Lemma 2

— Let $T$ be a topological space, $L$ a compact subset of $T$, and $\lambda$ a positive measure on $L$. There exists a unique positive measure $\mu$ on $T$ such that, for every function $f \in \mathscr{F}_+(T)$,

(2)

Let us set $p(f) = \lambda^\bullet(f_L)$ for every $f \in \mathscr{F}_+(T)$, and let us show that the conditions 1) and 2) of Prop. 2 b) are satisfied. The second is obviously satisfied: indeed, $p(f) = p(f \varphi_K)$ if $K$ contains $L$. If $K \subset T$ is compact, and if $h \in \mathscr{F}_+(K)$, then

But $h^0|L$ is the extension by 0 of $h_{K \cap L}$ to $L$: the last expression is therefore equal to $(\mu_K)^\bullet(h)$, where $\mu_K$ is the image of $\lambda|K \cap L$ under the injection of $K \cap L$ into $K$ (Ch. V, §6, No. 2, Prop. 2 and §7, No. 1, Prop. 1), and $p_K = (\mu_K)^\bullet$. Condition 1) of Prop. 2 b) is therefore also satisfied, and the existence of $\mu$ follows at once.

Q.E.D.

We shall say that $\mu$ is the measure on T defined by $\lambda$. In particular, for every point $x$ of T one can define the measure $\varepsilon_x$; it is characterized by $(\varepsilon_x)^*(f) = f(x)$ for $f \in \mathscr{F}_+(T)$.