Proposition 9

— Suppose that E is Hausdorff; let $f$ be a continuous mapping of X into E with compact support. When the space $\mathscr{M}(X;\mathbf{C})$ is equipped with the topology of strictly compact convergence (§1, No. 10), the mapping $\mu \mapsto \int f d\mu$ of $\mathscr{M}(X;\mathbf{C})$ into $\widehat{E}$ is the unique continuous linear mapping $\Psi$ such that $\Psi(\varepsilon_x) = f(x)$ for all $x \in X$.

For every $z' \in E'$,

whence $\int f\, d\varepsilon_x = f(x)$. We know, moreover, that the set of point measures is total in $\mathscr{M}(X; \mathbf{C})$ for the topology of strictly compact convergence (§ 2, No. 4, Cor. 4 of Th. 1). Thus it all comes down to proving the continuity of the linear mapping $u : \mu \mapsto \int f\, d\mu$. For this, consider the linear mapping $v : z' \mapsto \langle f, z' \rangle$ of $E'$ into $\mathscr{K}(X; \mathbf{C})$, and let us show that the image under $v$ of an equicontinuous subset $H$ of $E'$ is contained in a strictly compact subset of $\mathscr{K}(X; \mathbf{C})$. For, if $K$ is the support of $f$, the functions $\langle f, z' \rangle$ for $z' \in H$ have support contained in $K$; on the other hand, these functions form an equicontinuous set, and for each $x \in X$ the set of $z'(f(x))$ is bounded; our assertion therefore follows from Ascoli’s theorem (GT, X, § 2, No. 5, Cor. 3 of Th. 2). Now, it follows from formula (1) of No. 1 that $u$ is none other than the restriction to $\mathscr{M}(X; \mathbf{C})$ of the transpose $^t v$ (in the algebraic sense); its continuity therefore follows from the foregoing (TVS, IV, § 1, No. 3, Prop. 6).