Fuzzy measure spaces and fuzzy integrals: an overview and an application

Abstract

Fuzzy measure theory is a generalization of classical measure theory. It was first introduced by Lotfi Zadeh in 1965 in his famous paper ”Fuzzy Sets”. After more than 50 years of the existence and development of classical measure theory, mathematicians felt that the additivity property is, in some applications, too restrictive. It is also unrealistic under real and physical conditions where measurement errors are unavoidable. According to Sugeno, fuzzy measures are obtained by replacing the additivity condition of classical measures with weaker conditions of monotonicity and continuity. Chapter 1 defines fuzzy measures, semi-continuous fuzzy measures, and λ-fuzzy measures. We show that a λ-fuzzy measure naturally exists on a finite set X . Then, we prove that a non-additive measure is induced from a classical measure by a transformation of range of the classical measure. It is called quassimeasure. Then, other non-additive measures are constructed in different ways. These measures include belief, plausibility, possibility, and necessity measures. We end Chapter 1 by giving some properties of finite fuzzy measures. In Chapter 2, we define measurable functions on fuzzy measure spaces. Also, we explain what it means for a sequence of measurable functions to converge almost everywhere, pseudo-almost everywhere, almost uniformly, pseudo-almost uniformly, in measure or pseudo in measure to a function. In Chapter 3, we define a fuzzy integral and give some of its properties. Moreover, we discuss several convergence theorems of fuzzy integral sequences, in addition to the transformation theorem of fuzzy integrals. We end the chapter by defining fuzzy measures using the fuzzy integral. Finally, in Chapter 4, we give an application of fuzzy measure theory in real life. We apply this theory in areas where human decision-making plays an important role. Students’ failure is one of the issues that all academic institutes face. For this problem, there are many interactive and interdependent criteria. As a result, the most important reasons for students’ failure are given.