Exercise 13, § 4

a) Let $X$ be a locally compact space, and let $\Gamma$ be a base for the topology of $X$ consisting of relatively compact sets. Let $\mathscr{U}$ be a uniform structure compatible with the topology of $X$, and let $\mathscr{S}$ be a fundamental system of entourages for this structure. Let $M \mapsto \lambda(M)$ be a finite numerical function $\geqslant 0$ defined on $\Gamma$. For every compact set $K \subset X$ and every entourage $V \in \mathscr{S}$, let $\alpha_V(K)$ be the infimum of the numbers $\sum_i \lambda(U_i)$ for all finite coverings $(U_i)$ of $K$ formed by sets in $\Gamma$ that are small of order $V$; suppose that as $V$ runs over $\mathscr{S}$, the supremum $\alpha(K)$ of the numbers $\alpha_V(K)$ is finite. Show that there exists a measure $\mu$ on $X$ (and only one) such that $\mu(K) = \alpha(K)$ for every compact set $K$ (make use of Th. 5).

b) Let $\Psi$ be the set of Borel subsets of $X$, $\alpha$ a mapping of $\Psi$ into $[0,+\infty]$ satisfying the following conditions: (i) If $B_1, B_2$ are two disjoint Borel subsets of $X$, then $\alpha(B_1 \cup B_2) = \alpha(B_1) + \alpha(B_2)$. (ii) If $B$ is a compact subset of $X$ then $\alpha(B) < +\infty$. (iii) If $B$ is a Borel subset of $X$ then $\alpha(B) = \inf \alpha(U)$, where $U$ runs over the set of open subsets of $X$ containing $B$.

Then, there exists one and only one positive measure $\mu$ on $X$ such that $\alpha(B) = \mu^*(B)$ for every Borel subset $B$ of $X$ that can be covered by a sequence of compact sets.

c) For every $B \in \Phi$, set $\alpha(B) = 0$ if $B$ can be covered by a sequence of compact sets, and $\alpha(B) = +\infty$ in the contrary case. Then, the conditions (i), (ii), (iii) of b) are satisfied. One has $\mu = 0$, therefore $\alpha(B) \neq \mu^*(B)$ if $B$ cannot be covered by a sequence of compact sets.