Three types of generalized Choquet integral
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Three types of generalized Choquet integral Endre Pap1 Received: 25 May 2020 / Accepted: 24 June 2020 © Unione Matematica Italiana 2020
Abstract In this paper we present three recently obtained important generalizations of the Choquet integral. First generalization is based on different collection of finite systems. Second generalization of the Choquet integral uses sublinear means. Third generalization is based on two fuzzy measures, one of which is pseudo-additive. Keywords Monotone (fuzzy) measure · Choquet integral · Sublinear mean · Supedecomposition integral · Pseudo-additive measure Mathematics Subject Classification 28E10 · 28A25 · 28B10
1 Introduction The Choquet integral is an important aggregation function, which enable non-probabilistic modeling. One of its advantage is to enable modeling dependent events. Choquet integral [5], see also [2,3,6,27], is given by ∞ I (μ, f ) = μ({ f ≥ t}) dt, (1) 0
where f : X → [0, ∞[ is a bounded measurable function and μ : A → [0, ∞] is a monotone measure (fuzzy measure) satisfying the following conditions on a σ -algebra A of a given set X : (i) μ(∅) = 0, (ii) μ(A) ≤ μ(B) whenever A ⊆ B and A, B ∈ A. Many of its important applications require further generalizations. In this paper there are presented three ways for generalizing the Choquet integral. First generalization is based on different collection of finite systems [16,20] (Sect. 2). Second generalization of the Choquet integral is based on sublinear means [7,30] (Sect. 3). Third generalization is based on two fuzzy measures, one of which is pseudo-additive [36] (Sect. 4).
The paper is dedicated to my good friend and mathematician with original ideas Domenico Candeloro. This research was partially supported by the project Artificial Intelligence ATLAS by Science Found Serbia.
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Endre Pap [email protected] Department of postgraduate studies, University Singidunum, Belgrade, Serbia
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2 Superdecomposition integral This section is based on [8,15,16,20]. For a fixed measurable space (X , A) denote by X the set of all collection systems H, where C ⊆ A \{∅} is a collection whenever it is finite. Taking (X , A) we denote by M the set of all monotone measures on (X , A) and with F the set of all bounded measurable functions f : X → [0, ∞[. Definition 2.1 Let H ∈ X be fixed. Then the mapping I H : M × F → [0, ∞] given by H a A · μ(A) | C ∈ H, a A ≥ 0 for each A ∈ C , aA1A ≥ f I (μ, f ) = inf A∈C
A∈C
is called a superdecomposition integral. It is obvious that each superdecomposition integral I H is positively homogeneous and increasing in each coordinate. Example 2.1 (i) Taking H1 = {{A} | A ∈ A} we obtain I H1 (μ, f ) = inf {a · μ(A) | A ∈ A, a · 1 A ≥ f } = sup{ f (x) | x ∈ X } · μ({ f > 0}). Remark that if μ has no non-trivial null-sets, i.e., m(A) = 0 only if A = ∅, then I H1 coincide with the greatest universal integral (see [13]) related to the product as underlying multiplication. (ii) Taking H2 = {C | C is a finite chain in A}, i.e, C is a finite chain in A if and only if there is a
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