Proposition 15

Let $T$ be a locally compact space, $F$ a Hausdorff locally convex space that is quasi-complete, and $f$ a continuous mapping of $T$ into $F$ such that $f(T)$ is bounded. If $\mathscr{M}^1(T)$ is equipped with its Banach space topology, then the linear mapping $\lambda \mapsto \int f\, d\lambda$ of $\mathscr{M}^1(T)$ into $F$ is continuous.

For every closed, balanced, convex neighborhood $V$ of 0 in $F$, there exists a $\rho > 0$ such that $f(T) \subset \rho V$; the closed, balanced, convex envelope $B$ of $f(T)$ is therefore contained in $\rho V$, and it is complete by hypothesis. If then $\| \lambda \| \leq 1/\rho$, it follows from No. 2, Prop. 8, and the relation $\| \lambda \| = \lambda^+(T) + \lambda^-(T)$, that $\int f\, d\lambda \in B/\rho \subset V$.