Research On The Developments Of Fuzzy Integrals

As people know, the precise Probability and Mathematical Statistics Theory is based on the Measure Theory. Firstly this essay reviewed the present development situation of the Measure Theory. Furthermore, it gave a summary on the current development situation of the Fuzzy Integral. And it improved the Fuzzy Integral Theory on the basis of its existing in the end.In 1965, Professor L.A.Zadeh, the automatic control expert, proposed the Fuzzy Set Theory[1] which means the birth of Fuzzy Mathematics. About 50 years, this theory has aroused great international attentions, penetrated into various fields, and formed many new branches of mathematics, such as Fuzzy Analytics, Fuzzy Topology, Fuzzy Algebra, Fuzzy System, Fuzzy Procedures, etc. In 1974, a Japanese Professor Sugeno proposed the concepts of Fuzzy Measure and Fuzzy Integral for the first time in his doctoral thesis. Compared with the Probability Measure, the Fuzzy Measure abandoned the additivity. However, it got a wider range of monotonicity. Therefore, the Fuzzy Measure is a special case of the Probability Measure. The Fuzzy Integral has material difference with the Lebesgue Integral. The Fuzzy Integral mainly changed the operators "+," in Lebesgue Integral to "∨,∧". Therefore, it abandoned the additivity. The Fuzzy Integral was first used by Professor Sugeno to the subjective evaluation process that achieved very good results. This theory earned widespread respect as a result. It showed abnormal vitality as soon as the Fuzzy Mathematics was proposed. Its applications include clustering analysis, image recognition, data structure, evaluation system, automatic control, decision making, optimization, the humanities, social science, and many other fields. It has showed great superiority in dealing with vagueness. This essay mainly did the following work on the basis of the original Fuzzy Integral Theory:(1) Fuzzy Measure and Fuzzy Integral in the Fuzzy Mathematics is summarized systematically.(2) The existing Fuzzy Integral Theory is improved. The Fubini theorem of set-valued functions, the fuzzy set-valued function control convergence theorem, the generalized Fatou theorem, and the interval set-valued function on the interval fuzzy measure integral monotone convergence theorem are given and proved.(3) Under the existing Fuzzy Integral Theory, the relationship between the various fuzzy functions is given and summarized.(4) As the real field application of Fuzzy Integral is very extensive, this paper attempted to give an application of (Y) complex Fuzzy Integral in Information Fusion.