Proposition 13

Let X be a locally compact space and, for every $x \in X$, let $\varepsilon_x$ be the Dirac measure at the point x. The mapping $x \mapsto \varepsilon_x$ is a homeomorphism of X onto a subspace of the space $\mathscr{M}(X; \mathbf{C})$ of measures on X, equipped with the vague topology. Moreover, if $X'$ denotes the compact space obtained by adjoining to X a point at infinity $\omega$, then $\varepsilon_x$ tends to 0 as x tends to $\omega$.

For every function $f \in \mathscr{K}(X; \mathbf{C})$, $\langle f, \varepsilon_x \rangle = f(x)$; since $f$ is continuous, this proves that the mapping $x \mapsto \varepsilon_x$ is continuous. If $x, y$ are two distinct points of X, there exists a function $f \in \mathscr{K}(X; \mathbf{C})$ such that $f(x) = 1, f(y) = 0$ (No. 2, Lemma 1), which proves that $\varepsilon_x \neq \varepsilon_y$; the mapping $x \mapsto \varepsilon_x$ is therefore injective. Moreover, for every function $f \in \mathscr{K}(X; \mathbf{C})$, $\langle f, \varepsilon_x \rangle$ tends to 0 by definition as $x$ tends to $\omega$, therefore $x \mapsto \varepsilon_x$ may be extended by continuity to $X' = X \cup \{ \omega \}$ by assigning to it the value 0 at the point $\omega$. This extended mapping is also injective, since $\varepsilon_x \neq 0$ for all $x \in X$. It is therefore a homeomorphism of the compact space $X'$ onto a subspace of $\mathscr{M}(X; \mathbf{C})$, since $\mathscr{M}(X; \mathbf{C})$ is Hausdorff for the vague topology (GT, I, §9, No. 4, Cor. 2 of Th. 2).