On set-valued fuzzy measures

Abstract

So far, there have been many discussions on Sugeno’s fuzzy measures and fuzzy integrals, but most of them are concentrated on single-valued functions. Motivated by Aumann’s set-valued integral [J. Math. Anal. Appl. 12 (1965) 1], we have introduced the fuzzy integral of set-valued functions [Fuzzy Sets Syst. 56 (1993) 237; Fuzzy Sets Syst. 76 (1995) 365; Fuzzy Sets Syst. 78 (1996) 341; Ph.D. Dissertation, Harbin Institute of Technology, 1998]. It is a well-behaved extension of fuzzy integrals of single-valued functions. It is well known that Arstein’s set-valued measure [Trans. Am. Math. Soc. 165 (1972) 103] is an important branch in set-valued analysis [Set-valued Analysis, Birkhauser, Berlin, 1990] or theory of correspondences [Theory of Correspondences, Wiley, New York, 1984]. Compared to it, the present paper will try to establish the basic idea of set-valued fuzzy measures, which are monotone set-valued set-functions. It is also a natural generalization of (single-valued) fuzzy measures, as well as an extension of set-valued measures in the case of one-dimension. These works include the concept of set-valued fuzzy measures, set-valued fuzzy measures defined by set-valued fuzzy integrals and set-valued pseudo-additive measures. It can be viewed as a continuation of previous work [Fuzzy Sets Syst. 56 (1993) 237; Fuzzy Sets Syst. 76 (1995) 365; Fuzzy Sets Syst. 78 (1996) 341].