Some Characterizations of the Choquet Integral with Respect to a Monotone Interval-Valued Set Function

  • cc icon
  • ABSTRACT

    Intervals can be used in the representation of uncertainty. In this regard, we consider monotone interval-valued set functions and the Choquet integral. This paper investigates characterizations of monotone interval-valued set functions and provides applications of the Choquet integral with respect to monotone interval-valued set functions, on the space of measurable functions with the Hausdorff metric.


  • KEYWORD

    Interval-valued set function , Choquet integral , Hausdorff metric , Subadditivity of the Choquet integral

  • 1. Introduction

    Axiomatic characterizations of the Choquet integral have been introduced by Choquet [1], Murofushi et al [2,3], Wang [4] and Campos-Bolanos [5] as an interesting extension of the Lebesgue integral. Other researchers have studied various convergence problems on monotone set functions and, on sequences of measurable functions, as well as applications. For example, the convergence in the (C) mean [6], on decision-making problems [7,8], on the Choquet weak convergence [9], on the monotone expectation [10], and on the aggregation approach [11].

    In the past decade, it has been suggested to use intervals in order to represent uncertainty, for example, for economic uncertainty [12], for fuzzy random variables [13], in intervalprobability [14], for martingales of multi-valued functions [15], in the integrals of set-valued functions [16], in the Choquet integrals of interval-valued (or closed set-valued) functions [17-22], and for interval-valued capacity functions [23]. Couso-Montes-Gil [24] studied applications under the sufficient and necessary conditions on monotone set functions, i.e., the subadditivity of the Choquet integral with respect to monotone set functions.

    Intervals are useful in the representation of uncertainty. We shall consider monotone intervalvalued set functions and the Choquet integral with respect to a monotone interval-valued set function of measurable functions. Based on the results of Couso-Motes-Gil [24], we shall provide characterizations of monotone interval-valued set functions as well as applications of the Choquet integral regarding a monotone interval-valued set function in the space of measurable functions with the Hausdorff metric.

    In Section 2, we list definitions and basic properties for the monotone set functions, the Choquet integrals and for the various convergence notions in the space of measurable functions.

    In Section 3, we define a monotone interval-valued set function and the Choquet integral with respect to a monotone interval-valued set function of measurable functions, and we discuss their properties. We also investigate various convergences in the Hausdorff metric on the space of intervals as well as the characterizations of the Choquet integral with respect to a monotone interval-valued set function of measurable functions.

    In Section 4, we give a brief summary of our results and conclusions.

    2. Preliminaries and Definitions

    In this section, we consider monotone set functions, also called fuzzy measures, and the Choquet integral defined by Choquet [1]. The Choquet integral [1] generalizes the Lebesgue integral to the case of monotone set functions. Let X be a non-empty set, and let A denote a σ-algebra of subsets of X. Let

    image

    and

    image

    First we define, monotone set functions, the Choquet integral, the different types of convergences, and the uniform integrability of measurable functions as follows:

    Definition 2..1. [2,3,5,24] (1) A mapping

    image

    is said to be a set function if μ(Ø)=0.

    (2) A set function μ is said to be monotone if

    image

    (3) A set function μ is said to be continuous from below (or lower semi-continuous) if for any sequence {An} ⊂ A and AA such that

    image

    (4) A set function μ is said to be continuous from above (or upper semi-continuous) if for any sequence {An} ⊂ A and AA such that

    image

    (5) A set function μ is said to be continuous if it is continuous from above and continuous from below.

    (6) A set function μ is said to be subadditive if A, BA and A ∩ B = Ø, then

    image

    (7) A set function μ is said to be submodular if A, BA, then

    image

    (8) A set function μ is said to be null-additive if

    image

    Definition 2..2. [2,3,5,24] Let μ be a monotone set function on A. (1) If f : X → ?+ is a non-negative measurable function, then the Choquet integral of f with respect to μ is defined by

    image

    where

    image

    for all α ∈ 2 ?+ and the integral on the right-hand side is the Lebesgue integral of μf .

    (2) If f : X → ? is a real-valued measurable function, then the Choquet integral of f with respect to μ is defined by

    image

    where f+ = max{f,0}, f=max{?f,0}, Ac is the complementary set of A, and μ* is the conjugate of μ, that is,

    image

    (3) A measurable function f is said to be μ-integrable if the Choquet integral of f on X exists.

    We note that

    image

    for all α ∈ ?+ and

    image

    for all α ∈ ? = (?∞, 0). Thus, we have

    image

    We introduce almost everywhere convergence, convergence in μ-mean, and uniform μ-integrability as follows:

    Definition 2..3. Let μ be a monotone set function on a measurable space (X, A), {fn} a sequence of measurable functions from X to ?, and f a measurable function from X to ?.

    (1) A sequence {fn} almost everywhere converges to f if there exists a measurable and null subset NA, μ(N) = 0 such that

    image

    (2) A sequence {fn} converges in μ-mean to f if

    image

    where | ? | is the absolute value on ?.

    Definition 2..4. [24] Let μ be a monotone set function on A and I ⊂ ? an index set. A class of real-valued measurable functions {fn}nI is said to be uniform μ-integrable if

    image
    image

    Now, we recall from [24] the subadditivity of the Choquet integral, the equivalence between the convergence in mean and the uniform integrability of a sequence of measurable functions.

    Theorem 2..5. (Subadditivity for the Choquet integral) Let (X, A) be a measurable space. If a monotone set function μ : A → ?+ is submodular and f, g : X → ? are realvalued measurable functions, then we have

    image

    Theorem 2..6. Let (X, A) be a measurable space. If a monotone set function μ : A → ?+ is subadditive and f, g : X → ? are measurable functions with disjoint support, that is, {xX | f(x) > 0}∩{xX | g(x) > 0} = Ø, then we have

    image

    3. Main Results

    In this section, we consider intervals, interval-valued functions, and the Aumann integral of measurable interval-valued functions. Let I(?) be the class of all bounded and closed intervals (intervals, for short) in ? as follows:

    image

    For any a ∈ ?, we define a = [a, a]. Obviously, aI(?) [18-21].

    Recall that if (?, , m) is the Lebesgue measure space and C(?) is the set of all closed subsets of ?, then the Aumann integral of a closed set-valued function F : ? → C(?) is defined by

    image

    where S(F) is the set of all integrable selections of F, that is,

    image

    where ma.e. means almost everywhere in the Lebesgue measure m, and |g| is the absolute value of g [15,16]. In [13,23], we can see that (A) ? Fdm is a nonempty bounded and closed interval in ? whenever F is an interval-valued function as in the following theorem.

    Theorem 3..1. If an interval-valued function F = [gl, gr] : ? → I(?) is measurable and integrably bounded, then gl, grS(F) and

    image

    where the two integrals on the right-hand side are the Lebesgue integral with respect to m.

    Note that we write

    image

    for all bounded continuous function g. Let C(?) be the class of all closed ?. We recall that the Hausdorff metric dH : C(?) × C(?) → ?+ is defined by

    image

    for all A, BC(?). It is well-known that for all = [al,ar],

    image
    image

    Next, we shall define monotone interval-valued set functions and discuss their characterization.

    Definition 3..2. (1) A mapping

    image

    is said to be an interval-valued set function if

    image

    (2) An interval-valued set function

    image

    is said to be monotone if

    image

    (3) An interval-valued set function

    image

    is said to be continuous from below if for any sequence {An}⊂ A and AA such that AnA, then

    image

    that is,

    image

    (4) An interval-valued set function

    image

    is said to be continuous from above if for any sequence {An}⊂A and AA such that

    image

    is a bounded interval and AnA, then

    image

    (5) An interval-valued set function

    image

    is said to be continuous if it is both continuous from above and continuous from below.

    (6) An interval-valued set function

    image

    is said to be subadditive if A, BA, then

    image

    (7) An interval-valued set function

    image

    is said to be submodular if A, BA, then

    image

    (8) An interval-valued set function

    image

    is said to be nulladditive if

    image

    From Definition 3.2 and Eq. (25), we can directly derive the following theorem [23,25].

    Theorem 3..3. (1) A mapping

    image

    is an intervalvalued set function if only only if μl and μr are set functions, and μlμr.

    (2) An interval-valued set function

    image

    is monotone if only only if the set functions μl and μr are monotone.

    (3) An interval-valued set function

    image

    is continuous from below if only only if the set functions μl and μr are continuous from below, and μlμr.

    (4) An interval-valued set function

    image

    is continuous from above if only only if the set functions μl and μr are continuous from above, and μlμr.

    (5) An interval-valued set function

    image

    is subadditive if and only if the set functions μl and μr are subadditive, and μlμr.

    (6) An interval-valued set function

    image

    is submodular if and only if the set functions μl and μr are submodular, and μlμr.

    (7) An interval-valued set function

    image

    is null-additive if and only if the set functions μl and μr are null-additive, and μlμr.

    By using Definition 2.2 and Theorem 3.3, we define the Choquet integral of a non-negative measurable function with respect to a continuous from below and monotone intervalvalued set function as follows:

    Definition 3..4. (1) The Choquet integral of a non-negative measurable function f : X → ?+, with respect to a monotone interval-valued set function

    image

    is defined by

    image

    where m is the Lebesgue measure on ? and the integral on the right-hand side is the Aumann integral with respect to m of

    image

    (2) The Choquet integral of a real-valued measurable function f : X → ?, with respect to a monotone interval-valued set function

    image

    is defined by

    image

    where f+ = max{f,0} and f=max{?f,0}, and

    image

    is the conjugate of

    image

    that is,

    image

    (3) A measurable function f is said to be

    image

    -integrable if

    image

    We note that Eq. (36) implies

    image

    where

    image

    for all α ∈ ?+. By the definition of

    image

    we easily get the following theorem.

    Theorem 3..5. (1) A monotone interval-valued set function

    image

    is continuous from below (resp. from above) if and only if

    image

    is continuous from above (resp. from below).

    (2) If

    image

    is a monotone interval-valued set function and μl(X) = μr(X), then

    image

    where

    image

    In [21], we can find the theorem below. This gives a useful and interesting tool for the application of the Choquet integral of a non-negative measurable function f, with respect to a monotone interval-valued set function

    image

    Theorem 3..6. ([23, Lemma 2.5 (i) and (v)]) Let f be a nonnegative measurable function and

    image

    a monotone interval-valued function. If

    image

    is continuous from above and we take

    image

    for all α ∈ ?+, then we have

    (1) F is continuous from above, and

    (2)

    image

    where m is the Lebesgue measure and μlf (α)= μl({xX|f(x) >α}) and μrf (α)= μr({xX|f(x) >α}).

    Note that Theorem 3.6(2) implies the following equation (36) under the same condition of f and

    image
    image

    By using Theorem 3.5 and Eq. (36), we can obtain the following theorem, which is a useful formula for the Choquet integral of a measurable function f : X → ?, with respect to a continuous monotone interval-valued set function.

    Theorem 3..7. Let f be a measurable function and

    image

    a monotone interval-valued set function. If

    image

    is continuous and μl(X) = μr(X), then we have

    image

    Proof. Let f+ = max{f, 0} and f = max{?f, 0}. Since

    image

    is continuous from above, by (40), we have

    image

    Since

    image

    is continuous from below and μl(X) = μr(X), by Theorem 3.5(2),

    image

    is continuous from above. Thus, by (36), we have

    image

    By Definition 3.4(2), Eq. (38), and Eq. (39), we have the result.

    Next, we present the following theorems which give characterizations of the Choquet integral with respect to a monotone interval-valued set function.

    Theorem 3..8. Let

    image

    be a monotone interval-valued set function and let AA. If

    image

    is continuous from above, then we have

    image

    Proof. If a ≥ 0, then by Eq. (36), we have

    image

    If a < 0, then by Eq. (36), we have

    image

    Theorem 3..9. Let a monotone interval-valued set function

    image

    be continuous from above and let f a non-negative

    image

    -integrable function. If

    image

    is continuous from above and A, BA with AB, then we have

    image

    Proof. Since

    image

    is a monotone interval-valued set function, by Theorem 3.3 (1) and (2), μl and μr are monotone interval-valued set functions. Thus,

    image

    By Eq. (36) and Eq. (42), we have the result.

    We remark that if we take a

    image

    -integrable function f which is f+ = 0 and f > 0, then

    image

    is not monotone, that is, for each pair A, BA with AB,

    image

    Theorem 3..10. Let

    image

    be a monotone intervalvalued set function which is continuous from above, and let AA. If f and g are non-negative

    image

    -integrable functions with fg, then we have

    image

    Proof. The proof is similar to the proof of Theorem 3.10.

    Theorem 3..11. Let

    image

    be monotone interval-valued set functions, f a non-negative

    image

    -integrable and

    image

    -integrable function, and AA.

    (1) If

    image

    then we have

    image

    (2) If

    image

    then we have

    image

    Proof. (1) Note that

    image

    if and only if μ1lμ2l and μlrμ2r. Thus, we have

    image

    By (36) and (47), we have the result.

    (2) Note that

    image

    if and only if μ2lμ1l and μ1rμ2r. Thus, we have

    image

    By Eq. (36) and eq. (48), we have the result.

    Finally, we investigate the subadditivity of the Choquet integral under some conditions for the monotone interval-valued set functions.

    Theorem 3..12. Let (X, A) be a measurable space. If a continuous monotone interval-valued set function

    image

    with μl(X) = μr(X), is submodular and

    image

    are measurable functions, then we have

    image

    Proof. Since

    image

    is a submodular monotone interval-valued set function, by Theorem 3.3(6), μl and μr are submodular monotone set functions.

    By Theorem 2.5, we have

    image

    and

    image

    By Eq. (36), eq. (50), and eq. (51), we have the result.

    Theorem 3..13. Let (X, A) be a measurable space. If a continuous monotone interval-valued set function

    image

    with μl(X) = μr(X), is subadditive, and

    image

    are measurable functions with disjoint support, then

    image

    Proof. Since

    image

    is a subadditive monotone interval-valued set function, by Theorem 3.3(5), μl and μr are subadditive monotone set functions. By Theorem 2.6, we have

    image

    and

    image

    By Eq. (36), Eq. (53), and Eq. (54), we have the result.

    4. Conclusions

    In this paper, we introduced the concept of a monotone intervalvalued set function and, the Aumann integral of a measurable function, with respect to the Lebesgue measure. By using the two concepts, we define the Choquet integral with a monotone interval-valued set function of measurable functions.

    From Theorem 3.2, Definition 3.3(3), and the condition that μl(X) = μr(X) of a continuous monotone set function, we can deal with the new concept of the Choquet integral of a monotone interval-valued set function

    image

    of measurable functions f : X → ?. Theorems 3.3, 3.5, 3.6, 3.7, 3.8, 3.9, and 3.10 are important characterizations of the Choquet integral with respect to a monotone interval-valued set function on the space of non-negative

    image

    -integrable functions. Theorem 3.12 and Theorem 3.13 are both, useful and interesting tools, in the application of the Choquet integral with respect to a continuous monotone interval- valued set function.

    In the future, by using the results in this paper, we shall investigate various problems and models, for representing monotone uncertain set functions, and for the application of the bi-Choquet integral with respect to a monotone interval-valued set function

      >  Conflict of Interest

    No potential conflict of interest relevant to this article was reported.

  • 1. Choquet G. 1953 “Theory of capacities” [Annales de l’institut Fourier] Vol.5 P.131-295 google
  • 2. Murofushi T., Sugeno M. 1991 “A theory of fuzzy measures: representations, the Choquet integral, and null sets” [Journal of Mathematical Analysis and Applications] Vol.159 P.532-549 google doi
  • 3. Murofushi T., Sugeno M., Suzaki M. 1997 “Autocontinuity, convergence in measure, and convergence in distribution” [Fuzzy Sets and Systems] Vol.92 P.197-203 google
  • 4. Wang Z. 2007 “Convergence theorems for sequences of Choquet integral” [International Journal of General Systems] Vol.26 P.133-143 google doi
  • 5. de Campos L. M., Jorge M. 1992 “Characterization and comprison of Sugeno and Choquet integrals” [Fuzzy Sets and Systems] Vol.52 P.61-67 google doi
  • 6. Pedrycz W., Yang L., Ha M. 2009 “On the fundamental convergence in the (C) mean in problems of information fusion” [Journal of Mathematical Analysis and Applications] Vol.358 P.203-222 google doi
  • 7. Bolanos M.J., Lamata M.T., Moral S. 1988 “Decision making problems in a general environment” [Fuzzy Sets and Systems] Vol.25 P.135-144 google doi
  • 8. Merigo J.M., Cassanovas M. 2011 “Decision-making with distance measures and induced aggregation operators” [Computers and Industrial Engineering] Vol.60 P.66-76 google doi
  • 9. Feng D., Nguyen H.T. 2007 “Choquet weak convergence of capacity functionals of random sets” [Information Sciences] Vol.177 P.3239-3250 google
  • 10. Bolanos Carmona M. J., de Campose Ibanez L. M., Gonzalez Munoz A. 1999 “Convergence properties of the monotone expectation and its applications to the extension of fuzzy measures” [Fuzzy Sets and Systems] Vol.33 P.201-212 google doi
  • 11. Buyukozkan G., Buyukozkan G., Duan D. 2010 “Choquet integral based aggregation approach to software development risk assessment” [Information Sciences] Vol.180 P.441-451 google doi
  • 12. Schjaer-Jacobsen H. 2002 “Representation and calculation of economic uncertainties: intervals, fuzzy numbers, and probabilities” [International Journal of Production Economics] Vol.78 P.91-98 google doi
  • 13. Li L., Zhaohan S. 1998 “The fuzzy set-valued measures generated by fuzzy random variables” [Fuzzy Sets and Systems] Vol.97 P.203-209 google doi
  • 14. Weichselberger K. 2000 “The theory of interval-probability as a unifying concept for uncertainty” [International Journal of Approximate Reasoning] Vol.24 P.149-170 google doi
  • 15. Hiai F., Umegaki H. 1977 “Integrals, conditional expectations, and martingales of multivalued functions” [Journal of Multivariate Analysis] Vol.7 P.149-182 google doi
  • 16. Aumann R. J. 1965 “Integrals of set-valued functions” [Journal of Mathematical Analysis and Applications] Vol.12 P.1-12 google doi
  • 17. Jang L. C., Kil B. M., Kim Y. K., Kwon J. S. 1997 “Some properties of Choquet integrals of set-valued functions” [Fuzzy Sets and Systems] Vol.91 P.95-98 google doi
  • 18. Jang L. C., Kwon J. S. 2000 “On the representation of Choquet integrals of set-valued functions and null sets” [Fuzzy Sets and Systems] Vol.112 P.233-239 google doi
  • 19. Jang L. C. 2004 “Interval-valued Choquet integrals and their applications” [Journal of Applied Mathematics and Computing] Vol.16 P.429-443 google
  • 20. Jang L. C. 2007 “A note on the monotone interval-valued set function defined by the interval-valued Choquet integral” [Communications of the Korean Mathematical Society] Vol.22 P.227-234 google
  • 21. Jang L. C. 2011 “On properties of the Choquet integral of interval-valued functions” [Journal of Applied Mathematics] Vol.2011 google doi
  • 22. Zhang D., Guo C., Liu D. 2004 “Set-valued Choquet integrals revisited” [Fuzzy Sets and Systems] Vol.147 P.475-485 google doi
  • 23. Jang L. C. 2012 “A note on convergence properties of intervalvalued capacity functionals and Choquet integrals” [Information Sciences] Vol.183 P.151-158 google
  • 24. Couso I., Montes S., Gil P. 2002 “Stochastic convergence, uniform integrability and convergence in mean on fuzzy measure spaces” [Fuzzy Sets and Systems] Vol.129 P.95-104 google doi
  • 25. Gavrilut A. C. 2010 “A Lusin type theorem for regular monotone uniformly autocontinuous set multifunctions” [Fuzzy Sets and Systems] Vol.161 P.2909-2918 google doi