Proposition 9

For a mapping f of T into F to belong to $\overline{\mathscr{F}}_F^p$ (resp. to $\overline{\mathscr{L}}_F^p$), if is necessary and sufficient that (resp. that f be measurable and that)

One then has $\overline{N}_p(f) = (\mu^\bullet(|f|^p))^{1/p}$.

We may clearly limit ourselves to the assertion concerning $\overline{\mathscr{F}}_F^p$. If f belongs to $\overline{\mathscr{F}}_F^p$, let g be a function belonging to $\mathscr{F}_F^p$ that is equal to f locally almost everywhere; then $|f|^p = |g|^p$ locally almost everywhere, therefore

(Prop. 1, a) and Prop. 7), and, on the other hand, by the definition of $\overline{N}_p$,

Conversely, suppose that $\mu^\bullet(|f|^p) < +\infty$; then there exists a moderated set A such that f is zero locally almost everywhere in T − A (Prop. 7). The function $f \varphi_A$, equal locally almost everywhere to f, is such that $N_p(f \varphi_A) = \overline{N}_p(f) < +\infty$, therefore it belongs to $\mathscr{F}_F^p$, and $f \in \overline{\mathscr{F}}_F^p$.