Exercise 14, § 1

a) Let $X$ be a locally compact space with a countable base. Show that the space $\mathscr{M}_+(X)$, equipped with the vague topology, is metrizable.

b) Let $L$ be the set of increasing sequences $(k_n)_{n \in \mathbf{Z}}$ of integers $\geqslant 0$, tending to $+\infty$ with $n$, and zero for $n \leqslant 0$ except for a finite number of values of $n$; let $P$ be the set of equivalence classes formed by the translates $(k_{n+h})_{n \in \mathbf{Z}}$ of a same sequence $(k_n)_{n \in \mathbf{Z}}$ (where $h$ runs over $\mathbf{Z}$); $L$ and $P$ are uncountable. Let $X$ be the discrete topological sum space of $P$ and $\mathbf{Z}$; let $H$ be the set of positive measures on $X$ defined in the following way: for every $\alpha \in P$ and every sequence $g \in \alpha$, $\nu_{\alpha, g}$ is the measure defined by the mass $+1$ at the point $\alpha$, mass $0$ at the other points of $P$, and mass $g(n)$ at every point $n \in \mathbf{Z}$; take $H$ to be the cone generated by the measures $\nu_{\alpha, g}$. Now let $\mu$ be the measure on $X$ defined by the mass $+1$ at each point of $P$ and mass $0$ at each point of $\mathbf{Z}$. Show that $\mu$ is in the vague closure of $H$ but is not in the vague closure of any vaguely bounded subset of $H$. (Make use of the fact that for every mapping $f$ of $\mathbf{Z}$ into $\mathbf{N}$, there exists an increasing sequence $(k_n) \in L$ such that for every $h \in \mathbf{Z}$, $\lim_{n \to \infty} k_{n+h}/f(n) = +\infty$.)