Generalizing Expected Values To The Case Of L *-Fuzzy Events

Starting with [Goguen, J.A. 1967. Journal of Mathematical Analysis and Applications], several generalizations of the original definition of a fuzzy set have been proposed. In one popular case, one considers as truth values the points in the lower left triangle of the unit square, where their first coordinate is interpreted as "degree of membership", and their second coordinate as "degree of non-membership". Generalizing ideas in [Zadeh, L.A. 1968. Journal of Mathematical Analysis and Applications], [Grzegorzewski, P. and E. Mrowka. 2002. In: Soft Methods in Probability, Statistics and Data Analysis, Heidelberg: Physica], [Grzegorzewski, P. 2013. Information Sciences] and [Klement, E.P. and R. Mesiar. 2015. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems], the concept of expected values (based on capacities) of fuzzy events in this general sense is introduced and investigated. Expected values satisfying additional properties such as positive-linearity, comonotone additivity and comonotone maxitivity are studied, as well as an extension to real-valued expected values.