Proposition 12

— Let $a_i$ ($1 \leq i \leq n$) be distinct points in a locally compact space $X$. Every measure on $X$ whose support is contained in the set of the $a_i$ is a linear combination of the measures $\varepsilon_{a_i}$ ($1 \leq i \leq n$).

For, such a measure $\mu$ is zero for every function $f \in \mathscr{K}(X; \mathbf{C})$ satisfying the $n$ relations $f(a_i) = 0$ (No. 3, Prop. 8); since these relations may be written $\varepsilon_{a_i}(f) = 0$, $\mu$ is a linear combination of the $\varepsilon_{a_i}$ (A, II, §7, No. 5, Cor. 1 of Th. 7).

In particular, every measure whose support is either empty or reduced to a single point $x$ is of the form $\alpha \varepsilon_x$, where $\alpha$ is a complex number; such a measure is said to be a point measure; thus every measure whose support is finite is a sum of point measures.