Proposition 10

— Let $U$ be a mapping of $T$ into $F = \mathscr{L}_s(G; H)$. In each of the following three cases, $U$ is scalarly essentially $\mu$-integrable and $\int U d\mu \in \mathscr{L}_s(G; H)$:

a) $H$ is quasi-complete, $\mu$ is bounded and, if $S$ is its support, $U$ is $\mu$-measurable and $U(S)$ is equicontinuous.

b) $H$ is semi-reflexive, $\mu$ is bounded and, if $S$ is its support, $U$ is scalarly $\mu$-measurable and $U(S)$ is equicontinuous.

c) $H$ is semi-reflexive, $G$ is barreled, $U$ is scalarly essentially $\mu$-integrable, and, for every compact subset $K$ of $T$, $U(K)$ is bounded.

The fact that $U$ is scalarly essentially integrable is obvious in all three cases; by virtue of Prop. 9 and its corollary, it suffices in each case to verify condition a) of Prop. 9. Now, this condition follows from Prop. 8 of No. 2 in the first case, and from the Cor. of Prop. 7 of No. 2 in the other two cases.