Exercise 7, § 1

a) Let $(\mu_n)$ be an increasing sequence of measures $\geq 0$ on a locally compact space X; assume that the sequence is bounded above in $\mathscr{M}_+(X)$ and denote by $\mu$ its supremum. Let $f$ be a function $\geq 0$ defined on X and zero on the complement of a countable union of compact sets. Show that $\mu^*(f) = \sup_n \mu_n^*(f)$ (cf. Exer. 6 b)).

b) Let X and $\mu$ be the locally compact space and the measure defined in Exer. 5. Let $\alpha_n$ be the numerical function equal to $\alpha$ (in the notation of Exer. 5) for every point $(1/m, k/m^2)$ such that $m \leq n$, and equal to 0 at the other points of X; let $\mu_n$ be the measure defined by the masses $\alpha_n(x)$. Show that $\mu$ is the supremum of the sequence $(\mu_n)$ in $\mathscr{M}_+(X)$ and that $\mu_n^*(D) = 0$ for all $n$, but $\mu^*(D) = +\infty$.