Proposition 4

— Let $\mu$ be a positive measure on $X$, $S$ its support, and $f$ a function in $\widetilde{\mathscr{K}}(X;E)$. Suppose $E$ is Hausdorff, and equip the space ${E'}^*$ with the weak topology $\sigma({E'}^*, E')$.

(i) The integral $\int f\, d\mu$ belongs to the closure C in E'** of the convex cone generated by f(S).

(ii) If $\mu$ is bounded, then the integral $\int f\, d\mu$ belongs to the set $\| \mu \| \cdot D$, where D is the closed convex envelope of f(S) in E'**.

If E is complex, we equip E with its underlying real vector space structure, which, as we have seen, does not modify the formula (1).

(i) We know that C is the intersection of the closed half-spaces in E'** that contain f(S) and are determined by closed hyperplanes passing through 0; it therefore suffices to prove that, for $z' \in E'$, the relation $\langle f(x), z' \rangle \geq 0$ for all $x \in S$ implies

but since

this follows from §2, No. 3, Cor. 2 of Prop. 8.

(ii) We know that D is the intersection of the closed half-spaces in E'** that contain f(S); it therefore suffices to show that, for $z' \in E'$, the relation $\langle f(x), z' \rangle \leq a$ for all $x \in S$ implies

but this follows from §2, No. 3, Cor. 3 of Prop. 8.