Fuzzy Sets and Systems 112 (2000) 207–216 www.elsevier.com/locate/fss Pseudometric generating property and autocontinuity of fuzzy measures Qingshan Jiang a; ∗ , Shengrui Wang a , Djemel Ziou a , Zhenyuan Wang b , George J. Klir b a Department of Mathematics and Computer Sciences, University of Sherbrooke, Sherbrooke, Queb ec, Canada J1K 2R1 b Department of Systems Science and Industrial Engineering, Thomas J. Watson School of Engineering and Applied Science, State University of New York at Binghamton, Binghamton, NY 13902-6000, USA Received December 1996; received in revised form November 1997 Abstract Several necessary and sucient conditions for the pseudometric generating property of non-additive set functions are given. The relation among the pseudometric generating property and autocontinuity of fuzzy measures is depicted by using the absolute continuity and exhaustivity. Moreover, any uniformly autocontinuous nite fuzzy measure is equivalent to a subadditive nite fuzzy measure. A further research on purely atomic fuzzy measures is also made. ? 2000 Elsevier Science B.V. All rights reserved. Keywords: Fuzzy measure; Exhaustivity; Pseudometric-generating property; Autocontinuity; Subadditivity; Absolute conti- nuity; Atom 1. Introduction A variety of structural characteristics of non- additive set functions are introduced and discussed by Dobrakov [2,3], Drewnowski [4], Wang [20], Wang and Klir [22], Pap [13,14], and Denneberg [1], and the relevant theories are developed by them separately. The most noticeable two structural char- acteristics are the pseudometric generating property [3] and the autocontinuity [20]. These concepts play important roles in fuzzy measure theory [9,10,12,22]. The rst version of the manuscript was written while the author was visiting State University of New York at Bingham- ton. And this work was partially supported by the ONR Grant No. N00014-94-1-0263 and the NSERC Grant No. OGP0121680. * Corresponding author. To make a progress toward establishing a unied fuzzy measure theory, it is valuable to investigate the relations among these structural characteristics and some additional characteristics or restrictions such as subadditivity, exhaustivity, and absolute continuity. The main results are stated in Sections 3 and 4. They include several necessary and sucient conditions for the pseudometric-generating property in various situ- ations. We also show that any uniformly autocontinu- ous nite fuzzy measure is equivalent to a subadditive nite fuzzy measure in the sense of absolute conti- nuity to each other. In Section 5, we discuss -nite and purely atomic fuzzy measures. In this case, the uniform autocontinuity can be replaced with the null- additivity and the exhaustivity in the above-mentioned equivalence. 0165-0114/00/$ - see front matter c 2000 Elsevier Science B.V. All rights reserved. PII:S0165-0114(97)00395-3