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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.

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Uniform absolute continuity in spaces of set functions
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by James D. Stein PDF
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 $\mu \in K$.
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Additional Information
  • © Copyright 1975 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 51 (1975), 137-140
  • MSC: Primary 28A32
  • DOI: https://doi.org/10.1090/S0002-9939-1975-0440012-0
  • MathSciNet review: 0440012