Corollary 1

— Let E be a Hausdorff locally convex space; for every function f ∈ $\mathscr{K}(X; E)$, $\int f d\mu$ belongs to the completion $\widehat{E}$ of E.

Since the duals of E and $\widehat{E}$ are identical, it suffices to apply Prop. 7 while regarding f as taking its values in $\widehat{E}$.