Uniform absolute continuity in spaces of set functions
James D. Stein
Proc. Amer. Math. Soc. 51 (1975), 137-140
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Abstract: Let be a regular topological space, a collection of bounded regular measures defined on the Borel sets of . The following conditions are equivalent.
(1) Let denote the Borel measures, the nonnegative members of . There is a such that is uniformly -continuous.
(2) If is a disjoint sequence of open sets, then uniformly for .
(3) If is a Borel subset of and , there is a compact set such that for .
(4) If is a disjoint sequence of Borel sets, then uniformly for .
Grothendieck, Sur les applications linéaires faiblement
compactes d’espaces du type 𝐶(𝐾), Canadian J.
Math. 5 (1953), 129–173 (French). MR
- A. Grothendieck, Sur les applications linéares faiblement compactes d' espaces du type , Canad. J. Math. 5 (1953), 129-173. MR 15, 438. MR 0058866 (15:438b)
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