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A Multi-Valued Choquet Integral with Respect to a Multisubmeasure

A Multi-Valued Choquet Integral with Respect to a Multisubmeasure Jg, Kim d Kwon introduced a multi-valued Choquet integral for multifunctions with respect to real fuzzy measures d Zhg, Guo d Liu established for this kind of integral some convergence theorems. The aim of this paper is to present other type of set-valued Choquet integral, called by us the Aumn-Choquet integral, for non-negative measurable functions with respect to multisubmeasures taking values in the class of all non-empty,compact d convex sets of R+ on which we use the order relation considered by Guo d Zhg. For this kind of integral, we study some importt properties d we prove that if we add some supplementary properties to the multisubmeasure then they are also preserved by the set-valued function defined as Aumn-Choquet integral. Mathematics Subject Classification 2010: 28B20, 28C15, 49J53. Key words: real Choquet integral, set-valued Choquet integral, multisubmeasure, absolute continuity. 1. Introduction In recent years, there have appeared numerous works concerning Choquet integral which has drawn much attention due to the multiple applications in mathematics, economics, theory of control, decision making, risk alysis d my other fields. Motivated by some problems in potential theory, in 1953/1954, Choquet ([2]) had been introduced integral for non-negative measurable functions with respect to non-additive measures which has the monotonicity d continuity properties on certain sequences of sets. Later, the theory of Choquet integral was extended for the fuzzy measure, that is monotonic FLOAREA-NICOLETA SOFI-BOCA set functions. For this kind of Choquet integral was proved several importt properties such as monotonicity d comonotonical additivity. Other properties of this integral was obtained under supplementary conditions imposed to the fuzzy measures ([14,15,21]). Using Aumn type procedure ([1]), Jg et al. [10] introduced the concept of set-valued Choquet integral for a multifunction taking values in the class of all closed, nonempty sets of R+ with respect to a nonegative fuzzy measure,for which the selectors are real Choquet integrals. For this integral, in [9], [10], [11] were obtained some properties d in [8] were proved some convergence theorems for sequences of multifunctions taking values in the class of all closed, nonempty sets of R+ with respect to autocontinuous fuzzy measures,under Hausdorff convergence. In [22], using the Kuratowski convergence, Zhg, Guo d Liu studied some properties of the set-valued Choquet integral of multifunctions with respect to real fuzzy measures d proved some convergence theorems for this kind of integral. In [4], Hongxia d Jun proved that if are added supplementary properties to the fuzzy measures then they are also preserved by the multifunctions defined as the set-valued Choquet integral. In [6], [7] Jg presented some applications of Choquet integral for closed bounded interval-valued multifunctions. Using the Aumn procedure, Precupu d Satco introduced d studied in [19], a set-valued integral of real functions with respect to finite additive multimeasures for which the selectors are Gould integrals with respect to finite additive measures taking their values in a Bach space (also see [17], [18]). Based on alogous construction as in [19], we present in this paper a new type of set-valued Choquet integral for non-negative measurable functions with respect to multisubmeasures taking values in the class of all nonempty, compact d convex sets of R+ . The orgisation of the paper is as follows: in section 2 we present notations d some basic concepts d in section 3 we recall some definitions d basic properties of classical Choquet integral of non-negative functions with respect to fuzzy measures ([14], [15], [16]). In section 4 we define other set-valued Choquet integral for non-negative bounded measurable functions with respect to multisubmeasures taking values in the class of all nonempty, compact d convex sets of R+ , on which we use order relation considered by Guo d Zhg in [3]. We also present some remarkable A MULTI-VALUED CHOQUET INTEGRAL properties of this kind of integral such as homogenity, monotonicity, linearity under some special conditions. In section 5, we proved that under supplementary conditions for the multisubmeasure such as null-additivity, converse null-additivity, pseudometric generating property or Darboux property then these are preserved by the corresponding Aumn Choquet integral. 2. Terminology d notations Let S be a nonempty set, A algebra of subsets of S d X = [0, +) = R+ . We denote by: P0 (X), the family of all nonempty subsets of X, Pk (X), the family of all nonempty compact subsets of X, Pf (X), the family of all nonempty, closed subsets of X, Pb (X), the family of all nonempty, bounded subsets of X, Pbf (X), the family of all nonempty, bounded d closed subsets of X, Pkc (X), the family of nonempty, compact convex subsets of X. For every A, B P0 (X) we denote by e(A, B) = supxA inf yB |x - y| d by h : P0 (Y ) × P0 (Y ) R+ the Hausdorff pseudometric defined by h(A, B) = max{e(A, B), e(B, A)}. We observe that h becomes a metric on Pbf (X) ([5], ch.I.1, [13]). We also denote by |A| = h(A, {0}), for every A P0 (X). By the definition of the Hausdorff metric, we have immediately the following: Lemma 2.1. For every A, B Pkc (R+ ), with A = [a, b], B = [c, d] we have h(A, B) = max{|a - c| , |b - d|}. In [3], Guo d Zhg consider the following order relation on Pkc (X): Definition 2.2. Let A, B P0 (X). 1) A B if: (i) for each x0 A, there exists y0 B such that x0 (ii) for each y0 B there exists x0 A such that x0 2) A B if: y0 ; y0 . (i') for each x0 A, there exists y0 B such that x0 < y0 ; (ii') for each y0 B there exists x0 A such that x0 < y0 . FLOAREA-NICOLETA SOFI-BOCA It is easy to prove the following: Proposition 2.3. If A, B Pkc (R+ ) where A = [a, b] d B = [c, d], a, b, c, d R+ , we have A B if d only if a c d b d. Remark 2.4. 1) If A, B Pkc (R+ ), where A = [a, b] d B = [c, d], a, b, c, d R+ , then A B mes c a b d. 2) It is easy to see that if A is a singleton, A Pkc (R+ ), the relation becomes the usual order relation on R+ . Definition 2.5 ([5]). For a net (Ai )iI P0 (R+ ), where I is a filtering set, we define lim supiI Ai = {x R+ : x = limk xik , xik Aik } d lim inf iI Ai = {x R+ : x = limi xi , xi Ai }. We say that A is the Kuratowski limit (briefly K-limit) of (Ai ) if lim supiI Ai = lim inf iI Ai = A d we denote it by Ai A (or, simply Ai A). This kind of convergence is called Kuratowski convergence (briefly Kconvergence). Lemma 2.6 ([23]). If Ai = [ai , bi ] Pkc (R+ ),i I, then Ai A = [a, b] if d only if ai a, bi b. Definition 2.7 ([22]). Let ( )nN Pf (R+ ) be a sequence d A Pf (R+ ). We say that: 1) ( ) is increasing K-convergent to A if A1 d we denote it by A; 2) ( ) is decreasing K-convergent to A if A1 d we denote it by A. A2 A2 . . . d A . . . d A K K K K Remark 2.8. From [5], ch.1, we have that if (Ai )iI Pkc (R) then Kconvergence of (Ai )iI is equivalent to its convergence in Hausdorff metric. It is easy to prove that the same is true for Pkc (R+ ). Definition 2.9 ([7]). If [a, b], [c, d] Pkc (R+ ) d k R+ , then we define: [a, b] + [c, d] = [a + c, b + d], k[a, b] = [ka, kb], [a, b][c, d] = [ac, bd], [a, b] [c, d] = [a c, b d] d [a, b] [c, d] = [a c, b d], where a b = min{a, b} d a b = min{a, b}. A MULTI-VALUED CHOQUET INTEGRAL Remark 2.10. For all A, B Pkc (R+ ) we have: i) A A B d B A B; B. ii) A B A d A B Definition 2.11 ([14]). Let be f, g : S R two functions. We say that f d g are comonotonic d we denote it by f g, if f (s) < f (s ) g(s) g(s ), for every s, s S. Definition 2.12. The set function m : A R+ with m() = 0 is said to be: 1) a fuzzy measure if m(A) m(B), for every A, B A with A B; 2) a submeasure in Drewnowski sense if: (i) m(A B) m(A) + m(B), for every A, B A with A B = ; (ii) m(A) m(B), for every A, B A with A B. 3) a strict submeasure if: (i') m(A B) < m(A) + m(B), for every A, B A with A B = ; (ii') m(A) < m(B), for every A, B A with A B. 4) additive measure (shortly a measure) if m(A B) = m(A) + m(B), for every A, B A with A B = . The order in the Definition 2.12. is the usual order on R+ . Definition 2.13. A set-valued function µ:A Pkc (R+ ) with µ() = {0} is said to be: 1) a fuzzy multimeasure if µ(A) µ(B), for every A, B A with A B; 2) a monotone set-function if µ(A) A B; µ(B), for every A, B A with 3) a multisubmeasure if the following conditions are satisfied: (i) µ(A B) (ii) µ(A) µ(A) + µ(B), for every A, B A with A B = ; µ(B), for every A, B A with A B; FLOAREA-NICOLETA SOFI-BOCA 4) a strict multisubmeasure if: (i') µ(A B) (ii') µ(A) µ(A)+ µ(B), for every A, B A with A B = ; µ(B), for every A, B A with A B. 5) additive multimeasure (shortly a multimeasure) if µ(A B) = µ(A) + µ(B), for every A, B A with A B = , where we denoted by E + F = E + F , E, F A in which E mes the closure of E. Unless stated otherwise, in what follows we consider that µ : A Pkc (X) is a multisubmeasure. Definition 2.14 ([4]). 1) The set multifunction µ : A P0 (X) is called: i) null-additive if for every A, B A with µ(A) = {0} we have µ(A B) = µ(B); ii) weakly null-additive if for every A, B A with µ(A) = µ(B) = {0} we have µ(A B) = {0}; iii) converse null-additive if for every A, B A with A B such that µ(A) = µ(B) we have µ(B - A) = {0}; iv) order continuous (briefly o-continuous) if for every ( )nN A, A with µ(A) = {0} we have limn µ( ) = {0}; v) lower semi-continuous if for ( )nN A, A we have h(µ( ), µ(A)) 0. Definition 2.15. The set multifunction µ : A P0 (X) is said to have: i) the pseudometric generating property (briefly p.g.p.) if for every > 0 there exists > 0 such that for A, B A with µ(A) [0, ], µ(B) [0, ] we have µ(A B) [0, ]; ii) the Darboux property (briefly d.p.) if for every A A with µ(A) {0}, there exists p (0, 1) d a set B A such that B A d µ(B) = pµ(A). iii) the property (S) if for y ( )nN A such that limn µ( )={0} there exists a subsequence (k ) ( ) such that µ(lim inf k k ) = {0}. A MULTI-VALUED CHOQUET INTEGRAL Definition 2.16 ([4]). Let be µ : A Pkc (R+ ) a multisubmeasure d : A R+ a fuzzy measure. We say that: 1) µ is absolutely continuous of type I with respect to , denoted by µ I , if for every A A with (A) = 0 we have µ(A) = {0}; 2) µ is absolutely continuous of type II with respect to , denoted by µ II , if for every B A, A, B A for which (A - B) = 0 we have µ(A) = µ(B); 3) µ is absolutely continuous of type III with respect to , denoted by µ III , if for every ( )nN A with limn ( ) = 0 we have limn µ( ) = {0}; 4) µ is absolutely continuous of type IV with respect to , denoted by µ IV , if for every > 0 there exists > 0 such that for y A A with (A) we have µ(A) [0, ]. Remark 2.17. If µ : A Pkc (R+ ) is a null-additive multisubmeasure d is a fuzzy measure we have that µ I µ II . Indeed, let be µ I d B A, A A, with (A-B) = 0. Then from µ I we obtain that µ(A - B) = {0}. Because µ is a multisubmeasure null-additive we have that µ(A) = µ((A - B) B) = µ(B) which assures that µ II . Conversely, let be µ II d A A with (A) = 0. If we particularly take B = , B A then (A - B) = (A) = 0. Since µ II we have that µ(A) = µ(B) = {0}, so µ I . Definition 2.18. We say that a property P holds µ-almost everywhere (µ-a.e.) on S if the property P is true on S - N, with µ(N ) = {0}. Definition 2.19. Let be a set multifunction µ : A P0 (R+ ). The set function m : A X is a selector of µ if for every A A we have m(A) µ(A). We denote by Sµ the set of all selectors of the set multifunction µ. Remark 2.20. If µ : A Pkc (R+ ) is a multisubmeasure, we c associate to µ two nonnegative set functions: (1) m1 (A) = inf µ(A), A A d m2 (A) = sup µ(A), for every A A. FLOAREA-NICOLETA SOFI-BOCA According to [23], Proposition 2.1, m1 , m2 are submeasures in Drewnowski sense, such that m1 (A) µ(A) d m2 (A) µ(A), for every A A. Thus, every multisubmeasure has selectors that are fuzzy measures. Moreover, for µ : A Pkc (R+ ) we have that µ(A) = [m1 (A),m2 (A)], for every A A. In what follows we shall also use: Lemma 2.21. If A, B, C, D Pkc (R+ ) with A B, C D d B then A C. D Proof. Since A, B, C, D Pkc (R+ ) then we c write A = [a1 , a2 ], B = [b1 , b2 ], C = [c1 , c2 ], D = [d1 , d2 ], where a1 , a2 , b1 , b2 , c1 , c2 , d1 , d2 R+ . From A B d C D we have b1 a1 a2 b2 d, respectively, d1 c1 c2 d2 . Since B D we obtain b1 d1 d b2 d2 . We have the following cases: a) b1 a1 d1 c1 a2 b2 c2 d2 ; b) b1 d1 a1 a2 b2 c1 c2 d2 d respectively, c) b1 a1 a2 d1 c1 b2 c2 d2 . It is easy to see that in all cases, we obtain that a1 c1 d a2 c2 , so A C. Definition 2.22 ([12]). Let be µ, :A Pkc (R+ ) two multisubmeasures. We define, for every A A, (µ)(A) = µ(A)(A) d (µ)(A) = µ(A) (A). 3. The Choquet integral In this section we recall the definition d some basic properties of the real-valued Choquet integral. Let be f : S R+ a measurable function d m : A R+ a fuzzy measure. Definition 3.1 ([14, 15, 16]). 1) The Choquet integral of the function f with respect to the fuzzy measure m on A A is defined by (2) (C) f dm = m({s : f (s) t} A)dt, A MULTI-VALUED CHOQUET INTEGRAL where the integral on the right-hd is the Lebesgue integral. 2) The function f is called Choquet integrable on A if (C) A f dm < . If A = S then we write (C) S f dm or (C) f dm. We denote by LC (m, A) the set of all Choquet integrable functions on A A d by LC (m, S) (or shortly LC (m)) the set of all Choquet integrable functions on S. We mention the following properties for Choquet integral defined by (2): Theorem 3.2 ([14, 15, 16]). Let be m : A R+ a fuzzy measure d f, g : S R+ Choquet integrable functions on S. The following statements hold: i) if f = 0 then (C) f dm = 0; ii) for every A A, (C) A dm = m(A), where A is the characteristic function of the set A; iii) for y 0 we have (C) f dm = (C) iv) if > 0, m is a fuzzy measure d (C) f dm; f dm; f d(m) = (C) v) if m(S) < then for every R+ we have (C) (f + )dm = (C) f dm + m(S); vi) for every A, B A with A B, f is Choquet integrable on A d B d we have (C) A f dm (C) B f dm. Theorem 3.3 ([14, 15, 16]). Let be m : A R+ a fuzzy measure d f, g:S R+ Choquet integrable functions on S. If f g on S, then (C) f dm (C) gdm. Proposition 3.4 ([14]). Let be m : A R+ a fuzzy measure d f, g : S R+ Choquet integrable functions on S. Then the following statements hold: i) (C) (f g)dm (C) f dm (C) gdm d (C) (f g)dm (C) f dm (C) gdm, where (f g)(s) = max{f (s), g(s)} d (f g)(s) = min{f (s), g(s)}, for all s S; ii) for every A, B A we have (C) AB f dm (C) d (C) AB f dm (C) A f dm (C) B f dm. A f dm(C) B f dm FLOAREA-NICOLETA SOFI-BOCA Proposition 3.5 ([12, 14]). Let be m, n : A R+ two fuzzy measures such that m(S) = n(S) d f : S R+ a measurable function, Choquet integrable on S with respect to m d n. Then: i) (C) f d(m + n) = (C) f dm + (C) f dn; ii) if m n then (C) f dm (C) f dn. One of the most importt property of Choquet integral is the comonotonic additivity that is the Choquet integral is additive for functions which are comonotone. Theorem 3.6 ([16]). Let be m : A R+ a fuzzy measure d f, g : S R+ measurable functions Choquet integrable on S such that f, g are comonotonic on S. Then (C) (f + g)dm = (C) f dm + (C) gdm. 4. A multi-valued Choquet integral with respect to a multisubmeasure In [10], Jg et al. introduced a multi-valued Choquet integral for a multifunction taking values in the class of all closed, nonempty sets of R+ with respect to a nonegative fuzzy measure, for which the selectors are real Choquet integrals d in [6], [7] Jg presented some properties of the setvalued Choquet integral for closed bounded interval-valued multifunctions. In this section, we introduce d study other multivalued Choquet integral, this time for non-negative functions with respect to multisubmeasures taking values in Pkc (R+ ). In what follows we consider that µ:A Pkc (R+ ) is a multisubmeasure. From Remark 2.20 we have that µ(A) = [m1 (A), m2 (A)], for every A A,d Sµ = because contains the submeasures m1 d m2 . Let be f : C C S R+ a measurable function. We denote by Sµ (f ) or Sµ the set of all fuzzy measures m : A R+ which are selectors of the multisubmeasure µ with respect to which the function f is Choquet integrable. Using a similar procedure as in [19] for Aumn-Gould integral we give: Definition 4.1. Let be µ : A Pkc (R+ ) a multisubmeasure, f : S R+ a measurable function d A A. 1) The Aumn-Choquet integral (shortly -integral) of f on A with respect to the multisubmeasure µ is the set (3) f dµ = {(C) C f dm, m Sµ (f )}, A MULTI-VALUED CHOQUET INTEGRAL where (C) A f dm is the Choquet integral defined by (2). 2) The function f is said to be Aumn-Choquet integrable on A if A f dµ = . If A = S we denote the integral by S f dµ or simply f dµ. We also denote by LAC (µ, A) the set of all Aumn-Choquet integrable functions with respect to µ on A d by LAC (µ, S) or LAC (µ) the set of all Aumn-Choquet integrable functions with respect to µ on S. Proposition 4.2. Let be µ : A Pkc (R+ ) a multisubmeasure d f : S R+ a measurable function such that f LC (m2 ), where m2 is defined by (1). Then the function f is AC-integrable on every A A. Proof. From Remark 2.20 we have that µ(A) = [m1 (A), m2 (A)], for every A A, d since f LC (m2 ), then f LC (m2 , A). Hence, from Definition 3.1,2), we have that (C) A f dm2 < . Then, for every m Sµ (f ) we have that m(A) m2 (A), for every A A. From Theorem 3.5, ii) we obtain (C) A f dm < (C) A f dm2 < , which assures that f LAC (µ, A). Proposition 4.3. Let be µ : A Pkc (R+ ) a multisubmeasure d C f : S R+ AC-integrable function on S. Then Sµ (f )is a convex set. C Proof. Let be Sµ (f ) = {m Sµ ; (C) f dm < } = d m, n C C Sµ (f ). Let be (0, 1). Since m Sµ (f ) we have (C) f dm < , d C (f ) we have (C) f dn < . If we denote by = m + (1- )n from n Sµ then, evidently, Sµ . According to Theorems 3.2, iv) d 3.5 we obtain f d = (C) f d(m + (1 - )n) = (C) f dm + (1 - )(C) f dn < C C d, consequently = m + (1 - )n Sµ (f ) d hence Sµ (f ) is a convex set. In the following, we point out some immediate properties of the AumnChoquet integral. Proposition 4.4. Let be µ : A Pkc (R+ ) a multisubmeasure d f : S R+ a measurable function. Then: i) if f=0 on S then f is AC integrable on S d ii) if µ(A) = {0} then A f dµ f dµ = {0}; = {0}. FLOAREA-NICOLETA SOFI-BOCA C Proof. i) Since f = 0 on S then for every m Sµ (f ) d all t 0 we have from Theorem 3.2, i) that (C) f dm = 0 m({s : f (s) t})dt = 0 hence f dµ = {0}. For ii) let us observe that since µ(A) = {0} then from Remark 2.20 d µ(A) Pkc (R+ ) we have that µ(A) = [m1 (A), m2 (A)] = {0} hence C m1 (A) = m2 (A) = 0. Then, for every m Sµ , we have m(A) = 0. Hence, (C) A f dm = 0 m({s : f (s) t} A)dt = 0 d we have that A f dµ = {0}. Proposition 4.5. Let be µ : A Pkc (R+ ) a multisubmeasure d f, g : S R+ measurable functions, -integrable with respect to µ on S, such that f = g(µ - a.e). Then f dµ = gdµ. Proof. Since f, g LAC (µ) then f dµ = {(C) f dm, m C = d gdµ = {(C) gdm, m Sµ (g)} = . Let be y C (f ) such that x = (C) f dm. x0 f dµ. Then there exists m Sµ 0 Since f = g(µ-a.e.) then there exists N A with µ(N ) = {0} such that f (s) = g(s), for every s cN d at the same time we have f = g( m-a.e.). Thus x0 = (C) cN f dm = (C) cN gdm d consequently (C) gdm gdµ. Hence f dµ gdµ. alogously, we c prove that gdµ f dµ. C Sµ (f )} Proposition 4.6. Let be µ:A Pkc (R+ ) a multisubmeasure d f : S R+ AC-integrable function on S. Then f is Aumn-Choquet integrable on every E A. Proof. Since f LAC (µ) we have that C f dµ={(C) f dm, m Sµ (f )} = . Let be y y0 f dµ. C (f ) such that y = (C) f dm < . Let be y Then there exists m Sµ 0 E A. We have, for all t 0, that {s : f (s) t} E {s : f (s) t} d since m : A R+ is a fuzzy measure selector of multisubmeasure µ we obtain that m({s : f (s) t} E) m({s : f (s) t}) d hence we have that C (C) E f dm (C) f dm < . Therefore m Sµ (f, E) d consequently, E f dµ = , that is f LAC (µ, E). Theorem 4.7. Let be µ:A Pkc (R+ ) a multisubmeasure, f : S R+ AC- integrable function with respect to µ on S d > 0. The following statements hold: i) f is AC -integrable with respect to µ on S d (f )dµ = a f dµ; A MULTI-VALUED CHOQUET INTEGRAL ii) f is AC-integrable with respect to µ on S d f d(µ) = a f dµ. Proof. i) From AC-integrability of f on S we have that f dµ = C {(C) f dm, m Sµ (f )} = . Let be y0 f dµ. Then there exists C m Sµ (f ) such that y0 = (C) f dm < . From Theorem 3.2, iii) we have that (C) (f )dm = (C) f dm = C y0 . Then m Sµ (f ) d we obtain (f )dµ = {(C) (f )dm, C C m Sµ (f )} = {(C) f dm, m Sµ (f )} = {y, y f dµ} = a f dµ. Hence (f )dµ = a f dµ. In order to prove ii), let us observe that we have f d(µ) = C {(C) f dm, m Sµ (f )}. Since f is AC-integrable, for y z0 f dµ there exists n C (f ) such that z = (C) f dn < . From Theorem 3.2, iv) we obtain Sµ 0 C that (C) f d(n) = (C) f dn < . Hence n = m Sµ (f ) d (C) f dm < . Then f LAC (µ) d f d(µ) = {(C) f dm , C C C m Sµ (f )} = {(C) f d(n), n Sµ (f )} = {(C) f dn, n Sµ (f )} = a f dµ. Theorem 4.8. Let be µ, : A Pkc (R+ ) two multisubmeasures d f : S R+ a function such that f LAC (µ) LAC (). Then, for every , > 0, f is AC-integrable with respect to µ+ on S d f d(µ+) = a f dµ + f d. Proof. Let be y , > 0. From AC-integrability of f with respect to µ, respectively to , we have f dµ = d f d = . From Theorem 4.7, ii) we have that f d(µ) = a f dµ, respectively f d() = f d. Let be x1 f d(µ) d x2 f d(), arbitrarily chosen. C C Then there exists m Sµ (f ) d n S (f ) such that x1 = (C) f dm C C d x2 = (C) f dn . Since m Sµ (f ) d n S (f ) then we obtain = m, where m S C (f ) d, respectively, n = n, where n that m µ C S (f ). Using Theorem 3.2, iv) d Theorem 3.5, i) we have that m + n C Sµ+ (f ) d (C) f d(m + n ) = (C) f d(m + n) = (C) f d(m) + (C) f d(n) = x1 + x2 , hence f LAC (µ + ). Moreover, using Theorems 3.2 d 3.5 we obtain that (C) f d(m + n ) = (C) f d(m + n) = (C) f dm + (C) f dn a f dµ + FLOAREA-NICOLETA SOFI-BOCA f d, hence f d(µ + ) a f dµ + f d. Conversely, let be z a f dµ + f d. Then we c write z = z1 + z2 , where z1 a f dµ d z2 f d. Hence there C C exists m Sµ (f ) d n S (f ) such that z1 = a(C) f dm , d 1 1 1 . From properties of real Choquet integral (Theorems 3.2, z2 = (C) f dn1 iv) d 3.5, i)) we obtain that z = z1 + z2 = (C) f d(m ) + (C) 1 f d(n ) 1 (n )({s : f (s) t})dt 1 f d(m + n ) 1 1 (m )({s : f (s) t})dt + 1 (m + n )({s : f (s) t})dt = (C) 1 1 d hence z f d(µ+). Consequently we have that f d(µ +) = a f dµ + f d. Theorem 4.9. Let be µ : A Pkc (R+ ) a multisubmeasure d f, g : S R+ two AC- integrable functions with respect to µ on S. If f d g are comonotonic functions, then f +g is AC integrable d (f + g)dµ f dµ + gdµ. Proof. Since f d g are AC-integrable functions we have that C f dµ = {(C) f dm, m Sµ (f )} = d gdµ = {(C) gdn, C C C C n Sµ (g)} = . Firstly, we observe that Sµ (f + g) = Sµ (f ) Sµ (g). C C Indeed, Sµ (f +g) = {m Sµ : (C) (f +g)dm < }. But, since f d g are comonotonic functions then (C) (f +g)dm = (C) f dm+(C) gdm < . C Hence (C) f dm < d (C) gdm < d then m Sµ (f ) C C C C Sµ (g). Thus Sµ (f + g) Sµ (f ) Sµ (g). alogously, we c prove the conversely inclusion. C Now, let be y y0 (f + g)dµ. Then there exists m Sµ (f + g) such that we have y0 = (C) (f + g)dm. Since f d g are comonotonic functions then using Theorem 3.6 we obtain that y0 = (C) (f + g)dm = (C) f dm + (C) gdm = y1 + y2 , where y1 = (C) f dm f dµ d y2 = (C) gdm gdµ. Hence, we have that (f + g)dµ f dµ + gdµ. Proposition 4.10. Let be µ:A Pkc (R+ ) a multisubmeasure d f : S R+ AC- integrable function with respect to µ on S. Then S f dµ is a convex set. A MULTI-VALUED CHOQUET INTEGRAL Proof. Since f is AC integrable with respect to µ on S we have S f dµ = . Let be y1 , y2 S f dµ, arbitrarily chosen, such that C y1 < y2 . Then there exists m, n Sµ (f ) such that y1 = (C) f dm < d y2 = (C) f dn < . From Proposition 4.3, for every (0, 1) we C have that m = m + (1 - )n Sµ (f ) d from Theorems 3.2, iv) d 3.3 we obtain y = (C) = (C) = (C) f dm = (C) f d(m) + (C) f d(m + (1 - )n) f d((1 - )n) f dn = y1 + (1 - )y2 < . f dm + (1 - )(C) Then y = y1 + (1 - )y2 (y1 , y2 ) is a convex set. f dµ d hence f dµ Theorem 4.11. Let be µ:A Pkc (R+ ) a multisubmeasure d f : S R+ a measurable function. If f LC (m2 ), where m2 : A R+ is defined by (1), then for every A A, f LAC (µ, A) d (4) Proof. Let be A A fixed. From Remark 2.20, since µ(A) = [m1 (A), m2 (A)] then for every m : A R+ with m Sµ we have (5) m1 (A) m(A) m2 (A). Since f LC (m2 ), then f LC (m2 , A), d hence (C) A f dm2 < . From (5) we have that (C) A f dm < d hence A f dµ = . Obviously, C for every m Sµ (f ) we have 0 m1 ({s : f (s) t} A)dt 0 m({s : f (s) t} A)dt 0 m2 ({s : f (s) t} A)dt d from 0 m({s : f (s) t} A)dt = (C) A f dm A f dµ we obtain that A A A If m : A R+ is a strict submeasure then we proved in [21], ex. 2.15 that µ:A Pkc (R+ ) defined by µ(A) = [0, m(A)] is a strict multisubmeasure, called the strict multisubmeasure induced by m. Now, we c easily obtain the following: FLOAREA-NICOLETA SOFI-BOCA Corollary 4.12. Let be m:A R+ a strict submeasure, µ:A Pkc (R+ ) the strict multisubmeasure induced by the submeasure m d f : S R+ a measurable function. If f LC (m), then f LAC (µ) d for every A A. A f dµ [0, (C) A f dm], Proposition 4.13. Let be µ:A Pkc (R+ ) a multisubmeasure d f, g : S R+ measurable functions such that f, g LC (m2 ), where m2 is defined by (1). If f g on S then, for every A A, we have that A f dµ A gdµ. Proof. Since f LC (m2 ) from Theorem 4.11 we have that, for every A A, f LAC (µ, A) d (6) alogously we obtain from g LC (m2 ) that g LAC (µ, A) d (7) gdµ [(C) gdm1 , (C) gdm2 ]. Since f g on S we obtain from Theorem 3.3 that for every A A (C) A f dm1 (C) A gdm1 d (C) A f dm2 (C) A gdm2 . Then from Proposition 2.3 we have that [(C) A A f dm2 ] [(C) A gdm1 , (C) A gdm2 ]. Using Lemma 2.21 we obtain that A f dµ A gdµ, for every A A. Theorem 4.14. Let be µ:A Pkc (R+ ) a multisubmeasure d f : S R+ a measurable functions such that f LC (m2 ), where m2 is defined by (1). Then, for every A, B A with A B we have A f dµ B f dµ. Proof. Since f LC (m2 ) then for A, B A with A B we have that f LC (m2 , A), respectively f LC (m2 , B). From (C) A f dm1 (C) A f dm2 < , respectively from (C) B f dm1 (C) B f dm2 < , we obtain that f LC (m1 , A) d f LC (m1 , B). Then, since f LC (m1 , A) LC (m2 , A) d f LC (m1 , B) LC (m2 , B) we have, from Theorem 4.11, that f LAC (µ, A) d f LAC (µ, B) d (8) f dm2 ], 17 respectively (9) A MULTI-VALUED CHOQUET INTEGRAL A f dm1 (C) B Since AB, from Theorem 3.2, vi) we obtain that (C) d (C) A f dm2 (C) B f dm2 . From Proposition 2.3 we easily see that (10) [(C) f dm1 f dm2 ] [(C) Using (8), (9), (10) d Lemma 2.21 we obtain that for every A, B A with A B we have A f dµ B f dµ. Proposition 4.15. Let be µ, :A Pkc (R+ ) multisubmeasures such that µ(A) = [m1 (A), m2 (A)] d (A) = [n1 (A), n2 (A)], for every A A, where m1 , m2 , n1 , n2 are defined by (1). If f : S R+ is a measurable function such that f LC (m2 ) d f LC (n2 ), then the following statements hold: i) if f LC (m2 n2 ) then ), for every A A; ii) if f LC (m2 n2 ) then for every A A. A f dµ A f d f d(µ f d(µ) A f dµ A d, Proof. Firstly, let us observe that since for every A A, µ(A) = [m1 (A), m2 (A)], (A) = [n1 (A), n2 (A)] we have from Definition 2.22 that (µ )(A) = µ(A) (A) = [m1 (A) n1 (A), m2 (A) n2 (A)], respectively (µ )(A) = µ(A) (A) = [m1 (A) n1 (A), m2 (A) n2 (A)]. Let be A A arbitrarily chosen.To prove i) let us observe that since f LC (m2 ) from Theorem 4.11 we have f LAC (µ, A) d A A A alogously from f LC (n2 ) we obtain that f LAC (µ, A) d A f d [(C) A f dn1 , (C) A f dn2 ], for every A A. Then A f dµ A f d [(C) A f dm1 (C) A f dn1 , (C) A f dm2 (C) A f dn2 ]. C Let be y y0 A f d(µ ). Then there exists m Sµ (f ) such that y0 = (C) A f dm. C Since m Sµ (f ) we have m1 n1 m m2 n2 . FLOAREA-NICOLETA SOFI-BOCA C Because m1 m1 n1 , m1 Sµ (f ) we obtain (11) x0 = (C) f dm1 f dµ f d d from Theorem 3.5, iii) we have x0 = (C) A f dm1 (C) A f dm = y0 . Hence, the condition i) from Definition 2.2 is true. alogously we c prove ii) from Definition 2.2 d so we have that A f dµ A f d A f d(µ ), for every A A, d hence i) is true. alogously, we c prove ii). 5. Some properties of the multifunction defined as Aumn-Choquet integral In this section we shall prove that for µ:A Pkc (R+ ) a multisubmeasure d f : S R+ AC-integrable function if we impose supplementary conditions to µ, the same properties are also preserved by the multifunction I defined by (12) I(E) = f dµ, E A. From Theorem 4.15 we c easily obtain: Proposition 5.1. The multifunction I defined by (12) is a monotone set multifunction. Proposition 5.2. Let be µ : A Pkc (R+ ) a multisubmeasure d f : S R+ a function such that f LAC (µ). If µ is weakly null-additive, then the multifunction defined by (12) is weakly null-additive. Proof. Since µ is a weakly null-additive multisubmeasure then for every A,B A with µ(A) = µ(B) = {0} we have µ(A B) = {0}. Let be A, B A with I(A) = I(B) = {0}. Firstly, let us observe that every fuzzy measure selector m of µ is weakly null-additive. Indeed, since µ(A) = [m1 (A), m2 (A)], where m1 , m2 are defined by (1) then, for every A, B A with µ(A) = µ(B) = {0} we have [m1 (A), m2 (A)] = {0} d [m1 (B), m2 (B)] = {0}. Hence m1 (A)=0=m2 (A) d m1 (B)=m2 (B)=0 d for every m Sµ we have m(A) = m(B) = 0. From µ(AB) = [m1 (AB), m2 (AB)] = {0} we easily obtain that m(A B) = 0. Consequently, m is null-additive. A MULTI-VALUED CHOQUET INTEGRAL Since f LAC (µ) we have from Proposition 4.6 that I(A B) = C AB f dµ = {(C) AB f dm, m Sµ (f )} = . C Let be y y I(A B). Then there exists m Sµ (f ), weakly nulladditive, such that y = (C) AB f dm. Hence, from [24], th.7 we obtain that y = (C) AB f dm = 0 d so I(A B) = {0}, which assures that I(·) is weakly null-additive. Proposition 5.3. Let be µ:A Pkc (R+ ) a multisubmeasure d f : S R+ a function such that f LAC (µ). If µ is a null-additive multisubmeasure then the multifunction defined by (12) is null-additive. Proof. We shall prove that if A, B A d I(A) = {0} we obtain I(A B) = I(B). As in the proof of Proposition 5.2, we c show that if µ is null-additive then every m Sµ is a null-additive fuzzy measure. From C C Sµ Sµ we have that every m Sµ is null-additive. C Now, let be x0 I(AB) arbitrarily chosen. Then there exists m Sµ , null-additive, such that x0 = (C) AB f dm . Moreover, since m is null-additive we obtain x0 = (C) AB f dm = 0 m ({s : f (s) t} (A B))dt = 0 m ({s : f (s) t} B)dt = (C) B f dm I(B), which assures that I(A B) I(B). alogously, we c prove the converse inclusion. Indeed, if y0 C I(B) is arbitrarily chosen then there exists m Sµ , null-additive such . Since m is null-additive then we have that y = that y0 = (C) B f dm 0 (C) B f dm = 0 m ({s : f (s) t} B)dt= 0 m ({s : f (s) t} (A B)dt = (C) AB f dm , hence y0 I(A B) d the proof is completed. In what follows we shall prove that if we impose supplementary conditions for the multisubmeasure µ, such as converse null-additivity, pseudo generating property or Darboux property, then they are also preserved by the corresponding Aumn-Choquet integral. Indeed we have: Proposition 5.4. Let be µ : A Pkc (R+ ) a multisubmeasure such that every m Sµ is converse null-additive d f : S R+ AC- integrable function with respect to µ on S. Then the multifunction defined by (12) is converse null-additive. Proof. To prove that I is converse null-additive let be y A, B A with A B d I(A) = I(B). Firstly, we prove that µ is converse null-additive. FLOAREA-NICOLETA SOFI-BOCA Indeed, since every m Sµ is a converse null-additive fuzzy measure, C C from Sµ (f ) Sµ we obtain that every m Sµ (f ) is converse null-additive. Since from (1) we have µ(E) = [m1 (E), m2 (E)], for every E A, where m1 , m2 are converse null-additive, if µ(A) = µ(B) then [m1 (A), m2 (A)] = [m1 (B), m2 (B)] d thus mi (A) = mi (B), i = 1, 2. Because mi , i = 1, 2, are converse null-additive we have m1 (B - A) = 0 = m2 (B - A), d consequently µ(B-A) = {0} which assures that µ is a converse null-additive multisubmeasure. From f LAC (µ) d Proposition 4.6 we have that f LAC (µ, B - A) C d hence I(B - A) = B-A f dµ = {(C) B-A f dm, m Sµ (f )} = . From [24], th.8, since m is a converse null-additive fuzzy measure then the set function defined by (A) = (C) A f dm, A A, is converse nullC additive. Hence (C) B-A f dm = 0, for every m Sµ (f ), d we have that I(B - A) = B-A f dµ = {0} d consequently I is converse null-additive. Proposition 5.5. Let be µ:A Pkc (R+ ) a multisubmeasure d f : S R+ AC-integrable function with respect to µ on S. If µ has the pseudometric generating property then the same is true for the multifunction defined by (12). Proof. Firstly, let us observe that since µ has p.g.p then every fuzzy measure m Sµ has p.g.p. Indeed, since µ has p.g.p, then for every > 0 there exists > 0 such that for every A, B A with µ(A) [0, ] d µ(B) [0, ] we have µ(A B) [0, ]. From (1) we have that µ(A) = [m1 (A), m2 (A)] [0, ] d µ(B) = [m1 (B), m2 (B)] [0, ] d for every m Sµ we have m(A) m2 (A) d m(B) m2 (B) . From µ(A B) = [m1 (A B), m2 (A B)] [0, ] we obtain that m(A B) m2 (A B) , hence m Sµ has p.g.p. C C Since Sµ (f ) Sµ , then every m Sµ (f ) has p.g.p. Now, for every > 0 we consider > 0 from the definition of p.g.p. of µ d let be A, B A with I(A) [0, ] d I(B) [0, ]. Let be y y I(A B), C where I(A B) = AB f dµ = {(C) AB f dm, m Sµ (f )}. Then C (f ) with p.g.p. such that y = (C) there exists m Sµ AB f dm d from [20], th.5 the set function defined as Choquet integral has p.g.p. d hence we obtain that y . Hence I(A B) = AB f dµ [0, ] d I has p.g.p. Theorem 5.6. Let be µ : A Pkc (R+ ) a multisubmeasure d f : S R+ a function AC- integrable with respect to µ on S such that f (s) = 0, A MULTI-VALUED CHOQUET INTEGRAL for every s S. If every m Sµ has the Darboux property, then the multifunction defined by (12) has the Darboux property. Proof. Since f LAC (µ) we have from Proposition 4.6, that I(A) = C A f dµ = , for every A A. Let be y m Sµ (f ) with Darboux property. Since m(A) = 0 d f (s) = 0, for every s S we have I(A) = {0}. We shall prove that for y p (0, 1) there exists A f dµ B A, B A such that I(B) = pI(A). C Let be y y0 I(A). Then there exists m Sµ (f ) with Darboux = m ({s : f (s) t} A)dt. property, such that y0 = (C) A f dm 0 Since m has Darboux property then for y p (0, 1) there exists B A such that m (B) = pm (A). 1 Thus, we have that y0 = 0 m ({s : f (s) t} A)dt = 0 p m ({s : 1 f (s) t} B)dt d so y0 = p yo , where y0 I(B). Consequently, we have pI(A) I(B). It is easy to see that I(B) pI(A), hence the multifunction defined by (12) has the Darboux property. In what follows, we prove some absolutely continuity properties for the multifunction I with respect to µ. Proposition 5.7. If µ : A Pkc (R+ ) is a multisubmeasure d f : S R+ is AC- integrable function with respect to µ on S then I I µ, where I(A) = A f dµ, for every A A. Proof. Let be y A A with µ(A) = {0}. Then we have µ(A) = [m1 (A), m2 (A)] = {0} d, consequently, m1 (A) = 0 d m2 (A) = 0. C Then, for every m Sµ , we have m(A) = 0 d we obtain that C A f dµ = {(C) A f dm, m Sµ (f )} = {0}, hence I I µ. Remark 5.8. From Remark 2.17 d Proposition 5.3 we obtain that if µ is a null-additive multisubmeasure then I I µ I II µ where I is given by (12). Hence we have: Proposition 5.9. If µ : A Pkc (R+ ) is a null-additive multisubmeasure d f : S R+ is AC- integrable function with respect to µ on S then I II µ, where I(A) = A f dµ, for every A A. Denoting by the fuzzy measure defined by (A) = |µ(A)| , for every A A we have: FLOAREA-NICOLETA SOFI-BOCA Proposition 5.10. If µ:A Pkc (R+ ) is a multisubmeasure d f : S R+ is a measurable function such that f LC (), then I IV , where I is the multifunction given by I(A) = A f dµ, for every A A. Proof. According to the Definition 2.16 we shall prove that for every > 0 there exists > 0 such that for y A A with (A) < we have I(A) [0, ]. Let be A A, arbitrarily chosen. From hypothesis we obtain that f LC (m2 , A) d from Theorem 4.11, f is -integrable with respect to µ on A. From f LC (m2 ) we have (C) f dm2 = 0 m2 ({s : f (s) t})dt < d from [24], th.13, [4], th.4.4, for every > 0 there exists 0 < a < b such that (13) m2 ({s : f (s) t})dt + m2 ({s : f (s) t})dt < . 2 Let us consider = 2(b-a) d (A) < . Then (A) = |µ(A)| = m2 (A) < . To prove that I(A) [0, ] let be y y I(A) = A f dµ = C C {(C) A f dm, m Sµ (f )}. Then there exists m Sµ (f ) such that y = (C) A f dm d m(A) m2 (A). We obtain y = (C) f dm = m({s : f (s) t} A)dt (14) m2 ({s : f (s) t} A)dt. a b From 0 m2 ({s : f (s) t}A)dt = 0 m2 ({s : f (s) t}A)dt+ a m2 ({s : f (s) t} A)dt + b m2 ({s : f (s) t} A)dt d using (14) we have b that y = (C) A f dm < 2 + a m2 ({s : f (s) t} A)dt < 2 + (b - a) = 2 + 2(b-a) (b - a) = . Hence, for every y I(A) we have y < , that is I(A) [0, ] d I IV . Proposition 5.11. Let be µ : A Pkc (R+ ) a multisubmeasure d f : S R+ a measurable function such that f LC (). If the multisubmeasure µ is o-continuous then, I III where I is defined by (12). Proof. We shall prove that for y ( )n A with ( ) 0 we have |I( )| 0. Since for every A A we have that µ(A) = [m1 (A), m2 (A)] then µ( ) = [m1 ( ), m2 ( )]. Because ( ) = |µ( )| = m2 ( ) 0 A MULTI-VALUED CHOQUET INTEGRAL d since m1 ( ) m2 ( ), n N we have that m1 ( ) 0. From [24], th.3,4 we have that 1 ( ) 0 d 2 ( ) 0, where we denote by i (A) = (C) A f dmi , i = 1, 2. Since f LC () we easily obtain that f LC (m1 ) d from Theorem 4.11 we have that f LAC (µ, ) d (15) f dm2 ], for every n N. From (15) we have h(I( ), {0}) h([(C) max{(C) f dm2 } = (C) f dm2 0. Hence h(I( ), {0}) 0 d we have I III . f dm2 ], [0, 0]) = Acknowledgement. The author is indebted to Prof. Dr. A.M. Precupu for the careful reading of the paper d for the useful suggestions. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Annals of the Alexandru Ioan Cuza University - Mathematics de Gruyter

A Multi-Valued Choquet Integral with Respect to a Multisubmeasure

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Abstract

Jg, Kim d Kwon introduced a multi-valued Choquet integral for multifunctions with respect to real fuzzy measures d Zhg, Guo d Liu established for this kind of integral some convergence theorems. The aim of this paper is to present other type of set-valued Choquet integral, called by us the Aumn-Choquet integral, for non-negative measurable functions with respect to multisubmeasures taking values in the class of all non-empty,compact d convex sets of R+ on which we use the order relation considered by Guo d Zhg. For this kind of integral, we study some importt properties d we prove that if we add some supplementary properties to the multisubmeasure then they are also preserved by the set-valued function defined as Aumn-Choquet integral. Mathematics Subject Classification 2010: 28B20, 28C15, 49J53. Key words: real Choquet integral, set-valued Choquet integral, multisubmeasure, absolute continuity. 1. Introduction In recent years, there have appeared numerous works concerning Choquet integral which has drawn much attention due to the multiple applications in mathematics, economics, theory of control, decision making, risk alysis d my other fields. Motivated by some problems in potential theory, in 1953/1954, Choquet ([2]) had been introduced integral for non-negative measurable functions with respect to non-additive measures which has the monotonicity d continuity properties on certain sequences of sets. Later, the theory of Choquet integral was extended for the fuzzy measure, that is monotonic FLOAREA-NICOLETA SOFI-BOCA set functions. For this kind of Choquet integral was proved several importt properties such as monotonicity d comonotonical additivity. Other properties of this integral was obtained under supplementary conditions imposed to the fuzzy measures ([14,15,21]). Using Aumn type procedure ([1]), Jg et al. [10] introduced the concept of set-valued Choquet integral for a multifunction taking values in the class of all closed, nonempty sets of R+ with respect to a nonegative fuzzy measure,for which the selectors are real Choquet integrals. For this integral, in [9], [10], [11] were obtained some properties d in [8] were proved some convergence theorems for sequences of multifunctions taking values in the class of all closed, nonempty sets of R+ with respect to autocontinuous fuzzy measures,under Hausdorff convergence. In [22], using the Kuratowski convergence, Zhg, Guo d Liu studied some properties of the set-valued Choquet integral of multifunctions with respect to real fuzzy measures d proved some convergence theorems for this kind of integral. In [4], Hongxia d Jun proved that if are added supplementary properties to the fuzzy measures then they are also preserved by the multifunctions defined as the set-valued Choquet integral. In [6], [7] Jg presented some applications of Choquet integral for closed bounded interval-valued multifunctions. Using the Aumn procedure, Precupu d Satco introduced d studied in [19], a set-valued integral of real functions with respect to finite additive multimeasures for which the selectors are Gould integrals with respect to finite additive measures taking their values in a Bach space (also see [17], [18]). Based on alogous construction as in [19], we present in this paper a new type of set-valued Choquet integral for non-negative measurable functions with respect to multisubmeasures taking values in the class of all nonempty, compact d convex sets of R+ . The orgisation of the paper is as follows: in section 2 we present notations d some basic concepts d in section 3 we recall some definitions d basic properties of classical Choquet integral of non-negative functions with respect to fuzzy measures ([14], [15], [16]). In section 4 we define other set-valued Choquet integral for non-negative bounded measurable functions with respect to multisubmeasures taking values in the class of all nonempty, compact d convex sets of R+ , on which we use order relation considered by Guo d Zhg in [3]. We also present some remarkable A MULTI-VALUED CHOQUET INTEGRAL properties of this kind of integral such as homogenity, monotonicity, linearity under some special conditions. In section 5, we proved that under supplementary conditions for the multisubmeasure such as null-additivity, converse null-additivity, pseudometric generating property or Darboux property then these are preserved by the corresponding Aumn Choquet integral. 2. Terminology d notations Let S be a nonempty set, A algebra of subsets of S d X = [0, +) = R+ . We denote by: P0 (X), the family of all nonempty subsets of X, Pk (X), the family of all nonempty compact subsets of X, Pf (X), the family of all nonempty, closed subsets of X, Pb (X), the family of all nonempty, bounded subsets of X, Pbf (X), the family of all nonempty, bounded d closed subsets of X, Pkc (X), the family of nonempty, compact convex subsets of X. For every A, B P0 (X) we denote by e(A, B) = supxA inf yB |x - y| d by h : P0 (Y ) × P0 (Y ) R+ the Hausdorff pseudometric defined by h(A, B) = max{e(A, B), e(B, A)}. We observe that h becomes a metric on Pbf (X) ([5], ch.I.1, [13]). We also denote by |A| = h(A, {0}), for every A P0 (X). By the definition of the Hausdorff metric, we have immediately the following: Lemma 2.1. For every A, B Pkc (R+ ), with A = [a, b], B = [c, d] we have h(A, B) = max{|a - c| , |b - d|}. In [3], Guo d Zhg consider the following order relation on Pkc (X): Definition 2.2. Let A, B P0 (X). 1) A B if: (i) for each x0 A, there exists y0 B such that x0 (ii) for each y0 B there exists x0 A such that x0 2) A B if: y0 ; y0 . (i') for each x0 A, there exists y0 B such that x0 < y0 ; (ii') for each y0 B there exists x0 A such that x0 < y0 . FLOAREA-NICOLETA SOFI-BOCA It is easy to prove the following: Proposition 2.3. If A, B Pkc (R+ ) where A = [a, b] d B = [c, d], a, b, c, d R+ , we have A B if d only if a c d b d. Remark 2.4. 1) If A, B Pkc (R+ ), where A = [a, b] d B = [c, d], a, b, c, d R+ , then A B mes c a b d. 2) It is easy to see that if A is a singleton, A Pkc (R+ ), the relation becomes the usual order relation on R+ . Definition 2.5 ([5]). For a net (Ai )iI P0 (R+ ), where I is a filtering set, we define lim supiI Ai = {x R+ : x = limk xik , xik Aik } d lim inf iI Ai = {x R+ : x = limi xi , xi Ai }. We say that A is the Kuratowski limit (briefly K-limit) of (Ai ) if lim supiI Ai = lim inf iI Ai = A d we denote it by Ai A (or, simply Ai A). This kind of convergence is called Kuratowski convergence (briefly Kconvergence). Lemma 2.6 ([23]). If Ai = [ai , bi ] Pkc (R+ ),i I, then Ai A = [a, b] if d only if ai a, bi b. Definition 2.7 ([22]). Let ( )nN Pf (R+ ) be a sequence d A Pf (R+ ). We say that: 1) ( ) is increasing K-convergent to A if A1 d we denote it by A; 2) ( ) is decreasing K-convergent to A if A1 d we denote it by A. A2 A2 . . . d A . . . d A K K K K Remark 2.8. From [5], ch.1, we have that if (Ai )iI Pkc (R) then Kconvergence of (Ai )iI is equivalent to its convergence in Hausdorff metric. It is easy to prove that the same is true for Pkc (R+ ). Definition 2.9 ([7]). If [a, b], [c, d] Pkc (R+ ) d k R+ , then we define: [a, b] + [c, d] = [a + c, b + d], k[a, b] = [ka, kb], [a, b][c, d] = [ac, bd], [a, b] [c, d] = [a c, b d] d [a, b] [c, d] = [a c, b d], where a b = min{a, b} d a b = min{a, b}. A MULTI-VALUED CHOQUET INTEGRAL Remark 2.10. For all A, B Pkc (R+ ) we have: i) A A B d B A B; B. ii) A B A d A B Definition 2.11 ([14]). Let be f, g : S R two functions. We say that f d g are comonotonic d we denote it by f g, if f (s) < f (s ) g(s) g(s ), for every s, s S. Definition 2.12. The set function m : A R+ with m() = 0 is said to be: 1) a fuzzy measure if m(A) m(B), for every A, B A with A B; 2) a submeasure in Drewnowski sense if: (i) m(A B) m(A) + m(B), for every A, B A with A B = ; (ii) m(A) m(B), for every A, B A with A B. 3) a strict submeasure if: (i') m(A B) < m(A) + m(B), for every A, B A with A B = ; (ii') m(A) < m(B), for every A, B A with A B. 4) additive measure (shortly a measure) if m(A B) = m(A) + m(B), for every A, B A with A B = . The order in the Definition 2.12. is the usual order on R+ . Definition 2.13. A set-valued function µ:A Pkc (R+ ) with µ() = {0} is said to be: 1) a fuzzy multimeasure if µ(A) µ(B), for every A, B A with A B; 2) a monotone set-function if µ(A) A B; µ(B), for every A, B A with 3) a multisubmeasure if the following conditions are satisfied: (i) µ(A B) (ii) µ(A) µ(A) + µ(B), for every A, B A with A B = ; µ(B), for every A, B A with A B; FLOAREA-NICOLETA SOFI-BOCA 4) a strict multisubmeasure if: (i') µ(A B) (ii') µ(A) µ(A)+ µ(B), for every A, B A with A B = ; µ(B), for every A, B A with A B. 5) additive multimeasure (shortly a multimeasure) if µ(A B) = µ(A) + µ(B), for every A, B A with A B = , where we denoted by E + F = E + F , E, F A in which E mes the closure of E. Unless stated otherwise, in what follows we consider that µ : A Pkc (X) is a multisubmeasure. Definition 2.14 ([4]). 1) The set multifunction µ : A P0 (X) is called: i) null-additive if for every A, B A with µ(A) = {0} we have µ(A B) = µ(B); ii) weakly null-additive if for every A, B A with µ(A) = µ(B) = {0} we have µ(A B) = {0}; iii) converse null-additive if for every A, B A with A B such that µ(A) = µ(B) we have µ(B - A) = {0}; iv) order continuous (briefly o-continuous) if for every ( )nN A, A with µ(A) = {0} we have limn µ( ) = {0}; v) lower semi-continuous if for ( )nN A, A we have h(µ( ), µ(A)) 0. Definition 2.15. The set multifunction µ : A P0 (X) is said to have: i) the pseudometric generating property (briefly p.g.p.) if for every > 0 there exists > 0 such that for A, B A with µ(A) [0, ], µ(B) [0, ] we have µ(A B) [0, ]; ii) the Darboux property (briefly d.p.) if for every A A with µ(A) {0}, there exists p (0, 1) d a set B A such that B A d µ(B) = pµ(A). iii) the property (S) if for y ( )nN A such that limn µ( )={0} there exists a subsequence (k ) ( ) such that µ(lim inf k k ) = {0}. A MULTI-VALUED CHOQUET INTEGRAL Definition 2.16 ([4]). Let be µ : A Pkc (R+ ) a multisubmeasure d : A R+ a fuzzy measure. We say that: 1) µ is absolutely continuous of type I with respect to , denoted by µ I , if for every A A with (A) = 0 we have µ(A) = {0}; 2) µ is absolutely continuous of type II with respect to , denoted by µ II , if for every B A, A, B A for which (A - B) = 0 we have µ(A) = µ(B); 3) µ is absolutely continuous of type III with respect to , denoted by µ III , if for every ( )nN A with limn ( ) = 0 we have limn µ( ) = {0}; 4) µ is absolutely continuous of type IV with respect to , denoted by µ IV , if for every > 0 there exists > 0 such that for y A A with (A) we have µ(A) [0, ]. Remark 2.17. If µ : A Pkc (R+ ) is a null-additive multisubmeasure d is a fuzzy measure we have that µ I µ II . Indeed, let be µ I d B A, A A, with (A-B) = 0. Then from µ I we obtain that µ(A - B) = {0}. Because µ is a multisubmeasure null-additive we have that µ(A) = µ((A - B) B) = µ(B) which assures that µ II . Conversely, let be µ II d A A with (A) = 0. If we particularly take B = , B A then (A - B) = (A) = 0. Since µ II we have that µ(A) = µ(B) = {0}, so µ I . Definition 2.18. We say that a property P holds µ-almost everywhere (µ-a.e.) on S if the property P is true on S - N, with µ(N ) = {0}. Definition 2.19. Let be a set multifunction µ : A P0 (R+ ). The set function m : A X is a selector of µ if for every A A we have m(A) µ(A). We denote by Sµ the set of all selectors of the set multifunction µ. Remark 2.20. If µ : A Pkc (R+ ) is a multisubmeasure, we c associate to µ two nonnegative set functions: (1) m1 (A) = inf µ(A), A A d m2 (A) = sup µ(A), for every A A. FLOAREA-NICOLETA SOFI-BOCA According to [23], Proposition 2.1, m1 , m2 are submeasures in Drewnowski sense, such that m1 (A) µ(A) d m2 (A) µ(A), for every A A. Thus, every multisubmeasure has selectors that are fuzzy measures. Moreover, for µ : A Pkc (R+ ) we have that µ(A) = [m1 (A),m2 (A)], for every A A. In what follows we shall also use: Lemma 2.21. If A, B, C, D Pkc (R+ ) with A B, C D d B then A C. D Proof. Since A, B, C, D Pkc (R+ ) then we c write A = [a1 , a2 ], B = [b1 , b2 ], C = [c1 , c2 ], D = [d1 , d2 ], where a1 , a2 , b1 , b2 , c1 , c2 , d1 , d2 R+ . From A B d C D we have b1 a1 a2 b2 d, respectively, d1 c1 c2 d2 . Since B D we obtain b1 d1 d b2 d2 . We have the following cases: a) b1 a1 d1 c1 a2 b2 c2 d2 ; b) b1 d1 a1 a2 b2 c1 c2 d2 d respectively, c) b1 a1 a2 d1 c1 b2 c2 d2 . It is easy to see that in all cases, we obtain that a1 c1 d a2 c2 , so A C. Definition 2.22 ([12]). Let be µ, :A Pkc (R+ ) two multisubmeasures. We define, for every A A, (µ)(A) = µ(A)(A) d (µ)(A) = µ(A) (A). 3. The Choquet integral In this section we recall the definition d some basic properties of the real-valued Choquet integral. Let be f : S R+ a measurable function d m : A R+ a fuzzy measure. Definition 3.1 ([14, 15, 16]). 1) The Choquet integral of the function f with respect to the fuzzy measure m on A A is defined by (2) (C) f dm = m({s : f (s) t} A)dt, A MULTI-VALUED CHOQUET INTEGRAL where the integral on the right-hd is the Lebesgue integral. 2) The function f is called Choquet integrable on A if (C) A f dm < . If A = S then we write (C) S f dm or (C) f dm. We denote by LC (m, A) the set of all Choquet integrable functions on A A d by LC (m, S) (or shortly LC (m)) the set of all Choquet integrable functions on S. We mention the following properties for Choquet integral defined by (2): Theorem 3.2 ([14, 15, 16]). Let be m : A R+ a fuzzy measure d f, g : S R+ Choquet integrable functions on S. The following statements hold: i) if f = 0 then (C) f dm = 0; ii) for every A A, (C) A dm = m(A), where A is the characteristic function of the set A; iii) for y 0 we have (C) f dm = (C) iv) if > 0, m is a fuzzy measure d (C) f dm; f dm; f d(m) = (C) v) if m(S) < then for every R+ we have (C) (f + )dm = (C) f dm + m(S); vi) for every A, B A with A B, f is Choquet integrable on A d B d we have (C) A f dm (C) B f dm. Theorem 3.3 ([14, 15, 16]). Let be m : A R+ a fuzzy measure d f, g:S R+ Choquet integrable functions on S. If f g on S, then (C) f dm (C) gdm. Proposition 3.4 ([14]). Let be m : A R+ a fuzzy measure d f, g : S R+ Choquet integrable functions on S. Then the following statements hold: i) (C) (f g)dm (C) f dm (C) gdm d (C) (f g)dm (C) f dm (C) gdm, where (f g)(s) = max{f (s), g(s)} d (f g)(s) = min{f (s), g(s)}, for all s S; ii) for every A, B A we have (C) AB f dm (C) d (C) AB f dm (C) A f dm (C) B f dm. A f dm(C) B f dm FLOAREA-NICOLETA SOFI-BOCA Proposition 3.5 ([12, 14]). Let be m, n : A R+ two fuzzy measures such that m(S) = n(S) d f : S R+ a measurable function, Choquet integrable on S with respect to m d n. Then: i) (C) f d(m + n) = (C) f dm + (C) f dn; ii) if m n then (C) f dm (C) f dn. One of the most importt property of Choquet integral is the comonotonic additivity that is the Choquet integral is additive for functions which are comonotone. Theorem 3.6 ([16]). Let be m : A R+ a fuzzy measure d f, g : S R+ measurable functions Choquet integrable on S such that f, g are comonotonic on S. Then (C) (f + g)dm = (C) f dm + (C) gdm. 4. A multi-valued Choquet integral with respect to a multisubmeasure In [10], Jg et al. introduced a multi-valued Choquet integral for a multifunction taking values in the class of all closed, nonempty sets of R+ with respect to a nonegative fuzzy measure, for which the selectors are real Choquet integrals d in [6], [7] Jg presented some properties of the setvalued Choquet integral for closed bounded interval-valued multifunctions. In this section, we introduce d study other multivalued Choquet integral, this time for non-negative functions with respect to multisubmeasures taking values in Pkc (R+ ). In what follows we consider that µ:A Pkc (R+ ) is a multisubmeasure. From Remark 2.20 we have that µ(A) = [m1 (A), m2 (A)], for every A A,d Sµ = because contains the submeasures m1 d m2 . Let be f : C C S R+ a measurable function. We denote by Sµ (f ) or Sµ the set of all fuzzy measures m : A R+ which are selectors of the multisubmeasure µ with respect to which the function f is Choquet integrable. Using a similar procedure as in [19] for Aumn-Gould integral we give: Definition 4.1. Let be µ : A Pkc (R+ ) a multisubmeasure, f : S R+ a measurable function d A A. 1) The Aumn-Choquet integral (shortly -integral) of f on A with respect to the multisubmeasure µ is the set (3) f dµ = {(C) C f dm, m Sµ (f )}, A MULTI-VALUED CHOQUET INTEGRAL where (C) A f dm is the Choquet integral defined by (2). 2) The function f is said to be Aumn-Choquet integrable on A if A f dµ = . If A = S we denote the integral by S f dµ or simply f dµ. We also denote by LAC (µ, A) the set of all Aumn-Choquet integrable functions with respect to µ on A d by LAC (µ, S) or LAC (µ) the set of all Aumn-Choquet integrable functions with respect to µ on S. Proposition 4.2. Let be µ : A Pkc (R+ ) a multisubmeasure d f : S R+ a measurable function such that f LC (m2 ), where m2 is defined by (1). Then the function f is AC-integrable on every A A. Proof. From Remark 2.20 we have that µ(A) = [m1 (A), m2 (A)], for every A A, d since f LC (m2 ), then f LC (m2 , A). Hence, from Definition 3.1,2), we have that (C) A f dm2 < . Then, for every m Sµ (f ) we have that m(A) m2 (A), for every A A. From Theorem 3.5, ii) we obtain (C) A f dm < (C) A f dm2 < , which assures that f LAC (µ, A). Proposition 4.3. Let be µ : A Pkc (R+ ) a multisubmeasure d C f : S R+ AC-integrable function on S. Then Sµ (f )is a convex set. C Proof. Let be Sµ (f ) = {m Sµ ; (C) f dm < } = d m, n C C Sµ (f ). Let be (0, 1). Since m Sµ (f ) we have (C) f dm < , d C (f ) we have (C) f dn < . If we denote by = m + (1- )n from n Sµ then, evidently, Sµ . According to Theorems 3.2, iv) d 3.5 we obtain f d = (C) f d(m + (1 - )n) = (C) f dm + (1 - )(C) f dn < C C d, consequently = m + (1 - )n Sµ (f ) d hence Sµ (f ) is a convex set. In the following, we point out some immediate properties of the AumnChoquet integral. Proposition 4.4. Let be µ : A Pkc (R+ ) a multisubmeasure d f : S R+ a measurable function. Then: i) if f=0 on S then f is AC integrable on S d ii) if µ(A) = {0} then A f dµ f dµ = {0}; = {0}. FLOAREA-NICOLETA SOFI-BOCA C Proof. i) Since f = 0 on S then for every m Sµ (f ) d all t 0 we have from Theorem 3.2, i) that (C) f dm = 0 m({s : f (s) t})dt = 0 hence f dµ = {0}. For ii) let us observe that since µ(A) = {0} then from Remark 2.20 d µ(A) Pkc (R+ ) we have that µ(A) = [m1 (A), m2 (A)] = {0} hence C m1 (A) = m2 (A) = 0. Then, for every m Sµ , we have m(A) = 0. Hence, (C) A f dm = 0 m({s : f (s) t} A)dt = 0 d we have that A f dµ = {0}. Proposition 4.5. Let be µ : A Pkc (R+ ) a multisubmeasure d f, g : S R+ measurable functions, -integrable with respect to µ on S, such that f = g(µ - a.e). Then f dµ = gdµ. Proof. Since f, g LAC (µ) then f dµ = {(C) f dm, m C = d gdµ = {(C) gdm, m Sµ (g)} = . Let be y C (f ) such that x = (C) f dm. x0 f dµ. Then there exists m Sµ 0 Since f = g(µ-a.e.) then there exists N A with µ(N ) = {0} such that f (s) = g(s), for every s cN d at the same time we have f = g( m-a.e.). Thus x0 = (C) cN f dm = (C) cN gdm d consequently (C) gdm gdµ. Hence f dµ gdµ. alogously, we c prove that gdµ f dµ. C Sµ (f )} Proposition 4.6. Let be µ:A Pkc (R+ ) a multisubmeasure d f : S R+ AC-integrable function on S. Then f is Aumn-Choquet integrable on every E A. Proof. Since f LAC (µ) we have that C f dµ={(C) f dm, m Sµ (f )} = . Let be y y0 f dµ. C (f ) such that y = (C) f dm < . Let be y Then there exists m Sµ 0 E A. We have, for all t 0, that {s : f (s) t} E {s : f (s) t} d since m : A R+ is a fuzzy measure selector of multisubmeasure µ we obtain that m({s : f (s) t} E) m({s : f (s) t}) d hence we have that C (C) E f dm (C) f dm < . Therefore m Sµ (f, E) d consequently, E f dµ = , that is f LAC (µ, E). Theorem 4.7. Let be µ:A Pkc (R+ ) a multisubmeasure, f : S R+ AC- integrable function with respect to µ on S d > 0. The following statements hold: i) f is AC -integrable with respect to µ on S d (f )dµ = a f dµ; A MULTI-VALUED CHOQUET INTEGRAL ii) f is AC-integrable with respect to µ on S d f d(µ) = a f dµ. Proof. i) From AC-integrability of f on S we have that f dµ = C {(C) f dm, m Sµ (f )} = . Let be y0 f dµ. Then there exists C m Sµ (f ) such that y0 = (C) f dm < . From Theorem 3.2, iii) we have that (C) (f )dm = (C) f dm = C y0 . Then m Sµ (f ) d we obtain (f )dµ = {(C) (f )dm, C C m Sµ (f )} = {(C) f dm, m Sµ (f )} = {y, y f dµ} = a f dµ. Hence (f )dµ = a f dµ. In order to prove ii), let us observe that we have f d(µ) = C {(C) f dm, m Sµ (f )}. Since f is AC-integrable, for y z0 f dµ there exists n C (f ) such that z = (C) f dn < . From Theorem 3.2, iv) we obtain Sµ 0 C that (C) f d(n) = (C) f dn < . Hence n = m Sµ (f ) d (C) f dm < . Then f LAC (µ) d f d(µ) = {(C) f dm , C C C m Sµ (f )} = {(C) f d(n), n Sµ (f )} = {(C) f dn, n Sµ (f )} = a f dµ. Theorem 4.8. Let be µ, : A Pkc (R+ ) two multisubmeasures d f : S R+ a function such that f LAC (µ) LAC (). Then, for every , > 0, f is AC-integrable with respect to µ+ on S d f d(µ+) = a f dµ + f d. Proof. Let be y , > 0. From AC-integrability of f with respect to µ, respectively to , we have f dµ = d f d = . From Theorem 4.7, ii) we have that f d(µ) = a f dµ, respectively f d() = f d. Let be x1 f d(µ) d x2 f d(), arbitrarily chosen. C C Then there exists m Sµ (f ) d n S (f ) such that x1 = (C) f dm C C d x2 = (C) f dn . Since m Sµ (f ) d n S (f ) then we obtain = m, where m S C (f ) d, respectively, n = n, where n that m µ C S (f ). Using Theorem 3.2, iv) d Theorem 3.5, i) we have that m + n C Sµ+ (f ) d (C) f d(m + n ) = (C) f d(m + n) = (C) f d(m) + (C) f d(n) = x1 + x2 , hence f LAC (µ + ). Moreover, using Theorems 3.2 d 3.5 we obtain that (C) f d(m + n ) = (C) f d(m + n) = (C) f dm + (C) f dn a f dµ + FLOAREA-NICOLETA SOFI-BOCA f d, hence f d(µ + ) a f dµ + f d. Conversely, let be z a f dµ + f d. Then we c write z = z1 + z2 , where z1 a f dµ d z2 f d. Hence there C C exists m Sµ (f ) d n S (f ) such that z1 = a(C) f dm , d 1 1 1 . From properties of real Choquet integral (Theorems 3.2, z2 = (C) f dn1 iv) d 3.5, i)) we obtain that z = z1 + z2 = (C) f d(m ) + (C) 1 f d(n ) 1 (n )({s : f (s) t})dt 1 f d(m + n ) 1 1 (m )({s : f (s) t})dt + 1 (m + n )({s : f (s) t})dt = (C) 1 1 d hence z f d(µ+). Consequently we have that f d(µ +) = a f dµ + f d. Theorem 4.9. Let be µ : A Pkc (R+ ) a multisubmeasure d f, g : S R+ two AC- integrable functions with respect to µ on S. If f d g are comonotonic functions, then f +g is AC integrable d (f + g)dµ f dµ + gdµ. Proof. Since f d g are AC-integrable functions we have that C f dµ = {(C) f dm, m Sµ (f )} = d gdµ = {(C) gdn, C C C C n Sµ (g)} = . Firstly, we observe that Sµ (f + g) = Sµ (f ) Sµ (g). C C Indeed, Sµ (f +g) = {m Sµ : (C) (f +g)dm < }. But, since f d g are comonotonic functions then (C) (f +g)dm = (C) f dm+(C) gdm < . C Hence (C) f dm < d (C) gdm < d then m Sµ (f ) C C C C Sµ (g). Thus Sµ (f + g) Sµ (f ) Sµ (g). alogously, we c prove the conversely inclusion. C Now, let be y y0 (f + g)dµ. Then there exists m Sµ (f + g) such that we have y0 = (C) (f + g)dm. Since f d g are comonotonic functions then using Theorem 3.6 we obtain that y0 = (C) (f + g)dm = (C) f dm + (C) gdm = y1 + y2 , where y1 = (C) f dm f dµ d y2 = (C) gdm gdµ. Hence, we have that (f + g)dµ f dµ + gdµ. Proposition 4.10. Let be µ:A Pkc (R+ ) a multisubmeasure d f : S R+ AC- integrable function with respect to µ on S. Then S f dµ is a convex set. A MULTI-VALUED CHOQUET INTEGRAL Proof. Since f is AC integrable with respect to µ on S we have S f dµ = . Let be y1 , y2 S f dµ, arbitrarily chosen, such that C y1 < y2 . Then there exists m, n Sµ (f ) such that y1 = (C) f dm < d y2 = (C) f dn < . From Proposition 4.3, for every (0, 1) we C have that m = m + (1 - )n Sµ (f ) d from Theorems 3.2, iv) d 3.3 we obtain y = (C) = (C) = (C) f dm = (C) f d(m) + (C) f d(m + (1 - )n) f d((1 - )n) f dn = y1 + (1 - )y2 < . f dm + (1 - )(C) Then y = y1 + (1 - )y2 (y1 , y2 ) is a convex set. f dµ d hence f dµ Theorem 4.11. Let be µ:A Pkc (R+ ) a multisubmeasure d f : S R+ a measurable function. If f LC (m2 ), where m2 : A R+ is defined by (1), then for every A A, f LAC (µ, A) d (4) Proof. Let be A A fixed. From Remark 2.20, since µ(A) = [m1 (A), m2 (A)] then for every m : A R+ with m Sµ we have (5) m1 (A) m(A) m2 (A). Since f LC (m2 ), then f LC (m2 , A), d hence (C) A f dm2 < . From (5) we have that (C) A f dm < d hence A f dµ = . Obviously, C for every m Sµ (f ) we have 0 m1 ({s : f (s) t} A)dt 0 m({s : f (s) t} A)dt 0 m2 ({s : f (s) t} A)dt d from 0 m({s : f (s) t} A)dt = (C) A f dm A f dµ we obtain that A A A If m : A R+ is a strict submeasure then we proved in [21], ex. 2.15 that µ:A Pkc (R+ ) defined by µ(A) = [0, m(A)] is a strict multisubmeasure, called the strict multisubmeasure induced by m. Now, we c easily obtain the following: FLOAREA-NICOLETA SOFI-BOCA Corollary 4.12. Let be m:A R+ a strict submeasure, µ:A Pkc (R+ ) the strict multisubmeasure induced by the submeasure m d f : S R+ a measurable function. If f LC (m), then f LAC (µ) d for every A A. A f dµ [0, (C) A f dm], Proposition 4.13. Let be µ:A Pkc (R+ ) a multisubmeasure d f, g : S R+ measurable functions such that f, g LC (m2 ), where m2 is defined by (1). If f g on S then, for every A A, we have that A f dµ A gdµ. Proof. Since f LC (m2 ) from Theorem 4.11 we have that, for every A A, f LAC (µ, A) d (6) alogously we obtain from g LC (m2 ) that g LAC (µ, A) d (7) gdµ [(C) gdm1 , (C) gdm2 ]. Since f g on S we obtain from Theorem 3.3 that for every A A (C) A f dm1 (C) A gdm1 d (C) A f dm2 (C) A gdm2 . Then from Proposition 2.3 we have that [(C) A A f dm2 ] [(C) A gdm1 , (C) A gdm2 ]. Using Lemma 2.21 we obtain that A f dµ A gdµ, for every A A. Theorem 4.14. Let be µ:A Pkc (R+ ) a multisubmeasure d f : S R+ a measurable functions such that f LC (m2 ), where m2 is defined by (1). Then, for every A, B A with A B we have A f dµ B f dµ. Proof. Since f LC (m2 ) then for A, B A with A B we have that f LC (m2 , A), respectively f LC (m2 , B). From (C) A f dm1 (C) A f dm2 < , respectively from (C) B f dm1 (C) B f dm2 < , we obtain that f LC (m1 , A) d f LC (m1 , B). Then, since f LC (m1 , A) LC (m2 , A) d f LC (m1 , B) LC (m2 , B) we have, from Theorem 4.11, that f LAC (µ, A) d f LAC (µ, B) d (8) f dm2 ], 17 respectively (9) A MULTI-VALUED CHOQUET INTEGRAL A f dm1 (C) B Since AB, from Theorem 3.2, vi) we obtain that (C) d (C) A f dm2 (C) B f dm2 . From Proposition 2.3 we easily see that (10) [(C) f dm1 f dm2 ] [(C) Using (8), (9), (10) d Lemma 2.21 we obtain that for every A, B A with A B we have A f dµ B f dµ. Proposition 4.15. Let be µ, :A Pkc (R+ ) multisubmeasures such that µ(A) = [m1 (A), m2 (A)] d (A) = [n1 (A), n2 (A)], for every A A, where m1 , m2 , n1 , n2 are defined by (1). If f : S R+ is a measurable function such that f LC (m2 ) d f LC (n2 ), then the following statements hold: i) if f LC (m2 n2 ) then ), for every A A; ii) if f LC (m2 n2 ) then for every A A. A f dµ A f d f d(µ f d(µ) A f dµ A d, Proof. Firstly, let us observe that since for every A A, µ(A) = [m1 (A), m2 (A)], (A) = [n1 (A), n2 (A)] we have from Definition 2.22 that (µ )(A) = µ(A) (A) = [m1 (A) n1 (A), m2 (A) n2 (A)], respectively (µ )(A) = µ(A) (A) = [m1 (A) n1 (A), m2 (A) n2 (A)]. Let be A A arbitrarily chosen.To prove i) let us observe that since f LC (m2 ) from Theorem 4.11 we have f LAC (µ, A) d A A A alogously from f LC (n2 ) we obtain that f LAC (µ, A) d A f d [(C) A f dn1 , (C) A f dn2 ], for every A A. Then A f dµ A f d [(C) A f dm1 (C) A f dn1 , (C) A f dm2 (C) A f dn2 ]. C Let be y y0 A f d(µ ). Then there exists m Sµ (f ) such that y0 = (C) A f dm. C Since m Sµ (f ) we have m1 n1 m m2 n2 . FLOAREA-NICOLETA SOFI-BOCA C Because m1 m1 n1 , m1 Sµ (f ) we obtain (11) x0 = (C) f dm1 f dµ f d d from Theorem 3.5, iii) we have x0 = (C) A f dm1 (C) A f dm = y0 . Hence, the condition i) from Definition 2.2 is true. alogously we c prove ii) from Definition 2.2 d so we have that A f dµ A f d A f d(µ ), for every A A, d hence i) is true. alogously, we c prove ii). 5. Some properties of the multifunction defined as Aumn-Choquet integral In this section we shall prove that for µ:A Pkc (R+ ) a multisubmeasure d f : S R+ AC-integrable function if we impose supplementary conditions to µ, the same properties are also preserved by the multifunction I defined by (12) I(E) = f dµ, E A. From Theorem 4.15 we c easily obtain: Proposition 5.1. The multifunction I defined by (12) is a monotone set multifunction. Proposition 5.2. Let be µ : A Pkc (R+ ) a multisubmeasure d f : S R+ a function such that f LAC (µ). If µ is weakly null-additive, then the multifunction defined by (12) is weakly null-additive. Proof. Since µ is a weakly null-additive multisubmeasure then for every A,B A with µ(A) = µ(B) = {0} we have µ(A B) = {0}. Let be A, B A with I(A) = I(B) = {0}. Firstly, let us observe that every fuzzy measure selector m of µ is weakly null-additive. Indeed, since µ(A) = [m1 (A), m2 (A)], where m1 , m2 are defined by (1) then, for every A, B A with µ(A) = µ(B) = {0} we have [m1 (A), m2 (A)] = {0} d [m1 (B), m2 (B)] = {0}. Hence m1 (A)=0=m2 (A) d m1 (B)=m2 (B)=0 d for every m Sµ we have m(A) = m(B) = 0. From µ(AB) = [m1 (AB), m2 (AB)] = {0} we easily obtain that m(A B) = 0. Consequently, m is null-additive. A MULTI-VALUED CHOQUET INTEGRAL Since f LAC (µ) we have from Proposition 4.6 that I(A B) = C AB f dµ = {(C) AB f dm, m Sµ (f )} = . C Let be y y I(A B). Then there exists m Sµ (f ), weakly nulladditive, such that y = (C) AB f dm. Hence, from [24], th.7 we obtain that y = (C) AB f dm = 0 d so I(A B) = {0}, which assures that I(·) is weakly null-additive. Proposition 5.3. Let be µ:A Pkc (R+ ) a multisubmeasure d f : S R+ a function such that f LAC (µ). If µ is a null-additive multisubmeasure then the multifunction defined by (12) is null-additive. Proof. We shall prove that if A, B A d I(A) = {0} we obtain I(A B) = I(B). As in the proof of Proposition 5.2, we c show that if µ is null-additive then every m Sµ is a null-additive fuzzy measure. From C C Sµ Sµ we have that every m Sµ is null-additive. C Now, let be x0 I(AB) arbitrarily chosen. Then there exists m Sµ , null-additive, such that x0 = (C) AB f dm . Moreover, since m is null-additive we obtain x0 = (C) AB f dm = 0 m ({s : f (s) t} (A B))dt = 0 m ({s : f (s) t} B)dt = (C) B f dm I(B), which assures that I(A B) I(B). alogously, we c prove the converse inclusion. Indeed, if y0 C I(B) is arbitrarily chosen then there exists m Sµ , null-additive such . Since m is null-additive then we have that y = that y0 = (C) B f dm 0 (C) B f dm = 0 m ({s : f (s) t} B)dt= 0 m ({s : f (s) t} (A B)dt = (C) AB f dm , hence y0 I(A B) d the proof is completed. In what follows we shall prove that if we impose supplementary conditions for the multisubmeasure µ, such as converse null-additivity, pseudo generating property or Darboux property, then they are also preserved by the corresponding Aumn-Choquet integral. Indeed we have: Proposition 5.4. Let be µ : A Pkc (R+ ) a multisubmeasure such that every m Sµ is converse null-additive d f : S R+ AC- integrable function with respect to µ on S. Then the multifunction defined by (12) is converse null-additive. Proof. To prove that I is converse null-additive let be y A, B A with A B d I(A) = I(B). Firstly, we prove that µ is converse null-additive. FLOAREA-NICOLETA SOFI-BOCA Indeed, since every m Sµ is a converse null-additive fuzzy measure, C C from Sµ (f ) Sµ we obtain that every m Sµ (f ) is converse null-additive. Since from (1) we have µ(E) = [m1 (E), m2 (E)], for every E A, where m1 , m2 are converse null-additive, if µ(A) = µ(B) then [m1 (A), m2 (A)] = [m1 (B), m2 (B)] d thus mi (A) = mi (B), i = 1, 2. Because mi , i = 1, 2, are converse null-additive we have m1 (B - A) = 0 = m2 (B - A), d consequently µ(B-A) = {0} which assures that µ is a converse null-additive multisubmeasure. From f LAC (µ) d Proposition 4.6 we have that f LAC (µ, B - A) C d hence I(B - A) = B-A f dµ = {(C) B-A f dm, m Sµ (f )} = . From [24], th.8, since m is a converse null-additive fuzzy measure then the set function defined by (A) = (C) A f dm, A A, is converse nullC additive. Hence (C) B-A f dm = 0, for every m Sµ (f ), d we have that I(B - A) = B-A f dµ = {0} d consequently I is converse null-additive. Proposition 5.5. Let be µ:A Pkc (R+ ) a multisubmeasure d f : S R+ AC-integrable function with respect to µ on S. If µ has the pseudometric generating property then the same is true for the multifunction defined by (12). Proof. Firstly, let us observe that since µ has p.g.p then every fuzzy measure m Sµ has p.g.p. Indeed, since µ has p.g.p, then for every > 0 there exists > 0 such that for every A, B A with µ(A) [0, ] d µ(B) [0, ] we have µ(A B) [0, ]. From (1) we have that µ(A) = [m1 (A), m2 (A)] [0, ] d µ(B) = [m1 (B), m2 (B)] [0, ] d for every m Sµ we have m(A) m2 (A) d m(B) m2 (B) . From µ(A B) = [m1 (A B), m2 (A B)] [0, ] we obtain that m(A B) m2 (A B) , hence m Sµ has p.g.p. C C Since Sµ (f ) Sµ , then every m Sµ (f ) has p.g.p. Now, for every > 0 we consider > 0 from the definition of p.g.p. of µ d let be A, B A with I(A) [0, ] d I(B) [0, ]. Let be y y I(A B), C where I(A B) = AB f dµ = {(C) AB f dm, m Sµ (f )}. Then C (f ) with p.g.p. such that y = (C) there exists m Sµ AB f dm d from [20], th.5 the set function defined as Choquet integral has p.g.p. d hence we obtain that y . Hence I(A B) = AB f dµ [0, ] d I has p.g.p. Theorem 5.6. Let be µ : A Pkc (R+ ) a multisubmeasure d f : S R+ a function AC- integrable with respect to µ on S such that f (s) = 0, A MULTI-VALUED CHOQUET INTEGRAL for every s S. If every m Sµ has the Darboux property, then the multifunction defined by (12) has the Darboux property. Proof. Since f LAC (µ) we have from Proposition 4.6, that I(A) = C A f dµ = , for every A A. Let be y m Sµ (f ) with Darboux property. Since m(A) = 0 d f (s) = 0, for every s S we have I(A) = {0}. We shall prove that for y p (0, 1) there exists A f dµ B A, B A such that I(B) = pI(A). C Let be y y0 I(A). Then there exists m Sµ (f ) with Darboux = m ({s : f (s) t} A)dt. property, such that y0 = (C) A f dm 0 Since m has Darboux property then for y p (0, 1) there exists B A such that m (B) = pm (A). 1 Thus, we have that y0 = 0 m ({s : f (s) t} A)dt = 0 p m ({s : 1 f (s) t} B)dt d so y0 = p yo , where y0 I(B). Consequently, we have pI(A) I(B). It is easy to see that I(B) pI(A), hence the multifunction defined by (12) has the Darboux property. In what follows, we prove some absolutely continuity properties for the multifunction I with respect to µ. Proposition 5.7. If µ : A Pkc (R+ ) is a multisubmeasure d f : S R+ is AC- integrable function with respect to µ on S then I I µ, where I(A) = A f dµ, for every A A. Proof. Let be y A A with µ(A) = {0}. Then we have µ(A) = [m1 (A), m2 (A)] = {0} d, consequently, m1 (A) = 0 d m2 (A) = 0. C Then, for every m Sµ , we have m(A) = 0 d we obtain that C A f dµ = {(C) A f dm, m Sµ (f )} = {0}, hence I I µ. Remark 5.8. From Remark 2.17 d Proposition 5.3 we obtain that if µ is a null-additive multisubmeasure then I I µ I II µ where I is given by (12). Hence we have: Proposition 5.9. If µ : A Pkc (R+ ) is a null-additive multisubmeasure d f : S R+ is AC- integrable function with respect to µ on S then I II µ, where I(A) = A f dµ, for every A A. Denoting by the fuzzy measure defined by (A) = |µ(A)| , for every A A we have: FLOAREA-NICOLETA SOFI-BOCA Proposition 5.10. If µ:A Pkc (R+ ) is a multisubmeasure d f : S R+ is a measurable function such that f LC (), then I IV , where I is the multifunction given by I(A) = A f dµ, for every A A. Proof. According to the Definition 2.16 we shall prove that for every > 0 there exists > 0 such that for y A A with (A) < we have I(A) [0, ]. Let be A A, arbitrarily chosen. From hypothesis we obtain that f LC (m2 , A) d from Theorem 4.11, f is -integrable with respect to µ on A. From f LC (m2 ) we have (C) f dm2 = 0 m2 ({s : f (s) t})dt < d from [24], th.13, [4], th.4.4, for every > 0 there exists 0 < a < b such that (13) m2 ({s : f (s) t})dt + m2 ({s : f (s) t})dt < . 2 Let us consider = 2(b-a) d (A) < . Then (A) = |µ(A)| = m2 (A) < . To prove that I(A) [0, ] let be y y I(A) = A f dµ = C C {(C) A f dm, m Sµ (f )}. Then there exists m Sµ (f ) such that y = (C) A f dm d m(A) m2 (A). We obtain y = (C) f dm = m({s : f (s) t} A)dt (14) m2 ({s : f (s) t} A)dt. a b From 0 m2 ({s : f (s) t}A)dt = 0 m2 ({s : f (s) t}A)dt+ a m2 ({s : f (s) t} A)dt + b m2 ({s : f (s) t} A)dt d using (14) we have b that y = (C) A f dm < 2 + a m2 ({s : f (s) t} A)dt < 2 + (b - a) = 2 + 2(b-a) (b - a) = . Hence, for every y I(A) we have y < , that is I(A) [0, ] d I IV . Proposition 5.11. Let be µ : A Pkc (R+ ) a multisubmeasure d f : S R+ a measurable function such that f LC (). If the multisubmeasure µ is o-continuous then, I III where I is defined by (12). Proof. We shall prove that for y ( )n A with ( ) 0 we have |I( )| 0. Since for every A A we have that µ(A) = [m1 (A), m2 (A)] then µ( ) = [m1 ( ), m2 ( )]. Because ( ) = |µ( )| = m2 ( ) 0 A MULTI-VALUED CHOQUET INTEGRAL d since m1 ( ) m2 ( ), n N we have that m1 ( ) 0. From [24], th.3,4 we have that 1 ( ) 0 d 2 ( ) 0, where we denote by i (A) = (C) A f dmi , i = 1, 2. Since f LC () we easily obtain that f LC (m1 ) d from Theorem 4.11 we have that f LAC (µ, ) d (15) f dm2 ], for every n N. From (15) we have h(I( ), {0}) h([(C) max{(C) f dm2 } = (C) f dm2 0. Hence h(I( ), {0}) 0 d we have I III . f dm2 ], [0, 0]) = Acknowledgement. The author is indebted to Prof. Dr. A.M. Precupu for the careful reading of the paper d for the useful suggestions.

Journal

Annals of the Alexandru Ioan Cuza University - Mathematics – de Gruyter

Published: Jan 1, 2015

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