Fuzzy probability measures

In 4] Höhle has defined fuzzy measures on G-fuzzy sets 2] where G stands for a regular Boolean algebra. Consequently, since the unit interval is not complemented, fuzzy sets in the sense of Zadeh 8] do not fit in this framework in a straightforward manner. It is the purpose of this paper to continue the work started in 5] which deals with 0,1]-fuzzy sets and to give a natural definition of a fuzzy probability measure on a fuzzy measurable space 5]. We give necessary and sufficient conditions for such a measure to be a classical integral as in 9] in the case the space is generated. A counterexample in the general case is also presented. Finally it is shown that a fuzzy probability measure is always an integral (if the space is generated) if we replace the operations ∧ and ∨ by the t-norm T_o and its dual S₀ (see 6]).