Proposition 7

Let $E$ be a Hausdorff locally convex space, and $f \in \mathscr{K}(X; E)$. If $f(X)$ is contained in a complete convex subset $A$ of $E$, then $\int f d\mu \in E$.

Let K be the support of f, which is compact by hypothesis. Since f is zero on X − K, f(X) is equal to f(K) or to f(K) ∪ {0}, therefore is compact since f is continuous and E is Hausdorff; the closed convex envelope C of f(X) in E is then precompact (for the uniform structure induced by that of E) (TVS, II, §4, No. 1, Prop. 3). But since C is a closed subset of the complete space A, C is complete and therefore compact; a fortiori, C is compact for the weakened topology σ(E, E′); but since that topology is induced by σ(E′*, E′), C is the closed convex envelope of f(X) in E′* for the topology σ(E′*, E′); the proof is therefore concluded by the Corollary of Prop. 4 of No. 2.