Abstract
This paper gives a new necessary condition and a new sufficient condition for the Egoroff theorem in non-additive measure theory. The new necessary condition is condition (M), which is newly defined in this paper, and the new sufficient condition is the conjunction of null continuity and condition (M). The new sufficient condition is strictly weaker than both of known two sufficient conditions: continuity and the conjunction of strong order continuity and property (S). The new necessary condition is strictly stronger than the known necessary condition: strong order continuity.
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References
Denneberg, D.: Non-additive Measure and Integral, 2nd edn. Kluwer, Dordrecht (1997)
Li, J.: Order continuous of monotone set function and convergence of measurable functions sequence. Appl. Math. Comput. 135, 211–218 (2003)
Li, J.: A further investigation for Egoroff’s theorem with respect to monotone set functions. Kybernetika 39, 753–760 (2003)
Li, J.: On Egoroff’s theorems on fuzzy measure spaces. Fuzzy Sets and Systems 135, 367–375 (2003)
Li, J., Yasuda, M.: Egoroff’s theorem on monotone non-addditive measure spaces. Internat. J. Uncertain. Fuzziness Knowledge-Based Systems 12, 61–68 (2004)
Murofushi, T., Uchino, K., Asahina, S.: Conditions for Egoroff’s theorem in non-additive measure theory. Fuzzy Sets and Systems 146, 135–146 (2004)
Pap, E.: Null-additive Set Functions. Kluwer, Dordrecht (1995)
Sugeno, M.: Theory of Fuzzy Integrals and Its Applications, Doctoral Thesis, Tokyo Institute of Technology (1974)
Sun, Q.: Property (S) of fuzzy measure and Riesz’s theorem. Fuzzy Sets and Systems 62, 117–119 (1994)
Uchino, K., Murofushi, T.: Relations between mathematical properties of fuzzy measures. In: Proc. 10th Int. Fuzzy Syst. Assoc. World Congr., pp. 27–30 (2003)
Wang, Z., Klir, G.J.: Fuzzy Measure Theory. Plenum, New York (1992)
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Takahashi, M., Murofushi, T. (2010). New Conditions for the Egoroff Theorem in Non-additive Measure Theory. In: Huynh, VN., Nakamori, Y., Lawry, J., Inuiguchi, M. (eds) Integrated Uncertainty Management and Applications. Advances in Intelligent and Soft Computing, vol 68. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11960-6_9
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DOI: https://doi.org/10.1007/978-3-642-11960-6_9
Publisher Name: Springer, Berlin, Heidelberg
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