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Conditions for Egoroff’s theorem in non-additive measure theory. (English) Zbl 1046.28013
Summary: This paper gives necessary and/or sufficient conditions for Egoroff ’s theorem in non-additive measure theory: a necessary and sufficient condition described without measurable functions, two sufficient conditions, and a necessary condition. One of the two sufficient conditions is strong order total continuity (continuity at measurable sets of measure zero with respect to net convergence), and the other is strong order continuity (sequential continuity at measurable sets of measure zero) together with property (S). The necessary condition is strong order continuity. In addition, the paper shows the following: continuity from above and below, which is a known sufficient condition, and the above-mentioned two sufficient conditions are independent of each other; the disjunction of these three sufficient conditions is not a necessary condition; if the underlying set is at most countable, then strong order continuity is necessary and sufficient; and generally, strong order continuity is not sufficient.

28E10Fuzzy measure theory
Full Text: DOI
[1] Denneberg, D.: Non-additive measure and integral. (1997) · Zbl 0927.28011
[2] Dobrakov, I.: On submeasures I. Dissertationes math. 112, 1-35 (1974)
[3] Ghirardato, P.: On independence for non-additive measures, with a Fubini theorem. J. econom. Theory 73, 261-291 (1997) · Zbl 0934.28012
[4] Gilboa, I.; Schmeidler, D.: Additive representations of non-additive measures and the Choquet integral. Ann. oper. Res. 52, 43-65 (1994) · Zbl 0814.28010
[5] Li, J.: Order continuous of monotone set function and convergence of measurable functions sequence. Appl. math. Comput. 135, 211-218 (2003) · Zbl 1025.28012
[6] Li, J.: On egoroff’s theorems on fuzzy measure spaces. Fuzzy sets and systems 135, 367-375 (2003) · Zbl 1014.28015
[7] Murofushi, T.; Sugeno, M.: A theory of fuzzy measuresrepresentations, the Choquet integral, and null sets. J. math. Anal. appl. 159, 532-549 (1991) · Zbl 0735.28015
[8] Murofushi, T.; Sugeno, M.; Suzaki, M.: Autocontinuity, convergence in measure, and convergence in distribution. Fuzzy sets and systems 92, 197-203 (1997) · Zbl 0927.28012
[9] Pap, E.: Null-additive set functions. (1995) · Zbl 0856.28001
[10] Puri, M. L.; Ralescu, D.: A possibility measure is not a fuzzy measure. Fuzzy sets and systems 7, 311-313 (1982) · Zbl 0543.28002
[11] M. Sugeno, Theory of fuzzy integrals and its applications, Doctoral Thesis, Tokyo Institute of Technology, 1974.
[12] Sun, Q.: Property (S) of fuzzy measure and Riesz’s theorem. Fuzzy sets and systems 62, 117-119 (1994) · Zbl 0824.28014
[13] K. Uchino, T. Murofushi, Relations between mathematical properties of fuzzy measures, Proc. 10th IFSA World Congr., Istanbul, Turkey, 2003, pp. 27--30.
[14] Wagner, E.; Wilczynski, W.: Convergence almost everywhere of sequences of measurable functions. Colloq. math. 45, 119-124 (1981) · Zbl 0497.28006
[15] Wang, Z.; Klir, G. J.: Fuzzy measure theory. (1992) · Zbl 0812.28010