Calculus of fuzzy functions

Abstract

This chapter is concerned with the necessary concepts and results related to the calculus of fuzzy set-valued mappings, which we call, for short, fuzzy-valued functions or fuzzy functions. These are essentially a family of set-valued mappings and therefore we utilize the results of set-valued mappings. Section 2.2 deals with convergence properties of fuzzy sets. In this section we adopt, for convenience, the notation u" instead of [uIa for the level sets of u : Rn + I. Section 2.3 discusses the measurability of fuzzy functions. In Sections 2.4 and 2.5, we develop the necessary concepts of integral and differential calculus for fuzzy functions respectively. The definition given in Section 2.4 for the integral of fuzzy functions generalizes that of Aumanri [2] for set-valued mappings. Notes and comments form the content of Section 2.6. 2.2 Convergence of Let (Y, p) be a metric space and A and B two nonempty compact subsets of Y. The Hausdorff distance between A and B is where