Corollary

— For a bounded measure $\mu$ on $X$, every bounded continuous mapping $f$ of $X$ into $F$ belongs to each of the $\mathscr{L}_F^p$ ($1 \leq p < +\infty$).

For every compact subset $K$ of $X$, let $M_K$ be the set of mappings of $X$ into $F$ of the form $hf$, where $h$ is a continuous mapping of $X$ into $[0, 1]$ equal to 1 on $K$ and with compact support. It is clear that the sets $M_K$ form a filter base $\mathscr{B}$ on $\mathscr{L}_F^p$, that the functions belonging to $M_K$ are uniformly bounded, and that $\mathscr{B}$ converges uniformly to $f$ on every compact subset of $X$, whence the corollary.

In particular, the function $f$ is integrable and its integral $\int f\, d\mu$ is the limit with respect to $\mathscr{B}$ of the integrals $\int h f\, d\mu$.

We will obtain anew the Cor. of Prop. 13 as a consequence of a general criterion for integrability in §5, No. 6.

In the notations of Ch. III, §1, No. 2, $|f| \leq \|f\| \cdot 1$ for every function $f \in \mathscr{C}^b(X; F)$, whence, by the formulas (3) and (4) of No. 2,

In particular, for $p = 1$, formula (5) of No. 2 yields

consequently the mapping $f \mapsto \int f\, d\mu$ is continuous on the Banach space $\mathscr{C}^b(X; F)$; its restriction to the closure $\mathscr{C}^0(X; F)$ of $\mathscr{K}(X; F)$ in $\mathscr{C}^b(X; F)$, that is, to the space of continuous functions tending to 0 at the point at infinity (Ch. III, §1, No. 2, Prop. 3), is therefore the extension by continuity of the integral to $\mathscr{C}^0(X; F)$.