Proposition 14

Let T be a locally compact space, F a quasi-complete, Hausdorff locally convex space, and f a continuous mapping of T into F. If the space $\mathscr{C}'(T)$ of measures on T with compact support is equipped with the topology of uniform convergence in the compact subsets of $\mathscr{C}(T)$, then the mapping $\lambda \mapsto \int f d\lambda$ is the unique continuous linear mapping $\tilde{f}$ of $\mathscr{C}'(T)$ into F such that $\tilde{f}(\varepsilon_t) = f(t)$ for every $t \in T$.

To establish the uniqueness of the extension, it suffices to see that the point measures $\varepsilon_t$ form a total set in $\mathscr{C}'(T)$; since the dual of $\mathscr{C}'(T)$ is $\mathscr{C}(T)$ (TVS, IV, §1, No. 1, Th. 1), it suffices to observe that every function $g \in \mathscr{C}(T)$ that is orthogonal to all of the measures $\varepsilon_t$ is equal to 0 by definition (TVS, IV, §1, No. 2, Prop. 2).

Let us now show that $\lambda \mapsto \int f d\lambda$ is continuous. Let V be a closed, balanced convex neighborhood of 0 in F; it suffices to prove that there exists a relatively compact subset L of $\mathscr{C}(T)$ such that the relations $\lambda \in L^\circ$ and $z' \in V^\circ$ imply $|\langle \int f d\lambda, z' \rangle| \leq 1$, or again that $|\int \langle f, z' \rangle d\lambda| \leq 1$. To this end, we are going to show that as $z'$ runs over $V^\circ$, the set L of numerical functions $\langle f, z' \rangle$ is relatively compact in $\mathscr{C}(T)$. Since $V^\circ$ is bounded for $\sigma(F', F)$, the supremum of the numbers $|\langle f(t), z' \rangle|$, for $t \in T$ fixed and $z'$ running over $V^\circ$, is finite; by virtue of Ascoli’s theorem (GT, X, §2, No. 5, Cor. 2 of Th. 2), it therefore suffices to show that the set of $\langle f, z' \rangle$ ($z' \in V^\circ$) is equicontinuous. But, for every $t_0 \in T$ and every $\delta > 0$, there exists by hypothesis a neighborhood W of $t_0$ in T such that $f(t) - f(t_0) \in \delta V$ for all $t \in W$; it follows that $|\langle f(t), z' \rangle - \langle f(t_0), z' \rangle| \leq \delta$ for all $t \in W$ and all $z' \in V^\circ$, which completes the proof.