Exercise 7, § 5

Let H be a set of bounded positive measures on a topological space T. Assume that $\sup_{\mu \in H} \mu^\bullet(T)$ is finite, and that for every $\varepsilon > 0$, there exists a compact subset K of T such that $\sup_{\mu \in H} \mu^\bullet(T - K) < \varepsilon$. Show that the condition (V) of the preceding exercise is verified. Let $\mathscr{U}$ be an ultrafilter on H. The measure $\pi$ is defined as in the preceding exercise.

a) Show that $\pi^\bullet(g) \geq \lim_{\mu, \mathscr{U}} \mu^\bullet(g)$ for every upper semi-continuous bounded positive function g on T. Deduce from this that $\pi^\bullet(f) = \lim_{\mu, \mathscr{U}} \mu^\bullet(f)$ for every bounded function f on T whose set of points of discontinuity is $\pi$-negligible.

b) When T is completely regular, deduce from a) that H is relatively compact for the tight topology (this furnishes a new proof of Th. 1 of No. 5 for the case of positive measures).