| - |f_2|) \leq N_p(f_1 - f_2)$, which shows that $f \mapsto |f|$ is a uniformly continuous mapping.