Exercise 6, § 5

Let T be a topological space and H a set of positive measures on T; assume that the following condition is fulfilled: (V) Every point of T admits an open neighborhood W such that $\sup_{\mu \in H} \mu^*(W)$ is finite.

Finally, let $\mathscr{U}$ be an ultrafilter on H.

a) Let K be a compact subset of T. Show that the induced measures $\mu_K$ ($\mu \in H$) converge vaguely with respect to $\mathscr{U}$ to a measure $\pi^K$ on K.

b) Let K and L be two compact subsets of T such that $K \subset L$; show that $(\pi^L)_K \geq \pi^K$.

c) For every compact subset K of T, let $i^K$ be the canonical injection of K into T. Show that the family of measures $i^K(\pi^K)$ admits a supremum $\pi$ in $\mathscr{M}_+(T)$.

d) Let $f$ be a lower semi-continuous positive function on T, and $g$ an upper semi-continuous positive function on T with compact support. Show that $\pi^*(f) \leq \lim_{\mu, \mathscr{U}} \mu^*(f)$ and $\pi^*(g) \geq \lim_{\mu, \mathscr{U}} \mu^*(g)$.