On Fuzzy Random Variables: Examples and Generalizations

There are random experiments in which the notion of a classical random variable, as a map sending each elementary event to a real number, does not capture their nature. This leads to fuzzy random variables in the Bugajski-Gudder sense. The idea is to admit variables sending the set Ω of elementary events not into the real numbers, but into the set M 1 (R) of all probability measures on the real Borel sets (each real number r ∈ R is considered as the degenerated probability measure δr concentrated at r). We start with four examples of random experiments (Ω is finite); the last one is more complex, it generalizes the previous three, and it leads to a general model. A fuzzy random variable is a map φ of M 1 (Ω) into M + 1 (Ξ), where M + 1 (Ξ) is the set of all probability measures on another measurable space (Ξ,B(Ξ)), satisfying certain measurability condition. We show that for discrete spaces the measurability condition holds true. We continue in our effort to develop a suitable theory of ID-posets, ID-random variables, and ID-observables. Fuzzy random variables and Markov kernels become special cases.