Proposition 5

— Let $\mu$ be a bounded positive measure on $T$, $S$ a $\mu$-measurable set carrying $\mu$ (Ch. V, §5, No. 7, Def. 4), $f$ a scalarly $\mu$-integrable () function with values in $F$. Let $D$ be the closed convex envelope of $f(S)$ in the space $F'$ equipped with the topology $\sigma(F', F')$. Then $\int f\, d\mu \in \mu(T)D$.

Since $D$ is the intersection of the closed half-spaces containing $f(S)$ (TVS, II, §5, No. 3, Cor. 1 of Prop. 4), it suffices to prove that the relation $\langle f(t), z' \rangle \leq a$ for all $t \in S$ (where $z' \in F', a \in \mathbf{R}$) implies $\langle z', \int f\, d\mu \rangle \leq a \cdot \mu(T)$; but since $\int f\, d\mu = \int_S f\, d\mu$, this follows from Ch. IV, §4, No. 2, Cor. 1 of Th. 1.