Absolutely Monotone Real Set Functions Biljana Mihailovi´ c Faculty of Engineering, University of Novi Sad Trg Dositeja Obradovi´ ca 6, 21000 Novi Sad, Serbia Email: lica@uns.ac.rs Endre Pap Department of Mathematics and Informatics, University of Novi Sad Trg Dositeja Obradovi´ ca 4, 21000 Novi Sad, Serbia Email: pape@eunet.yu Ljubo Nedovi´ c Faculty of Engineering, University of Novi Sad Trg Dositeja Obradovi´ ca 6, 21000 Novi Sad, Serbia Email: nljubo@uns.ac.rs Abstract—We present a class of absolutely monotone and signed stable set functions with m(∅) = 0, AMSS. The representation of a set function from AMSS as a symmetric maximum of two monotone set function is obtained. We present three integrals of a real-valued measurable function based on m ∈ AMSS. Keywords: symmetric maximum, absolutely monotone set function, sign stable set function I. I NTRODUCTION A generalization of the classical probability measure, a fuzzy measure (capacity), together with fuzzy integrals, has many applications in economics, pattern recognition, and decision analysis [3], [4], [5], [20], [21]. It is proven in [1], [16] that a real-valued set function m, m(∅), belongs to the space of set functions with the bounded chain variation, BV , if it can be represented as a difference of two finite fuzzy measures, vanishing at the empty set, i.e. m = ν 1 - ν 2 . The most important integral defined with respect to a fuzzy measure is introduced by G. Choquet in [2]. The Choquet integral of a non-negative real-valued function f based on a fuzzy measure m : A→ [0, ∞] is defined by C m (f )=  ∞ 0 m({x|f (x) ≥ t})dt. (1) The Choquet integral is defined with respect to non- monotonic, real-valued set functions with the bounded chain variation [14], and it is also known as the asymmetric Choquet integral [3], [14], [16]. The asymmetric Choquet integral is linear with respect to m. An another important integral based on a fuzzy measure m : A→ [0, 1], introduced in [18] by M. Sugeno, is the Sugeno integral defined by S m (f )= sup t∈[0,1[ (t ∧ m({x|f (x) ≥ t})). (2) The symmetric Sugeno integral of a real-valued function, introduced in [5], is also defined with respect to a fuzzy measure. The Choquet-like integral related to some non-decreasing function g : [0, 1] → [0, ∞], g(0) = 0, defined for a non- negative function and a fuzzy measure m is given by C g m (f )= g -1 (C g◦m (g ◦ f )) . (3) This integral is introduced in [9] and it is also defined for a real-valued function f , if for g is taken its odd extension on the real line [9]. In [13] the authors introduced an absolutely monotone and sign stabile set function m : A→ [-1, 1], m(∅)=0. It is shown that m can be represented by a pseudo-difference of two fuzzy measures. The class of such set functions is denoted by AMSS. The aim of this paper is to present the class AMSS, and to propose different types of monotonic integrals based on a set function from AMSS. Our previous papers [12], [13] are devoted to non-monotonic integrals. The paper is organized as follows. In Section 2 the pre- liminary notions and definitions are given. In Section 3 we consider the class of absolutely monotone and sign stabile set functions, AMSS. In Section 4 we propose definitions of three integrals based on m ∈ AMSS, m(X) < 1 and present their basic properties. II. PRELIMINARIES The symmetric maximum is originally introduced in [5], [6]. Definition 1: The symmetric maximum  :[-a, a] 2 → [-a, a], a ∈ R + is given by xy := sign(x + y)(|x|∨|y|). The symmetric maximum is a commutative, non-decreasing operation with neutral element 0 and anihilator a. It is not associative, nor continious. 1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 Figure 1. Symmetric maximum 115 978-1-4244-5349-8/09/$26.00 ©2009 IEEE