Corollary

— Let $\mu$ be a bounded positive measure on $T$, $S$ a $\mu$-measurable set carrying $\mu$, and $f$ a mapping of $T$ into $F$, scalarly $\mu$-measurable and such that $f(S)$ is contained in a weakly compact convex subset $A$ of $F$. Then $f$ is scalarly $\mu$-integrable, and $\int f\, d\mu \in \mu(T)A \subset F$.

For every $z' \in F'$, $\langle z', f \rangle$ is $\mu$-measurable and bounded in $S$, hence integrable, which proves that $f$ is scalarly integrable. Moreover, since $A$ is compact in $F_\sigma$, it is closed in $F'$*, and the closed convex envelope of $f(S)$ in $F'$* is contained in $A$, whence the corollary.