Concepts of probabilistic sets

In the field of pattern recognition or decision making theory, the important subjects are as follows: (1) ambiguity of property of objects, (2) variety of character of objects, (3) subjectivity of observers, (4) evolution of knowledge of observers (i.e. learning). Considering these points, the concept of probabilistic set is proposed. It is based on both probability theory and fuzzy concepts. A probabilistic set on a total space is defined by a point wise measurable function from a parameter space (which is a probability space) to a characteristic space (which is a measurable space). It is shown that the family of all probabilistic sets constitutes a complete pseudo-Boolean algebra. Moment analysis is possible by using a probability measure of the parameter space. Other useful concepts are also mentioned such as probabilistic mappings and expected cardinal numbers.