Sometimes, it is difficult to determine an event. We call an ill-defined event a fuzzy event. For a given fuzzy event, we always use a fuzzy set to represent it . Formally, probability of fuzzy event is defined as follows:
Definition 2.1 . Let (Rn, , P) be a probability space. Then, a fuzzy event is a fuzzy set A on Rn whose membership function,
μA(x): Rn→[0,1] is measurable. The probability of fuzzy event A is defined by Lebesgue-Stieltjes integral:
P x dPA R
( )A ( ) . = ∫ μ Here, Rn is a n-dimensional Euclidean space,
is a Borel field in Rn and P is a probability measure on .