A note on a paper of J. D. Stein, Jr.: “Sequence of regular finitely additive set functions” (Trans. Amer. Math. Soc. 192 (1974), 59–66)
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Abstract:
Among other results, it is proved that if a sequence $\{ {\mu _n}\}$ of regular measures on a Hausdorff space, with values in a normed group, is convergent to zero for all $\sigma$-compact sets or all open sets, then there exists a maximal open set U such that ${\dot \mu _n}(U) \to 0,\{ {\dot \mu _n}\}$ being the associated submeasures.References
- L. Drewnowski, Topological rings of sets, continuous set functions, integration. I, II, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 20 (1972), 269–276; ibid. 20 (1972), 277–286 (English, with Russian summary). MR 306432
- L. Drewnowski, Equivalence of Brooks-Jewett, Vitali-Hahn-Saks and Nikodym theorems, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 20 (1972), 725–731 (English, with Russian summary). MR 311869
- 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
- J. D. Stein Jr., Sequence of regular finitely additive set functions, Trans. Amer. Math. Soc. 192 (1974), 59–66. MR 379791, DOI 10.1090/S0002-9947-1974-0379791-3
Additional Information
- © Copyright 1977 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 67 (1977), 74-76
- MSC: Primary 28A10
- DOI: https://doi.org/10.1090/S0002-9939-1977-0492155-5
- MathSciNet review: 0492155