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Regularities of Riesz space-valued non-additive measures with applications to convergence theorems for Choquet integrals. (English) Zbl 1183.28032

Summary: A deeper investigation of Radonness and \(\tau \)-smoothness properties of Riesz space-valued Borel non-additive measures is carried out. To this end, due to a lack of \(\varepsilon \)-argument in a Riesz space, the multiple Egoroff property is introduced and enforced on the involved Riesz space. The established regularity properties of Borel non-additive measures are instrumental when formulating certain types of monotone convergence theorems for Choquet integrals.

MSC:

28E10 Fuzzy measure theory
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[1] Duchoň, M.; Haluška, J.; Riečan, B., On the Choquet integral for Riesz space valued measure, Tatra Mt. Math. Publ., 19, 75-89 (2000) · Zbl 0992.46036
[2] Denneberg, D., Non-Additive Measure and Integral (1997), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht
[3] Jiang, Q.; Wang, S.; Ziou, D., A further investigation for fuzzy measures on metric spaces, Fuzzy Sets and Systems, 105, 293-297 (1999) · Zbl 0954.28009
[4] Kawabe, J., The Egoroff theorem for non-additive measures in Riesz spaces, Fuzzy Sets and Systems, 157, 2762-2770 (2006) · Zbl 1106.28005
[5] Kawabe, J., The Egoroff property and the Egoroff theorem in Riesz space-valued non-additive measure theory, Fuzzy Sets and Systems, 158, 50-57 (2007) · Zbl 1117.28013
[6] Kawabe, J., Regularity and Lusin’s theorem for Riesz space-valued fuzzy measures, Fuzzy Sets and Systems, 158, 895-903 (2007) · Zbl 1121.28021
[7] Kawabe, J., The Alexandroff theorem for Riesz space-valued non-additive measures, Fuzzy Sets and Systems, 158, 2413-2421 (2007) · Zbl 1145.28012
[8] Kawabe, J., The Choquet integral in Riesz space, Fuzzy Sets and Systems, 159, 629-645 (2008) · Zbl 1179.28025
[9] J. Kawabe, Some properties on the regularity of Riesz space-valued non-additive measures, in: M. Kato, L. Maligranda (Eds.), Proc. Banach and Function Spaces II (ISBFS 2006), Yokohama Publishers, 2008, pp. 337-348.; J. Kawabe, Some properties on the regularity of Riesz space-valued non-additive measures, in: M. Kato, L. Maligranda (Eds.), Proc. Banach and Function Spaces II (ISBFS 2006), Yokohama Publishers, 2008, pp. 337-348. · Zbl 1168.28310
[10] Kawabe, J., Continuity and compactness of the indirect product of two non-additive measures, Fuzzy Sets and Systems, 160, 1327-1333 (2009) · Zbl 1180.28008
[11] Li, J.; Yasuda, M., Lusin’s theorem on fuzzy measure spaces, Fuzzy Sets and Systems, 146, 121-133 (2004) · Zbl 1046.28012
[12] Luxemburg, W. A.J.; Zaanen, A. C., Riesz Spaces I (1971), North-Holland: North-Holland Amsterdam · Zbl 0231.46014
[13] Narukawa, Y.; Murofushi, T.; Sugeno, M., Regular fuzzy measure and representation of comonotonically additive functional, Fuzzy Sets and Systems, 112, 177-186 (2000) · Zbl 0954.28007
[14] Pap, E., Null-Additive Set Functions (1995), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0856.28001
[15] Riečan, B.; Neubrunn, T., Integral, Measure, and Ordering (1997), Kluwer Academic Publishers: Kluwer Academic Publishers Bratislava · Zbl 0916.28001
[16] Schwartz, L., Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures (1973), Oxford University Press: Oxford University Press Oxford · Zbl 0298.28001
[17] Šipoš, J., Integral with respect to a pre-measure, Math. Slovaca, 29, 141-155 (1979) · Zbl 0423.28003
[18] Vakhania, N. N.; Tarieladze, V. I.; Chobanyan, S. A., Probability Distributions on Banach Spaces (1987), D. Reidel Publishing Company: D. Reidel Publishing Company Dordrecht · Zbl 0698.60003
[19] Wang, Z.; Klir, G. J., Fuzzy Measure Theory (1992), Plenum Press: Plenum Press New York · Zbl 0812.28010
[20] Wu, J.; Wu, C., Fuzzy regular measures on topological spaces, Fuzzy Sets and Systems, 119, 529-533 (2001) · Zbl 0983.28009
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