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J. Math. Anal. Appl. 282 (2003) 201–211 www.elsevier.com/locate/jmaa
Conditions for Choquet integral representation of the comonotonically additive and monotone functional
Yasuo Narukawa a,∗ and Toshiaki Murofushi b
a Toho Gakuen, 3-1-10 Naka, Kunitachi, Tokyo 186-0004, Japan b Department of Computational Intelligence and Systems Science, Tokyo Institute of Technology, 4259 Nagatuta, Midori-ku, Yokohama 226-8502, Japan Received 25 December 2000 Submitted by J. Horvath
Abstract If the universal set X is not compact but locally compact, a comonotonically additive and mon- otone functional (for short c.m.) on the class of continuous functions with compact support is not represented by one Choquet integral, but represented by the difference of two Choquet integrals. The conditions for which a c.m. functional can be represented by one Choquet integral are discussed. 2003 Elsevier Science (USA). All rights reserved.
Keywords: Nonadditive measure; Fuzzy measure; Cooperative game; Choquet integral; Comonotonic additivity
1. Introduction
The Choquet integral with respect to a nonadditive measure is one of the nonlinear functionals defined on the class B of measurable functions on a measurable space (X, B). It was introduced by Choquet [1] in potential theory with the concept of capacity. Then, in the field of economic theory, it has been used for utility theory [17], and has been used for image processing and recognition [4,5], in the context of fuzzy measure theory [9,20]. Essential properties characterizing this functional are comonotonic additivity and monotonicity. Let f and g be measurable functions. We say that f and g are comonotonic if f(x) * Corresponding author. E-mail address: [email protected] (Y. Narukawa). 0022-247X/03/$ – see front matter 2003 Elsevier Science (USA). All rights reserved. doi:10.1016/S0022-247X(03)00139-2 202 Y. Narukawa, T. Murofushi / J. Math. Anal. Appl. 282 (2003) 201–211 for x,x ∈ X. We denote f ∼ g when f and g are comonotonic. Let I be a real valued functional on B.WesayI is comonotonically additive iff f ∼ g ⇒ I(f + g) = I(f)+ I(g) for f,g ∈ B,andI is monotone iff f g ⇒ I(f) I(g) for f,g ∈ B. We say the functional I is a c.m. functional if I is comonotonically additive and monotone. Schmeidler [16] proves that a c.m. functional I on B can be represented by a Choquet integral. Parker [14] fully discusses the Choquet integral representations of comonotoni- cally additive functionals on B and related concepts. Skala [19] provides a generalization of the Choquet integral and uses this for an integral representation of comonotonically additive operators. Concerning the problem of whether or not the c.m. functional I can be represented by a Choquet integral when the domain of I is smaller than B, Greco [6] proved it when I has some continuity. Sugeno et al. [21] proved that a c.m. functional I can be represented by a Choquet integral with respect to a regular nonadditive measure (in the fuzzy context we say a nonadditive measure is a fuzzy measure) when the domain of I is the class K+ of non- negative continuous functions with compact support on a locally compact Hausdorff space. In [11], it is proved that a c.m. functional is a rank- and sign-dependent functional, that is, the difference of two Choquet integrals, if the domain of I is the class K of continuous functions with compact support on a locally compact Hausdorff space X. This functional is used in utility theory [7] and cumulative prospect theory [22,23]. It is also proved in [11] that a rank- and sign-dependent functional is a c.m. functional if X is not compact. From this fact, in order to represent I by one Choquet integral, further conditions are needed. In this paper, we present the conditions for which a c.m. functional can be represented by one Choquet integral. We call these conditions the conjugate conditions. We define the conjugate conditions and show their basic properties in Section 3. The conjugate conditions are stronger than the boundedness. A c.m. functional I is represented by one Choquet integral when I satisfies one of the conjugate conditions. Conversely if a c.m. functional I is represented by one Choquet integral, I satisfies the conjugate condition when the universal set X is separable. 2. Preliminaries In this section, we define the nonadditive measure, the Choquet integral and the rank- and sign-dependent functional, and show their basic properties. Throughout the paper we assume that X is a locally compact Hausdorff space, B is the class of Borel subsets, O is the class of open subsets, and C is the class of compact subsets. Definition 2.1. A nonadditive measure µ is an extended real valued set function + µ : B → R¯ with the following properties: Y. Narukawa, T. Murofushi / J. Math. Anal. Appl. 282 (2003) 201–211 203 (1) µ(∅) = 0; (2) µ(A) µ(B) whenever A ⊂ B, A,B ∈ B, where R¯+ =[0, ∞] is the set of extended nonnegative real numbers. When µ(X) < ∞,wedefinethe conjugate µc of µ by µc(A) = µ(X) − µ(Ac) for A ∈ B. Definition 2.2. Let µ be a nonadditive measure on measurable space (X, B). µ is said to be outer regular if µ(B) = inf µ(O) | O ∈ O,O ⊃ B for all B ∈ B. An outer regular nonadditive measure µ is said to be regular,ifforallO ∈ O, µ(O) = sup µ(C) | C ∈ C,C ⊂ O . K denotes the class of continuous functions with compact support, K+ denotes the + class of nonnegative continuous functions with compact support, and K1 denotes the class of nonnegative continuous functions with compact support that satisfy 0 f 1. supp(f ) denotes the support of f ∈ K. Definition 2.3 [1,8]. Let µ be a nonadditive measure on (X, B). (1) The Choquet integral of f ∈ K+ with respect to µ is defined by ∞ (C) fdµ= µf (r) dr, 0 where µf (r) = µ({x | f(x) r}). (2) Suppose µ(X) < ∞. The Choquet integral of f ∈ K with respect to µ is defined by (C) fdµ= (C) f + dµ− (C) f − dµc, where f + = f ∨ 0andf − =−(f ∧ 0). When the right-hand side is ∞−∞,the Choquet integral is not defined. If µ is not additive, the Choquet integral with respect to µ is nonlinear functional. In general, the Choquet integral is comonotonically additive and monotone (for short c.m.) [2,3,15]. Suppose that I is a c.m. functional, then we have I(af)= aI(f) for a 0andf ∈ K, that is, I is positively homogeneous. 204 Y. Narukawa, T. Murofushi / J. Math. Anal. Appl. 282 (2003) 201–211 Definition 2.4. Let I be a real valued functional on K. I is said to be a rank- and sign- dependent functional (for short a r.s.d. functional)onK, if there exist two nonadditive measures µ+, µ− such that for every f ∈ K, + + − − I(f)= (C) f dµ − (C) f dµ , where f + = f ∨ 0andf − =−(f ∧ 0). The r.s.d. functional is used in utility theory [7] and cumulative prospect theory [22,23]. When µ+ = µ−, we say that the r.s.d. functional is a Šipoš functional [18]. If the r.s.d. functional is a Šipoš functional, we have I(−f)=−I(f). If µ+(X) < ∞ and µ− = (µ+)c, we say that the r.s.d. functional is a Choquet func- tional. Theorem 2.1 [11]. Let I be a c.m. functional on K. (1) We put + = | ∈ + ⊂ µI (O) sup I(f) f K1 , supp(f ) O and + = + | ∈ O ⊃ µI (B) inf µI (O) O ,O B ∈ O ∈ B + for O and B .ThenµI is a regular nonadditive measure. (2) We put − = − − | ∈ + ⊂ µI (O) sup I( f) f KI , supp(f ) O and − = − | ∈ O ⊃ µI (B) inf µI (O) O ,O B ∈ O ∈ B − for O and B .ThenµI is a regular nonadditive measure. (3) A c.m. functional is a r.s.d. functional, that is, there exist unique regular nonadditive + − measures µ and µ such that I I = ∨ + − − ∧ − I(f) (C) (f 0)dµI (C) (f 0)dµI for f ∈ K. (4) If X is compact, then a c.m. functional can be represented by one Choquet integral. (5) If X is locally compact but not compact, then a r.s.d. functional is a c.m. functional. + Let I be a c.m. functional on K. We say that µ defined in Theorem 2.1 is the regular I − nonadditive measure induced by the positive part of I and µI the regular nonadditive measure induced by the negative part of I. Definition 2.5. Let I be a real valued functional on K. Y. Narukawa, T. Murofushi / J. Math. Anal. Appl. 282 (2003) 201–211 205 (1) I is said to be bounded above if there exists M>0 such that I(f) Mf for all f ∈ K. (2) I is said to be bounded below if there exists M>0 such that −Mf I(f) for all f ∈ K. (3) I is said to be bounded if I is bounded above and below. + − Proposition 2.1 [13]. Let I be a c.m. functional on K and µI and µI the regular nonad- ditive measures induced by I. + ∞ (1) I is bounded above iff µI (X) < . − ∞ (2) I is bounded below iff µI (X) < . Proposition 2.2 [12]. Let X be separable and I be a c.m. functional on K that is bounded, + − and µI and µI the regular nonadditive measure induced by I. + = − c ∈ + + = − c (1) If (C) fdµI (C) fd(µI ) for all f K ,thenµI (C) (µI ) (C) for all C ∈ C. − = + c ∈ − = + c (2) If (C) fdµI (C) fd(µI ) for all f K,thenµI (C) (µI ) (C) for all C ∈ C. + Proposition 2.2 says that if a c.m. functional I is Choquet integral with respect to µI − = + c ∈ C + c then we have µI (C) (µI ) (C) for every C .Since(µI ) is not always regular, it − = + c is not always true that µI (µI ) .Thatis,I is not always a Choquet functional. See the example in [10]. 3. Conjugate condition for compact sets We have stated in Theorem 2.1 that a c.m. functional is not always represented by one Choquet integral if the universal space X is not compact. In this section we present the con- ditions for a c.m. functional being represented by one Choquet integral. In the following, we assume that X is not compact. Definition 3.1. Let I be a c.m. functional and C ∈ C. (1) We say that I satisfies the positive conjugate condition for C if there exists a positive ∈ + real number M such that for any >0thereexistf1,f2 K1 satisfying the following condition: c 1C g1 f1 and f2 g2 1Cc with supp(f2) ⊂ supp(g2) ⊂ C imply I(−g1) − I(g2) + M < ∈ + for g1,g2 K1 . 206 Y. Narukawa, T. Murofushi / J. Math. Anal. Appl. 282 (2003) 201–211 (2) We say that I satisfies the negative conjugate condition for C if there exists a positive ∈ + real number M such that for any >0thereexistf1,f2 K1 satisfying the following condition: c 1C g1 f1 and f2 g2 1Cc with supp(f2) ⊂ supp(g2) ⊂ C imply −I(g1) + I(−g2) + M < ∈ + for g1,g2 K1 . The conjugate conditions are stronger than the boundedness. Proposition 3.1. Let I be a c.m. functional. (2) If I satisfies the positive conjugate condition for ∅,thenI is bounded above. (2) If I satisfies the negative conjugate condition for ∅,thenI is bounded below. Proof. Suppose that a c.m. functional I satisfies the positive conjugate condition for ∅. Let g1(x) = 0forallx ∈ X.Since∅⊂supp(g1) and I(g1) = 0, there exists M>0andfor + any >0 there exists f ∈ K such that supp(f ) ⊂ supp(g ) ⊂ X implies 2 1 2 2 −I(g2) + M <. + ∈ + ∈ + Then we have I(g2) h f2 ∨ h g2. It follows from monotonicity of I that I(h) I(g2) We need the next two lemmas to show that these conditions are necessary and sufficient conditions for representation by one Choquet integral. Lemma 3.1. Let A ∈ B and f ∈ K+. Suppose that A ⊂{x | f 1}; then we have + − − − µI (A) I(f) and µI (A) I( f). Proof. Let Oε ={x | f(x)>1 − ε},where0<ε<1. Then Oε is open and A ⊂ Oε. ∈ + ⊂ ∈ − Let g K1 such that supp(g) Oε.Thenx supp(g) implies f(x)/(1 ε) 1. Therefore we have g f/(1 − ε). It follows from the monotonicity and positive homo- geneity of I that I(g) (1/(1 − ε))I (f ). Therefore 1 µ(A) µ(Oε) I(f). 1 − ε Since ε is arbitrary, we have µ(A) I(f). − ✷ The part of µI (A) is much the same. Y. Narukawa, T. Murofushi / J. Math. Anal. Appl. 282 (2003) 201–211 207 Applying Lemma 3.1, we have the next lemma. The details of the proof are in Section 4. ∈ C + − Lemma 3.2. Let C ,letI be a c.m. functional and µI and µI the regular fuzzy measure induced by I. (1) I satisfies the positive conjugate condition for every C ∈ C if and only if − = + c µI (C) µI (C) for every C ∈ C. (2) I satisfies the negative conjugate condition for C if and only if + = − c µI (C) µI (C) for every C ∈ C. Suppose that a c.m. functional I satisfies the positive conjugate condition for all C ∈ C. It follows from Lemma 3.2 that − − + µ (X) = sup µ (C) | C ⊂ X = sup µ c(C) | C ⊂ X I I I = + − + c | ⊂ + sup µI (X) µI (C ) C X µI (X). Therefore we have the next corollary. Corollary 3.1. If a c.m. functional I satisfies the positive or negative conjugate condition for all C ∈ C,thenI is bounded. Applying Lemma 3.2, we have the next theorem. Theorem 3.1. Let I be a c.m. functional. (1) If I satisfies the positive conjugate condition for all C ∈ C, we have = + I(f) (C) fdµI for all f ∈ L. (2) If I satisfies the negative conjugate condition for all C ∈ C, we have =− − − I(f) (C) fdµI for all f ∈ K. Proof. We prove (1). The proof of (2) is much the same. Let f ∈ K. It follows from + − Theorem 2.1 that there exist unique regular nonadditive measures µI and µI such that = ∨ + − − ∧ − I(f) (C) (f 0)dµI (C) (f 0)dµI . 208 Y. Narukawa, T. Murofushi / J. Math. Anal. Appl. 282 (2003) 201–211 Let r be a real number such that 0 Therefore we have = ∨ + − − ∧ + + = ✷ I(f) (C) (f 0)dµI (C) (f 0)d µI (C) fdµ. If X is separable, the converse of Theorem 3.1 is valid. Theorem 3.2. Let X be separable and I be a c.m. functional on K that is bounded, and + − µI and µI the regular fuzzy measure induced by I. = + ∈ (1) If I(f) (C) fdµI for all f K,thenI satisfies the positive conjugate condition for all C ∈ C. =− − − ∈ (2) If I(f) (C) fdµI for all f K,thenI satisfies the negative conjugate con- dition for all C ∈ C. Proof. (1) Let f ∈ K+. It follows from Theorem 2 that − =− − I( f) (C) fµI . Since −f ∈ K,wehave − =− + c I( f) (C) fd µI from Definition 2.3. Therefore we have − = + c (C) fdµI fd µI for all f ∈ K+. Applying Proposition 2.2, we have − = + c µI (C) µI (C) for every C ∈ C. It follows from Lemma 3.2 that I satisfies the positive conjugate condition for every C ∈ C. (2) It is much the same as (1). ✷ Y. Narukawa, T. Murofushi / J. Math. Anal. Appl. 282 (2003) 201–211 209 4. Proof of Lemma 3.2 In this section, the proof of Lemma 3.2(1) is shown. (2) can be proved in the same way. Let >0andC ∈ C. First suppose that a c.m. functional I satisfies the positive conjugate condition for every compact set C. That is, there exists a positive real number M such that ∀>0, ∃f1, f2 c ∈ K1,1C g1 f1,andf2 g2 1Cc with supp(f2) ⊂ supp(g2) ⊂ C imply M − I(g2) − <−I(−g1) Since supp(h1) ⊂ O,wehave − − − µI (O) I( h1) (4) − from the definition of µI . Then it follows from (2) and (4) that − + − − µI (C) I( h1). (5) Since Cc is an open set, it follows from the definition of the induced regular fuzzy + ∈ + ⊂ c measure µI that there exists h2 K1 such that supp(h2) C and + c − I(h2) µI (C ) . (6) We may suppose that f2 h2 1Cc . Then applying (5) and (6), we have − + − − µI (C) M I(h2) . (7) + c ⊂ c Since we have I(h2) µI (C ) from supp(h2) C ,wehave − + − + c − µI (C) M µI (C ) . (8) ∅ = + Since I satisfies the conjugate condition for ,wehaveM µI (X). Therefore we have + c − − 2 µI (C) µI (C) (9) from (8). On the other hand, it follows from (1), (2), and (6) that − − − + − + c − + + c + I( h1) M I(h0) M µI (C ) µI (C) 2. Therefore we have − − + µI (C) µI (C) 2. − = + c Since is an arbitrary, we have µI (C) (µI ) (C). 210 Y. Narukawa, T. Murofushi / J. Math. Anal. Appl. 282 (2003) 201–211 − + + Next suppose that µ (C) = (µ )c(C).DefineM = µ (X). Then it follows from the I − I I definition of the conjugate of µI that − = − − c µI (C) M µI (C ). (10) − ⊃ Since µI is regular, there exists an open set O such that O C and − + − µI (C) ε µI (O). (11) ∈ + Using Uryson’s lemma, there exists f1 K1 such that 1C f1 1O . Then for every ∈ + g1 K1 such that 1C g1 f1,wehave − − − − µI (O) I( g1) µI (C) (12) from Lemma 3.1. It follows from the definition of the induced regular nonadditive measure + ∈ + ⊂ c µI that there exists f2 K1 such that supp(f2) C and + c − µI (C ) I(f2). 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