Uniform absolute continuity in spaces of set functions

Abstract

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