Generalizations of the Vitali-Hahn-Saks theorem on vector measures
HTML articles powered by AMS MathViewer
- by Richard A. Oberle PDF
- Proc. Amer. Math. Soc. 67 (1977), 227-235 Request permission
Abstract:
In this paper an extension of the Brook’s control measure existence theorem for families of vector measures is established. This result is applied to pointwise convergent sequences of finitely additive vector measures to obtain a generalization of the Vitali-Hahn-Saks theorem.References
- Tsuyoshi Andô, Convergent sequences of finitely additive measures, Pacific J. Math. 11 (1961), 395–404. MR 137806
- R. G. Bartle, N. Dunford, and J. Schwartz, Weak compactness and vector measures, Canadian J. Math. 7 (1955), 289–305. MR 70050, DOI 10.4153/CJM-1955-032-1
- James K. Brooks, On the existence of a control measure for strongly bounded vector measures, Bull. Amer. Math. Soc. 77 (1971), 999–1001. MR 286971, DOI 10.1090/S0002-9904-1971-12834-3
- James K. Brooks and Robert S. Jewett, On finitely additive vector measures, Proc. Nat. Acad. Sci. U.S.A. 67 (1970), 1294–1298. MR 269802, DOI 10.1073/pnas.67.3.1294
- 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
- I. Labuda, Sur quelques généralisations des théorèmes de Nikodym et de Vitali-Hahn-Saks, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 20 (1972), 447–456 (French, with English and Russian summaries). MR 340538
- Otton Nikodym, Sur les suites convergentes de fonctions parfaitement additives d’ensemble abstrait, Monatsh. Math. Phys. 40 (1933), no. 1, 427–432 (French). MR 1550217, DOI 10.1007/BF01708880 R. A. Oberle, Theory of a class of vector measures on topological rings of sets and generalizations of the Vitali-Hahn-Saks theorem, Catholic Univ., 1971.
- Richard Alan Oberle, A note on equicontinuous families of volumes with an application to vector measures, Canadian J. Math. 26 (1974), 273–280. MR 335744, DOI 10.4153/CJM-1974-028-2
- R. S. Phillips, On linear transformations, Trans. Amer. Math. Soc. 48 (1940), 516–541. MR 4094, DOI 10.1090/S0002-9947-1940-0004094-3
- C. E. Rickart, Decomposition of additive set functions, Duke Math. J. 10 (1943), 653–665. MR 9977
- Stanislaw Saks, On some functionals, Trans. Amer. Math. Soc. 35 (1933), no. 2, 549–556. MR 1501701, DOI 10.1090/S0002-9947-1933-1501701-7
- Stanislaw Saks, Addition to the note: “On some functionals” [Trans. Amer. Math. Soc. 35 (1933), no. 2, 549–556; 1501701], Trans. Amer. Math. Soc. 35 (1933), no. 4, 965–970. MR 1501728, DOI 10.1090/S0002-9947-1933-1501728-5
Additional Information
- © Copyright 1977 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 67 (1977), 227-235
- MSC: Primary 28A45
- DOI: https://doi.org/10.1090/S0002-9939-1977-0460585-3
- MathSciNet review: 0460585