Exercise 9, § 1

Let $G, H$ be two Hausdorff locally convex spaces, $U$ a mapping of $T$ into $F = \mathscr{L}_s(G;H)$. Assume that $G$ is barreled, $H$ is quasi-complete, $U$ is $\mu$-measurable and $U(K)$ is bounded for every compact subset $K$ of $T$; assume, moreover, that for every continuous semi-norm $q$ on $H$ and every $x \in G$, $\int^* q(U(t)\cdot x)\,d\mu(t) < +\infty$. Under these conditions, show that $U$ is scalarly essentially $\mu$-integrable and that $\int U(t)\,d\mu(t) \in F$ (make use of Exercise 8 b)).