Fuzzy Probability Theory

Probability of fuzzy event has always been a hotspot in research fields. In this theme after introducing the probability put forward by Zadeh, the author argues the necessity to study the relationship between involved fuzzy events: the conformation of a membership function which is the base of the definition proposed by Zadeh is inaccurate because a fuzzy statistic experiment is affected by the psychological process in human's brain. A membership function needs more emendation than dependence. And the essence of random phenomenon determines the necessity of research of the correlation between involved fuzzy events. The main purpose of this theme is to establish a fuzzy probability space.At first, the most important task is to ascertain the research object of this theme. During the process of study in fuzzy events, the author gives the definition from three angles: background in real life, intuitionistic meaning and definition in mathematics. Examples in real life make us realize that due to different purposes and realizing angles of different researchers, even for the same random experiment, there are different random events. As a result of human's characteristic, a large number of random events involves in fuzzy uncertainty.The following task is to study the relationship between involved fuzzy events. In the same way, to study from two sides: correlation between fuzzy events, operation between fuzzy sets. And at last, there is a proof of the one to one correspondence between the operation of fuzzy events and it of fuzzy sets.In order to find a field of events that is closed to the above operations, based on the introduction of fuzzy measure, the author propose two notions: fuzzy sample space, field of events. Probability, as a normal measure, is put forward in the following study. Finally, a fuzzy probability space has been established.A natural thing comes into one's thought is the transformation between different probability spaces. In this theme the author achieves the conclusion that the definition of probability of fuzzy events put forward by Zadeh has been the bridge for the transformation from classical probability space to fuzzy probability space and several theorems.