Exercise 17, § 3

Let $T$ be a topological space. a) Let $m$ be a countably additive mapping of $\mathscr{B}(T)$ into $\mathbf{R}_+$. Assume that there exists a sequence $(T_n)_{n \in \mathbf{N}}$ of Borel subsets of $T$ such that $T = \bigcup_{n \in \mathbf{N}} T_n$, $m(T_0) = 0$ and the subspace $T_n$ of $T$ is Radon for every $n \geq 1$. Show that there exists a bounded positive measure $\mu$ on $T$ such that $m(A) = \mu^*(A)$ for every Borel subset $A$ of $T$ (one can reduce to the case that the $T_n$ are pairwise disjoint on applying the Cor. of Prop. 2 of No. 3). b) Extend the preceding result to the case that the sets $T_n$ for $n \geq 1$ are no longer assumed Borel but merely universally measurable in $T$ (argue as in Prop. 2 of No. 3). c) Every topological space that is the union of a sequence of metrizable compact subspaces is Radon.