sugeno integral based on absolutely monotone real set functions

Fuzzy Sets and Systems 161 (2010) 2857–2869www.elsevier.com/locate/fss

Sugeno integral based on absolutely monotone real set functions

Biljana Mihailovica, Endre Papb,∗aFaculty of Technical Sciences, University of Novi Sad, Trg Dositeja Obradovica 6, 21000 Novi Sad, Serbia

bDepartment of Mathematics and Informatics, University of Novi Sad, Trg Dositeja Obradovica 4, 21000 Novi Sad, Serbia

Received 24 March 2009; received in revised form 14 December 2009; accepted 3 March 2010Available online 10 March 2010

Abstract

The class of all absolutely monotone and sign stable set functions with m(∅) = 0, denoted by AMSS, is introduced. Necessaryand sufficient conditions are obtained for a set function m, m(∅) = 0, to be a member of AMSS. A Sugeno type integral of anA- measurable real-valued function is defined with respect to an absolutely monotone and sign stable set function and some of itsproperties are shown. A representation of a comonotone–cosigned -additive functional by Sugeno integral based on m ∈ AM SSis obtained.© 2010 Elsevier B.V. All rights reserved.

Keywords: Absolutely monotone set function; Sign stable set function; Sugeno integral; Symmetric Sugeno integral

1. Introduction

Sugeno and Choquet integrals are very useful in several applications in mathematics, economics, pattern recognition,decision making and aggregation problems, see [4,7,11,18,24,25,27,28]. They are usually defined with respect to afuzzy measure. A fuzzy measure m : A → [0, 1] (or [0, ∞]), with m(∅) = 0 is a non-decreasing set function, definedon �-algebra A. In decision theory, loosely speaking, a utility functional is a rule for assigning a utility to a prospect. Auniversal set X is a state of nature ((X,A) is a measurable space) and functions from X to R are prospects (A-measurablefunctions). M is the class of all prospects f : X → R, i.e., the class of all A-measurable functions. The functionf + = f ∨ 0 is the gain part of prospect f, and −f− is its loss part, where f − = (− f ) ∨ 0. An integral-based utilitymodel holds when a utility functional L : M → R is represented by L( f ) = I( f ), where I( f ) is an integral of thefunction f. Integral-based utility models are studied in [8,11,16,19,24,25,28]. One of such models is CPT (cumulativeprospect theory) introduced by Tversky and Kahneman [28]. CPT holds if there exist two fuzzy measures, m+ and m−,such that a utility functional L is a difference of two Choquet integrals, i.e.,

L( f ) = Cm+ ( f +) − Cm− ( f −).

In [8], Grabisch et al. introduced a framework for ∨-CPT, i.e., the cumulative prospect theory in an ordinal context,based on a Symmetric Sugeno integral.

∗ Corresponding author. Tel.: +381216350458.E-mail addresses: [email protected] (B. Mihailovic), [email protected], [email protected] (E. Pap).

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2858 B. Mihailovic, E. Pap / Fuzzy Sets and Systems 161 (2010) 2857–2869

There are many generalizations of the concept of a classical (�-additive) measure studied by various authors[1,6,12,13,30]. An extension of the classical measure by allowing that m can take also negative values, leads tothe notion of a signed measure, see [12]. On the other hand, in [8,15] a generalization of a fuzzy measure has beenintroduced. A generalized fuzzy measure, a signed fuzzy measure, has been introduced by Liu in [15]. As an anothergeneralization of fuzzy measures, the concepts of bi-capacities and bi-cooperative games have been introduced in [3,8].S-measures, i.e., S-decomposable fuzzy measures, where S is a t-conorm and (S,U)-integrals, based on a t-conorm S andan appropriate uninorm U have been introduced in [14], see also [20–22,29,30]. A special type of S-measures, maxitivemeasures, i.e., ∨-decomposable fuzzy measures can be extended, as it is proposed in [17], to so-called R-measures.The operation R is the symmetric maximum, introduced in [9], given in Definition 1, below.

Sugeno integral has been introduced by Sugeno in [26] for an A-measurable function f : X → [0, 1] and a fuzzymeasure m : A → [0, 1]. It is well known that the Sugeno integral is one of a non-linear functional on the class ofmeasurable functions which is comonotone-maxitive, monotone and ∧-homogeneous [2,21,29]. A symmetric extensionof the Sugeno integral on the class of functions f : X → [−1, 1] defined on a finite set X is proposed in [9]. Forf : X → [−1, 1], the Symmetric Sugeno integral is defined by

Sm( f ) = Sm( f +)R(−Sm( f −)). (1)

For more details see [9–11,23].In [23] the authors considered a pseudo-difference representation by two Sugeno integrals of a functional L defined

on the class of functions f : X → [−1, 1] with finite supports. In the case of a countable set X it has been proven thata comonotone-R-additive, weakS-homogeneous and monotone functional L can be represented in the following way:

L( f ) = Sm+ ( f +)R(−Sm− ( f −)),

where the fuzzy measures m+ and m− are not necessary equal. The comonotone-R-additivity of the Symmetric Sugenointegral has been obtained.

The aim of this paper is to present a Sugeno type integral of an A-measurable function f : X → [−a, a], a > 0,with respect to real-valued set functions. We propose this definition in the context when the underlining set functionis absolutely monotone and fulfills one additional property, named the sign stable property. The class of set functionswith these properties is denoted by AMSS. Several properties of absolutely monotone set functions and the Sugenointegral based on m ∈ AM SS are proven. The paper is organized as follows. The next section is devoted to preliminarynotions and definitions. In Section 3 we recall definitions of the Sugeno integral and Symmetric Sugeno integral, andwe present some properties of the Symmetric Sugeno integral using the concept of cosigned and comonotone functions.In Section 4 we introduce absolutely monotone set functions. The representation theorem of absolutely monotone andsign stable set functions is proven and corresponding examples are given. The relationship of a set function from AMSSand aR-measure is established. Finally, in Section 5 we introduce a Sugeno type integral of an A-measurable functionf : X → [−a, a] with respect to an absolutely monotone and sign stable set function and we prove some of itsproperties. We consider the necessary conditions for a functional L to be representable by Sugeno integral with respectto m ∈ AM SS.

2. Preliminaries

Symmetric maximum and symmetric minimum are originally introduced in [9–11]. Analogously as it is done in [9],we define R,S : [−a, a]2 → [−a, a], a ∈ R

+, in the following way:

Definition 1. Symmetric maximum R : [−a, a]2 → [−a, a], is defined by

xRy = sign(x + y)(|x | ∨ |y|).

Symmetric minimum S : [−a, a]2 → [−a, a], is defined by

xSy = sign(x · y)(|x | ∧ |y|).

B. Mihailovic, E. Pap / Fuzzy Sets and Systems 161 (2010) 2857–2869 2859

Symmetric maximum is commutative, with neutral element 0 and absorbing element a. Symmetric minimum is com-mutative, associative, with neutral element a and absorbing element 0.

Since the symmetric maximum is not an associative operation, one of the computational rules, introduced and studiedin [9,11], will be adopted. Namely, on an arbitrary set I of elements from the interval [−a, a], a ∈ R

+, the symmetric

maximum is given by

Ri∈I

xi =(R

xi ≥0xi

)R(R

xi <0xi

)= sup

xi ≥0xiR inf

xi <0xi .

Distributivity of the operation S with respect to R does not hold in general. However, distributivity is satisfied forx, y ≥ 0 and z ≤ 0 in some cases, as it was shown in the following lemma from [23].

Lemma 1. Let x, y ≥ 0 and z ≤ 0. If xSy � xS(−z) then

xS(yRz) = (xSy)R(xSz).

We assume that X is a non-empty universal set. Let A be a �-algebra of subsets of X. The signed fuzzy measures areintroduced in [15], see [17,21].

Definition 2. A set function m : A → [−a, a], a ∈ R+

, m(∅) = 0, is

(i) non-negative if m(A) ≥ 0 for all A ∈ A;(ii) a fuzzy measure if it is non-negative and monotone, i.e., if for all A, B ∈ A, A ⊂ B, we have 0 ≤ m(A) ≤ m(B);

(iii) revised monotone (a signed fuzzy measure) if for all A, B ∈ A, A ∩ B = ∅, we have(a) m(A) ≥ 0, m(B) ≥ 0, m(A) ∨ m(B) > 0

⇒ m(A ∪ B) ≥ m(A) ∨ m(B),

(b) m(A) ≤ 0, m(B) ≤ 0, m(A) ∧ m(B) < 0

⇒ m(A ∪ B) ≤ m(A) ∧ m(B),

(c) m(A) > 0, m(B) < 0 ⇒ m(B) ≤ m(A ∪ B) ≤ m(A);(iv) continuous from below if for An ∈ A, An ↗ A, we have m(An) ↗ m(A).

Important special fuzzy measures are maxitive fuzzy measures, see [21,29].

Definition 3. A fuzzy measure m is completely maxitive if for any family {Ai }i∈I from A such that⋃

i∈I Ai ∈ A wehave

m

(⋃i∈I

Ai

)= sup

i∈Im(Ai ).

As an extension of maxitive measures, the R-measures are introduced in [17].

Definition 4. A set function, m : A → [−a, a], is a R-measure if it satisfies

m(A ∪ B) = m(A)Rm(B) for all A,B ∈ A, A ∩ B = ∅.

Note that each R-measure is revised monotone (Definition 2(iii)).


Example 1. Let X={x1,x2,. . ., xn}. Let m be a set function m : P(X ) −→ [−1, 1] with m(∅) = 0, defined by

m(A) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

1

minxi ∈A

iif min

xi ∈Ai = 2k,

− 1

minxi ∈A

iif min

xi ∈Ai = 2k + 1.

m is a R-measure.

3. Comonotone functions and Symmetric Sugeno integral

Let (X,A) be a measurable space, where X is a non-empty universal set. Let f : X → [−a, a], a > 0, be anA-measurable function, i.e., f −1(B) ∈ A, for all Borel sets B ∈ B([−a, a]).

Measurable functions f and g on X are comonotone, see [5,21], if there are no x, x1 ∈ X such that f (x) < f (x1) andg(x) > g(x1).

Let f and g be functions defined on X with values in [−a, a]. Then, we define for any x ∈ X , ( fRg)(x) = f (x)Rg(x),( fSg)(x) = f (x)Sg(x), and for each c ∈ [0, a], (cS f )(x) = cS f (x), f +(x) = f (x) ∨ 0, f −(x) = −( f (x) ∧ 0) =− f (x) ∨ 0.

Let M be the class of all A-measurable functions f : X → [−a, a], a ∈ R+. We denote M+ = { f ∈ M| f ≥ 0}.Let S be the subclass of M of simple functions s with finite range, i.e., Ran(s) = {s1, s2, . . . , sn}. For c ∈ [0, a] andA ⊆ X , basic functions are defined by

(c · 1A)(x) ={

c if x ∈ A,0 if x /∈ A,

(c · (−1A))(x) ={−c if x ∈ A,

0 if x /∈ A.

Definition 5. f and g are cosigned functions if for all x ∈ X we have f (x) · g(x) ≥ 0. A couple of comonotone andcosigned functions (co-comonotone functions) f and g is denoted by f �s g.

Lemma 2. For any f, g ∈ M, such that f �s g we have

(i) f +�s g+; f −�s g−; f +�s − f −;(ii) ( fRg)+ = f +Rg+;

(iii) ( fRg)− = f −Rg−;(iv) cS f �sdSg for every c, d ∈ [0, a];(v) f �s fRg.

Proof. Let f, g ∈ M such that f �s g. Since f and g are cosigned, for all x ∈ X we have

f (x) < 0 ⇒ g(x) ≤ 0 and f (x) > 0 ⇒ g(x) ≥ 0. (2)

Functions f + and g+ are non-negative, hence they are cosigned functions and the same holds for f− and g−.For all f ∈ M and for all x ∈ X we have f +(x) · (− f −)(x) = 0, hence f + and −f − are cosigned.By definitions of R and S, for c, d ≥ 0 we have cS f and dSg are cosigned and the same holds for f and fRg.Therefore, by (2) we obtain (ii) and (iii). f + and g+, f − and g−, f + and −f − are comonotone (see [5,19]). By

definitions of comonotone functions and S, for c, d ≥ 0, we obtain that cS f and dSg are comonotone, and bymonotonicity of R we have f and fRg are comonotone. Hence, we obtain (i), (iv) and (v). �

Remark 1.(i) Each function f ∈ M can be represented by symmetric maximum of comonotone and cosigned functions, f + ≥ 0

and − f − ≤ 0, i.e., f = f +R(− f −).


(ii) For each c ∈ [0, a] we have cS f +�scS(− f −) and

cS f = cS( f +R(− f −)) = (cS f +)R(cS(− f −)).

Let s ∈ S. Its comonotone–cosigned R-additive representation is given by: s = s+R(−s−), where

s+ = nR

i=1ci 1Ai , − s− = n

Ri=1

di (−1Bi ), (3)

and ci = s+�(i), di = s−

�(n+1−i), Ai = A�(i), Bi = A1 \ A�(n+2−i), and � is a permutation, such that −a ≤ s�(1) ≤ · · · ≤s�(n) ≤ a, A�(i) = {x ∈ X |s(x) ≥ s�(i)}, A�(n+1) = ∅.

The Sugeno integral, introduced in [26], is defined for all non-negative measurable functions. This integral wasextended on the class of all measurable functions under the name Symmetric Sugeno integral, introduced in [9].

Definition 6. Let m : A → [0, a], a > 0, be a fuzzy measure on the measurable space (X,A).

(i) The Sugeno integral of f ∈ M+, with respect to m is defined by

Sm( f ) = supt∈[0,a]

(t ∧ m({x | f (x) ≥ t})).

(ii) The Symmetric Sugeno integral of f ∈ M with respect to m is defined by

Sm( f ) = supt∈[0,a]

(t ∧ m({ f + ≥ t}))R− supt∈[0,a]

(t ∧ m({ f − ≥ t}))

= Sm( f +)R(−Sm( f −)).

where f + = f ∨ 0 and f − = (− f ) ∨ 0 = −( f ∧ 0).

Definition 7. Let L : M → [−a, a] be a functional. We say that L is:

(i) symmetric if for all f ∈ M we have L(− f ) = −L( f ),(ii) monotonic if for all f, g ∈ M, f ≤ g ⇒ L( f ) ≤ L(g),

(iii) weak S-homogeneous if for all basic function c · 1A, A ∈ A, c ∈ [−a, a], we have L(c · 1A) = cSL(1A),(iv) comonotone–cosigned R-additive if for all f �s g, f, g ∈ M we have L( fRg) = L( f )RL(g).(v) continuous from below if for each non-decreasing sequence { fn}n∈N, fn ∈ M we have L(supn fn) = supn L( fn),

(vi) monotone ∨-separable if there exist monotone functionals L1, L2 : M → [−a, a], such that for all f ∈ M wehave L( f ) = L1( f )R(−L2( f )).

(vii) signed monotone ∨-separable if it is monotone ∨-separable and for every A ∈ A it fulfils the next conditions(a) if L1(1A) < L2(1A), then L1(1A) = sup{L1(1E )|E ⊂ A, L1(1E ) > L2(1E )},(b) if L1(1A) > L2(1A), then L2(1A) = sup{L2(1E )|E ⊂ A, L2(1E ) > L1(1E )}.

Obviously, if L is comonotonoR-additive, i.e., we have that (7) holds for every comonotone functions f, g ∈ M, thenL is comonotone–cosignedR-additive. ∨-Separable aggregation functions are originally introduced in [11], symmetric∨-separable aggregation functions are considered in [16].

Definition 8. Let L : M → [−a, a] be a functional. The support of L, denoted by supp(L), is given by supp(L) ={ f ∈ M| L( f ) � 0}.

It was shown in [9] that the Symmetric Sugeno integral of function f : X → [−1, 1], where X is a finite set,is monotone and symmetric. In [23], the class K1(X ) of measurable functions f : X → [−1, 1] with finite support isconsidered and it is shown that the Symmetric Sugeno integral is comonotone-R-additive on K1(X ) ∩ supp(Sm) if X isa countable set.

We have the following properties of the Symmetric Sugeno integral on M summarized in the next proposition.

Proposition 1. The Symmetric Sugeno integral Sm : M → [−a, a] based on a fuzzy measure m satisfies:

(S1) symmetry on M,


(S2) monotonicity on M,(S3) weak S-homogeneity on M,(S4) comonotone–cosigned R-additivity on supp(Sm).

Proof. By monotonicity and ∧-homogeneity of the original Sugeno integral, Definition 6(ii) and definition of symmetricmaximum we obtain (S1)–(S3). To verify (S4) let us suppose f, g ∈ supp(Sm) such that f �s g. By Lemma 2 andcomonotone ∨-additivity (hence, comonotono–cosigned ∨-additivity) of the original Sugeno integral we obtain

Sm( fRg) = (Sm(( fRg)+))R(−Sm(( fRg)−))

= (Sm(( f +Rg+)))R(−Sm(( f −Rg−)))

= (Sm( f +))R(Sm(g+))R(−(Sm( f −)RSm(g−)))

= (Sm( f +))R(Sm(g+))R(−Sm( f −))R(−Sm(g−))

= (Sm( f +))R(−Sm( f −))R(Sm(g+))R(−Sm(g−))

= Sm( f )RSm(g). �

The Symmetric Sugeno integral is comonotone–cosigned R-additive on supp(Sm), but comonotone–cosigned R-additivity is not fulfilled on M.

Example 2. Let X be an arbitrary universal non-empty set, such that X = A ∪ B, A ∩ B = ∅, A �∅, B �∅, A, B ∈ A.Let m be a fuzzy measure defined on A by

m(E) ={

1 if E �∅,0 if E = ∅.

Since Sm( f ) = Sm( f +)R(−Sm( f −)). For comonotone and cosigned f, g ∈ M, defined by f(x)=0.5, for x ∈ A,else f(x)=0 and g(x)=0.5, for x ∈ A, else g(x)=−0.5, we have Sm( f ) = 0.5 and Sm(g) = 0, but Sm( fRg) =0 � Sm( f )RSm(g).

4. Absolutely monotone set function

In this section we shall introduce conditions for a representation of a real-valued set function as symmetric maximumof two monotone set functions. First, we introduce the notion of absolutely monotone set functions.

Definition 9. A set function m : A → R is

(i) absolutely monotone if for all A, B ∈ A, A ⊂ B, we have |m(A)| ≤ |m(B)|.(ii) strictly absolutely monotone if for all A, B ∈ A, A ⊂ B, A � B, we have |m(A)| < |m(B)|.

Obviously, any fuzzy measure is an absolutely monotone set function.

Example 3. Let X = N = {1, 2, . . .} and let m : P(N) → R be defined by

m(A) =⎧⎨⎩

−k, card(A) = 2k,k + 1, card(A) = 2k + 1,+∞, card(A) = ℵ0,

where ℵ0 = card(N). The set function m is absolutely monotone, but it is not strictly absolutely monotone.

Example 4. Let X={1,2} and let m be a signed fuzzy measure defined on P(X ) by: m(∅) = 0, m({1})=3, m({2})=−2,m({1,2})=1. We have m({1, 2}) < m({1}), hence m is not absolutely monotone, but it is a signed fuzzy measure.

However, some classes of signed fuzzy measures, as it will be shown in Theorem 3, are absolutely monotone. Onesuch signed fuzzy measure is given in the next example.


Example 5. Let X={1,2,3,4} and m be a R-measure such that m(∅) = 0, defined on P(X ) by:

m({1})=0 m({2})=−3 m({3})=−4m({4})=5 m({1,2})=−3 m({1,3})=−4m({1,4})=5 m({2,3})=−4 m({2,4})=5m({3,4})=5 m({1,2,3})=−4 m({1,2,4})=5m({2,3,4})=5 m({1,3,4})=5 m(X)=5

m is absolutely monotone.

Lemma 3. Let m1, m2 : A → [0, a] be fuzzy measures such that for each A ⊂ B, A, B ∈ Awe have m1(A) = m2(A),whenever m1(B) = m2(B). Then a set function m : A → [−a, a] defined by m(A) = m1(A)R(−m2(A)) for A ∈ A,is absolutely monotone.

Proof. For A ⊂ B, A, B ∈ A we have

|m(A)| = |m1(A)R(−m2(A))|= |sign(m1(A) − m2(A))|(|m1(A)| ∨ | − m2(A)|)≤ |sign(m1(B) − m2(B))|(|m1(B)| ∨ | − m2(B)|)= |m(B)|. �

Let m : A → [−a, a], a ∈ R+

be a set function. Let m+, m− : A → [0, a] be given by

m+(A) = m(A) ∨ 0,

m−(A) = (−m(A)) ∨ 0.

Definition 10. Let m : A → [−a, a].

(i) We say that m : A → [−a, a] is sign stable if for all A ∈ A it fulfils:

supE⊂A

m+(E) < m−(A) if m(A) < 0,

supE⊂A

m−(E) < m+(A) if m(A) > 0,

for all E ⊂ A m+(E) = m−(E) if m(A) = 0.

(ii) If m is sign stable and absolutely monotone, we say that m belongs to the class AMSS.

Obviously, every fuzzy measure belongs to AMSS.

Proposition 2. Each strictly absolutely monotone set function m, m(∅) = 0, defined on a finite A belongs to AMSS.

Proof. Let m : A → [−a, a], m(∅) = 0, be strictly absolutely monotone. Therefore, m is an absolutely monotone setfunction. Since A is finite, for all A ∈ A we have

supE⊂A

m+(E) = maxE⊂A

m+(E), supE⊂A

m−(E) = maxE⊂A

m−(E).

Let us show that m is sign stable. As m is strictly absolutely monotone, m(A)=0 if and only if A = ∅.

(1) m(A) < 0. Since m is strictly monotone, we obtain maxE⊂A m+(E) < m−(A).(2) m(A) > 0. We have maxE⊂A m−(E) < m+(A), hence m is sign stable, therefore m ∈ AM SS. �

In the next theorem a decomposition of absolutely monotone and sign stable set functions is established.


Theorem 1. Let m : A → [−a, a], a ∈ R+

, m(∅) = 0. If m ∈ AM SS then the set functions m1, m2 : A → [0, a],defined by

m1(A) ={

m+(A), m(A) ≥ 0,supE⊂A

m+(E), m(A) < 0. (4)

m2(A) ={

m−(A), m(A) ≤ 0,supE⊂A

m−(E), m(A) > 0 (5)

are fuzzy measures such that m = m1R(−m2) and |m| = m1 ∨ m2.If there exist fuzzy measures m1, m2 : A → [0, a], such that for each A ⊂ B, A, B ∈ A, we have m1(A) = m2(A)

whenever m1(B) = m2(B) and m = m1R(−m2), then m ∈ AM SS and m1 ≤ m1 and m2 ≤ m2.

Proof. Let m : A → [−a, a], a ∈ R+

, m(∅) = 0, be an absolutely monotone and sign stable set function and let m1and m2 be two set functions given by (4) and (5) respectively.

Clearly, mi (∅) = 0, for i=1,2 and for any A ∈ A we have

m(A) = m1(A)R(−m2(A)),

|m(A)| = m1(A) ∨ m2(A).

We shall prove now that m1 and m2 are monotone. Let us suppose A, B ∈ A and A ⊂ B. We have the following cases:

(i) m(A) ≥ 0, m(B) > 0. Then m1(A) = m+(A) ≤ m+(B) = m1(B).(ii) m(A) ≥ 0, m(B) < 0. Then by definition of m1 we have

m1(A) = m+(A) ≤ supE⊂B

m+(E) = m1(B).

(iii) m(A) < 0, m(B) > 0. Then by definition of m1 and absolutely monotonicity and sign stability of m we have

m1(A) = supE⊂A

m+(E) < m−(A) ≤ m+(B) = m1(B).

(iv) m(A) < 0, m(B) < 0. Then

m1(A) = supE⊂A

m+(E) ≤ supE⊂B

m+(E) = m1(B).

(v) for m(B)=0 we have m1(A) = m+(A) = m−(A) = 0.

Analogously, m2 is a monotone set function. Hence m1 and m2 are fuzzy measures, and this part of theorem is proven.Let m1 and m2 be fuzzy measures such that for each A ⊂ B, A, B ∈ A, we have m1(A) = m2(A), whenever

m1(B) = m2(B). By Lemma 3 we have that the set function m = m1R(−m2) is absolutely monotone. We will provethat m is sign stable. For m(A) < 0 we have m1(A) < m2(A) and hence

supE⊂A

m+(E) ≤ supE⊂A

m1(E) ≤ m1(A) < m2(A) = m−(A).

Analogously, for m(A) > 0 we have m1(A) > m2(A) and this imply

supE⊂A

m−(E) ≤ supE⊂A

m2(E) ≤ m2(A) < m1(A) = m+(A).

Finally, if m(A)=0 we have m1(A) = m2(A) and this imply

m+(E) = m1(E) = m2(E) = m−(E) = 0,

for all E ⊂ A, E ∈ A.In order to complete the proof we have to show m1 ≤ m1 and m2 ≤ m2. Let A ∈ A. If m(A) ≥ 0, then we have

m1(A) = m1(A). For m(A) < 0 we have m1(A) = supE⊂A m+(E) ≤ supE⊂A m1(E) ≤ m1(A). Hence m1 ≤ m1.Analogously we obtain m2 ≤ m2. �


Example 6. Let m be a set function defined on the subclass of Borel subsets of the interval [−a, a], and for each[c, d] ∈ B([−a, a]), −a ≤ c ≤ d ≤ a, a ∈ R

+, m is defined by

m([c, d]) ={

d − c, d > 0,c − d , d ≤ 0.

m is an absolutely monotone set function. If a < ∞, then m is sign stable on the class of all Borel subsets having theform [c, d]. Hence, there exist m1 and m2 defined by (4), (5), given by

m1([c, d]) ={

d − c, d > 0,0, d ≤ 0,

m2([c, d]) =⎧⎨⎩

0, d > 0, c > 0,−c, d > 0, c ≤ 0,d − c, d ≤ 0.

Obviously, m1 and m2 are fuzzy measures with extensions m1 and m2 on B([−a, a]). This fact by Theorem 1 implies

m(B) = m1(B)R(−m2(B)) for each B ∈ B([−a, a]).

Example 7. Let m be theR-measure from Example 5. m is absolutely monotone and sign stable, and the lowest fuzzymeasures (in the sense of Theorem 1) such that m = m1R(−m2) are m1 and m2 given by (4) and (5):

m1(A) = m+(A) for all A ∈ P(X )

and

m2({1})=0 m2({2})=3 m2({3})=4m2({4})=0 m2({1,2})=3 m2({1,3})=4m2({1,4})=0 m2({2,3})=4 m2({2,4})=3m2({3,4})=4 m2({1,2,3})=4 m2({1,2,4})=3m2({2,3,4})=4 m2({1,3,4})=4 m(X)=4

The relationship of m ∈ AM SS and a R-measure is established in the next theorems. In the sequel of this sectionwe suppose that A = P(X ).

Theorem 2. Let m ∈ AM SS, m(∅) = 0, be such that m = m1R(−m2). If m1 and m2 are completely maxitive fuzzymeasures then m is a R-measure.

Proof. Let m ∈ AM SS, m(∅) = 0, be such that m = m1R(−m2), where m1 and m2 are completely maxitive fuzzymeasures defined by (4) and (5). First, we shall prove that for all A, B ∈ A, A ⊂ B we have m1(A)=m2(A) wheneverm1(B)=m2(B).

Let us suppose m1(A) � m2(A) and m1(B)=m2(B), for some A ⊂ B, A, B ∈ A. Then m(B)=0 and m(A) � 0 what isin a contradiction with the absolutely monotonicity of m.

Now, we shall prove that m is a R-measure.Let A, B ∈ A, A ∩ B = ∅. First, let us suppose m(A ∪ B) = 0. Then m1(A ∪ B) = m2(A ∪ B), therefore we have

m1({x})=m2({x}) for all x ∈ A ∪ B. Since m1 and m2 are completely maxitive, we obtain

supx∈A

m1({x}) = supx∈A

m2({x}), supx∈B

m1({x}) = supx∈B

m2({x}).

Hence, m(A)=0, m(B)=0, and therefore m(A ∪ B) = m(A)Rm(B). For m(A ∪ B) � 0 we have

m(A ∪ B) = m1(A ∪ B)R(−m2(A ∪ B))

= (m1(A) ∨ m1(B))R(−(m2(A) ∨ m2(B)))

= (m1(A)Rm1(B))R((−m2(A))R(−m2(B)))


= (m1(A)R(−m2(A)))R(m1(B)R(−m2(B)))

= m(A)Rm(B),

because R is commutative and associativity of R is fulfilled for all c1, c2, . . . , cn ∈ [0, a] such that∨i ci � −∧

i ci . �

Theorem 3. Let m be a R-measure such that each subset of m-null-set is m-null-set, i.e., for each A ⊂ B, A, B ∈ Awe have m(A)=0, whenever m(B)=0. Then m ∈ AM SS.

Proof. Let m be a R-measure such that each subset of m-null-set is m-null-set. m is a R-measure, hence for all A ∈ Awe have m(A) = Rx∈A m({x}).

We define m1 and m2 on A by

m1(A) = supx∈A

m+({x}), m2(A) = supx∈A

m−({x}).

Obviously, m1 and m2 are fuzzy measures such that m = m1R(−m2). If m1(B) = m2(B) for B ∈ A, then for eachA-measurable subset A of set B we have m1(A) = m2(A). By Theorem 1 we obtain m ∈ AM SS. �

5. Sugeno integral based on absolutely monotone and sign stable set functions

Keeping the same notation given in Sections 2 and 3, we introduce the Sugeno integral of f ∈ M, based onm ∈ AM SS.

Definition 11. The Sugeno integral of function f ∈ M with respect to an absolutely monotone and sign stable setfunction m : A → [−a, a], m(∅) = 0 is defined by

Sm( f ) = Sm1 ( f )R(−Sm2 ( f )), (6)

where the integrals on the right-hand side of the equality are Symmetric Sugeno integrals with respect to the fuzzymeasures m1 and m2, given by (4) and (5).

Remark 2. Since each fuzzy measure belongs to AMSS, specially, for a fuzzy measure m and any f ∈ M we haveSm( f ) = Sm( f ). Obviously, for a fuzzy measure m and a non-negative function f ∈ M+ it holds Sm( f ) = Sm( f ) =Sm( f ).

Example 8. Let m ∈ AM SS be from Example 5. Let f be defined by f(1)=−3, f(2)=−2, f(3)=1, f(4)=3. For i=1,2, wehave Smi ( f ) = ((1 ∧ mi ({3, 4})) ∨ (3 ∧ mi ({4})))R(−((2 ∧ mi ({2, 1})) ∨ (3 ∧ mi ({1})))). Therefore,

Sm( f ) = Sm1 ( f )R(−Sm2 ( f )) = (3R0)R(−(1R(−2)) = 3.

In this special case, also, we obtain

Sm( f ) = Sm( f +)R(−Sm( f −)) = (3R(−1))R(−(0R(−2))) = 3.

For m ∈ AM SS and for each f ∈ supp(Sm1 ) ∩ supp(Sm2 ), where Sm1 and Sm2 are Symmetric Sugeno integrals withrespect to fuzzy measures m1 and m2, given by (4) and (5), respectively, we can express (6) in the following form:

Sm( f ) = Sm( f +)R(−Sm( f −)).

Some of the properties of the Sugeno integral based on m ∈ AM SS are given in the next propositions.

Proposition 3. The Sugeno integral Sm based on m ∈ AM SS fulfils the next properties:

(GS1) symmetry: for all f ∈ M we have

Sm(− f ) = −Sm( f ),


(GS2) absolutely monotonicity: for all f ∈ M+, g ∈ supp(Sm) we have

f ≤ g ⇒ |Sm( f )| ≤ |Sm(g)|,(GS3) weak S-homogeneity on supp(Sm): for all basic function c · 1A ∈ supp(Sm), A ∈ A, c ∈ [−a, a], we have

Sm(c · 1A) = cSm(A),

(GS4) comonotone–cosignedR-additivity: for all f, g ∈ supp(Sm1 ) ∩ supp(Sm2 ) ∩ supp(Sm) such that f �s g we have

Sm( fRg) = Sm( f )RSm(g).

Proof. Directly from (6) and (S1) we obtain (GS1). To verify (GS2), it is enough to remark that

(a) for each f ∈ M+ we have |Sm( f )| ≤ |Sm1 ( f )| ∨ |Sm2 ( f )|,(b) for each g ∈ supp(Sm) we have |Sm(g)| = |Sm1 (g)| ∨ |Sm2 (g)|.

Let us suppose f ≤ g and g ∈ supp(Sm). By (S2), (a) and (b) we obtain |Sm( f )| ≤ |Sm(g)|.By (S3) we have for each c ∈ [0, a] and A ∈ A, c · 1A ∈ supp(Sm)

Sm(c · 1A) = (cSm1(A))R(cS(−m2(A)) � 0. (7)

Therefore, by Lemma 1 and (GS1) we obtain (GS3).Let f �s g, f, g ∈ supp(Sm1 ) ∩ supp(Sm2 ). Using the comonotone–cosigned R-additivity of Sm1 and Sm2 , for f, g ∈

supp(Sm) we obtain

Sm( fRg) = Sm1 ( fRg)R(−Sm2 ( fRg))

= (Sm1 ( f )RSm1 (g))R(−(Sm2 ( f )RSm2 (g)))

= Sm1 ( f )R(−Sm2 ( f ))RSm1 (g)R(−Sm2 (g))

= Sm( f )RSm(g). �

Sm is not monotone, neither absolutely monotone on the whole class M. For the property (GS2) it is necessary thatg ∈ supp(Sm).

Example 9. Let m ∈ AM SS be from Example 5. Let f, g : X → [−5, 5] be simple functions, defined by f = 2 · 1{4}and g = 3 · 1X . Then by (GS3) we have Sm( f ) = 2Sm({4}) = 2S5 = 2. We obtain Sm(g) = (3Sm1(X ))R(3S(−m2(X ))) = (3S5)R(3S(−4)) = 3R(−3) = 0. Obviously, g /∈ supp(Sm). We have f ≤ g and |Sm( f )| >

|Sm(g)|.

Proposition 4. Let m ∈ AM SS, with m(∅) = 0. If f ∈ supp(Sm1 ) ∩ supp(Sm2 ), where m1 and m2 are given by (4) and(5), respectively, then

S|m|( f ) = Sm1 ( f )RSm2 ( f ),

where the integrals on the right-hand side of the equality are the Symmetric Sugeno integrals of f.

Proof. A set function |m| : A → [0, a], defined by |m|(A) = |m(A)|, for A ∈ A is monotone if m is absolutelymonotone and sign stable, and |m| = m1 ∨ m2, where m1 and m2 are defined by (4) and (5), respectively. By themaxitivity with respect to an underlying fuzzy measure of the ordinary Sugeno integral we have

S|m|( f ) = Sm1∨m2 ( f )

= Sm1∨m2 ( f +)R(−Sm1∨m2 ( f −))

= (Sm1 ( f +) ∨ Sm2 ( f +))R(−(Sm1 ( f −) ∨ Sm2 ( f −)))

= Sm1 ( f +)R(−Sm1 ( f −))RSm2 ( f +)R(−Sm2 ( f −))

= Sm1 ( f )RSm2 ( f ). �


In the sequel, we consider conditions for representations of a functional L : M → [−a, a] by Sugeno integral basedon m ∈ AM SS.

Theorem 4. Let L : M → [−a, a] be a functional which fulfils the properties (i)–(iv) on M and (v) on M+ given inDefinition 7. Then there exists a fuzzy measure m continuous from below, such that, for all f ∈ M, we have

L( f ) = Sm( f ),

where Sm is the Symmetric Sugeno integral of f.

Proof. Let a set function m be defined for all A ∈ A by m(A) = L(1A). For A ⊂ B, A, B ∈ A, we have m(A) =L(1A) ≤ L(1B) = m(B), i.e., m is a fuzzy measure. Since L is a functional continuous from below, m is a fuzzymeasure continuous from below. Using the properties (i), (iii), and (iv) of the functional L, when applying it on thecomonotone–cosigned R-representation (3) of a simple function s ∈ S we obtain

L(s) = L(s+)R(−L(s−)) = Sm(s+)R(−Sm(s−)) = Sm(s).

The classical Sugeno integral is the only monotone functional onM+ which is comonotone ∨-additive, ∧-homogeneousand continuous from below. L and Sm are identical on S+ and hence, for f ∈ M we have

L( f +) = L(supn

sn)

= supn

L(sn)

= supn

Sm(sn)

= Sm( f +),

where {sn}n∈N is the non-decreasing sequence of simple non-negative functions converging to f +, defined by

sn =n·2n∨i=1

i

2n· 1Cn,i , n ∈ N,

where Cn,i = {x ∈ X | f +(x) ≥ i/2n}, for i = 0, . . . , n · 2n . Analogously, we obtain L( f− )=Sm( f− ). Therefore, weobtain

L( f ) = L( f +)R(−L( f −)) = Sm( f +)R(−Sm( f −)) = Sm( f ). �

Theorem 5. Let L : M → [−a, a] be a signed monotone ∨-separable functional such that monotone functionals L1and L2 from Definition 7(vii) fulfil the properties (i), (iii), (iv) on M and (v) on M+ given in Definition 7, and for allA ⊂ B, A, B ∈ A, the equality L1(1B) = L2(1B) implies L1(1A) = L2(1A). Then, there exists m ∈ AM SS such thatfor all f ∈ M, we have

L( f ) = Sm( f ),

where Sm is the Sugeno integral based on m.

Proof. Let L : M → [−a, a] be a monotone ∨-separable functional, i.e.,

L( f ) = L1( f )R(−L2( f )) for f ∈ M.

By Theorem 4 there exist fuzzy measures m1 and m2 such that for any f ∈ M we have

L1( f ) = Sm1 ( f ), L2( f ) = Sm2 ( f ). (8)

For each A ∈ A, we have Li (1A) = mi (A), i=1,2. Therefore, m1 and m2 satisfy the condition given in Theorem 1 andhence m = m1R(−m2) and m ∈ AM SS. L is signed monotone ∨-separable and by Theorem 1 for m(A) < 0, we have

m1(A) = L1(1A) = supE⊂A,

L1(1E )>L2(1E )

L1(1E ) = supE⊂A

m+(E) = m1(A),


and for m(A) ≥ 0, we have m1(A) = m1(A). Therefore m1 = m1. Analogously, we obtain m2 = m2 and by (8), forall f ∈ M, we have

L( f ) = Sm1 ( f )R(−Sm2 ( f )) = Sm( f ). �

Acknowledgments

The work has been supported by the project MNTRS 144012 and the project “Mathematical Models for DecisionMaking under Uncertain Conditions and Their Applications” supported by Vojvodina Provincial Secretariat for Scienceand Technological Development. The second author is supported by grant of MTA of HTMT.

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sugeno integral based on absolutely monotone real set functions

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