Borel-Cantelli Lemma for Sugeno Measure

Abstract:

Sugeno measure is a fuzzy measure. In this paper, we derive the Borel-Cantelli lemma for Sugeno measure. This result is a natural extension of the classical Borel-Cantelli lemma to the case where the measure tool is fuzzy. The properties of Sugeno measure are further discussed. Then the Borel-Cantelli lemma will be proven on Sugeno measure space. This work generalizes the research and applications of the Borel-Cantelli lemma.

• Periodical: Applied Mechanics and Materials (Volume 614)
• Pages: 367-370
• DOI: https://doi.org/10.4028/www.scientific.net/AMM.614.367
• Online since: September 2014
• Authors: Chun Qin Zhang*, Hui Zhang
• Keywords: Borel-Cantelli Lemma, G_λ- Random Variable, Sugeno Measure

References

[1] M. Sugeno, Theory of fuzzy integrals and its applications, PH.D. dissertation, Tokyo: Tokyo Insitute of Technology(1974).

[2] L.A. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems, vol l00, no 1, (1978), pp.9-34.

DOI: 10.1016/0165-0114(78)90029-5

[3] L.A. Zadeh, A theory of approximate reasoning, in: Machine Intelligence, J. Hayes, D. Michie and R. M. Thrall, eds, Mathematical Frontiers of the Social and Policy Sciences, Westview Press, Boulder, Cororado (1979).

[4] Z. Wang and G. Klir, Fuzzy measure theory , New York: Plenum Press, (1992).

[5] M. Sugeno, Industrial applications of fuzzy control, New York: Elsevier Science Inc, (1985).

[6] Minghu Ha, Yan Li, Jia Li, Dazeng Tian. The key theorem and the bounds on the rate of uniform convergence of learning theory on sugeno measure space [J]. Science in China Series F: Information Sciences, 2006, 49(3): 372-385.

DOI: 10.1007/s11432-006-0372-8

[7] Minghu Ha, Jia Li, Jing Tian , Xizhao Wang, The key theorem of learning theory on measure spaces, Proceedings of the 3th international conference on Machine Learning and Cybernetics. Shanghai, pp.1904-1907, Aug (2004).

DOI: 10.1109/icmlc.2004.1382089

[8] Chunqin Zhang, Minghu Ha, Jing Tian, The key theorem of learning theory about Samples corrupted by noise on Sugeno measure space, Information Processing and Management of uncertainty in Knowledge-based Systems. Baoding, pp.2491-2496, Sep. (2006).

[9] Zhishun Liang, Jixian Deng, Probability Theory and Mathematical Statistics, Beijing: Higher Education Press (1988) in Chinese.

[10] V. V. Petrov, A generalization of Borel-Cantelli lemma, Statistic and Probability Letters, vol67, no3( 2004), pp.233-239.

DOI: 10.1016/j.spl.2004.01.008

[11] .

[11] T. K. Chandra, The Borel-Cantelli lemma under dependence conditions, Statistic and Probability Letters, vol 78, no 4(2008), pp.390-395.

DOI: 10.1016/j.spl.2007.07.023