Exercise 8, § 1

a) Let $(\mu_n)$ be a decreasing sequence of measures $\geq 0$ on a locally compact space X, and let $\mu$ be the infimum of the sequence in $\mathscr{M}_+(X)$. Show that if $f$ is a function $\geq 0$ such that $\mu_n^*(f) < +\infty$ from some index onward, then $\mu^*(f) = \inf_n \mu_n^*(f)$ (when $g$ is lower semi-continuous, positive, and satisfies $\mu^*(g) < +\infty$, note that there exists a sequence $(h_m)$ of continuous functions $\geqslant 0$ with compact support, such that

b) On the discrete space $X = \mathbf{N}$, let $\mu_n$ be the measure defined by placing mass +1 at each point $m \geqslant n$. Show that the infimum of the decreasing sequence $(\mu_n)$ in $\mathscr{M}_+(X)$ is 0, but that $\mu_n^*(X) = +\infty$ for all $n$.