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Possibility and necessity integrals
Gert de CoomanUniversiteit Gent, Vakgroep Elektrische Energietechniek, Technologiepark 9, 9052 Zwijnaarde, Belgium
Etienne E. KerreUniversiteit Gent, Vakgroep Toegepaste Wiskunde en Informatica, Krijgslaan 281–S9, 9000 Gent, Bel-gium
Abstract: In this paper, we introduce seminormed and semiconormed fuzzy integrals associated withconfidence measures. These confidence measures have a field of sets as their domain, and a completelattice as their codomain. In introducing these integrals, the analogy with the classical introduction ofLegesgue integrals is explored and exploited. It is amongst other things shown that our integrals are themost general integrals that satisfy a number of natural basic properties. In this way, our dual classes offuzzy integrals constitute a significant generalization of Sugeno’s fuzzy integrals.
A large number of important general properties of these integrals is studied. Furthermore, and mostimportantly, the combination of seminormed fuzzy integrals and possibility measures on the one hand,and semiconormed fuzzy integrals and necessity measures on the other hand, is extensively studied.It is shown that these combinations are very natural, and have properties which are analogous to thecombination of Lebesgue integrals and classical measures. Using these results, the very basis is laid fora unifying measure- and integral-theoretic account of possibility and necessity theory, in very much thesame way as the theory of Lebesgue integration provides a proper framework for a unifying and formalaccount of probability theory.
Keywords: Operators; measure theory; triangular (semi)norms and (semi)conorms; confidence measures;seminormed and semiconormed fuzzy integrals; possibility and necessity theory.
1 Introduction
In his doctoral dissertation, Sugeno [20] introduced fuzzy measures and fuzzy integrals associated withthem. Although Sugeno’s definition of a fuzzy integral is clearly a fuzzification of the Choquet integral[3], his integral also bears a formal resemblance to the Lebesgue integral, that plays a very important rolein classical measure and probability theory. This was shown in 1980 by Ralescu and Adams [15] in animportant paper which explores the relation between Sugeno’s fuzzy integral and the Lebesgue integral.Weber has thoroughly studied integrals associated with special types of decomposable fuzzy measures[24, 25, 26]. He has also introduced a number of new integrals associated with these fuzzy measures.Interesting extensions of the fuzzy integral in the same spirit as Weber’s work have been introducedby Sugeno and Murofushi [21]. Murofushi and Sugeno [13] have also proposed the Choquet integralas an alternative for the fuzzy integral. In a recent paper [10], Grabisch, Murofushi and Sugeno haveintroduced a new type of integral that is a generalization of the above-mentioned types. On the otherhand, Sugeno’s fuzzy integral has been generalized by Suarez Garcıa and Gil Alvarez towards seminormedand semiconormed fuzzy integrals [19]. These generalizations do not fit into the general class of integralsintroduced by Grabish et. al.
In the theory of the above-mentioned integrals, the chain ([0, 1],≤) plays the important part of thecodomain of the fuzzy measures they are associated with. In [4], we have introduced possibility integralsassociated with (L,≤)-possibility measures. In his doctoral dissertation [5], one of us has extended thesenotions towards seminormed and semiconormed (L,≤)-fuzzy integrals associated with a very generalclass of measures, namely (L,≤)-confidence measures having a complete lattice (L,≤) as their codomain
1
and a field of sets as their domain. He has also convincingly shown that, on the one hand, seminormed(L,≤)-fuzzy integrals and (L,≤)-possibility measures, and, on the other hand, semiconormed (L,≤)-fuzzyintegrals and (L,≤)-necessity measures, are a perfect match. This paper reports on these results.
Our seminormed and semiconormed (L,≤)-fuzzy integrals are a generalization of existing integrals,and many properties of Sugeno’s fuzzy integrals, and of Suarez and Gil’s seminormed and semiconormedfuzzy integrals remain valid for our integrals. This generalization, and especially the transition towardsmore general codomains for the associated measures, involves a number of complications. In many casesthe proofs of the above-mentioned properties must be changed or completely reformulated. We shalltherefore in the sequel as often as reasonable explicitly state these proofs.
Our integrals are, in a sense, the most general types of integrals that can be associated with (L,≤)-confidence measures and that satisfy at the same time a number of natural properties. Their importancelies in the fact that they can be harmoniously associated with (L,≤)-possibility and (L,≤)-necessitymeasures, and help provide a unifying measure- and integral-theoretic foundation to possibility andnecessity theory, in very much the same way as the Lebesgue integral provides the basis for a systematictreatment of probability theory.
In section 2, we give a brief summary of the preliminary definitions and notational conventions, nec-essary for a proper understanding of the main material of the paper. In section 3, we introduce ourgeneralized seminormed and semiconormed fuzzy integrals in a way that is formally analogous to thatin which traditionally Lebesgue integrals are introduced (see, for instance, [2, 11, 22]). It is shown herethat the definition of these classes of integrals—which are, in a sense, each other’s duals—is a general aspossible: integrals satisfying a certain number of natural properties must belong to one of these classes.We also deduce a number of important formulas that will facilitate the subsequent use of these integrals.In section 4, general properties of our integrals are proven. In section 5 we take a conceptually veryimportant step: we deduce interesting results and properties for our generalized types of seminormedand semiconormed fuzzy integrals, when they are associated with our generalized possibility respectivelynecessity measures. Section 6 concludes this paper with a brief discussion of the most important resultsand their relevance to fuzzy set, possibility and necessity theory. Indeed, using the results discussedin this paper, a general measure- and integral-theoretic treatment can be given for Zadeh’s possibilitytheory [5, 28], which includes a consistent account of product possibility measures and integrals, and ofconditional possibility and possibilistic independence.
2 Preliminary definitions
Let us begin with a few preliminary definitions and notational conventions. We shall denote by X anarbitrary universe, that contains at least two different elements, so that there exist proper subsets of X.
By (L,≤) we shall mean a complete lattice that is arbitrary but fixed throughout the whole text. Thesmallest element of (L,≤) will be denoted by ` and the greatest element by u. We shall also assume that` 6= u. The meet of (L,≤) will be denoted by _, the join of (L,≤) by ^.
With an arbitrary subset A of a universe X, we can associate its characteristic X − L mapping χA,defined by
χA(x) def=
u ; x ∈ A` ; x ∈ coA.
An arbitrary X − L mapping will be called a (L,≤)-fuzzy set on X, in accordance with the terminologyintroduced by Goguen [9]. The set of the (L,≤)-fuzzy sets on X will be denoted by F(L,≤)(X). Weshall also need the partial order relation v on F(L,≤)(X), defined as follows: for arbitrary h1 and h2 inF(L,≤)(X),
h1 v h2 ⇔ (∀x ∈ X)(h1(x) ≤ h2(x)).
Of course, the structure (F(L,≤)(X),v) is a complete lattice. The supremum in this complete lattice isthe pointwise supremum, and the infimum the pointwise infimum. The complete lattice (P(X),⊆) can
2
be embedded in this complete lattice by the injection that maps a subset of X to its characteristic X−Lmapping.
For arbitrary λ in L, λ will denote the constant X −λ mapping. Furthermore, let h be an arbitrary(L,≤)-fuzzy set on X, and let λ be an arbitrary element of L. We shall use the following notations:
Shλ
def= h−1([λ, u]) = x | x ∈ X and h(x) ≥ λ Dh
λdef= h−1([`, λ]) = x | x ∈ X and h(x) ≤ λ
Nhλ
def= h−1(λ) = x | x ∈ X and h(x) = λ .
Shλ will be called the λ-cut of h and is a generalization of the level sets of standard fuzzy set theory. Dh
λis called the dual λ-cut of h and is related to the strict level sets of standard fuzzy set theory. Indeed, if(L,≤)=([0, 1],≤), the complement of Dh
λ is a strict level set of h [27]. Nhλ will be called the λ-level of h.
We shall denote by V an arbitrary proper field of subsets of a universe X, i.e., a set of subsets ofX that contains ∅, is closed under complementation and finite unions, and also contains at least oneproper subset of X. We shall say that a subset A of X is V-measurable iff A ∈ V. Furthermore, anarbitrary X − L mapping h is called V-cut measurable iff (∀λ ∈ L)(Sh
λ ∈ V), dually V-cut measurable iff(∀λ ∈ L)(Dh
λ ∈ V) and V-level measurable iff (∀λ ∈ L)(Nhλ ∈ V).
An isotonic (V,⊆)− (L,≤) mapping v will be called a (L,≤)-confidence measure on (X,V) [5], i.e.,
(∀(A,B) ∈ V2)(A ⊆ B ⇒ v(A) ≤ v(B)).
The triple (X,V, v) is called a (L,≤)-confidence space. If v(∅) = `, then v is called normalized from below;and v is called normalized from above if v(X) = u. Finally, v is called normalized if it is normalized fromabove and from below. Whenever we want to omit reference to the structure (L,≤), we shall also ingeneral speak about confidence measures and confidence spaces. Evidently, confidence measures are ageneralization of Sugeno’s fuzzy measures [20] and the mesures de confiance of Dubois and Prade [8]. Fora more detailed account of our theory of confidence measures, we refer to [5].
3 Seminormed and semiconormed fuzzy integrals
In Lebesgue integration theory, simple mappings—there defined as measurable real mappings with afinite range—play an important role. We want to give an extended definition of simple mappings inthe same vein as the above-mentioned classical definition, that is however better adapted to the moregeneral framework studied here. We shall define a simple mapping as a mapping with a finite range thatfurthermore satisfies certain conditions of measurability.
Definition 1 Let X be an arbitrary universe and let V be an arbitrary proper field on X. A X −L mapping s is called V-simple iff it has a finite range s(X) = a1, . . . , an (n ∈ N∗) and (∀k ∈1, . . . , n)(Dk
def= s−1(ak) ∈ V).
In other words, a X − L mapping with finite range is simple if and only if it is V-level measurable.Moreover, it can be proven [5] that for X−L mappings with finite range the notions V-cut measurability,dual V-cut measurability and V-level measurability coincide.
In order to define an integral in classical integration theory, certain types of functionals on the set ofsimple mappings are first defined. The definition of these functionals draws its inspiration from the formof possible decompositions of the simple mappings. More explicitly, when (Ω,S) is a measurable space,m is a classical measure on (Ω,S) and s is a non-negative (classical) simple mapping, we may write withobvious notations (see, for instance, [2])
(∀ω ∈ Ω)(s(ω) =n
∑
k=1
akχDk(ω)), (1)
3
with n ∈ N∗, s(Ω) = a1, . . . , an, ak ∈ R and χDk the characteristic Ω − 0, 1 mapping of Dkdef=
s−1(ak) ∈ S (k ∈ 1, . . . , n). Let furthermore E be an arbitrary element of S. The value of thefunctional IE in s is, by definition, given by
IE(s) def=n
∑
k=1
akm(E ∩Dk). (2)
The formal analogy between (1) and (2) is striking. Let us also remark that decompositions of the simplemapping s other than (1) are possible. Indeed, if we furthermore assume that the ak are ordered in sucha way that
(∀(k, l) ∈ 1, . . . , n2)(k ≤ l ⇒ ak ≤ al),
we have that
(∀ω ∈ Ω)(s(ω) =n
∑
l=1
(al − al−1)χFl(ω)), (3)
with
Fldef=
n⋃
k=l
Dk = s−1([al, 1]), l ∈ 1, . . . , n
and aodef= 0. Using this decomposition, the functional CE can be introduced as:
CE(s) def=n
∑
l=1
(al − al−1)m(E ∩ Fl).
It is interesting to note that, while the decomposition (1), and therefore also the functional IE , areconnected with the definition of the Lebesque integral, decomposition (3), and therefore also the functionalCE , essentially lead to the definition of the Choquet integral1 (see, for instance, [3, 13]).
This observation is the starting point of our attempt to generalize the seminormed and semiconormedfuzzy integrals of Suarez Garcıa and Gil Alvarez. Let us first investigate which decompositions of theform (1)—or of a related form—are possible in this more general context.
Theorem 1 Let X be an arbitrary universe. Let φ and ξ be binary operators on L that are commutativeand associative. Let ψ and ζ be binary operators on L.
(i) The following proposition holds for all X − L mappings s with finite range:
(∀x ∈ X)(s(x) = φnk=1ψ(ak, χDk(x))), (4)
if and only if φ and ψ satisfy
(∀λ ∈ L)(ψ(λ, u) = λ)(∀(λ, µ) ∈ L2)(φ(λ, ψ(µ, `)) = λ). (5)
(ii) The following proposition holds for all X − L mappings s with finite range:
(∀x ∈ X)(s(x) = ξnk=1ζ(ak, χcoDk(x))), (6)
if and only if ξ and ζ satisfy
(∀λ ∈ L)(ζ(λ, `) = λ)(∀(λ, µ) ∈ L2)(ξ(λ, ζ(µ, u)) = λ). (7)
1For classical measures, the σ-additivity property holds, which implies that there is no difference between the Lebesgueand Choquet integrals associated with these measures. Differences between these integrals only arise for non-classicalmeasures that are not additive.
4
Proof. We shall give the proof of (ii). The proof of (i) is analogous. First, let us assume that (6) holdsfor arbitrary X − L mappings with finite range. Let λ and µ be arbitrary elements of L. For the X − Lmapping λ, we deduce from (6) in particular that λ = ζ(λ, `). Also consider the X−L mapping s, definedby
s(x) def=
λ ; x ∈ Aµ ; x ∈ coA,
where x is an arbitrary element of X and A an arbitrary proper subset of X. Since X is assumed tocontain at least two elements, this is always possible. Using (6) and putting x ∈ A, we deduce that
λ = ξ(ζ(λ, `), ζ(µ, u)) = ξ(λ, ζ(µ, u)).
Conversely, let us assume that (7) holds. Consider an arbitrary X−L mapping s with finite range. First,if s is a constant mapping, we immediately see that (6) holds. Let us therefore assume that s is not aconstant mapping, i.e., with the notations of definition 1, s(X) = a1, . . . , an with n ∈ N∗ and n > 1.Consider an arbitrary x in X and assume that x ∈ Dk, where k is an arbitrary element of 1, . . . , n.Then, using (7) and repeatedly taking into account the commutativity and associativity of ξ:
ξnm=1ζ(am, χcoDm(x)) = ξ(ζ(ak, χcoDk(x)), ξm 6=kζ(am, χcoDm(x)))
= ξ(ζ(ak, `), ξm6=kζ(am, u))= ξ(ak, ξm 6=kζ(am, u)) = ak = s(x).
We conclude that (6) holds. 2
In the rest of this section, we shall denote by φ a commutative and associative binary operator on L andby ψ a binary operator on L such that (5) holds for φ and ψ. Also, we shall denote by ξ a commutativeand associative binary operator on L and by ζ a binary operator on L such that (7) holds for ξ and ζ.This implies that for an arbitrary V-simple X − L mapping the decompositions (4) and (6) are valid.It should in this context also be noted that there always exist φ and ψ satisfying (5), e.g., φ = ^ andψ = _. Analogously, there always exist ξ and ζ satisfying (7), e.g., ξ = _ and ζ = ^. It is also veryimportant to note that the conditions (5) and (7) are dual in a order-theoretic sense. Our introductorydiscussion makes the following definition rather obvious.
Definition 2 Let (X,V, v) be an arbitrary (L,≤)-confidence space. Let s be a V-simple X − L mappingand let A be an arbitrary element of V. Then the following definitions (with the notations of definition 1)make sense.
(i) αvφψ(A ; s) def= φn
k=1ψ(ak, v(A ∩Dk)).
(ii) βvξζ(A ; s) def= ξn
k=1ζ(ak, v(A ∩ coDk)).
In the following propositions, we investigate the requirements that the operators φ and ψ must havein order that the functional αv
φψ(· ; ·) satisfy some natural properties. We say that αvφψ(· ; ·) is isotonic in
both arguments iff for arbitrary A1 and A2 in V and for arbitrary V-simple X − L mappings s1 and s2:
(A1 ⊆ A2 and s1 v s2) ⇒ αvφψ(A1 ; s1) ≤ αv
φψ(A2 ; s2).
Proposition 1 In order that for an arbitrary (L,≤)-confidence space (X,V, v) the functional αvφψ(· ; ·)
be isotonic in both arguments, it is necessary that φ and ψ be isotonic.
Proof. Let us assume that αvφψ(· ; ·) is isotonic in both arguments for arbitrary (L,≤)-confidence spaces.
Choose arbitrary λ, µ and ν in L and assume that λ ≤ µ. First, it is always possible to choose a (L,≤)-confidence space (X,V, v) such that there exists an element A of V for which v(A) = ν. Since λ ≤ µ andtherefore also λ v µ, it follows from the assumptions that
ψ(λ, ν) = αvφψ(A ;λ) ≤ αv
φψ(A ; µ) = ψ(µ, ν).
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We conclude that ψ is isotonic in its first argument.Next, it is always possible to choose a (L,≤)-confidence space (X,V, v) such that there exist elements
A1 and A2 in V for which v(A1) = λ and v(A2) = µ and A1 ⊆ A2. It now follows from the assumptionsthat
ψ(ν, λ) = αvφψ(A1 ; ν) ≤ αv
φψ(A2 ; ν) = ψ(ν, µ).
We conclude that ψ is also isotonic in its second argument. Hence, ψ is isotonic in both arguments, andtherefore also isotonic.
Finally, let λ1, λ2, µ1 and µ2 be arbitrary elements of L. Assume that λ1 ≤ λ2 and µ1 ≤ µ2. It isalways possible to choose a (L,≤)-confidence space (X,V, v) such that (∀A ∈ V \ ∅)(v(A) = u) andsuch that V contains at least one element B other than ∅ and X. Furthermore, let s1 and s2 be twoV-simple X − L mappings, defined by
sk(x) def=
λk ; x ∈ Bµk ; x ∈ coB
for arbitrary x in X and k in 1, 2. Of course, we have that s1 v s2, and it therefore follows from theassumptions that
φ(λ1, µ1) = φ(ψ(λ1, u), ψ(µ1, u))= αv
φψ(X ; s1)≤ αv
φψ(X ; s2)= φ(ψ(λ2, u), ψ(µ2, u)) = φ(λ2, µ2).
We conclude that φ is isotonic as well. 2
Proposition 2 In order that for an arbitrary (L,≤)-confidence space (X,V, v) the following propositionhold:
(∀A ∈ V)(αvφψ(A ; u) = v(A)),
it is necessary that(∀λ ∈ L)(ψ(u, λ) = λ). (8)
Proof. Let λ be an arbitrary element of L. It is always possible to choose a (L,≤)-confidence space(X,V, v) and an element A of V, such that v(A) = λ. It now follows from the assumption that
λ = αvφψ(A ; u) = ψ(u, v(A)) = ψ(u, λ). 2
These propositions lead us to consider φ that are isotonic, and ψ that are isotonic and satisfy (8). Ifwe summarize the requirements for ψ, we find that it must be an isotonic binary operator on L, thatsatisfies
(∀λ ∈ L)(ψ(λ, u) = ψ(u, λ) = λ).
Such an operator has been studied before (see [4, 6]). It is called a t-seminorm on (L,≤). It is ageneralization of the t-seminorms on ([0, 1],≤), that were introduced by Suarez Garcıa and Gil Alvarez[19]. It is easily shown that such a ψ also satisfies
(∀λ ∈ L)(ψ(λ, `) = ψ(`, λ) = `).
If we substitute this last expression in the second expression of (5), we find that φ satisfies
(∀λ ∈ L)(φ(λ, `) = λ). (9)
Summarizing the requirements for φ, we find that it must be a commutative, associative and isotonicbinary operator on L that satisfies (9). Again, such an operator has been studied before (see [4, 6]). It
6
is called a t-conorm on (L,≤). It is a generalization of the t-conorms on ([0, 1],≤), that were introducedby Schweizer and Sklar [17, 18]. In the sequel, we shall denote by P an arbitrary t-seminorm on (L,≤),by S an arbitrary t-conorm on (L,≤), and only concern ourselves with functionals of the type αv
SP (· ; ·).In a fairly similar way, we could investigate the properties that the operators ξ and ζ must have in order
that the functional βvξζ(· ; ·) satisfy similar natural properties, and thus arrive at similar, dual results. Due
to limitations of space, we shall in this paper omit this investigation and simply assume from the outsetthat the restrictions imposed on ξ and ζ are in a sense dual to those imposed on φ and ψ respectively.This means that from now on, we shall assume that ξ is a t-norm on (L,≤) and that ζ is a t-semiconormon (L,≤). This means that ζ is an isotonic binary operator on L, that satisfies
(∀λ ∈ L)(ζ(λ, `) = ζ(`, λ) = λ),
and that ξ is an isotonic, commutative and associative binary operator on L satisfying
(∀λ ∈ L)(ξ(λ, u) = ξ(u, λ) = λ).
In the sequel, we shall denote by Q an arbitrary t-semiconorm on (L,≤), by T an arbitrary t-norm on(L,≤), and only make use of functionals of the type βv
TQ(· ; ·). We shall not explicitly concern ourselveswith the discussion of the properties of the t-(semi)norms and t-(semi)conorms that in general can bedefined on bounded partially ordered sets. For a detailed discussion of these dual classes of operators, weagain refer to [4, 6]. It should however be explicitly mentioned here that _ is a t-(semi)norm on (L,≤)and that ^ is a t-(semi)conorm on (L,≤).
Let us now use the functionals αvSP (· ; ·) and βv
TQ(· ; ·) to try and define functionals (integrals) on theset F(L,≤)(X) and not just on the set of the V-simple X − L mappings. In definition 3 (i) an integral isintroduced in very much the same way as the Lebesgue integral is traditionally defined. The integral indefinition 3 (ii) is in a sense dual to the first one.
Definition 3 Let (X,V, v) be an arbitrary (L,≤)-confidence space. Let h be a X − L mapping and letA be a V-measurable set.
(i) (SP ) –∫
Ahdv def= sup αv
SP (A ; s) | s is V-simple and s v h is called the (L,≤)-fuzzy SP -integral of
h on A (associated with v).
(ii) (TQ) –∫
Ahdv def= inf βv
TQ(A ; s) | s is V-simple and h v s is called the (L,≤)-fuzzy TQ-integral of
h on A (associated with v).
In general, an integral of a mapping over a set can be considered as a ‘weighted aggregation’ of thevalues the mapping takes over the set. The ‘weights’ in this aggregation are expressed in terms of the‘measure’ the integral is associated with. It is in the first place very important to study the behaviour ofan integral when it acts on constant mappings. In the following theorem, we investigate this behaviourfor (L,≤)-fuzzy SP - and TQ-integrals.
Theorem 2 (i) In order that for an arbitrary (L,≤)-confidence space (X,V, v) with v normalized, foran arbitrary t-seminorm P on (L,≤) and for arbitrary µ in L the following hold:
(SP ) –∫
Xµdv = µ, (10)
it is necessary and sufficient that S = ^.
(ii) In order that for an arbitrary (L,≤)-confidence space (X,V, v) with v normalized, for an arbitraryt-semiconorm Q on (L,≤) and for arbitrary µ in L the following hold:
(TQ) –∫
Xµdv = µ, (11)
it is necessary and sufficient that T =_.
7
Proof. As an example, we shall prove (ii). The proof of (i) is analogous. We shall prove in corollary 1furtheron that for arbitrary µ in L
(_Q) –∫
Xµdv = µ.
Let us therefore now concentrate on the reverse implication. If L contains only two elements, there existsonly one t-norm on (L,≤), that is precisely the meet _ of (L,≤) [6], and therefore the proof is trivial.Let us now assume that L contains more than two elements, which implies that there exists more thanone t-norm on (L,≤) [6]. Consider an arbitrary t-norm T on (L,≤) that differs from _. This means thatthere exist λ and µ in L such that T (λ, µ) < λ _ µ [4, 6] and therefore also T (λ, µ) < λ and T (λ, µ) < µ.Remark that this implies that ` < λ and ` < µ. Choose Q = ^, X = x1, x2, A1 = x1, A2 = x2,V = ∅, A1, A2, X and the (L,≤)-confidence measure v on (X,V) such that v(A1) < λ and v(A2) < µ,which is always possible. Define the V-simple X − L mapping so by
so(x) def=
λ ; x ∈ A2µ ; x ∈ A1.
For these choices we have that λ _ µ v so, whence
(T ^) –∫
Xλ _ µdv = inf
λ_µvsβv
T^(X ; s)
≤ βvT^(X ; so)
= T (λ ^ v(coA2), µ ^ v(coA1))= T (λ ^ v(A1), µ ^ v(A2))= T (λ, µ) < λ _ µ. 2
Whenever the ‘weight’ v(X) of the universe X equals u, it seems very natural to demand that the‘weighted aggregation’ of a constant mapping on this universe be equal to the constant value of thatmapping. We shall therefore in the sequel only concern ourselves with those classes of integrals for whichformulas (10) and (11) hold. This leads to the following basic definition.
Definition 4 Let (X,V, v) be an arbitrary (L,≤)-confidence space. Let h be a X − L mapping and letA be a V-measurable set.
(i) (P ) –∫
Ahdv def= (^P ) –
∫
Ahdv is called the (L,≤)-fuzzy P -integral of h on A (associated with v).
(ii) (Q) –∫
Ahdv def= (_Q) –
∫
Ahdv is called the (L,≤)-fuzzy Q-integral of h on A (associated with v).
In general, we shall also call these integrals seminormed respectively semiconormed (L,≤)-fuzzy integrals.Both types of integral will be given the collective name ‘(L,≤)-fuzzy integral’. Whenever we do notwant to be explicit about the complete lattice (L,≤) we shall simply speak of seminormed fuzzy integrals,semiconormed fuzzy integrals and fuzzy integrals.
This definition marks the endpoint of our search for a fairly general definition of seminormed and semi-conormed fuzzy integrals. It can be verified that Sugeno’s fuzzy integrals are special seminormed fuzzyintegrals, indeed special instances of ([0, 1],≤)-fuzzy min-integrals. Furthermore, the semi(co)normedfuzzy integrals of Suarez Garcıa and Gil Alvarez are also generalized by our definition.
To conclude this section, we shall deduce a few formulas that facilitate the calculation and subsequenttreatment of fuzzy integrals. First of all, we want to stress that the (L,≤)-fuzzy P - and Q-integrals of anarbitrary X −L mapping on an arbitrary V-measurable subset of X always exist, since we have assumedfrom the outset that (L,≤) is a complete lattice.
Theorem 3 gives us formulas for the calculation of (L,≤)-fuzzy integrals of arbitrary X −L mappings.In theorem 4 we show that these formulas can be further simplified whenever the mappings considered
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satisfy certain measurability requirements. Let us point out that these measurability conditions aredifferent—indeed, dual—for the two types of fuzzy integrals.
Theorem 3 Let (X,V, v) be an arbitrary (L,≤)-confidence space. Let h be a X − L mapping and let Abe a V-measurable set. Then
(i) (P ) –∫
Ahdv = sup
B∈VP ( inf
x∈Bh(x), v(A ∩B));
(ii) (Q) –∫
Ahdv = inf
B∈VQ(sup
x∈Bh(x), v(A ∩ coB)).
Proof. As an example, we shall prove (ii). The proof of (i) is similar. On the one hand, consider an
arbitrary B in V and let λBdef= sup
x∈Bh(x). Also consider the V-simple X − L mapping sB , defined by
sB(x) def=
λB ; x ∈ Bu ; x ∈ coB,
for arbitrary x in X. Then h v sB , and therefore also by definition
(Q) –∫
Ahdv = inf
hvsβv
_Q(A ; s)
≤ βv_Q(A ; sB)
= Q(λB , v(A ∩ coB)) _ Q(u, v(A ∩B))= Q(λB , v(A ∩ coB))= Q(sup
x∈Bh(x), v(A ∩ coB)).
Taking into account the definition of infimum, this implies that
(Q) –∫
Ahdv ≤ inf
B∈VQ(sup
x∈Bh(x), v(A ∩ coB)).
On the other hand, consider an arbitrary V-simple X − L mapping s for which h v s. Then, with thenotations of definition 1, for k ∈ 1, . . . , n, (∀x ∈ Dk)(ak ≥ h(x)), and therefore also, taking into accountthe definition of supremum
(∀k ∈ 1, . . . , n)(ak ≥ supx∈Dk
h(x)),
whence, taking into account the isotonicity of Q and the V-measurability of all the sets taken intoconsideration,
(∀k ∈ 1, . . . , n)(Q(ak, v(A ∩ coDk)) ≥ Q( supx∈Dk
h(x), v(A ∩ coDk))).
Taking into account the isotonicity of infimum, this leads to
βv_Q(A ; s) =
ninfk=1
Q(ak, v(A ∩ coDk))
≥n
infk=1
Q( supx∈Dk
h(x), v(A ∩ coDk))
≥ infB∈V
Q(supx∈B
h(x), v(A ∩ coB)),
which implies that (Q) –∫
Ahdv ≥ inf
B∈VQ(sup
x∈Bh(x), v(A ∩ coB)). 2
9
Theorem 4 Let (X,V, v) be an arbitrary (L,≤)-confidence space. Let h be a X − L mapping and let Abe a V-measurable set.
(i) If h is V-cut measurable, then (P ) –∫
Ahdv = sup
λ∈LP (λ, v(A ∩ Sh
λ)).
(ii) If h is dually V-cut measurable, then (Q) –∫
Ahdv = inf
λ∈LQ(λ, v(A ∩ coDh
λ)).
Proof. Let us prove (ii). The proof of (i) is completely analogous. On the one hand, consider an arbitrary
element B of V and let λBdef= sup
x∈Bh(x). Then (∀x ∈ B)(h(x) ≤ λB) and therefore also coDh
λB⊆ coB.
This implies that A∩coDhλB⊆ A∩coB, and since all the sets considered are by assumption V-measurable,
also that v(A ∩ coDhλB
) ≤ v(A ∩ coB). Taking into account the isotoniticity of Q, this leads to
(∀B ∈ V)(Q(λB , v(A ∩ coDhλB
)) ≤ Q(λB , v(A ∩ coB))),
whence, by definition of λB and by definition of infimum,
(∀B ∈ V)( infλ∈L
Q(λ, v(A ∩ coDhλ)) ≤ Q(sup
x∈Bh(x), v(A ∩ coB))),
and therefore also, again taking into account the definition of infimum,
infλ∈L
Q(λ, v(A ∩ coDhλ) ≤ inf
B∈VQ(sup
x∈Bh(x), v(A ∩ coB))).
On the other hand, we have for arbitrary λ in L that (∀x ∈ Dhλ)(h(x) ≤ λ), and therefore also
supx∈Dh
λ
h(x) ≤ λ.
Since all the sets considered are V-measurable, it follows from the isotonicity of Q that
(∀λ ∈ L)(Q(λ, v(A ∩ coDhλ)) ≥ Q( sup
x∈Dhλ
h(x), v(A ∩ coDhλ))).
Taking into account the isotonicity of infimum this implies that
infλ∈L
Q(λ, v(A ∩ coDhλ)) ≥ inf
λ∈LQ( sup
x∈Dhλ
h(x), v(A ∩ coDhλ))
≥ infB∈V
Q(supx∈B
h(x), v(A ∩ coB)). 2
Let us briefly discuss the originality of the definitions and results of this section. The items (i) ofdefinitions 2, 3 and 4 and of theorems 3 and 4 are generalizations—towards more general codomains forthe mappings and towards more general types of measures—of definitions and theorems by Suarez Garcıaand Gil Alvarez [19], which in turn are based upon the work of Ralescu and Adams [15] and Sugeno [20].Our introduction of semiconormed fuzzy integrals—generalizations of the semiconormed fuzzy integralsof Suarez Garcıa and Gil Alvarez—is on the other hand completely new and exposes the dual analogybetween seminormed and semiconormed fuzzy integrals. Furthermore, since the items (ii) of the above-mentioned theorems, and definitions, for that matter, are entirely original, we have only given the proofsof these items. The proofs of the items (i) can be found by dual analogy, or by the not always trivialextension of the proofs found in the literature [15, 19, 20]. Theorem 2 deserves some extra attention. Itis a modified and at the same time generalized version of theorem 3.5 in [19]. As is shown in [5], thelatter theorem is not valid, and can only be proven in the modified form given above.
10
4 General properties
In this section, we have gathered a few important general properties of our fuzzy integrals. We stressthat in many cases these properties are analogous to properties of Lebesgue integrals. Proposition 3 andits corollaries 1 and 2 describe the behaviour of fuzzy integrals acting on constant mappings. We pointout that corollary 1 completes the proof of theorem 2.
Proposition 3 Let (X,V, v) be an arbitrary (L,≤)-confidence space. Let µ be an element of L and letA be a V-measurable set. Then
(i) (P ) –∫
Aµdv = P (µ, v(A)) ^ v(∅);
(ii) (Q) –∫
Aµdv = Q(µ, v(∅)) _ v(A).
Proof. We shall give the proof of (i). The proof of (ii) is fairly analogous. For arbitrary λ in L we haveby definition that
Sµλ = x | x ∈ X and λ ≤ µ =
X ; λ ≤ µ∅ ; λ 6≤ µ.
which implies that µ is V-cut measurable. Taking into account theorem 4 (i) we may write that
(P ) –∫
Aµdv = sup
λ∈LP (λ, v(A ∩ S
µλ ))
= sup(supλ≤µ
P (λ, v(A ∩X)), supλ6≤µ
P (λ, v(A ∩ ∅)))
= sup(supλ≤µ
P (λ, v(A)), supλ 6≤µ
P (λ, v(∅))).
Taking into account the properties of t-seminorms [4, 6], it is now easily verified that on the one hand
supλ≤µ
P (λ, v(A)) = P (µ, v(A)),
and on the other hand
supλ6≤µ
P (λ, v(∅)) =
v(∅) ; µ < u` ; µ = u.
This implies that
(P ) –∫
Aµdv =
P (µ, v(A)) ^ v(∅) ; µ < uv(A) ; µ = u = P (µ, v(A)) ^ v(∅),
since v(∅) ≤ v(A). 2
Corollary 1 Let (X,V, v) be an arbitrary (L,≤)-confidence space, with v normalized. For arbitrary µin L:
(i) (P ) –∫
Xµdv = µ;
(ii) (Q) –∫
Xµdv = µ.
Proof. At once from the proposition above, taking into account v(∅) = `, v(X) = u and the boundaryconditions for t-seminorms and t-semiconorms on (L,≤) [4, 6]. 2
11
Corollary 2 Let (X,V, v) be an arbitrary (L,≤)-confidence space. For arbitrary A in V:
(i) (P ) –∫
Adv def= (P ) –
∫
Audv = v(A);
(ii) (Q) –∫
Adv def= (Q) –
∫
Audv = v(A).
Proof. At once from the proposition above, taking into account v(∅) ≤ v(A), and the boundary conditionsfor t-seminorms and t-semiconorms on (L,≤) [4, 6]. 2
The following proposition shows that the value of a fuzzy integral of an arbitrary mapping on a setwith confidence measure ` is always equal to `.
Proposition 4 Let (X,V, v) be an arbitrary (L,≤)-confidence space. Let h be a X −L mapping and letA be a V-measurable set. Then
(i) v(A) = ` ⇒ (P ) –∫
Ahdv = `;
(ii) v(A) = ` ⇒ (Q) –∫
Ahdv = `.
Proof. Assume that v(A) = `. Taking into account the isonicity of v, we have that (∀B ∈ V)(v(A∩B) =`). Taking into account theorem 3 (i), this implies that
(P ) –∫
Ahdv = sup
B∈VP ( inf
x∈Bh(x), v(A ∩B)) = sup
B∈VP ( inf
x∈Bh(x), `) = `,
and analogously, taking into account theorem 3 (ii), we have that
(Q) –∫
Ahdv = inf
B∈VQ(sup
x∈Bh(x), v(A ∩ coB))
= infB∈V
Q(supx∈B
h(x), `)
= infB∈V
supx∈B
h(x) = `,
since sup ∅ = ` holds in the complete lattice (L,≤), and furthermore ∅ ∈ V. 2
Propositions 5, 6, and 7, and their corollaries 3 and 4 express the isotonicity of our fuzzy integrals.
Proposition 5 Let (X,V, v) be an arbitrary (L,≤)-confidence space. Let h1 and h2 be X −L mappingswith h1 v h2. Then for arbitrary A in V:
(i) (P ) –∫
Ah1dv ≤ (P ) –
∫
Ah2dv;
(ii) (Q) –∫
Ah1dv ≤ (Q) –
∫
Ah2dv.
Proof. As an example, we shall prove (i). The proof of (ii) is analogous. For arbitrary B in V, we deducefrom the assumptions and the isotonicity of infimum, that
infx∈B
h1(x) ≤ infx∈B
h2(x)
whence, since P is isotonic,
P ( infx∈B
h1(x), v(A ∩B)) ≤ P ( infx∈B
h2(x), v(A ∩B)).
12
Taking into account the isotonicity of supremum it follows that
supB∈V
P ( infx∈B
h1(x), v(A ∩B)) ≤ supB∈V
P ( infx∈B
h2(x), v(A ∩B)),
which, taking into account theorem 3 (i), proves (i). 2
Proposition 6 Let (X,V, v) be an arbitrary (L,≤)-confidence space. Let h1 and h2 be X −L mappings.Let A be an element of V, and assume that (∀x ∈ A)(h1(x) ≤ h2(x)). Then the following propositionshold.
(i) If h1 and h2 are V-cut measurable, then (P ) –∫
Ah1dv ≤ (P ) –
∫
Ah2dv.
(ii) If h1 and h2 are dually V-cut measurable, then (Q) –∫
Ah1dv ≤ (Q) –
∫
Ah2dv.
Proof. As an example, we shall prove (ii). The proof of (i) is analogous. Assume that h1 and h2 aredually V-cut measurable. Consider an arbitrary λ in L and an arbitrary x in A ∩ coDh1
λ . By definition,we have that h1(x) 6≤ λ. From the assumptions it now also follows that h2(x) 6≤ λ. Indeed, shouldh2(x) ≤ λ, then the validity of h1(x) ≤ h2(x) and the transitivity of ≤ would imply that h1(x) ≤ λ, acontradiction. This implies that x ∈ A ∩ coDh2
λ , whence A ∩ coDh1λ ⊆ A ∩ coDh2
λ . Taking into accountthe isotonicity of v and Q, and the fact that all the sets considered are V-measurable, we obtain
Q(λ, v(A ∩ coDh1λ )) ≤ Q(λ, v(A ∩ coDh2
λ )).
The isotonicity of infimum now implies that
infλ∈L
Q(λ, v(A ∩ coDh1λ )) ≤ inf
λ∈LQ(λ, v(A ∩ coDh2
λ )),
which, taking into account theorem 4 (ii), proves (ii). 2
Corollary 3 Let (X,V, v) be an arbitrary (L,≤)-confidence space. Let (hj | j ∈ J) be a family of X −Lmappings. Then for arbitrary A in V:
(i) (P ) –∫
Asupj∈J
hjdv ≥ supj∈J
(P ) –∫
Ahjdv;
(ii) (P ) –∫
Ainfj∈J
hjdv ≤ infj∈J
(P ) –∫
Ahjdv;
(iii) (Q) –∫
Asupj∈J
hjdv ≥ supj∈J
(Q) –∫
Ahjdv;
(iv) (Q) –∫
Ainfj∈J
hjdv ≤ infj∈J
(Q) –∫
Ahjdv.
Proposition 7 Let (X,V, v) be an arbitrary (L,≤)-confidence space. Let A and B be V-measurable sets,with A ⊆ B. Then for an arbitrary X − L mapping h:
(i) (P ) –∫
Ahdv ≤ (P ) –
∫
Bhdv;
(ii) (Q) –∫
Ahdv ≤ (Q) –
∫
Bhdv.
13
Proof. As an example, we shall prove (ii). The proof of (i) is analogous. Let C be an arbitrary elementof V, then A ∩ coC ⊆ B ∩ coC, whence, since v and Q are isotonic,
Q(supx∈C
h(x), v(A ∩ coC)) ≤ Q(supx∈C
h(x), v(B ∩ coC)).
Since infimum is isotonic, this implies that
infC∈V
Q(supx∈C
h(x), v(A ∩ coC)) ≤ infC∈V
Q(supx∈C
h(x), v(B ∩ coC)),
which, taking into account theorem 3 (ii), proves (ii). 2
Corollary 4 Let (X,V, v) be an arbitrary (L,≤)-confidence space. Let (Aj | j ∈ J) be a family ofV-measurable sets. Then for an arbitrary X − L mapping h:
(i) (P ) –∫
⋃
j∈J
Aj
hdv ≥ supj∈J
(P ) –∫
Aj
hdv;
(ii) (P ) –∫
⋂
j∈J
Aj
hdv ≤ infj∈J
(P ) –∫
Aj
hdv;
(iii) (Q) –∫
⋃
j∈J
Aj
hdv ≥ supj∈J
(Q) –∫
Aj
hdv;
(iv) (Q) –∫
⋂
j∈J
Aj
hdv ≤ infj∈J
(Q) –∫
Aj
hdv.
Propositions 8 and 9 describe the behaviour of fuzzy integrals when characteristic mappings appear inthe integrand.
Proposition 8 Let (X,V, v) be an arbitrary (L,≤)-confidence space. Let A be an element of V withcharacteristic X − L mapping χA. Then
(i) v(A) = (P ) –∫
XχAdv;
(ii) v(A) = (Q) –∫
XχAdv.
Proof. As an example, we shall prove (ii). The proof of (i) is analogous. Since A ∈ V it can easily beshown that χA is dually V-cut measurable. Taking into account theorem 4 (ii), we may write that
(Q) –∫
XχAdv = inf
λ∈LQ(λ, v(coDχA
λ ))
= inf( infλ<u
Q(λ, v(A)), Q(u, v(∅)))= inf( inf
λ<uQ(λ, v(A)), u)
= infλ<u
Q(λ, v(A))
≤ Q(`, v(A)) = v(A).
On the other hand, we have that (∀λ ∈ L)(Q(λ, v(A)) ≥ v(A)), whence, taking into account the definitionof infimum, inf
λ<uQ(λ, v(A)) ≥ v(A). 2
14
Proposition 9 Let (X,V, v) be an arbitrary (L,≤)-confidence space. Let A be an element of V withcharacteristic X − L mapping χA, and let h be a X − L mapping.
(i) If h is V-cut measurable, then (P ) –∫
Ahdv = (P ) –
∫
X(χA _ h)dv.
(ii) If h is dually V-cut measurable, then (Q) –∫
Ahdv = (Q) –
∫
X(χA _ h)dv.
Of course, χA _ h is the X − L mapping defined pointwise by
(∀x ∈ X)((χA _ h)(x) def= χA(x) _ h(x)).
Proof. As an example, we give the proof of (ii). The proof of (i) is analogous. For arbitrary λ in L wehave that
DχA_hλ = x | x ∈ X and χA(x) _ h(x) ≤ λ = coA ∪Dh
λ,
and therefore χA _ h is dually V-cut measurable as well. Theorem 4 (ii) therefore implies that
(Q) –∫
X(χA _ h)dv = inf
λ∈LQ(λ, v(coDχA_h
λ )) = infλ∈L
Q(λ, v(A ∩ coDhλ)) = (Q) –
∫
Ahdv. 2
In the next theorem, we explicitly show that our two types of fuzzy integrals are in a sense dual. Inorder to be able to formulate this theorem, we introduce a few preliminary notions. Let n be a dual order-automorphism on (L,≤), i.e., an order-isomorphism between the structures (L,≤) and (L,≥) (see, forinstance, [1]). In order that such a dual order-isomorphism exist, (L,≤) must of course be self-dual. Letus assume in the rest of this section that this is indeed the case. Consider an arbitrary (L,≤)-confidencespace (X,V, v). Then the V − L mapping vn, defined by
(∀A ∈ V)(vn(A) def= n−1(v(coA))),
is a (L,≤)-confidence measure on (X,V), and is called the dual (L,≤)-confidence measure of v w.r.t. n[5]. Also, consider an arbitrary t-seminorm P on (L,≤). The L2 − L mapping Pn, defined by
(∀(λ, µ) ∈ L2)(Pn(λ, µ) def= n−1(P (n(λ), n(µ)))),
is a t-semiconorm on (L,≤), called the dual t-semiconorm of P w.r.t. n [6]. Dually, consider an arbitraryt-semiconorm Q on (L,≤). Then the L2 − L mapping Qn, defined by
(∀(λ, µ) ∈ L2)(Qn(λ, µ) def= n−1(Q(n(λ), n(µ)))),
is a t-seminorm on (L,≤), called the dual t-seminorm of Q w.r.t. n [6].
Theorem 5 Let (X,V, v) be an arbitrary (L,≤)-confidence space. Also, let n be a dual order-automorph-ism of (L,≤) and let h be a X − L mapping. Then
(i) (P ) –∫
Xhdv = n((Pn) –
∫
X(n−1 h)dvn), or equivalently, (Pn) –
∫
Xhdvn = n−1((P ) –
∫
X(n h)dv);
(ii) (Q) –∫
Xhdv = n((Qn) –
∫
X(n−1 h)dvn), or equivalently, (Qn) –
∫
Xhdvn = n−1((Q) –
∫
X(n h)dv).
15
Proof. As an example, we give the proof of (i). The proof of (ii) is analogous. We deduce from theorem 3that
n((Pn) –∫
X(n−1 h)dvn) = n( inf
B∈VPn(sup
x∈B(n−1 h)(x), vn(coB)))
= supB∈V
n(Pn(supx∈B
(n−1 h)(x), vn(coB)))
= supB∈V
n(n−1(P (n(supx∈B
(n−1 h)(x)), n(vn(coB)))))
= supB∈V
P (n(supx∈B
(n−1 h)(x)), n(vn(coB)))
= supB∈V
P (n(supx∈B
(n−1 h)(x)), n(n−1(v(co(coB)))))
= supB∈V
P ( infx∈B
h(x), v(B))
= (P ) –∫
Xhdv.
This immediately implies that (Pn) –∫
Xhdvn = n−1((P ) –
∫
X(n h)dv). 2
5 Possibility and necessity integrals
In this section, we want to combine seminormed fuzzy integrals and possibility measures on the one hand,and semiconormed fuzzy integrals and necessity measures on the other hand. We shall show that thesecombinations are indeed very fruitful.
We start by introducing a few preliminary notions, and by discussing the assumptions made in thissection. From now on, whenever we speak about a t-seminorm P on (L,≤), we shall assume that P iscompletely distributive w.r.t. supremum, i.e., for arbitrary λ in L and an arbitrary family (µj | j ∈ J) ofelements of L:
P (λ, supj∈J
µj) = supj∈J
P (λ, µj)
P (supj∈J
µj , λ) = supj∈J
P (µj , λ).
Analogously, whenever we speak about a t-semiconorm Q on (L,≤), we shall assume that Q is completelydistributive w.r.t. infimum. In short, we assume that the structure (L,≤, P ) is a complete lattice witht-seminorm and that the structure (L,≥, Q) is a complete lattice with t-semiconorm [6].
By R, we shall mean an arbitrary ample field on the universe X, i.e., a set of subsets of X that is closedunder arbitrary unions and intersections, and under complementation. We shall furthermore assume thatR is a proper ample field, i.e., ∅, X ⊂ R. The atom of R containing the element x of X will be denotedby [ x ]R:
[ x ]Rdef=
⋂
A | x ∈ A and A ∈ R.
For a more detailed account of ample fields, we refer to [7, 23].By Π, we shall mean an arbitrary (L,≤)-possibility measure on (X,R), i.e., a complete join-morphism
between the complete lattices (R,⊆) and (L,≤). This means by definition that Π satisfies the followingrequirement: for an arbitrary family (Aj | j ∈ J) of elements of R
Π(⋃
j∈J
Aj) = supj∈J
Π(Aj).
This definition immediately implies that Π(∅) = `. Also, (X,R,Π) is a special kind of (L,≤)-confidencespace, that will henceforth be called a (L,≤)-possibility space. For obvious reasons, Π will be callednormalized iff Π(X) = u. The X − L mapping π, defined by
(∀x ∈ X)(π(x) def= Π([ x ]R))
16
will be called the distribution of Π. Remark that it completely determines Π, since for arbitrary A in R
Π(A) = supx∈A
π(x). (12)
By N, we shall mean an arbitrary (L,≤)-necessity measure on (X,R), i.e., a complete meet-morphismbetween the complete lattices (R,⊆) and (L,≤). This means by definition that N satisfies the followingrequirement: for an arbitrary family (Aj | j ∈ J) of elements of R
N(⋂
j∈J
Aj) = infj∈J
N(Aj).
This definition immediately implies that N(X) = u. Also, (X,R,N) is a special kind of (L,≤)-confidencespace, that will henceforth be called a (L,≤)-necessity space. For obvious reasons, N will be callednormalized iff N(∅) = `. The X − L mapping ν, defined by
(∀x ∈ X)(ν(x) def= N(co[ x ]R))
will be called the distribution of N. Remark that it completely determines N, since for arbitrary A in R
N(A) = infx∈coA
ν(x). (13)
(L,≤)-possibility and (L,≤)-necessity measures are generalizations towards more general domains andcodomains of Zadeh’s possibility measures [28], Dubois and Prade’s necessity measures [8], Wang’s fuzzycontactabilities [23], and the possibility and necessity measures we introduced in [4]. For a more detaileddiscussion of these generalizations, we refer to [4, 5, 7].
A X − L mapping h will be called R-measurable iff it is constant on the atoms of R. It can beproven that the notions R-measurability, R-cut measurability, dual R-cut measurability and R-levelmeasurability in fact coincide [7].
Definition 5 A X − L mapping h—a (L,≤)-fuzzy set in X—is called a (L,≤)-fuzzy variable in (X,R)iff h is R-measurable. The set of the (L,≤)-fuzzy variables in (X,R) is denoted by GR(L,≤)(X). Wheneverwe want to omit reference to the structures (L,≤) and (X,R), we shall simply speak of fuzzy variables.
Our fuzzy variables are generalizations towards more general codomains and measurability conditionsof the fuzzy variables introduced by Nahmias [14], and further refined by Wang [23]. They are to acertain extent also in spirit related to Ralescu’s fuzzy variables [16]. They are primarily meant to serve asa possibilistic equivalent of the real stochastic variables in probability theory. On the other hand, (L,≤)-fuzzy variables in (X,R) can also be considered as obvious extensions—or indeed “fuzzifications”—ofR-measurable2 sets: the characteristic X − L mapping χA of a subset A of X is a (L,≤)-fuzzy variablein (X,R) if and only if A is R-measurable. Moreover, the fuzzification (GR(L,≤)(X),v) of (R,⊆) isa complete sublattice of the fuzzification (F(L,≤)(X),v) of (P(X),⊆), in a similar way as (R,⊆) is acomplete sublattice of (P(X),⊆) [5, 7].
We shall now derive a few important formulas for seminormed and semiconormed fuzzy integrals, whenthe confidence measure they are associated with is a possibility, respectively a necessity measure. Theseformulas will be used furtheron to derive results that constitute the foundation of a general measure- andintegral-theoretic approach to possibility and necessity theory [5].
2We note that the fuzzy variables introduced by Zadeh [28] are more general than the fuzzy variables discussed here.The former are intended as possibilistic analoga of general, not necessarily real-valued, stochastic variables. Furthermore,they cannot in general be interpreted as measurable fuzzy sets. We therefore suggest that the more appropriate name‘possibilistic variable’ be used for Zadeh’s notion, and that the name ‘fuzzy variables’ be reserved for measurable fuzzysets. For a detailed discussion of possibilistic variables, conditional possibility and possibilistic independence of possibilisticvariables, we refer to the doctoral dissertation of one of us [5].
17
Definition 6 Let h be a X − L mapping and A a R-measurable set.
(i) (P ) –∫
AhdΠ is called the (L,≤, P )-possibility integral of h on A (associated with Π).
(ii) (Q) –∫
AhdN is called the (L,≥, Q)-necessity integral of h on A (associated with N).
Whenever we want to omit reference to the complete lattice (L,≤), the t-seminorm P and/or the t-semiconorm Q, we shall simple speak of possibility and necessity integrals.
Theorem 6 Let A be a R-measurable set and let h be a X − L mapping. Then
(i) (P ) –∫
AhdΠ = sup
x∈AP ( inf
y∈[ x ]Rh(y), π(x));
(ii) (Q) –∫
AhdN = N(A) _ inf
x∈XQ( sup
y∈[ x ]Rh(y), ν(x)).
Proof. We shall prove (ii). The proof of (i) is fairly analogous. We already know, taking into accounttheorem 3 (ii) and the fact that Q is completely distributive w.r.t. infimum, that
(Q) –∫
AhdN = inf
B∈RQ(sup
y∈Bh(y),N(A ∩ coB))
= infB∈R
Q(supy∈B
h(y),N(A) _ N(coB))
= infB∈R
(Q(supy∈B
h(y), N(A)) _ Q(supy∈B
h(y), N(coB)))
= infB∈R
Q(supy∈B
h(y),N(A)) _ infB∈R
Q(supy∈B
h(y),N(coB))
= Q( infB∈R
supy∈B
h(y), N(A)) _ infB∈R
Q(supy∈B
h(y),N(coB)).
Since ∅ ∈ R and supy∈∅
h(y) = sup ∅ = ` holds in the complete lattice (L,≤), it follows that
infB∈R
supy∈B
h(y) = `.
Therefore, we find that, taking into account the complete distributivity of Q w.r.t. infimum, the associa-tivity of infimum and the boundary conditions of t-semiconorms [4, 6],
(Q) –∫
AhdN = N(A) _ inf
B∈RQ(sup
y∈Bh(y),N(coB))
= N(A) _ infB∈R
Q(supy∈B
h(y), infx∈X
Q(χcoB(x), ν(x)))
= N(A) _ infx∈X
infB∈R
Q(supy∈B
h(y), Q(χcoB(x), ν(x)))
= N(A) _ infx∈X
infB∈R
Q(Q(supy∈B
h(y), χcoB(x)), ν(x))
= N(A) _ infx∈X
Q( infB∈R
Q(supy∈B
h(y), χcoB(x)), ν(x)).
The proof of (ii) is complete if we can show that for arbitrary x in X:
infB∈R
Q(supy∈B
h(y), χcoB(x)) = supy∈[ x ]R
h(y).
18
To this end, consider an arbitrary x in X. We have on the one hand that
infB∈R
Q(supy∈B
h(y), χcoB(x)) = inf supy∈B
h(y) | B ∈ R and x ∈ B ≤ supy∈[ x ]R
h(y),
taking into account the definition of infimum and the fact that x ∈ [x ]R and [ x ]R ∈ R [7]. On the otherhand, since by definition of [ x ]R, for every B in R for which x ∈ B, [ x ]R ⊆ B, and therefore also
supy∈[ x ]R
h(y) ≤ supy∈B
h(y),
it follows from the definition of infimum that
inf supy∈B
h(y) | B ∈ R and x ∈ B ≥ supy∈[ x ]R
h(y). 2
Corollary 5 Let A be a R-measurable set and let h be a X − L mapping. Then
(i) (Q) –∫
XhdN = inf
x∈XQ( sup
y∈[ x ]Rh(y), ν(x));
(ii) (Q) –∫
AhdN = N(A) _ (Q) –
∫
XhdN.
Proof. This is an immediate consequence of the previous theorem, since N(X) = u. 2
Corollary 6 Let A be an element of R and let h be a X − L mapping that is R-measurable. Then
(i) (P ) –∫
AhdΠ = sup
x∈AP (h(x), π(x));
(ii) (Q) –∫
AhdN = N(A) _ inf
x∈XQ(h(x), ν(x)).
Proof. This is an immediate consequence of theorem 6 and the fact that h is by assumption constanton the atoms of R. 2
Using these basic results, we are now able to investigate the meaning of the notions ‘possibility integral’and ‘necessity integral’. We shall do this by further exploring the analogy between fuzzy variables andmeasurable sets, already mentioned at the beginning of this section.
In corollary 6 we have shown that the possibility and necessity integral of a fuzzy variable assume a verysimple form. Theorems 7, 8 and 9 shed some light on other, related aspects of the combination of fuzzyvariables and possibility and necessity integrals. They give us an impression of the ‘empathy’ betweenseminormed fuzzy integrals and possibility measures, and semiconormed fuzzy integrals and necessitymeasures. In theorem 7 we investigate the commutativity of the supremum operator and the possibilityintegral3, and the infimum operator and the necessity integral. In theorem 8 we give a related result.A comparison between these theorems and corollaries 3 and 4 of the previous section, should give thereader a first indication that the combinations introduced in definition 6 are indeed rather special.
Theorem 7 Let (hj | j ∈ J) be an arbitrary family of (L,≤)-fuzzy variables in (X,R), and let A be anarbitrary R-measurable set.
3Weber [26] has proven an analogous result for finite suprema for the combination of Sugeno’s fuzzy integral and whathe calls ‘σ-^-decomposable measures’. Furthermore, Sugeno [20] has derived an analogous result for his fuzzy integral andwhat he calls ‘F-additive fuzzy measures.’
19
(i) (P ) –∫
Asupj∈J
hjdΠ = supj∈J
(P ) –∫
AhjdΠ.
(ii) (Q) –∫
Ainfj∈J
hjdN = infj∈J
(Q) –∫
AhjdN.
Proof. We shall give the proof of (i). The proof of (ii) is fairly analogous. Remark that supj∈J
hj is a
(L,≤)-fuzzy variable in (X,R). We may therefore write, taking into account corollary 6 (i) and thecomplete distributivity of P w.r.t. supremum,
(P ) –∫
Asupj∈J
hjdΠ = supx∈A
P ((supj∈J
hj)(x), π(x))
= supx∈A
P (supj∈J
hj(x), π(x))
= supx∈A
supj∈J
P (hj(x), π(x))
= supj∈J
supx∈A
P (hj(x), π(x)) = supj∈J
(P ) –∫
AhjdΠ. 2
Theorem 8 Let (Aj | j ∈ J) be an arbitrary familiy of elements of R, and let h be an arbitrary (L,≤)-fuzzy variable in (X,R).
(i) (P ) –∫
⋃
j∈J
Aj
hdΠ = supj∈J
(P ) –∫
Aj
hdΠ.
(ii) (Q) –∫
⋂
j∈J
Aj
hdN = infj∈J
(Q) –∫
Aj
hdN.
Proof. We shall give the proof of (i). The proof of (ii) is immediate, taking into account corollary 5 (ii)and the definition of necessity measures. Taking into account corollary 6 (i) and the associativity ofsupremum, we may write
(P ) –∫
⋃
j∈J
Aj
hdΠ = supx∈
⋃
j∈J
Aj
P (h(x), π(x)) = supj∈J
supx∈Aj
P (h(x), π(x)) = supj∈J
(P ) –∫
Aj
hdΠ. 2
Theorem 9 (i) expresses that for R-simple X − L mappings, which are of course also special (L,≤)-fuzzy variables in (X,R), the (L,≤)-possibility integral coincides with the functional used to define it.This result is an analogon of a well-known theorem4 in the theory of classical measures and integrals(see, for instance, [2]). This is a very interesting point, since Ralescu and Adams [15] have shown thatan analogous theorem is not necessarily valid in general5 for fuzzy integrals associated with arbitraryconfidence measures.
Theorem 9 Let A be a R-measurable set and let s be a R-simple X − L mapping.4Weber [26] has proven an analogous result for the combination of Sugeno’s fuzzy integral and what he calls σ-^-
decomposable measures.5As a matter of fact, Ralescu and Adams have proven that an analogous result is not valid for the special case of ([0, 1],≤)-
fuzzy min-integrals associated with Sugeno’s fuzzy measures, by giving a counterexample. Of course, this counterexampleremains valid for the more general result discussed here.
20
(i) (P ) –∫
AsdΠ = αΠ
^P (A ; s).
(ii) (Q) –∫
AsdN = N(A) _ βN
_Q(X ; s).
Proof. We shall give the proof of (ii). The proof of (i) is fairly analogous. Since the R-simple mappings is by definition R-measurable, we may write, taking into account corollary 6 (ii), using the notationsof definition 1, the associativity of infimum and the complete distributivity of Q w.r.t. infimum, that
(Q) –∫
AsdN = N(A) _ inf
x∈XQ(s(x), ν(x))
= N(A) _n
infk=1
infx∈Dk
Q(sk, ν(x))
= N(A) _n
infk=1
Q(sk, infx∈Dk
ν(x))
= N(A) _n
infk=1
P (sk,N(coDk))
= N(A) _ βN_Q(X ; s). 2
Item (ii) of theorem 9 is not completely analogous to item (i), but is on the other hand a necessaryconsequence of proposition 9, the fact that the infimum operator and the necessity integral operatorcommute and the following equality
(Q) –∫
XsdN = βN
_Q(X ; s).
We now have sufficient knowledge about the combination of fuzzy variables and possibility and necessityintegrals, to be able to make the next step in our investigation of the analogy between the structures(R,⊆) and (GR(L,≤)(X),v): the extension of the notions ‘possibility measure’ and ‘necessity measure’.We shall introduce a generalized kind of possibility and necessity measures, the arguments of which areno longer measurable sets, but measurable fuzzy sets, i.e., fuzzy variables.
Definition 7 A (L,≤)-possibility measure Π′ on (X,GR(L,≤)(X)) is a GR(L,≤)(X)− L mapping satisfyingthe following requirement: for an arbitrary family (hj | j ∈ J) of (L,≤)-fuzzy variables in (X,R)
Π′(supj∈J
hj) = supj∈J
Π′(hj).
A (L,≤)-necessity measure N′ on (X,GR(L,≤)(X)) is a GR(L,≤)(X) − L mapping satisfying the followingrequirement: for an arbitrary family (hj | j ∈ J) of (L,≤)-fuzzy variables in (X,R)
N′(infj∈J
hj) = infj∈J
N′(hj).
This means that the mapping Π′ respectively N′ is a complete join-morphism respectively complete meet-morphism between the complete lattices (GR(L,≤)(X),v) and (L,≤). We shall call Π′ normalized iffΠ′(χX) = u, and N′ normalized iff N′(χ∅) = `. Whenever we do not want to be specific about thedomain or the codomain of these morphisms, we shall simply speak of extended possibility and necessitymeasures.
Corollary 7 Π′(χ∅) = ` and N′(χX) = u.
Proposition 10 indicates in precisely what way the extended possibility and necessity measures areextensions of the ordinary possibility and necessity measures. Its proof is immediate, and is thereforeomitted.
21
Proposition 10 Let Π′ be an arbitrary (L,≤)-possibility measure on (X,GR(L,≤)(X)). Consider the R−Lmapping Π′′ defined by
(∀A ∈ R)(Π′′(A) def= Π′(χA)),
where χA is the characteristic X − L mapping of the set A. Then Π′′ is a (L,≤)-possibility measure on(X,R). Similarly, let N′ be an arbitrary (L,≤)-necessity measure on (X,GR(L,≤)(X)). Consider the R−Lmapping N′′ defined by
(∀A ∈ R)(N′′(A) def= N′(χA)).
Then N′′ is a (L,≤)-necessity measure on (X,R).
From proposition 10 we deduce that, starting from an arbitrary extended possibility (necessity) measure,we can easily construct an ordinary possibility (necessity) measure by ‘restriction of the domain’. This ofcourse raises the following question: can we construct an extended possibility (necessity) measure startingfrom an ordinary possibility (necessity) measure? In proposition 11 we show that this is indeed the case,whenever it is possible to define on the complete lattice (L,≤) a t-seminorm P (t-semiconorm Q), thatis completely distributive w.r.t. supremum (infimum).
Proposition 11 Consider the GR(L,≤)(X)− L mapping ΠP , defined as
(∀h ∈ GR(L,≤)(X))(ΠP (h) def= (P ) –∫
XhdΠ).
Then ΠP is a (L,≤)-possibility measure on (X,GR(L,≤)(X)), and ΠP (χA) = Π(A), for arbitrary A in R.Similarly, consider the GR(L,≤)(X)− L mapping NQ, defined as
(∀h ∈ GR(L,≤)(X))(NQ(h) def= (Q) –∫
XhdN).
Then NP is a (L,≤)-necessity measure on (X,GR(L,≤)(X)), and NQ(χA) = N(A), for arbitrary A in R.
ΠP can be considered as an extension of the (L,≤)-possibility measure Π, and NQ an extension of the(L,≤)-necessity measure N. This leads to the following definition.
Definition 8 The (L,≤)-possibility measure ΠP on the structure (X,GR(L,≤)(X)) will be called the P -extension of the (L,≤)-possibility measure Π on (X,R). The (L,≤)-necessity measure NQ on the structure(X,GR(L,≤)(X)) will be called the Q-extension of the (L,≤)-necessity measure N on (X,R). Also, for anarbitrary (L,≤)-fuzzy variable h in (X,R), ΠP (h) is called the (L,≤, P )-possibility of h, and NQ(h) the(L,≥, Q)-necessity of h.
6 Conclusion
From the results of the previous section we may conclude that possibility integrals enable us to extendthe domain of possibility measures from collections of measurable sets towards collections of measurablefuzzy sets. In a similar way, necessity integrals allow the extension of the domain of necessity measures.We are thus led to consider such notions as the possibility and the necessity of measurable fuzzy sets.Moreover, consider an arbitrary dual order-automorphism n on (L,≤). Also consider a t-seminorm P on
(L,≤), and its dual t-semiconorm Q def= Pn w.r.t. n. Finally, consider a (L,≤)-possibility measure Π on
(X,R), with distribution π. It is easily proven that the dual confidence measure N def= Πn, defined as
(∀A ∈ R)(N(A) def= n−1(Π(coA))), (14)
22
is a (L,≤)-necessity measure on (X,R), called the dual (L,≤)-necessity measure of Π w.r.t. n [5]. Thedistribution ν of N satisfies ν = n−1 π. Furthermore, P = Qn−1 and Π = Nn−1 . Then, taking intoaccount theorem 5, we find that for arbitrary h in GR(L,≤)(X), with the notations of proposition 11,
NQ(h) = (Q) –∫
XhdN = (Pn) –
∫
XhdΠn = n−1((P ) –
∫
X(n h)dΠ) = n−1(ΠP (n h)),
whence, putting conh def= n h,NQ(h) = n−1(ΠP (conh)). (15)
In this expression, con is a truth-functional complement operator for (L,≤)-fuzzy sets in X, an obviousextension of the well-known complement operators for ([0, 1],≤)-fuzzy sets (see, for instance, [5, 12]).Noticing the strong formal analogy between the formulas (14) and (15), we may say that NQ = (Πn)Pn
can be considered as the dual ‘extended’ possibility measure of the ‘extended’ necessity measure ΠP
w.r.t. n.Let us now look at a well-known special case in order to reveal the significance of the results given
above. Indeed, choose (L,≤) = ([0, 1],≤), R = P(X)—the power class of X—, P = min, Q = max andlet n be the involutive dual order-automorphism on ([0, 1],≤) defined by
(∀a ∈ [0, 1])(n(a) def= 1− a).
It is fairly well known that the t-norm min on the complete chain ([0, 1],≤) is completely distributivew.r.t. supremum, and that the t-conorm max on ([0, 1],≤) is completely distributive w.r.t. infimum.Also, maxn = min and minn = max. Let furthermore Π be an arbitrary ([0, 1],≤)-possibility measure on(X,P(X))—and therefore a possibility measure according to Zadeh [28]—with distribution π, i.e.
(∀x ∈ X)(π(x) def= Π(x)).
Finally, let N be the dual ([0, 1],≤)-necessity measure—and therefore a necessity measure according to
Dubois and Prade [8]—of Π w.r.t. n, i.e, N def= Πn, or since n is involutive
(∀A ∈ P(X))(N(A) = n(Π(coA)) = 1−Π(coA)).
The distribution ν of N satisfies, for arbitrary x in X,
ν(x) = N(cox) = n(Π(x)) = 1− π(x).
For an arbitrary ([0, 1],≤)-fuzzy variable h in (X,P(X))—the membership function of a fuzzy set ac-cording to Zadeh [28]—we have, with the obvious notations
Πmin(h) def= (min) –∫
XhdΠ and Nmax(h) def= (max) –
∫
XhdN (16)
that
Πmin(h) = supx∈X
min(h(x), π(x))
Nmax(h) = infx∈X
max(h(x), ν(x)) = infx∈X
max(h(x), 1− π(x)).(17)
Formulas (17) are precisely the ones used by Zadeh [28] to define possibility for his type of fuzzy sets,and by by Dubois and Prade to define the notion of necessity for Zadeh’s fuzzy sets (see [8] section 1.7,formula (1.65)). In order to arrive at these definitions, Zadeh, Dubois and Prade start with the well-knownexpressions6
Π(A) = supx∈A
π(x)
N(A) = infx∈coA
(1− π(x))(18)
6We want to stress that the notion of distribution for necessity measures is not used by Zadeh nor by Dubois and Prade,who only use distributions of possibility measures.
23
for arbitrary A in P(X). They remark that these expressions can alo be written as
Π(A) = supx∈X
min(χA(x), π(x))
N(A) = infx∈X
max(χA(x), 1− π(x)),(19)
where, of course, χA is the characteristic X − [0, 1] mapping of the set A. The step towards the formulas(17) is obvious.
Although our approach and the one followed by Zadeh, Dubois and Prade lead to the same result, wewant to stress here that their respective backgrounds are completely different. In the approach of Zadehet. al. the crucial step is the transition from (19) to (17). In our approach, possibility and necessityintegrals play a very important part. Moreover, it combines three important notions in a very naturalway: fuzzy sets, possibility-necessity measures and fuzzy integrals.
To conclude this paper, let us briefly discuss the analogy probability-possibility. Although our treat-ment of fuzzy variables was formally inspired by the theory of the real stochastic variables, there areobvious differences between both notions, especially from the point of view of their interpretation. Onthe one hand, fuzzy variables have in this paper been interpreted as generalizations, or fuzzifications, ofmeasurable sets—they could equally well have been called measurable fuzzy sets. On the other hand,real stochastic variables can be considered as a formal mathematical concretization of the abstract notion‘variable that takes real values’. In probability theory, Lebesgue integrals are used to define the notionof expectation or mean of real stochastic variables. In the discussion above, possibility (necessity) inte-grals were used to extend the domain of possibility (necessity) measures from measurable sets towardsmeasurable fuzzy sets.
It is nevertheless possible to introduce the notion of a possibilistic variable, which is a possibilisticinstance of the abstract notion of a variable, and is therefore a perfect possibilistic analogon of thestochastic variables, known in probability theory. Using these possibilistic variables and the possibilityand necessity integrals discussed here, a general measure- and integral-theoretic treatment of possibilityand necessity theory can be given, which unifies a diversity of results in fuzzy set and possibility theory.This treatment includes amongst many other things a consistent discussion of product possibility measuresand integrals, and of conditional possibility of possibilistic variables and possibilistic independence, untilnow an outstanding problem in this field. A detailed account of this theory can be found in the doctoraldissertation of one of us [5], and will be published in detail elsewhere. The groundwork for this unifyingtreatment of possibility and necessity theory has however been laid in this paper.
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25
Footnotes1 For classical measures, the σ-additivity property holds, which implies that there is no difference
between the Lebesgue and Choquet integrals associated with these measures. Differences betweenthese integrals only arise for non-classical measures that are not additive.
2 We note that the fuzzy variables introduced by Zadeh [28] are more general than the fuzzy variablesdiscussed here. The former are intended as possibilistic analoga of general, not necessarily real-valued, stochastic variables. Furthermore, they cannot in general be interpreted as measurablefuzzy sets. We therefore suggest that the more appropriate name ‘possibilistic variable’ be used forZadeh’s notion, and that the name ‘fuzzy variables’ be reserved for measurable fuzzy sets. For adetailed discussion of possibilistic variables, conditional possibility and possibilistic independenceof possibilistic variables, we refer to the doctoral dissertation of one of us [5].
3 Weber [26] has proven an analogous result for finite suprema for the combination of Sugeno’s fuzzyintegral and what he calls ‘σ-^-decomposable measures ’. Furthermore, Sugeno [20] has derived ananalogous result for his fuzzy integral and what he calls ‘F-additive fuzzy measures .’
4 Weber [26] has proven an analogous result for the combination of Sugeno’s fuzzy integral and whathe calls σ-^-decomposable measures.
5 As a matter of fact, Ralescu and Adams have proven that an analogous result is not valid for thespecial case of ([0, 1],≤)-fuzzy min-integrals associated with Sugeno’s fuzzy measures, by giving acounterexample. Of course, this counterexample remains valid for the more general result discussedhere.
6 We want to stress that the notion of distribution for necessity measures is not used by Zadeh norby Dubois and Prade, who only use distributions of possibility measures.
26
