Proposition 16

There exists one and only one continuous linear mapping $f \mapsto \int f d\mu$, of the space $\overline{\mathscr{L}}_F^1(\mu)$ into F, having the following property: If f is of the form $t \mapsto g(t)a$, where $a \in F$, and where g is a positive function, finite, $\mu$-measurable and such that $\mu^*(g) < +\infty$, then $\int f d\mu = \mu^*(g) \cdot a$.

For, the semi-normed spaces $\overline{\mathscr{L}}_F^1(\mu)$ and $\overline{\mathscr{L}}_F^1(\mu')$ are identical. Since $\mu^* = {\mu'}^*$, the mapping $f \mapsto \int f d\mu'$ satisfies the conditions of the statement. On the other hand, the set of functions of the form $f = g \cdot a$ considered in the statement is total in $\overline{\mathscr{L}}_F^1(\mu')$ (Ch. IV, §3, No. 5, Prop. 10), whence uniqueness.

One says that $\int f\, d\mu$ is the integral of $f$ with respect to $\mu$, and this vector is also denoted $\mu(f)$ or $\int f(t)\, d\mu(t)$.

Since $\int f\, d\mu = \int f\, d\mu'$ for every essentially integrable function $f$ with values in $F$, all of the theory of the essential integral extends to measures on Hausdorff spaces, without new proofs; from it, one deduces results relative to the ordinary integral by restricting oneself to moderated functions. We cite in particular the following results:

— Th. 3 of Ch. IV, §3, No. 4, its extension to $\overline{\mathscr{L}}_F^p$, and its two corollaries. — Th. 4 of Ch. IV, §3, No. 5 (composition with a continuous linear mapping) and its corollaries; Props. 9, 11 and 12 of the same No. — All of the results of Ch. IV, §3, No. 6, relative to the ordered vector space structure of $L^p$. — All of the results of Ch. IV, §3, No. 7, and in particular Lebesgue’s theorem. — All of the results of Ch. IV, §3, No. 8, on the relations between the spaces $L_F^p$. — Theorem 2 of Ch. IV, §4, No. 3 (the statement of Lebesgue’s theorem specific to $L_F^1$). — Hölder’s inequality (Ch. IV, §6, No. 4, Th. 2) and its corollaries. — The relations between the spaces $L_F^p$ established in Ch. IV, §6, No. 5. — The results on the duality of the spaces $L^p$ established in Ch. V, §5, No. 8. — The Dunford–Pettis theorem (Ch. VI, §2, No. 5, Th. 1), its Corollaries 1 and 2, and Prop. 10 of Ch. VI, §2, No. 6 (dual of $L_F^1$).