Two fuzzy probability measures

2 Zadeh-type fuzzy probability The paper deals with two methods of a fuzzification of the Borel field of events and too the probability measure. The first approach generalizes the Zadeh definition of a crisp probability of fuzzy event. The second method is based on the Yager definition of a fuzzy probability of fuzzy event. The theoretical results obtained can be applied to modeling stochastic phenomena with uncertain character. The first fuzzification is based on Zadeh ́s definition [11] of the probability of fuzzy event : A ( ) m A P A dP μ = ∫ R where P is a probability measure, and ( ) , m A A μ = is a fuzzy set. In this section, this definition is generalized by means of Zadeh ́s extension principle [2]. Definition 2.1. Let Ω ≠∅ be a universal set (basic space). A fuzzy random event is the fuzzy set A = ( ) , A μ = Ω with the membership function : A μ Ω→ [ ] 0;1 → . Ω is the certain event ( ) 1 A μ ≡ and ∅ is the impossible event ( ) 0 ≡ A μ . A fuzzy random event with a Borel measurable membership function A A μ is called a fuzzy event.