Fuzzy Sets and Systems 156 (2005) 365 – 370 www.elsevier.com/locate/fss Fuzzy measures and integrals Radko Mesiar a, b, ∗ a Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Slovak University ofTechnology, SK-813 68 Bratislava, Slovakia b Institute of Information Theory and Automation, Czech Academy of Sciences, Prague, Czech Republic Available online 20 June 2005 Abstract We present fuzzy measures and fuzzy integrals as special poset homeomorphisms. Besides general fuzzy integrals, regular fuzzy integrals are introduced and some of their properties are discussed. State of art of fuzzy measure and fuzzy integral theory are briefly summarized and some further streaming is sketched. © 2005 Elsevier B.V. All rights reserved. Keywords: Fuzzy measure; Fuzzy integral; Regular fuzzy integral 1. Introduction Classical measure theory, see, e.g., [12], is based on a measure space (X, A, m), where A is a -algebra of subsets of a non-empty set X and m is a non-negative -additive real set function defined on A. Recall that (X, A) is called a measurable space, sets from A are called events and m is called a measure. In special case when m(X) = 1 is required, m is called a probability measure. The Lebesgue integral L related to a measure space (X, A, m) is a special functional acting on measurable real functions on X which can be viewed as a monotone extension of the measure m (i.e., for all A ∈ A, L(1 A ) = m(A)). Classical measure and integral theory was extended, generalized and deeply examined in many directions. For an exhaustive state-of-art overview we recommend the recent handbook [22] edited by Pap. In this contribution we will deal with special generalization of measure theory unifying the notions of measures and integrals. Our main aim is to clarify the notion of fuzzy integral. ∗ Corresponding author. Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Slovak University of Technology, SK-813 68 Bratislava, Slovakia. Tel.: +421 2 52925787; fax: +421 2 52967027. E-mail address: mesiar@math.sk. 0165-0114/$ - see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2005.05.033