Lemma

Let $g$ and $g'$ be two $\mu$-moderated mappings with values in $F$; if $g$ and $g'$ are equal locally almost everywhere to a same function $f$, then $g = g'$ almost everywhere.

For, let $D$ be the set of $t \in T$ such that $g(t) \neq g'(t)$; $D$ is locally negligible and moderated, therefore negligible (Cor. 1 of Prop. 7).

We shall denote by $\overline{\mathscr{F}}_F^p(T, \mu)$ (or simply $\overline{\mathscr{F}}_F^p(\mu)$, $\overline{\mathscr{F}}_F^p$, if no confusion can result) the set of mappings $f$ of $T$ into $F$, such that there exists a function $g \in \mathscr{F}_F^p$ equal to $f$ locally almost everywhere. Since the number $N_p(g)$ depends only on $f$ by the Lemma, we will write $\overline{N}_p(f) = N_p(g)$. The function $\overline{N}_p$ is obviously a semi-norm on $\overline{\mathscr{F}}_F^p$, and we shall always assume that $\overline{\mathscr{F}}_F^p$ is equipped with the topology defined by $\overline{N}_p$. The closure of 0 for this topology is the space $\mathscr{N}_F^\infty$; the relations $\overline{\mathscr{F}}_F^p = \mathscr{F}_F^p + \mathscr{N}_F^\infty$, $\mathscr{N}_F^\infty \cap \mathscr{F}_F^p = \mathscr{N}_F$ (Lemma) show that the normed space $\overline{\mathscr{F}}_F^p / \mathscr{N}_F^\infty$ may be canonically identified with $\mathscr{F}_F^p / \mathscr{N}_F$, which is complete (Ch. IV, §3, No. 3, Prop. 5); therefore $\overline{\mathscr{F}}_F^p$ is itself complete.

We shall similarly denote by $\overline{\mathscr{L}}_F^p(T, \mu)$ (or $\overline{\mathscr{L}}_F^p(\mu)$, or $\overline{\mathscr{L}}_F^p$) the subspace $\mathscr{L}_F^p + \mathscr{N}_F^\infty$ of $\overline{\mathscr{F}}_F^p$; one can also characterize $\overline{\mathscr{L}}_F^p$ as the subspace of $\overline{\mathscr{F}}_F^p$ constituted by the measurable mappings (Ch. IV, §5, No. 6, Th. 5).

The normed space $\overline{\mathscr{L}}_F^p / \mathscr{N}_F^\infty$ may be canonically identified with $L_F^p$; $\overline{\mathscr{L}}_F^p$ is therefore complete. Its elements are called the p-th power essentially integrable functions, this terminology being justified by the following proposition: