# Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

## Uniform absolute continuity in spaces of set functionsHTML articles powered by AMS MathViewer

by James D. Stein
Proc. Amer. Math. Soc. 51 (1975), 137-140 Request permission

## 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 $\mu \in K$.
References
Similar Articles
• Retrieve articles in Proceedings of the American Mathematical Society with MSC: 28A32
• Retrieve articles in all journals with MSC: 28A32