Exercise 8, § 1

Let us take up again the notations of Exer. 5 of Ch. IV, §1. Recall that

Show that $\mu^*(D) = 0$; deduce from this that $\mu$ is not moderated, even though it is the sum of a series of measures with pairwise disjoint compact supports. Show that there exists a partition $(X_n)_{n \in \mathbf{N}}$ of X into Borel subsets $X_n$ such that $\mu^*(X_n)$ is finite for every $n \in \mathbf{N}$.