On conditions under which some generalized Sugeno integrals coincide: A solution to Dubois' problem

New monotone measure-based integrals inspired by scientific impact problem

“…Proof of Theorems 3.5 and 4.6. The arguments are similar to those of [6,Theorem 2]. Put Assume that S 1 > 0 as if S 1 = 0, then S n = 0 = Z n for all n (see Propositions 3.3 (a) and 4.3 (a)).…”

“…To this day, many researchers introduced numerous generalizations of the Sugeno integral like generalized upper Sugeno integral, pseudo-decomposition integral or q-integral for ȳ = µ(X) = 1, and studied their properties [6,7,14,25,30,36].…”

“…for any n 1 and all f, g ∈ F (X,Y ) such that f + g ∈ F (X,Y ) . Moreover, if there is n such that (6) holds for all f, g ∈ F (X,Y ) , then µ is a subadditive monotone measure.…”

New Directions in Ordinal Evaluation: Sugeno Integrals and Beyond

“…f is carried out in [5]. It turns out that there exist very few operations other than Kleene-Dienes implication and Kleene conjunction for which these integrals coincide (the conjunctions must be of the form φ(α) ∧ ψ(β) for local utility functions φ, ψ : L → L).…”

“…The motivation for its definition is that this is the only expression which is invariant by translation, i.e., C µ (f + h) = C µ (f ) + hµ([n]), h being a constant function of value h. In our context of ordinal scales, translation has no meaning, and consequently mimicking the definition of the (asymmetric) Choquet integral for the Sugeno integral ofL-valued functions is meaningless. 5 ϕ is anonymous if for every σ = (αi)i∈I ∈ S and every permutation π on I, R (σ) = R (σ •π), where σ •π = (απ i )i∈I 6 ϕ is internal if for every σ = (αi)i∈I ∈ S, mini∈I αi ⩽ ϕ(σ) ⩽ maxi∈I αi 7 ϕ is monotone if ϕ(σ) ⩽ ϕ(σ ′ ) whenever σ = (αi)i∈I ∈ S and σ ′ = (a ′ i )i∈I ∈ S are such that αi ⩽ α ′ i for every i ∈ I. On the other hand, the symmetric Choquet integral is defined by…”