The Research On Fuzzy Probability Theory Based On Fuzzy Events

Fuzzy probability falls into three categories according to fuzzy situation:fuzzy events-accurate probability; clear events-fuzzy probability; fuzzy events-fuzzy probability. For the first type fuzzy probability problem, in1968, Zadeh proposed the mathematical definition of fuzzy event and the probability of fuzzy event based on the two fundamental theorem of fuzzy mathematics-Decomposition Theorem and Representation Theorem, as well as on the basis of the traditional probability space. It manifested the relation between traditional probability and the probability of fuzzy event that the probability theory of fuzzy event proposed by Zadeh. In recent years, many scholars have made many achievements about the first kind of fuzzy probability. This paper conducts the research mainly based on the probability of fuzzy events proposed by Zadeh. One of the purposes of this paper is to establish the exact probability space of fuzzy events, for another purpose, this paper has a further inquiry of the probability of fuzzy events proposed by Zadeh and then gets some new conclusions.To begin with, this paper introduces the correlation knowledge of fuzzy sets, decomposition theorem and representation theorem. Then introduces categories of fuzzy sets, gives the concepts of fuzzy classification, additive category of fuzzy sets and monotone class of fuzzy sets and so on, and does the key research to some nature. All this is to prepare, the ground for the following work.Furthermore, we all know that the probability measure is a special measure defined on [0,1]. In order to verify the relation between fuzzy probability measure and fuzzy measure, this paper first introduces the real-valued fuzzy measure of fuzzy sets. Then based on the real-valued fuzzy measure of fuzzy sets, this paper constructed the fuzzy probability measure space, and then derive that this fuzzy probability measure space is natural fuzzification of a classic probability measure space, and then propose a new definition of fuzzy event domain.Finally, describes the practical background of fuzzy events and the probability of fuzzy events proposed by Zadeh. And then research from the following three steps:Firstly, this part verifies that fuzzy probability of fuzzy events proposed by Zadeh is a special real-valued fuzzy measure of fuzzy sets and draws the appropriate conclusions. Secondly, this part proposes the concept of fuzzy events with no common point, non-crossing fuzzy events and non-crossing fuzzy event series, thus obtain some new conclusions of the probability of fuzzy events proposed by Zadeh. In this step, some conclusions are premature, this still need pending further research.Thirdly, this part still obtains two new properties of fuzzy expectation based on fuzzy integral theory proposed by Sugeno.In addition, this paper gives the proof of some important conclusions and natures that have failed to given in the literature.