Corollary 2

— If E is a quasi-complete Hausdorff locally convex space, then $\int f d\mu \in E$ for every function f ∈ $\mathscr{K}(X; E)$.

As noted at the beginning of the proof of Prop. 7, f(X) is compact and its closed convex envelope C in E is precompact, hence bounded; but since the set C is closed and bounded, it is complete by hypothesis, and it suffices to apply Prop. 7.

We shall see, in Ch. VI, §1, No. 2, other criteria for $\int f d\mu$ to belong to E, which apply in particular to the functions in $\widetilde{\mathscr{K}}(X; E)$ and not just those in $\mathscr{K}(X; E)$.

Corollary 2 of Proposition 7 may be applied in the following two cases: 1° E is a Banach space; 2° E is the dual of a barreled Hausdorff locally convex space G, and E is equipped with an $\mathscr{S}$-topology, where $\mathscr{S}$ is a covering of G by bounded subsets (TVS, III, §4, No. 2, Cor. 4 of Th. 1). For example, Cor. 2 of Prop. 7 can be applied when E is the weak dual of a Banach space, or a space of measures $\mathscr{M}(Y; \mathbf{C})$ equipped with the vague topology.

If X = $\mathbf{R}$, $\mu$ is Lebesgue measure on $\mathbf{R}$, and E is a Banach space, then the integral $\int f d\mu$ of a function in $\mathscr{K}(X; E)$ is none other than the integral

defined in FRV, II, §2, No. 1; this follows from formula (1), and from FRV, I, §1, No. 2, Cor. of Prop. 2.