Uniform Absolute Continuity in Spaces of Set Functions

Abstract

Let $X$ be a regular topological space, $K$ a collection of bounded regular measures defined on the Borel sets of $X$. The following conditions are equivalent. (1) Let $M(X)$ denote the Borel measures, $M{(X)^ + }$ the nonnegative members of $M(X)$. There is a $\lambda \in M{(X)^ + }$ such that $K$ is uniformly $\lambda$-continuous. (2) If $\{ {U_n}|n = 1,2, \ldots \}$ is a disjoint sequence of open sets, then ${\lim _{{n^{ \to \infty }}}}\mu ({U_n}) = 0$ uniformly for $\mu \in K$. (3) If $E$ is a Borel subset of $X$ and $\epsilon > 0$, there is a compact set $F \subseteq E$ such that $|\mu |(E \sim F) < \epsilon$ for $\mu \in K$. (4) If $\{ {E_n}|n = 1,2, \ldots \}$ is a disjoint sequence of Borel sets, then ${\lim _{n \to \infty }}\mu ({E_n}) = 0$ uniformly for $\mu \in K$.