Corollary

Let $\mu$ be a positive measure on X, $f$ a mapping belonging to $\mathscr{K}(X;E)$, and D the closed convex envelope of f(X) in E'**. There exists a number $a > 0$ such that $\int f\, d\mu \in a \cdot D$.

If $\nu$ is any measure on X, there exist numbers $a_1, a_2, a_3, a_4 > 0$ such that $\int f\, d\nu \in a_1 D - a_2 D + i a_3 D - i a_4 D$.

Suppose first that $\mu$ is positive; by hypothesis, the support K of f is compact; if $\nu$ is the restriction of $\mu$ to a relatively compact open neighborhood of K, then $\nu$ is bounded and $\int f\, d\mu = \int f\, d\nu \in \| \nu \| \cdot D$ by Prop. 4, (ii). The second result follows from this, since any complex measure may be written as $\mu_1 - \mu_2 + i \mu_3 - i \mu_4$, where the $\mu_j$ are positive.