Theorem 1

— Let $X$ be a locally compact space, $\mu$ a positive measure on $X$, $F$ a real Banach space, $D$ a closed convex set in $F$, and $f$ a function on $X$ such that $f(X) \subset D$. For every non-negligible integrable numerical function $g \geqslant 0$ such that $fg$ is integrable, the point $\frac{\int fg\, d\mu}{\int g\, d\mu}$ belongs to $D$.

For, let $F'$ be the dual of $F$ and let $\langle z, a' \rangle \leqslant \alpha$ ($a' \in F', \alpha \in \mathbf{R}$) be a relation defining a closed half-space containing $D$. Since $fg$ is integrable, so is the numerical function $\langle fg, a' \rangle = \langle f, a' \rangle g$, and

(§ 4, No. 2, Cor. 1 of Th. 1); but, by hypothesis, $\langle f(x), a' \rangle \leqslant \alpha$ for all $x \in X$, therefore $\langle f(x)g(x), a' \rangle \leqslant \alpha g(x)$; on integrating, we have

This proves that the point $\frac{\int fg\, d\mu}{\int g\, d\mu}$ belongs to every closed half-space containing $D$; but, by the Hahn–Banach theorem, $D$ is the intersection of the closed half-spaces containing it (TVS, II, §5, No. 3, Cor. 1 of Prop. 4), whence the theorem.