Exercise 7, § 3

Let $T$ be a topological space, $\Phi$ a clan of subsets of $T$, and $m$ a relatively bounded additive mapping (cf. Exer. 1) of $\Phi$ into $\mathbf{R}$. Assume that there exists an open covering $\mathscr{U}$ of $T$ such that $\Phi$ consists of the Borel subsets of $T$ contained in the union of a finite number of elements of $\mathscr{U}$. Assume, moreover, that for every $A \in \Phi$ and every $\varepsilon > 0$, there exist a compact subset $K$ of $T$ and an open subset $U$ of $T$ such that $K \subset A \subset U$, $U \in \Phi$ and $|m|(U - K) < \varepsilon$. Show that there exists a measure $\mu$ on $T$ (not necessarily positive) such that $m(A) = \mu^*(A)$ for all $A \in \Phi$, and that such a measure $\mu$ is unique (on introducing the decomposition $m = m_+ - m_-$ of Exer. 1 c), reduce to the case that $m$ takes positive values; first treat the case that $T \in \Phi$ and observe that $m$ is then inner regular; finally, treat the general case by pasting together measures).