The convergence determining class of regular open sets
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- by Lothar Rogge PDF
- Proc. Amer. Math. Soc. 37 (1973), 581-585 Request permission
Abstract:
The purpose of this paper is to prove that every sequence of closed approximate measures defined on the Borel-field of a normal topological space with values in an abelian topological group is Cauchy convergent for all Borel sets if it is Cauchy convergent for all regular open sets. In particular every sequence of measures on the Borel-field of a perfectly normal topological space which is Cauchy convergent for all regular open sets is Cauchy convergent for all Borel sets, too.References
- Peter Gänssler, Compactness and sequential compactness in spaces of measures, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 17 (1971), 124–146. MR 283562, DOI 10.1007/BF00538864
- Peter Gänßler, A convergence theorem for measures in regular Hausdorff spaces, Math. Scand. 29 (1971), 237–244 (1972). MR 311862, DOI 10.7146/math.scand.a-11049
- Paul R. Halmos, Measure Theory, D. Van Nostrand Co., Inc., New York, N. Y., 1950. MR 0033869
- D. Landers and L. Rogge, The Hahn-Vitali-Saks and the uniform boundedness theorem in topological groups, Manuscripta Math. 4 (1971), 351–359. MR 283169, DOI 10.1007/BF01168702
- Dieter Landers and Lothar Rogge, Cauchy convergent sequences of regular measures with values in a topological group, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 21 (1972), 188–196. MR 310170, DOI 10.1007/BF00538391
- Benjamin B. Wells Jr., Weak compactness of measures, Proc. Amer. Math. Soc. 20 (1969), 124–130. MR 238067, DOI 10.1090/S0002-9939-1969-0238067-9
Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 37 (1973), 581-585
- MSC: Primary 28A45; Secondary 60B05
- DOI: https://doi.org/10.1090/S0002-9939-1973-0311872-7
- MathSciNet review: 0311872