Proposition 6

— Let $f$ be a scalarly essentially $\mu$-integrable function with values in $F$, such that $\int f\, d\mu \in F$. For every lower semi-continuous semi-norm $q$ on $F$,

Let $D$ be the set of $z \in F$ such that $q(z) \leq 1$; $D$ is closed, convex, and contains $0$, therefore $D = D^{oo}$ (TVS, II, §6, No. 3, Cor. 3 of Th. 1). It therefore suffices to prove that for every $z' \in D^\circ$, one has $|\langle z', \int f\, d\mu \rangle| \leq \int^\bullet (q \circ f)\, d\mu$; but this follows at once from the fact that $|\langle z', f(t) \rangle| \leq q(f(t))$ for every $t \in T$.

Note that the numerical function $q \circ f$ need not be $\mu$-measurable (Exer. 12).