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Drewnowski, Lech; Labuda, Iwo

The Orlicz-Pettis theorem for topological Riesz spaces. (English) Zbl 0885.40002

Proc. Am. Math. Soc. 126, No. 3, 823-825 (1998).
Summary: A finitely additive vector measure from a \(\sigma\)-ring to a Riesz space is countably additive (exhaustive) for all Hausdorff Lebesgue topologies on the range space, or for none of them. In particular, subseries convergent series are the same for all Hausdorff Lebesgue topologies on a Riesz space.

Cited in 1 Document

MSC:

40A99 Convergence and divergence of infinite limiting processes
46A40 Ordered topological linear spaces, vector lattices

Keywords:

subseries convergence; countably additive vector measure; exhaustive vector measure; topological Riesz space; Lebesgue topology
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\textit{L. Drewnowski} and \textit{I. Labuda}, Proc. Am. Math. Soc. 126, No. 3, 823--825 (1998; Zbl 0885.40002)
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References:

[1] Charalambos D. Aliprantis and Owen Burkinshaw, Locally solid Riesz spaces, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. Pure and Applied Mathematics, Vol. 76. · Zbl 0402.46005
[2] C. D. Aliprantis and O. Burkinshaw, On the structure of locally solid topologies, Canad. Math. Bull. 23 (1980), no. 2, 185 – 191. · Zbl 0436.46009
[3] 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). · Zbl 0243.28011
[4] Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, With the assistance of W. G. Bade and R. G. Bartle. Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1958. · Zbl 0084.10402
[5] N. J. Kalton, The Orlicz-Pettis theorem, Proceedings of the Conference on Integration, Topology, and Geometry in Linear Spaces (Univ. North Carolina, Chapel Hill, N.C., 1979) Contemp. Math., vol. 2, Amer. Math. Soc., Providence, R.I., 1980, pp. 91 – 100.
[6] K. Kuratowski, Topology. Vol. I, New edition, revised and augmented. Translated from the French by J. Jaworowski, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe, Warsaw, 1966. · Zbl 0158.40901
[7] Iwo Labuda, Submeasures and locally solid topologies on Riesz spaces, Math. Z. 195 (1987), no. 2, 179 – 196. · Zbl 0601.46006
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