Coherent lower previsions and Choquet integrals

Fuzzy Sets and Systems 138 (2003) 469– 484www.elsevier.com/locate/fss

Coherent lower previsions and Choquet integralsVolker Kr*atschmer∗

Statistik und Okonometrie, Rechts- und Wirtschaftswissenschaftliche Fakultat, Universitat des Saarlandes, Bau 31,Postfach 15 11 50, D-66041 Saarbrucken, Germany

Received 29 June 1999; received in revised form 12 November 2002; accepted 18 November 2002

Abstract

Walley (Statistical Reasoning with Imprecise Probabilities, Chapman & Hall, London, 1991) has givena helpful general framework of uncertain reasoning which is grounded on a behavioural foundation, andextends de Finetti’s work. For some important applications like the expected utility theory it might be a tool,alternatively to the techniques of Choquet integration. Therefore, this paper tries to clarify the relation betweenWalley’s central concept of coherent lower prevision and the Choquet integration theory. In one direction wedeal with the problem to represent coherent lower previsions by Choquet integrals, and obtain general resultsusing Greco’s representation theorem. Another aspect of investigation is to compare Walley’s concept to de<neintegrals of coherent fuzzy measures, the so-called natural extensions, with Choquet integrals. We succeed inexploring completely the relationship.c© 2002 Elsevier B.V. All rights reserved.

Keywords: Lower prevision; Coherence; Asymmetric Choquet integral; Symmetric Choquet integral; Greco’s representationtheorem; Natural extension

0. Introduction

The traditional method of uncertain reasoning is to express the beliefs or the knowledge of uncer-tainty by probability measures. However, in many situations probability measures are not appropriaterepresentations of knowledge, e.g. in the case of complete ignorance. From a general point of viewit might be recommendable to treat knowledge representation in a very @exible way. Uncertaintymeasures should satisfy only some basic properties, as few as possible.

An important class that has been considered in literature consists of the so-called fuzzy measures.They are de<ned by monotonicity and some normalizing conditions. However, what is missing is a

∗ Tel.: +49-681302-3169; fax: +49-681302-3551.E-mail address: [email protected] (V. Kr*atschmer).

0165-0114/03/$ - see front matter c© 2002 Elsevier B.V. All rights reserved.doi:10.1016/S0165-0114(02)00545-6

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470 V. Kratschmer / Fuzzy Sets and Systems 138 (2003) 469– 484

uni<ed approach of uncertain reasoning which relies on fuzzy measures, e.g. concepts of integralsand updating rules.

Recently, Walley has oHered a promising framework for a broader class of uncertainty modelswithin a behavioural foundation [23]. In his setting the beliefs of uncertainty are represented byso-called lower previsions which are in fact real-valued functionals and which also cover the fuzzymeasures as special cases. One of his crucial concepts is coherence of lower previsions. It expressesa minimal requirement of behaving consistently and includes de Finetti’s concept of coherence [6].As it is already known, de Finetti introduces his notion of coherence to give probabilities somebehavioural foundation, contrary to the axiomatic approach of KolmogoroH. In terms of abstractmeasure theory de Finetti’s concept of coherence leads to probability contents.

Probability contents have played an important role in treating the problem of decision making underuncertainty. Savage [19] and Anscombe=Aumann [1] established axiomatic settings for preferencestructures of economic agents to be representable by expected utility functions. The (asymmetric)Choquet integrals with respect to probability contents were used as the mathematical tool to describethe expected utility functions. Nowadays this approach is known as the additive expected utilitytheory respectively the expected utility theory with additive probabilities.

During the last decades the additive expected utility theory has been challenged by experimentaleconomical investigations. Ellsberg [9], e.g., found out that the participants in his experiments oftendid not behave like maximizers of expected utility functions with respect to probability contents.Gilboa [10], Schmeidler [20] and Wakker [22] succeeded in establishing a general axiomatic approachto describe equivalently preference structures of economic agents as maximizers of expected utilityfunctions in some wider sense. They referred to (asymmetric) Choquet integrals with respect to moregeneral fuzzy measures. In particular, the participants in Ellberg’s experiments could be consideredas maximizers of expected utility who represent their knowledge of the possible states by fuzzymeasures.

Taking the point of view of Walley, every fuzzy measure, which expresses some agent’s beliefsabout her situation of uncertainty, should be coherent. For coherent fuzzy measures Walley introducesa new concept of integral, the so-called natural extension, which is a coherent lower prevision. Thus,the behavioural foundation of coherence gives good reason to suggest the development of an expectedutility theory which relies on coherent fuzzy measures and their natural extensions. As shown byWalley such an expected utility theory would extend the additive expected utility theory. Thus,hypothetically, we have two alternative frameworks to construct a general expected utility theory onbasis of fuzzy measures: On one hand the Choquet integration theory, on the other hand Walley’sconcept of coherent lower previsions, in particular the notion of natural extensions of coherent fuzzymeasures.

This paper deals with the relationship between the framework of coherent lower previsions and theChoquet integration theory. According to fuzzy measures on lattices two ways to de<ne the Choquetintegrals of measurable functions are known in literature, the symmetric and the asymmetric. Theycoincide in nonnegative measurable functions leading to the original notion which was introduced byChoquet [2]. Recently, the Choquet integration theory has been systemized analogously to Lebesgue’sapproach. For references see the monographs of Denneberg [7], K*onig [13] and Pap [18]. Murofushi,Narukawa and Sugeno deal with the problem of representing Choquet integrals by Lebesgue integrals[16,17,21]. Choquet integrals have been compared with Sugeno (Fuzzy) integrals by deCampos etal. [4], deCampos=Bolanos [3] and Pap [18, Section 7.9].

V. Kratschmer / Fuzzy Sets and Systems 138 (2003) 469– 484 471

Our investigations will proceed in two directions. Firstly, we shall treat the problem of representinga coherent lower prevision by an integral of a Choquet type. Secondly, we shall compare the conceptof natural extensions with integrals of a Choquet type. In particular, we are interested in conditionswhen natural extensions coincide with integrals of a Choquet type.

The paper is organized as follows.In the <rst section we shall introduce the general notations, and gather some facts from abstract

measure theory.The subject of the second section is Walley’s concept of coherent lower previsions. It includes

the coherent previsions in the sense of de Finetti which are now called linear previsions. Coherentlower previsions are closely related to linear previsions since every coherent lower prevision is alower envelope of linear previsions.

In Section 3 we shall introduce the symmetric and asymmetric Choquet integrals and collect someresults of abstract integration theory dealing with the correspondences between the properties ofChoquet integrals and their fuzzy measures.

Afterwards, in Section 4, we shall investigate the problem of representing coherent lower previsionsby integrals of a Choquet type. As the basic tool to show the results we shall introduce Greco’stheorem to represent functionals by Choquet integrals.

The last section deals with the connection of Choquet integrals and natural extensions.The results in the fourth and <fth section will be proved separately within the sections in the

appendix.

1. Notations and preliminaries

Let � be a nonvoid universe, and let K be a set of [−∞;∞]-valued mappings on �. The set Kis called positive homogeneous if it is stable under scalar multiplications with positive real numbers.

For [−∞;∞]-valued mappings f; g on � we de<ne f∧ g := min{f; g}; f∨ g := max{f; g}. Ifg= 0, then we shall use the notations f+ for f∨ g and f− for (−f)∨ g. In the case that K containsonly nonnegative functions, K is said to be Stonean if f∧ t; (f − t)+ ∈K for f∈K; t ∈ ]0;∞[.The families L(�) and L+(�) which gather all bounded real-valued mappings on � respectivelyall nonnegative mappings from L(�) are important for the investigations in this paper. They arepositive homogeneous as well as stable under addition, and L+(�) is Stonean.

Let S denote a lattice on �, i.e. a nonvoid collection of subsets of � which is stable under <niteunions and intersections. Throughout this paper we shall also consider the family LS(�) consistingof all mappings from L(�) which are S-measurable, i.e. functions f with f−1([t;∞[); f−1(]t;∞[)belonging to S for every real number t. The family L+

S (�) contains the nonnegative mappingsfrom LS(�). Both families are positive homogeneous, and L+

S (�) is Stonean. Furthermore LS(�)as well as L+

S (�) are nonvoid if and only if S contains ∅ and �. In this case all <nite sums ofcharacteristic functions of subsets from S are included in L+

S (�).A function � :S→ [0;∞] is said to be a set function on S if ∅∈S and �(∅) = 0. Let � be

some real-valued set function on a lattice S which additionally contains �. Then the set systemS⊥ := {�\A |A∈S} is also a lattice on � which contains ∅; �, and the function P� :S⊥→ [0;∞],de<ned by P�(B) :=�(�) − �(�\B), is also a real-valued set function, the so-called conjugateof �.


Let us recall some terminology for set functions (cf. [7,13]). A set function � on S is de<ned tobe isotone if �(A)⊆�(B) holds for every pair A; B∈S with A⊆B. We shall call a set function �on S to be submodular (supermodular) if (A∪B) + (A∩B)6(¿)(A) + (B) for A; B∈S, andmodular if it is sub- and supermodular. If a set function � on S satis<es �(A∪B) =�(A) + �(B)for disjoint A; B∈S, then it is said to be additive. It can be shown easily that the modular setfunctions on algebras are exactly the additive ones.

Throughout this paper we shall deal with lattices S which contain ∅ as well as �, and we shallfocus on the so-called fuzzy measures on S, i.e. isotone set functions which hold the normalizingcondition (�) = 1. Every conjugate of a fuzzy measure is a fuzzy measure too. Additive fuzzymeasures on algebras are known as probability contents.

The subject of abstract measure and integration theory is the relationship between set functions andso-called functionals which are [−∞;∞]-valued mappings de<ned on nonvoid families of [−∞;∞]-valued mappings on �. The relationship is treated in two ways: Identifying subsets of � with theirindicator functions, one looks for functionals which extend set functions. Roughly speaking theseextending functionals are the objects that are imagined as integrals. Conversely, one starts withfunctionals and tries to represent them as integrals with respect to set functions. Further investigationsof the relationship between set functions and integrals deal with the problem of transferring propertiesof set functions to integrals and vice versa. Later on we shall introduce some concepts which de<neintegrals for set functions, namely the integrals of Choquet type and Walley’s natural extension. But<rst, let us mention some special types of functionals (cf. [7,13]).

A functional I on some K is called isotone if I(f)6I(g) for f; g∈K; f6g. Let K be positivehomogeneous, then functional I on K is said to be positive homogeneous if I(�f) = �I(f) forf∈K; �¿0. A functional I on some K, which is stable under addition, is de<ned to be subadditiveif I(f+g)6I(f)+I(g) for f; g in K, superadditive if the inequality turns the direction, and additiveif it is sub- and superadditive. For the remainder of this section let us concentrate on Stonean familiesK (especially they contain only nonnegative functions). Then a functional I on such a family Kis called Stonean if the equality I(f) = I(f∧ t) + I((f − t)+) holds for all f∈K and t ∈ ]0;∞[.Additionally, a functional I is said to be truncable if I(f) = sup0¡a¡b¡∞ I((f− a)+ ∧ (b− a)) forevery f.

2. Coherent lower previsions

Now,we want to introduce the concept of coherent lower previsions, developed by Walley [23].Let � be a nonvoid set. Then a real-valued functional on a nonvoid family K⊆L(�) is called a

lower prevision. If K contains only indicator functions of subsets, one speaks of a lower probability.As usual in literature, we identify lower probabilities with the functions on the corresponding setsystems on �. We say that a lower prevision I :K→R is coherent if

sup!∈�

[n∑

i=1

(fi(!) − I(fi)) − m(f0(!) − I(f0))

]¿ 0;

whenever m; n are nonnegative integers and f0; f1; : : : ; fn are in K.


This notion is justi<ed by Walley’s interpretation of bounded functions as gambles, and outcomesof lower previsions as upper limits an actor is disposed to pay for each gamble in question. Walleyestablishes a system of axioms for behavioural rationality. Within this framework he can justifycoherence as a minimal requirement for behaving consistently. For further discussion the reader canconsult Walley’s textbook [23].

Let us collect some properties of coherent lower previsions.

Proposition 2.1. Let I be some coherent lower prevision on some K⊆L(�) containing the con-stant function 0. Then we have the following properties:

(1) I is isotone.(2) I(�f) = �I(f) for f; �f∈K; �∈ ]0;∞[. Especially, I is positive homogeneous if K is positive

homogeneous.(3) I(f + g)¿I(f) + I(g) for f; g∈K such that f + g∈K. In particular, I is superadditive if

K is stable under addition.(4) If f∈K; t ∈R such that f + t ∈K, then I(f + t) = I(f) + t.(5) Let S denote some lattice on � which contains ∅ and �. If K consists of all indicator

functions of the sets from S, then the lower probability I can be identi:ed with a fuzzymeasure on S.

(6) If K∩L+(�) is Stonean, then the restriction I |K∩L+(�) is truncable.

Proof. Statements (1)–(4) can be found in [23, 2.6.1], (5) follows from (1), (2), (4). Therefore itremains to show statement (6).

Proof of (6). Let f∈K∩L+(�). The statement is trivial for f= 0. Therefore, let us assume thatf �= 0, especially supf¿0. By assumption K∩L+(�) is Stonean, and I |K∩L+(�) is isotonedue to (1). Then we obtain immediately

sup0¡a¡b¡∞

I((f − a)+ ∧ (b− a))

= sup0¡a¡supf¡b¡∞

I((f − a)+ ∧ (b− a))

= sup0¡a¡supf

I((f − a)+):

The coherence of I implies

sup!∈�

[f(!) − I(f) − ((f − a)+(!) − I((f − a)+))] ¿ 0

for every a∈]0; supf[, which means that the inequalities

I(f) 6 a + I((f − a)+) 6 a + sup0¡a¡supf

I((f − a)+)

hold. Thus I(f)6 sup0¡a¡b¡∞ I((f − a)+ ∧ (b− a)).On the other hand f¿(f− a)+ ∧ (b− a) for every pair a; b of positive real numbers with a¡b.

Hence I(f)¿ sup0¡a¡b¡∞ I((f − a)+ ∧ (b− a)) since I is isotone. This completes the proof.


The concept of coherence extends the correspondent one introduced by de Finetti [6]. WithinWalley’s terminology a coherent lower prevision in the sense of de Finetti is said to be a linearprevision ([6, p. 87] and [23, p. 86=87]). A lower prevision I :K→R is de<ned to be linear if

sup!∈�

n∑

i=1

(fi(!) − I(fi)) −m∑j=1

(gj(!) − I(gj))

¿ 0;

whenever m; n are nonnegative integers and f1; : : : ; fn; g1; : : : ; gm are in K. In particular, every linearprevision is coherent. Linear lower probabilities are called additive probabilities. The name emergesfrom the fact that the linear lower probabilities on algebras are exactly the probability contents [23,2.8.9].

Proposition 2.2. A lower probability on an algebra on � is linear if and only if it is a probabilitycontent.

Every linear prevision can be extended to a linear prevision on L(�) [23, 3.4.2].

Proposition 2.3. For every linear prevision on some K⊆L(�) there exists a linear prevision Ion L(�) which coincides with I on K.

It is interesting that we can embed the concept of linear previsions into the terminology offunctional analysis. Indeed linear previsions on L(�) are continuous real-valued linear forms onL(�) w.r.t. the supremum norm (apply [23, Theorem 2.8.3]). This property enables us later torepresent linear previsions by Choquet integrals, drawing on a known result from functional analysiswhich deals with the representation of continuous real-valued linear forms by Choquet integrals.

Proposition 2.4. Every linear prevision on L(�) is a real-valued continuous linear form on L(�)w.r.t. the norm

‖ · ‖∞ :L(�) → R; f �→ sup!∈�

|f(!)|

As an important result, Walley gives a characterization of coherent lower previsions in terms ofenvelopes of the dominating linear previsions ([23, 3.3.3]). He uses the convention that a lowerprevision I on K dominates a lower prevision I on K if K contains K, and I |K¿I holds forthe restriction I |K of I to K.

Proposition 2.5. Let I be some lower prevision on K⊆L(�). Associated with I is the set M(I)consisting of all linear previsions on L(�) which dominate I. Then I is coherent if and only if

I(f) = min{I(f) | I ∈ M(I)}

holds for arbitrary f∈K.


3. Properties of Choquet integrals

A very well known way to extend isotone set functions is the concept of Choquet integrals. Inorder to simplify the presentation we restrict ourselves to real-valued isotone set function on latticeswhich contain ∅ and �. Such a set function induces the lower prevision

I :L+S(�) → R; f �→

∞∫0

(f−1(]t;∞[)) dt;

called the Choquet integral with respect to . This is the original approach of Choquet [2]. Theliterature of abstract integration theory oHers two diHerent suggestions to extend the notion of Choquetintegrals to bounded real-valued, S-measurable functions, the so-called symmetric and asymmetricChoquet integral [7, Chaps. 5,7]; [18, Chap. 7].

The lower prevision I a on LS(�) with

I a (f) :=

0∫−∞

((f−1(]t;∞[)) − (�)) dt +

∞∫0

(f−1(]t;∞[)) dt

de<nes the asymmetric Choquet integral with respect to . Since for f∈LS(�) (f−1([t;∞[))and (f−1(]t;∞[)) diHer only for at most countable values t (cf. Corollary C.2 in Appendix C),we can write

I a (f) =

0∫−∞

((f−1([t;∞[)) − (�)) dt +

∞∫0

(f−1([t;∞[)) dt:

Every f∈LS(�) can be decomposed in positive and negative parts, i.e f=f+ − f−. Then wehave f+ ∈LS(�); f− ∈LS⊥(�), and

I a (f) = I(f+) − I P(f−) (3.1)

for f∈LS(�) [7, p. 87]. This form of the asymmetric Choquet integral gives rise to another conceptfor an extension of the Choquet integral to LS(�). In the case that S is an algebra we can de<ne,analogously to the Lebesgue integral, the lower prevision

I s :LS(�) → R; f �→ I(f+) − I(f−)

which is called the symmetric Choquet integral (Sipos integral) with respect to . In order to be wellde<ned we have to ensure that for every f∈LS(�) the mapping f− is bounded and S-measurable.For this purpose we have assumed S to be an algebra, which implies S⊥=S.

I; I a ; I

s extend to L+

S (�) respectively LS(�), whereas I a ; I

s coincide with I on L+

S (�). Theterminology is inspired by

I a (−f) = −I a

P (f); I s(−f) = −I s

(f) (3.2)

for f∈LS(�), and S being an algebra. Additionally we have

I a (f + �) = I a

(f) + �(�) (3.3)


for f∈LS(�); �∈R, which leads to the characterization

I a (f) = I(f − �) + �(�) (3.4)

for f∈LS(�); �∈R with �6f.Next, we want to gather some properties of a real-valued set function which can be transferred

to the Choquet integrals I; I a ; I

s .

Proposition 3.1. Let denote a real-valued isotone set function on a lattice S on � which contains∅ as well as �. Then we have

(1) I; I a are isotone and positive homogeneous lower previsions. I s

satis:es these properties too ifS is an algebra.

(2) The following conditions are equivalent:(i) I(f + g)6I(f) + I(g) for f; g in L+

S (�) with f + g∈L+S (�).

(ii) I a (f + g)6I a

(f) + I a (g) for f; g in LS(�) with f + g∈LS(�).

(iii) is submodular.(3) The following conditions are equivalent:

(i) I(f + g)¿I(f) + I(g) for f; g in L+S (�) with f + g∈L+

S (�).(ii) I a

(f + g)¿I a (f) + I a

(g) for f; g in LS(�) with f + g∈LS(�).(iii) is supermodular.

(4) If S is an algebra, then the following conditions are equivalent:(i) I s

(f + g)6I s (f) + I s

(g) for f; g in LS(�) with f + g∈LS(�).(ii) I s

(f + g)¿I s (f) + I s

(g) for f; g in LS(�) with f + g∈LS(�).(iii) I s

(f + g) = I s (f) + I s

(g) for f; g in LS(�) with f + g∈LS(�).(iv) is modular.

Proof.

Proof of (1). The results according to I are shown in [13, Properties 11.8]. They can be extendedto I a

using (3.4). The remainder of the statement follows from the properties of I and the de<nitionof I s

.

Proof of (2), (3). The equivalence of (i), (iii) can be found in [13, Theorem 11.11, Properties 11.8, 4].This implies the equivalence of (ii), (iii) applying (3.4).

Proof of (4). Let us assume that (i) holds. Then is submodular due to (2). For A; B∈S wedenote their indicator functions by 1A; 1B. Since S is an algebra, the functions −1A − 1B;−1A;−1B

are S-measurable. Moreover

−(A) − (B) = I s(−1A) + I s

(−1B)

¿ I s(−1A − 1B)

= −I(1A + 1B)

= −((A ∪ B) + (A ∩ B)):

Therefore is even supermodular, hence modular.


Analogously is modular if (ii) holds.If is modular, then = P, and I a

= I s according to (3.1). Drawing on (2), (3), I s

ful<lls (iii).This completes the proof because the implications (iii)⇒ (i) and (iii)⇒(ii) are trivial.

4. Representation of a coherent lower prevision by a Choquet integral

In this section, we shall turn our attention to the aspect of measure and integration theory whichdeals with the representation of functionals by integrals. Since we want to link Walley’s concept ofcoherent lower previsions with Choquet integration theory, we are interested in representing coherentlower previsions by Choquet integrals.

In the special case of linear previsions we can draw on a known result from functional analysis[8, Corollary IV.5.3].

Lemma 4.1. Let I :L(�)→R be an isotone lower prevision on L(�), and let P(�) denote thepowerset of �.Then there exists a unique bounded content P on P(�) such that I(f) = I a

p (f) for all f∈L(�)if and only if I is a continuous real-valued linear form on L(�) w.r.t. the norm

‖ · ‖∞ :L(�) → R; f �→ sup!∈�

|f(!)|

Remark. Genuinely, in [8] Dunford and Schwarz adapt the classical Lebesgue approach to developan integration theory w.r.t. contents. Indeed they obtain asymmetric Choquet integrals w.r.t. boundedcontents on powersets. This may be seen in the following way: First observe that every functionf∈L(�) can be approximated uniformly by a sequence (fn)n of simple functions fn ∈L(�). Thenapply Proposition 3.1, (1)–(3), as well as Propositions 8.8, 8.5 and Theorem 8.9 in [7].

As a consequence of Lemma 4.1 we can represent every linear prevision by an asymmetric andsymmetric Choquet integral with respect to a probability content on the powerset of �.

Proposition 4.2. Let P(�) be the powerset of �, and let I be a linear prevision on some K⊆L(�).Then there exists a probability content P on P(�) such that I(f) = I a

P(f) = I sP(f) for all f∈K.

The proof can be found in Appendix A.The basic tool to represent more general coherent lower previsions is Greco’s representation

theorem ([14, Theorem 2.10]; see also the original paper [12], and for a specialized version Theorem13.2 in [7]).

Proposition 4.3. Let S be a lattice on � which contains ∅ as well as �. Let us also assumethat some K⊆L+

S (�) contains the constant functions 0; 1 and is Stonean as well as positivehomogeneous. Additionally, let I be an isotone lower prevision on K with I(0) = 0. Thenthere exists a real-valued isotone set function on S such that I(f) = I(f) for all f∈K if andonly if I is Stonean and truncable. In this case is bounded by I(1).


Remark. As mentioned above Choquet integrals can be de<ned in an analogous way with respectto arbitrary set functions. In the more general framework Greco’s representation theorem can besimpli<ed by omitting the assumption 1∈K [14, Theorem 2.10]. But then we cannot ensure thatthe lower prevision can be represented by a Choquet integral with respect to a real-valued setfunction, whereas the additional condition 1∈K is suRcient.

The most important application of Greco’s theorem leads to the following representation theoremfor coherent lower previsions.

Theorem 4.4. Let I denote a coherent lower prevision on some K⊆L(�) that contains the con-stant functions 0; 1 and is positive homogeneous. Moreover, assume K∩L+(�) to be Stonean.Then we can state

(1) If I a |K= I for some real-valued set function on a lattice S such that ∅; � are contained

in S and K⊆LS(�), then the restriction I |K∩L+(�) is Stonean.(2) If I s

|K= I for some real-valued set function on an algebra S such that K is contained inLS(�), then the restriction I |K∩L+(�) is Stonean.

(3) If f + 1∈K for all f∈K and the restriction I |K∩L+(�) is Stonean, then there existssome fuzzy measure on the powerset P(�) of � such that I(f) = I a

(f) for every f∈K.

The proof is delegated to Appendix A.As a conclusion from Theorem 4.4 we can solve completely the problem of representing coherent

lower previsions on the spaces LS(�) of bounded real-valued S-measurable mappings by integralsof a Choquet type. Let us recall that bounded real-valued S-measurable mappings only exist withrespect to lattices which contain ∅ and �.

Corollary 4.5. Let S be a lattice on � which contains ∅ as well as �, and let I denote somecoherent lower prevision on LS(�). Then we can state

(1) There is some fuzzy measure on S such that I = I a if and only if I |L+

S (�) is Stonean.(2) In the case that S is an algebra, there is some fuzzy measure on S such that I = I s

if andonly if I is a linear prevision.

The corollary will be proved within Appendix A.

5. Natural extensions and Choquet integrals

Now let us turn to the problem of extending coherent fuzzy measures to integrals. Up to this pointwe have met the integrals of a Choquet type. But Walley has introduced another concept which isbasic within his framework, the so-called natural extension of coherent lower previsions. It relies onthe following result [23, Sections 3.1.1 and 3.1.2].

Proposition and De0nition 5.1. Let I be a coherent lower prevision on some K⊆L(�). Thenthere is a coherent lower prevision EI :L(�)→R which coincides with I on K, and is the minimalcoherent lower prevision on L(�) dominating I. EI is called the natural extension of I.


Next, we want to compare the concept of natural extensions with the integrals of a Choquet type.First we shall treat the relation of asymmetric Choquet integrals and natural extensions.

Theorem 5.2. Let S be a lattice on � which contains ∅ and �. Additionally denotes some fuzzymeasure on S with crude inner envelope ? which is de:ned on the powerset P(�) of � by?(B) := supB⊇A∈S (A).Then ? is a fuzzy measure which extends , and the following statements are equivalent:

(i) I is coherent.(ii) I a

is coherent.(iii) ; I are coherent, respectively, and the natural extension E of coincides with I on L+

S (�).(iv) ; I a

are coherent, respectively, and the natural extension E of coincides with I a on LS(�).

(v) ; I a? are coherent, respectively, and the natural extension E of coincides with I a

? .(vi) is supermodular.

The proof is deferred to Appendix B.

Remark 5.3. Let us address some points with respect to Theorem 5.2:

(1) As a by-product of Theorem 5.2 we obtain that every supermodular fuzzy measure is coherent, aresult that has been mentioned by Walley [24, p. 14] in the case that S is an algebra. Generally,not every coherent fuzzy measure is supermodular as the following example shows:Let P1; P2 denote discrete probability measures on the power set of �=: {!1; : : : ; !4}, charac-terized by

!1 !2 !3 !4

P1({!i}) 14

14

14

14

P2({!i}) 12 0 1

2 0

We de<ne the set function on the power set of � to be the lower envelope of P1; P2, i.e.(A) = minj=1;2 Pj(A) for A⊆�. is coherent as a lower envelope of additive probabilities (Propositions 2.2, 2.5). But is notsupermodular: Take for example A := {!1; !2}, B := {!2; !3}. Then

(A ∪ B) + (A ∩ B) = 34 ¡ 1 = (A) + (B):

(2) The equivalence of (vi) and (iv) has been mentioned by Walley [23, p. 502] in the case thatS is an algebra.

Theorem 5.2 can be applied to some important classes of special types of fuzzy measures.

Example 5.4. Let S be a lattice on � which contains ∅ and �.

(1) Let Bel :S→R be a belief measure, i.e. (cf. [11, p. 321]).(i) Bel(�) = 1,


(ii)∑

J⊆{1; :::; n}(−1)]JBel(B0 ∩⋂

j∈J Bj)¿0 for B0; B1; : : : ; Bn ∈S,⋂

j∈∅ Bj :=�, ]J cardinalityof J .

If additionally Bel(∅) = 0 holds, then, due to Theorem 5.2, Bel is coherent, and its naturalextension coincides with its asymmetric Choquet integral on LS(�).

(2) Let Poss :S→R be a possibility measure, i.e. (cf. [5]).(i) Poss(∅) = 0; Poss(�) = 1

(ii) Poss(⋃

j∈J Bj) = supj∈J Poss(Bj) whenever J is a nonvoid index set and⋃

j∈J Bj; Bj aremembers of S.

The associated necessity measure Nec is de<ned as the conjugate of Poss (cf. [5]).Nec is coherent according to Theorem 5.2, and its natural extension coincides with its asymmetricChoquet integral on LS(�).

Next, we want to compare natural extensions of coherent fuzzy measures with their symmetricChoquet integrals.

Theorem 5.5. Let be a fuzzy measure on an algebra S on �.Then the following statements are equivalent:

(i) ; I s are coherent, respectively, and the natural extension E of coincides with I s

on LS(�).(ii) is a probability content.

The proof can be found in Appendix B.

Remark. Essentially, the implication (ii)⇒ (i) has been shown by Walley [23, 3.2.2].

6. Concluding remarks

In this paper, we succeeded in exploring the relationship between coherent lower previsions andintegrals of a Choquet type very generally. One part is the comparison between the concepts ofnatural extensions of coherent lower probabilities and integrals of Choquet type. This aspect hasbeen clari<ed completely. It has turned out that natural extensions of coherent lower probabilitiesare nearly related to the respective asymmetric Choquet integrals, to be more precise: both conceptscoincide on the spaces of measurable functions if and only if the lower probability is a supermodularfuzzy measure. Whereas only in the case of probability contents are the symmetric Choquet integralsidentical with the respective restrictions of the natural extensions.

Another point is the representation of coherent lower previsions by integrals of Choquet type. Wehave established a general condition which describes equivalently the representation by asymmetricChoquet integrals. Moreover, the representation problem is completely solved for linear previsions,and for coherent lower previsions on the spaces of measurable functions with respect to lattices. Itis notable that only linear previsions can be represented by symmetric Choquet integrals.

Firstly, we can conclude that concerning linear previsions the concepts of natural extensions andasymmetric as well as symmetric Choquet integrals are equivalent. Moreover, the framework oflinear previsions is transformable to Choquet integration theory with identical asymmetric as well assymmetric Choquet integrals and vice versa.


Secondly, we can conclude that if we deal generally with coherent lower previsions we have amutual transformation into Choquet integration theory with asymmetric Choquet integrals.

Appendix A.

A.1. Proof of Proposition 4.2

According to Proposition 2.3 there exists an linear prevision I which extends I to L(�).Drawing on Proposition 2.4 and Lemma 4.1 there exists some content P on P(�) such that we

have I(f) = I aP(f) for all f∈L(�). Moreover, we have P(�) = I(1) = 1 since I is coherent. Hence

P is a probability content on P(�) which coincides with its conjugate. This implies I aP = I s

P by (3.1),and completes the proof.

A.2. Proof of Theorem 4.4

Statements (1), (2) follow immediately from Greco’s theorem.

Proof of (3). Since I |K∩L+(�) is coherent, it is truncable due to Proposition 2.1, (6). Furthermoreit is also Stonean by assumption. Hence we can apply Greco’s theorem to P(�); K∩L+(�)and I |K∩L+(�). Then there exists some real-valued isotone set function on P(�) such thatI(f) = I(f) for f∈K∩L+(�). In particular, the constant function 1 is contained in K∩L+(�)by assumption, and (�) = I(1) = 1 since I |K∩L+(�) is coherent. That means is a fuzzymeasure.

Now, let f∈K. Keeping things general we can assume inf f¡0 (otherwise I(f) = I a (f)).

Since K is positive homogeneous, −f= inf f is contained in K, and therefore, by assumption,−(f= inf f) + 1∈K. This leads to f − inf f= − inf f(−(f= inf f) + 1)∈K, drawing again onpositive homogeneity of K. Thus we obtain from (3.4) and Proposition 2.1, (4)

I a (f) = I(f − inf f) + inf f

= I(f − inf f) + inf f = I(f)

This completes the proof.

A.3. Proof of Corollary 4.5

Application of Theorem 4.4 leads immediately to statement (1). Additionally, the if part of state-ment (2) is a direct consequence of Proposition 4.2. So the only if part of statement (2) remainsto prove. For this purpose let us suppose denotes some fuzzy measure on S with I = I s

. SinceI is coherent, we obtain I s

(f + g)¿I s (f) + I s

(g) for f; g∈LS(�) with f + g∈LS(�) due toProposition 2.1. This also implies that the equality I s

(f + g) = I s (f) + I s

(g) holds for f; g fromLS(�) with f + g∈LS(�), and that is modular (cf. Proposition 3.1, (4)).

Furthermore is an additive probability (Proposition 2.2). Then, according to Proposition 2.3,there exists some linear prevision I which extends to L(�). Drawing on Proposition 4.2, we can


<nd some probability content P on the powerset P(�) such that I = I sP. This means P|S= , and

we obtain for f∈LS(�)

I(f) = I sP(f) = I s

(f) = I(f);

which completes the proof.

Appendix B.

B.1. Proof of Theorem 5.2

Let either I or I a be coherent. By de<nition of coherence it follows for A1; A2 ∈S with indicator

functions 1A1 ; 1A2 that (1A1 + 1A2)∈L+S (�), and that the supremum of the function

2∑i=1

(1Ai − I(1Ai)) −(

2∑i=1

1Ai − I

(2∑

i=1

1Ai

))

is nonnegative. This implies supermodularity of because I(∑2

i=1 1Ai) = (A1 ∪A2) + (A1 ∩A2) isobtained by using the de<nition of I. Since the implications (v)⇒ (iv), (iv)⇒ (iii), (iii)⇒ (i), and(iv)⇒ (ii) are trivial, it remains to show the implication (vi)⇒ (v).

For this purpose let be supermodular. Then ? can be shown as a supermodular fuzzy measureon the power set P(�) by routine procedures. Furthermore the asymmetric Choquet integral I a

? isde<ned on L(�) and is, due to Proposition 3.1, isotone, superadditive, positive homogeneous. Thenwe obtain for arbitrary nonnegative integers n; m and f;f1; : : : ; fn ∈L(�)

mI a?(f) + sup

!∈�

[n∑

i=1

fi(!) − mf(!)

](3:3)= I a

?

(mf + sup

!∈�

[n∑

i=1

fi(!) − mf(!)

])

¿ I a?

(n∑

i=1

fi

)¿

n∑i=1

I a?(fi):

Hence I a? is coherent. Since ? extends obviously, is coherent, and, in accordance with Propo-

sition and De<nition 5.1, E6I a? .

Now let f∈L(�). Drawing on Proposition 2.5 we can <nd a linear prevision I on L(�) whichdominates E and holds E(f) = I(f). Applying Proposition 4.2 there is some probability content Pon the powerset P(�) such that I = I a

P. Since I dominates E it follows 6P|S, and then ?6P.Therefore

E(f) = I(f) = I aP(f) ¿ I a

?(f):

Particularly, E(f) = I a?(f) because I a

? dominates E. Hence the proof is complete.


B.2. Proof of Theorem 5.5

The implication (i)⇒ (ii) follows from Proposition 2.1, (3), and Proposition 3.1, (4). Thereforethe implication (ii)⇒ (i) remains to show.

Proof of (ii)⇒ (i). Let be a probability content, in particular modular. Then by Theorem 5.2, is coherent, and the natural extension E coincides on LS(�) with the asymmetric Choquet integralI a . Moreover, = P, and then I a

= I s by (3.1). This proves statement (i), and completes the proof.

Appendix C.

Lemma C.1. Let h :I→R denote a monotone function on a compact interval I⊆R. Then thereexists some at most countable subset C⊆I such that h|I\C is continuous.

The result is known from Analysis, a proof can be found e.g. in [15, p. 193].

Corollary C.2. Let � be a nonvoid set, and let be a real-valued isotone set function on a latticeS on � which contains ∅ as well as �. Then for every function f∈LS(�) the set

{t ∈ R | (f−1([t;∞[)) �= (f−1(]t;∞[))}is at most countable.

Proof. Let f∈LS(�). Then f−1([t;∞[) =f−1(]t;∞[) =� for t¡ inf f, and f−1([t;∞[) =f−1

(]t;∞[) = ∅ for t¿ supf. Furthermore the mapping

h : [inf f; supf] → R; t �→ (f−1(]t;∞[))

is nonincreasing because is isotone. Applying Lemma C.1 there is some at most countable subsetC of [inf f; supf] such that h|[inf f; supf]\C is continuous. Since is isotone, we obtain forarbitrary t ∈ ] inf f; supf[\C

h(t) = (f−1(]t;∞[))6 (f−1([t;∞[))

6 lims→t− (f−1(]s;∞[))

= lims→t− h(s) = h(t)

Thus the set

{t ∈ R | (f−1([t;∞[)) �= (f−1(]t;∞[))}is contained in C∪{inf f; supf}, which completes the proof.


Acknowledgements

The author would like to thank the referees and the editors as well as the area editor, ProfessorMesiar, for helpful comments on how to improve earlier drafts.

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Coherent lower previsions and Choquet integrals