(Y) Fuzzy Integral And Its Properties Based On The Choquet Integral And (N) Integral

"Fuzzy set" is the foundation of fuzzy mathematics on the concept, Zadeh introduced the definition in 1965. According to the definition of Zadeh, fuzzy set is the promotion of the general set. From another perspective, fuzziness and randomness are uncertainties, and the randomness can be described by probability measure. Then whether a so-called "fuzziness" can be described by fuzzy measure? Based on this, Sugeno the Japanese scholar put forward the concept of fuzzy measure in his doctoral thesis in 1974. Fuzzy measure is defined as a formal, monotonous, continuous set functions. Compared with the fuzzy measure and the probability measure, we can see that the fuzzy measure given up the additive of probability measure, was replaced by a wider range of monotonicity. Therefore it is a special section of the probability measure, and more in line with the inference of human day-to-day activities. Making use of fuzzy measures, Sugeno also defines a corresponding functionals, known as the fuzzy integral. Compared with the fuzzy integral and Lebesgue integral, it is not difficult to discover the essence of the difference between the two. Mainly due to the fuzzy integral in the Lebesgue integral operator "+,·" replaced by "∨,∧", and therefore it also loss the nature of additivity. Sugeno fuzzy integral was used as early as the subjective evaluation process by Sugeno to obtain good results, so this theory is much attention.In this paper, we summarize the fuzzy measure and fuzzy integral base on original theory at first. Secondly, we define a (Y) fuzzy integral based on Choquet and (N) integral, and complements the fundamental nature of fuzzy integral; further more we define (Y) fuzzy integral by Lebesgue-Stieltjes form, the(Y)fuzzy integral of fuzzy measurable function values in(-∞,+∞), the (Y) fuzzy integral based on the measurable function of fuzzy sets, and (Y) fuzzy integral based on the self-dual measure. And we discuss the basic properties of the corresponding. Then we take the (Y) fuzzy integral as a set function, to induce a fuzzy measure and discuss its properties. Finally, we give applications in daily life of (Y) fuzzy integral and map of relationships of some kinds of fuzzy integrals.