Proposition 14

— Let X be a locally compact space, F a Banach space not reduced to 0; equip the space $\mathscr{C}(X; F)$ of all continuous mappings of X into F with the topology of compact convergence. For a measure $\mu$ on X to be such that the linear mapping $f \mapsto \int f d\mu$ of $\mathscr{K}(X; F)$ into F is extendible to a continuous linear mapping of $\mathscr{C}(X; F)$ into F, it is necessary and sufficient that $\operatorname{Supp}(\mu)$ be compact; such an extension is unique and coincides with the integral defined in No. 1.

We have just seen that if $\mu$ has compact support, then the integral $\int f d\mu$ is defined for every function $f \in \mathscr{C}(X; F)$ and that the mapping $f \mapsto \int f d\mu$ of $\mathscr{C}(X; F)$ into F is continuous for the topology of compact convergence. Conversely, suppose that $f \mapsto \int f d\mu$ is continuous in $\mathscr{K}(X; F)$ for the topology of compact convergence. Then, there is a compact set $K \subset X$ and a number $a > 0$ such that $|\mu(f)| \leq a \cdot \sup_{x \in K} |f(x)|$ for every function $f \in \mathscr{K}(X; F)$; in particular, if the support of $g \in \mathscr{K}(X; F)$ does not intersect K, then $\mu(g) = 0$. Taking $g = h a$, where $a \neq 0$ is a vector in F and $h \in \mathscr{K}(X; \mathbf{C})$, we see that $\mu(h) = 0$ for every function $h \in \mathscr{K}(X; \mathbf{C})$ whose support does not intersect K, which proves that $\operatorname{Supp}(\mu) \subset K$. Finally, the uniqueness of the extension follows from the fact that $\mathscr{K}(X; F)$ is dense in $\mathscr{C}(X; F)$ for the topology of compact convergence (Ch. III, §1, No. 2, Prop. 4).

Prop. 14 permits identifying a measure on X with compact support with its continuous extension to $\mathscr{C}(X; \mathbf{C})$. The set of measures on X with compact support may therefore be identified with the dual $\mathscr{C}'(X; \mathbf{C})$ of the Hausdorff locally convex space $\mathscr{C}(X; \mathbf{C})$. Recall that $\mathscr{C}(X; \mathbf{C})$ is complete (GT, X, §1, No. 6, Cor. 3 of Th. 2), but it is not necessarily barreled (Exer. 17). However, if X is countable at infinity, hence is the union of an increasing sequence of compact sets $K_n$ such that $K_n \subset \overset{\circ}{K}_{n+1}$, then the topology of $\mathscr{C}(X; \mathbf{C})$ can be defined by the countable family of semi-norms $p_n(f) = \sup_{x \in K_n} |f(x)|$, therefore $\mathscr{C}(X; \mathbf{C})$ is a Fréchet space in this case. Consequently, for every covering $\mathscr{G}$ of $\mathscr{C}(X; \mathbf{C})$ by bounded sets, the space $\mathscr{C}'(X; \mathbf{C})$ is then quasi-complete for the $\mathscr{G}$-topology (TVS, III, §4, No. 2, Cor. 4 of Th. 1).

We shall consider above all on $\mathscr{C}'(X; \mathbf{C})$ the topology of compact convergence (the topology of uniform convergence on the compact subsets of $\mathscr{C}(X; \mathbf{C})$). Recall that the relatively compact subsets H of $\mathscr{C}(X; \mathbf{C})$ are characterized by the following properties (GT, X, §2, No. 5, Cor. 3 of Th. 2):

$1^\circ$ H is equicontinuous;

$2^\circ$ for every $x \in X$, the set $H(x)$ of the $f(x)$, where $f$ runs over H, is bounded in $\mathbf{C}$.