Fuzzy Sets and Systems 138 (2003) 497–506 www.elsevier.com/locate/fss Space of fuzzy measures and convergence Yasuo Narukawa a ; ∗ , Toshiaki Murofushi b , Michio Sugeno c a Toho Gakuen, 3-1-10 Naka, Kunitachi, Tokyo 186-0004, Japan b Department of Comp. Intell. & Syst. Sci., Tokyo Institute of Technology, 4259 Nagatsuta, Midoriku, Yokohama 226-8502, Japan c Brain Science Institute, RIKEN 2-1, Hirosawa, Wako, Saitama 351-0198, Japan Received 6 May 2001; received in revised form 27 August 2002; accepted 29 October 2002 Abstract The convergences of the net (generalized sequence) of fuzzy measures are discussed. It is shown that three types of convergence are not equivalent in general case, however they are equivalent if the universal set is nite. c  2002 Elsevier B.V. All rights reserved. Keywords: Non-additive measures; Fuzzy measure; Choquet integral; Bounded variation; Convergence 1. Introduction A fuzzy measure is a monotone set function on a -algebra which vanishes at empty set. In the original denition by Sugeno [10] a fuzzy measure is assumed to be continuous. There have been discussions on the continuity of fuzzy measure [9,11]. But in this paper it is possible to discuss the fuzzy measure and the integral without continuity. Fuzzy measures and Choquet integral [3,7] with respect to a fuzzy measure are basic tools for multicriteria decision making, image processing and recognition [5,4]. Using the Choquet integral, we introduce the topologies T X and T B + in the space of fuzzy mea- sure FM. The concept of topology is equivalent to the concept of convergence. The notion of convergence is very important for both theory and application of uncertainty management. Since the Radon–Nikodym Theorem and Fubini’s Theorem are generally not valid in fuzzy measure theory without any type of additivity, it is dicult to consider the theory of conditional fuzzy measure, and even more, the theory of fuzzy measure process. But it is possible to consider that the fuzzy measure  t with a parameter t ∈ T express the degree of uncertainty (or certainty) that is * Corresponding author. Tel.: +81-42-577-2171; fax: +81-42-574-9898. E-mail address: narukawa@d4.dion.ne.jp (Y. Narukawa). 0165-0114/03/$ - see front matter c  2002 Elsevier B.V. All rights reserved. PII:S0165-0114(02)00511-0