Corollary

— Let X be a locally compact space with a countable base, $\Psi$ the set of Borel sets of X, $\beta$ a mapping of $\Psi$ into $[0, +\infty]$ satisfying the following conditions:

(i) If $(B_1, B_2, \ldots)$ is a sequence of pairwise disjoint Borel sets of X, then $\beta(B_1 \cup B_2 \cup \ldots) = \beta(B_1) + \beta(B_2) + \ldots$.

(ii) If B is a compact subset of X, then $\beta(B) < +\infty$.

Then, there exists one and only one positive measure $\mu$ on X such that $\beta(B) = \mu^*(B)$ for all $B \in \Psi$.

Let $\Phi$ be the set of compact subsets of $X$ and let $\alpha$ be the restriction of $\beta$ to $\Phi$. The conditions (PC$_I$), (PC$_{II}$), (PM$_I$), (PM$_{II}$), (PM$_{III}$) and (PM$_{IV}'$) are then satisfied. Let $K$ be a compact subset of $X$, and $\varepsilon > 0$. Then $K$ is the intersection of a decreasing sequence $(U_1, U_2, \ldots)$ of relatively compact open sets of $X$ (GT, IX, §2, No. 5, Prop. 7). We have $\sum_{n=1}^{\infty} \beta(U_n - U_{n+1}) = \beta(U_1 - K) < +\infty$, therefore

tends to 0 as $n$ tends to $\infty$. This proves that the condition $(\mathrm{PM}'_{\mathrm{IV}})$ is satisfied. By Th. 5, there exists a positive measure $\mu$ on $X$ such that $\mu(K) = \alpha(K)$ for every compact subset $K$ of $X$. Since every open set $U$ of $X$ is the union of an increasing sequence of compact subsets, we have $\mu^*(U) = \beta(U)$. Let $L$ be a compact subset of $X$. By Prop. 7 of No. 5, the $\mu$-integrable subsets of $L$ form a tribe of subsets of $L$. Therefore, if $B$ is an element of $\Psi$ contained in $L$, then $B$ is $\mu$-integrable; for every $\varepsilon > 0$, there then exist a compact set $K$ and an open set $U$ in $X$ such that $K \subset B \subset U$ and $\mu^*(U) - \mu(K) \leq \varepsilon$ (No. 6, Th. 4). Since $\beta(U) = \mu^*(U)$ and $\beta(K) = \mu(K)$, we see that $|\mu^*(B) - \beta(B)| \leq 2\varepsilon$. Therefore $\beta(B) = \mu^*(B)$. Finally, every Borel set $C$ of $X$ is the union of a sequence of pairwise disjoint, relatively compact Borel sets, whence $\beta(C) = \mu^*(C)$. The uniqueness of $\mu$ follows at once from Th. 5.