Concepts of probabilistic sets
Kaoru Hirota · 1977
Probabilistic set theory is discussed from a mathematical point of view. It is constructed considering ambiguity or variety of objects and subjectivity or evolution of knowledge of observers, and it includes fuzzy set theory. The background of the idea is first explained and some inconveniences of fuzzy set theory are pointed out. A probabilistic set on a total space is defined by a pointwise measurable function from a parameter space to a characteristic space. The main result is that the family of probabilistic sets constitutes a complete pseudo Boolean algebra. Some new concepts are shown such as probabilistic mappings, moment analysis, and expected cardinal numbers. The possibility of moment analysis is an important result and it cannot be seen in other theories.