Exercise 22, § 5

In the Banach space $E = \mathscr{M}^1(T)$ of bounded measures on a locally compact space $T$, let $H$ be a compact set for the weakened topology.

a) Show that, for every $\varepsilon > 0$, there exists a positive measure $\mu_\varepsilon$ on $T$ such that every measure $\nu \in H$ is the sum of a positive measure with base $\mu_\varepsilon$ and a measure $\lambda$ alien to $\mu_\varepsilon$ and of norm $\leq \varepsilon$. (Make use of the fact that the space $E$, equipped with its norm and its order structure, is isomorphic to a space $L^1(S, \rho)$, where $S$ is the union of compact Stone spaces and $\rho$ is a positive measure on $S$ (Ch. IV, §4, Exer. 10), and apply in $L^1(S, \rho)$ the criterion $\alpha_1$ of Exer. 15.)

b) Deduce from a) that there exists a positive measure $\mu$ on $T$ such that all of the measures $\nu \in H$ have base $\mu$ (use Exer. 4 b)).