Definition 13

— Let $p \in [1, +\infty[$; one denotes by $\overline{\mathscr{L}}^p(T, \mu)$ (resp. $\overline{\mathscr{L}}_F^p(T, \mu)$ if F is a Banach space) the set of mappings $\mathbf{f}$ of T into $\overline{\mathbf{R}}$ (resp. into F) that are $\mu$-measurable and satisfy $\mu^\bullet(|\mathbf{f}|^p) < +\infty$. One denotes by $\mathscr{L}^p(T, \mu)$ (resp. $\mathscr{L}_F^p(T, \mu)$*) the set of $\mu$-moderated elements of* $\overline{\mathscr{L}}^p(T, \mu)$ (resp. $\overline{\mathscr{L}}_F^p(T, \mu)$*).*

We will write $\overline{N}_p(\mathbf{f}) = \left( \mu^\bullet(|\mathbf{f}|^p) \right)^{1/p}$, $N_p(\mathbf{f}) = \left( \mu^*(|\mathbf{f}|^p) \right)^{1/p}$. We denote by $\overline{N}_\infty(\mathbf{f})$ the infimum of the numbers $k \geq 0$ such that $|\mathbf{f}| \leq k$ locally $\mu$-almost everywhere; if $\overline{N}_\infty(\mathbf{f}) < +\infty$, $\mathbf{f}$ is said to be essentially bounded. The set of measurable and essentially bounded mappings of T into $\overline{\mathbf{R}}$ (resp. into F) is denoted $\overline{\mathscr{L}}^\infty(T, \mu)$ (resp. $\overline{\mathscr{L}}_F^\infty(T, \mu)$). The elements of $\overline{\mathscr{L}}_F^1(T, \mu)$ (resp. $\mathscr{L}_F^1(T, \mu)$) are called essentially integrable functions (resp. integrable functions) with values in F.

If $\mu$ is a complex measure, one sets

The above notations are often abbreviated to $\overline{\mathscr{L}}_F^p(\mu)$, $\overline{\mathscr{L}}_F^p$ or $\mathscr{L}^p(\mu)$, $\mathscr{L}^p$, if this does not lead to any confusion.

We saw in No. 8 (Scholium) that one can construct a locally compact space T', having the same underlying set as T and a topology finer than that of T, and equip T' with a measure $\mu'$ such that the $\mu$-measurable functions and the $\mu'$-measurable functions are the same, and such that the essential upper integrals of positive functions for $\mu$ and $\mu'$ are equal. It follows that the sets $\overline{\mathscr{L}}_F^p(\mu)$ and $\overline{\mathscr{L}}_F^p(\mu')$ are identical for $1 \leq p \leq +\infty$ $^{(1)}$. This also implies without new proof that $\overline{\mathscr{L}}_F^p$ is a vector space, and that the function $\overline{N}_p$ is a semi-norm on $\overline{\mathscr{L}}_F^p(\mu)$, for which this space is complete.

Let $\mathbf{f}$ be an element of $\overline{\mathscr{L}}_F^p$ ($1 \leq p < +\infty$); since one has $\mu^\bullet(|\mathbf{f}|^p) = {\mu'}^\bullet(|\mathbf{f}|^p) < +\infty$, Prop. 7 of Ch. V, §1, No. 2 implies that $\mathbf{f}$ is zero outside

(1) Note that the space $\mathscr{L}_F^p(\mu)$ is contained in $\mathscr{L}_F^p(\mu')$, but is in general distinct from it.

the union of a sequence of compact subsets of T' and a locally $\mu'$-negligible set; the latter set being locally $\mu$-negligible, and every compact subset of T' being compact in T, it follows that f is equal locally $\mu$-almost everywhere to a $\mu$-moderated function. Let us denote by $\overline{\mathscr{N}}_F$ (resp. $\mathscr{N}_F$) the space of locally $\mu$-negligible (resp. $\mu$-negligible) functions; we thus have $\overline{\mathscr{L}}_F^p = \mathscr{L}_F^p + \overline{\mathscr{N}}_F$, and $\mathscr{N}_F = \mathscr{L}_F^p \cap \overline{\mathscr{N}}_F$ (No. 9, Cor. 1 of Prop. 14). The space $\overline{\mathscr{L}}_F^p / \overline{\mathscr{N}}_F$ may therefore be canonically identified with $\mathscr{L}_F^p / \mathscr{N}_F$, and one verifies immediately that this identification preserves norm; this quotient space is denoted $L_F^p(\mu)$. It can be interpreted as the normed space associated with each of the semi-normed spaces $\overline{\mathscr{L}}_F^p(\mu)$ and $\mathscr{L}_F^p(\mu)$; since $\overline{\mathscr{L}}_F^p$ is complete, the same is true of $L_F^p$ and $\mathscr{L}_F^p$.

The set of functions f with values in F, continuous on T' with compact support, is dense in $\overline{\mathscr{L}}_F^p(\mu') = \overline{\mathscr{L}}_F^p(\mu)$ (Ch. IV, §3, No. 4, Def. 2). Let us take up again the notations of the Scholium of No. 8. Since a compact subset of T' intersects only a finite number of the compact sets $K_\alpha$, every continuous function f on T' with compact support may be written as a sum

where $f_\alpha$ is, for every $\alpha$, the extension by 0 of a continuous function on $K_\alpha$, where $f_\alpha = 0$ except for a finite number of indices, and where g is locally $\mu$-negligible. We thus have the following result: