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Some Characterizations of the Choquet Integral with Respect to a Monotone Interval-Valued Set Function
Some Characterizations of the Choquet Integral with Respect to a Monotone Interval-Valued Set Function
International Journal of Fuzzy Logic and Intelligent Systems. 2013. Mar, 13(1): 83-90
Copyright ©2013, Korean Institute of Intelligent Systems
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
• Received : January 25, 2013
• Accepted : March 15, 2013
• Published : March 30, 2013
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About the Authors
Lee-Chae, Jang
leechae.jang@kku.ac.kr

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.
Keywords
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
PPT Slide
Lager Image
and
PPT Slide
Lager 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
PPT Slide
Lager Image
is said to be a set function if μ (Ø)=0.
(2) A set function μ is said to be monotone if
PPT Slide
Lager Image
(3) A set function μ is said to be continuous from below (or lower semi-continuous) if for any sequence { An } ⊂ A and A A such that
PPT Slide
Lager Image
(4) A set function μ is said to be continuous from above (or upper semi-continuous) if for any sequence { An } ⊂ A and A A such that
PPT Slide
Lager 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 , B A and A ∩ B = Ø, then
PPT Slide
Lager Image
(7) A set function μ is said to be submodular if A , B A , then
PPT Slide
Lager Image
(8) A set function μ is said to be null-additive if
PPT Slide
Lager Image
Definition 2..2. [2 , 3 , 5 , 24] Let μ be a monotone set function on A . (1) If f : X → ℝ + is a non-negative measurable function, then the Choquet integral of f with respect to μ is defined by
PPT Slide
Lager Image
where
PPT Slide
Lager Image
for all α ∈ 2 ℝ + and the integral on the right-hand side is the Lebesgue integral of μf .
(2) If f : X → ℝ is a real-valued measurable function, then the Choquet integral of f with respect to μ is defined by
PPT Slide
Lager 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,
PPT Slide
Lager Image
(3) A measurable function f is said to be μ -integrable if the Choquet integral of f on X exists.
We note that
PPT Slide
Lager Image
for all α ∈ ℝ + and
PPT Slide
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for all α ∈ ℝ = (–∞, 0). Thus, we have
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Lager 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 ℝ, and f a measurable function from X to ℝ.
(1) A sequence { fn } almost everywhere converges to f if there exists a measurable and null subset N A , μ ( N ) = 0 such that
PPT Slide
Lager Image
(2) A sequence { fn } converges in μ -mean to f if
PPT Slide
Lager Image
where | ᠊ | is the absolute value on ℝ.
Definition 2..4. [24] Let μ be a monotone set function on A and I ⊂ ℕ an index set. A class of real-valued measurable functions { fn } nI is said to be uniform μ -integrable if
PPT Slide
Lager Image
PPT Slide
Lager 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 → ℝ + is submodular and f , g : X → ℝ are realvalued measurable functions, then we have
PPT Slide
Lager Image
Theorem 2..6. Let ( X , A ) be a measurable space. If a monotone set function μ : A → ℝ + is subadditive and f , g : X → ℝ are measurable functions with disjoint support, that is, { x X | f ( x ) > 0}∩{ x X | g ( x ) > 0} = Ø, then we have
PPT Slide
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3. Main Results
In this section, we consider intervals, interval-valued functions, and the Aumann integral of measurable interval-valued functions. Let I (ℝ) be the class of all bounded and closed intervals (intervals, for short) in ℝ as follows:
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For any a ∈ ℝ, we define a = [ a , a ]. Obviously, a I (ℝ) [18 - 21] .
Recall that if (ℝ, , m ) is the Lebesgue measure space and C (ℝ) is the set of all closed subsets of ℝ, then the Aumann integral of a closed set-valued function F : ℝ → C (ℝ) is defined by
PPT Slide
Lager Image
where S ( F ) is the set of all integrable selections of F , that is,
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where m a.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 ) ʃ Fdm is a nonempty bounded and closed interval in ℝ whenever F is an interval-valued function as in the following theorem.
Theorem 3..1. If an interval-valued function F = [ gl , gr ] : ℝ → I (ℝ) is measurable and integrably bounded, then gl , gr S ( F ) and
PPT Slide
Lager Image
where the two integrals on the right-hand side are the Lebesgue integral with respect to m .
Note that we write
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for all bounded continuous function g . Let C (ℝ) be the class of all closed ℝ. We recall that the Hausdorff metric dH : C (ℝ) × C (ℝ) → ℝ + is defined by
PPT Slide
Lager Image
for all A , B C (ℝ). It is well-known that for all ā = [ al , ar ],
PPT Slide
Lager Image
PPT Slide
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Next, we shall define monotone interval-valued set functions and discuss their characterization.
Definition 3..2. (1) A mapping
PPT Slide
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is said to be an interval-valued set function if
PPT Slide
Lager Image
(2) An interval-valued set function
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Lager Image
is said to be monotone if
PPT Slide
Lager Image
(3) An interval-valued set function
PPT Slide
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is said to be continuous from below if for any sequence { An }⊂ A and A A such that An A , then
PPT Slide
Lager Image
that is,
PPT Slide
Lager Image
(4) An interval-valued set function
PPT Slide
Lager Image
is said to be continuous from above if for any sequence { An }⊂ A and A A such that
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Lager Image
is a bounded interval and An A , then
PPT Slide
Lager Image
(5) An interval-valued set function
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is said to be continuous if it is both continuous from above and continuous from below.
(6) An interval-valued set function
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is said to be subadditive if A , B A , then
PPT Slide
Lager Image
(7) An interval-valued set function
PPT Slide
Lager Image
is said to be submodular if A , B A , then
PPT Slide
Lager Image
(8) An interval-valued set function
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Lager Image
is said to be nulladditive if
PPT Slide
Lager Image
From Definition 3.2 and Eq. (25), we can directly derive the following theorem [23 , 25] .
Theorem 3..3. (1) A mapping
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is an intervalvalued set function if only only if μl and μr are set functions, and μl μr .
(2) An interval-valued set function
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Lager Image
is monotone if only only if the set functions μl and μr are monotone.
(3) An interval-valued set function
PPT Slide
Lager Image
is continuous from below if only only if the set functions μl and μr are continuous from below, and μl μr .
(4) An interval-valued set function
PPT Slide
Lager Image
is continuous from above if only only if the set functions μl and μr are continuous from above, and μl μr .
(5) An interval-valued set function
PPT Slide
Lager Image
is subadditive if and only if the set functions μl and μr are subadditive, and μl μr .
(6) An interval-valued set function
PPT Slide
Lager Image
is submodular if and only if the set functions μl and μr are submodular, and μl μr .
(7) An interval-valued set function
PPT Slide
Lager Image
is null-additive if and only if the set functions μl and μr are null-additive, and μl μr .
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 → ℝ + , with respect to a monotone interval-valued set function
PPT Slide
Lager Image
is defined by
PPT Slide
Lager Image
where m is the Lebesgue measure on ℝ and the integral on the right-hand side is the Aumann integral with respect to m of
PPT Slide
Lager Image
(2) The Choquet integral of a real-valued measurable function f : X → ℝ, with respect to a monotone interval-valued set function
PPT Slide
Lager Image
is defined by
PPT Slide
Lager Image
where f + = max{ f ,0} and f =max{– f ,0}, and
PPT Slide
Lager Image
is the conjugate of
PPT Slide
Lager Image
that is,
PPT Slide
Lager Image
(3) A measurable function f is said to be
PPT Slide
Lager Image
-integrable if
PPT Slide
Lager Image
We note that Eq. (36) implies
PPT Slide
Lager Image
where
PPT Slide
Lager Image
for all α ∈ ℝ + . By the definition of
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Lager Image
we easily get the following theorem.
Theorem 3..5. (1) A monotone interval-valued set function
PPT Slide
Lager Image
is continuous from below (resp. from above) if and only if
PPT Slide
Lager Image
is continuous from above (resp. from below).
(2) If
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Lager Image
is a monotone interval-valued set function and μl ( X ) = μr ( X ), then
PPT Slide
Lager Image
where
PPT Slide
Lager 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
PPT Slide
Lager Image
Theorem 3..6. ([23, Lemma 2.5 (i) and (v)]) Let f be a nonnegative measurable function and
PPT Slide
Lager Image
a monotone interval-valued function. If
PPT Slide
Lager Image
is continuous from above and we take
PPT Slide
Lager Image
for all α ∈ ℝ + , then we have
(1) F is continuous from above, and
(2)
PPT Slide
Lager Image
where m is the Lebesgue measure and μlf ( α )= μl ({ x X | f ( x ) > α }) and μrf ( α )= μr ({ x X | f ( x ) > α }).
Note that Theorem 3.6(2) implies the following equation (36) under the same condition of f and
PPT Slide
Lager Image
PPT Slide
Lager 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 → ℝ, with respect to a continuous monotone interval-valued set function.
Theorem 3..7. Let f be a measurable function and
PPT Slide
Lager Image
a monotone interval-valued set function. If
PPT Slide
Lager Image
is continuous and μl ( X ) = μr ( X ), then we have
PPT Slide
Lager Image
Proof. Let f + = max{ f , 0} and f = max{– f , 0}. Since
PPT Slide
Lager Image
is continuous from above, by (40), we have
PPT Slide
Lager Image
Since
PPT Slide
Lager Image
is continuous from below and μl ( X ) = μr ( X ), by Theorem 3.5(2),
PPT Slide
Lager Image
is continuous from above. Thus, by (36), we have
PPT Slide
Lager 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
PPT Slide
Lager Image
be a monotone interval-valued set function and let A A . If
PPT Slide
Lager Image
is continuous from above, then we have
PPT Slide
Lager Image
Proof. If a ≥ 0, then by Eq. (36), we have
PPT Slide
Lager Image
If a < 0, then by Eq. (36), we have
PPT Slide
Lager Image
Theorem 3..9. Let a monotone interval-valued set function
PPT Slide
Lager Image
be continuous from above and let f a non-negative
PPT Slide
Lager Image
-integrable function. If
PPT Slide
Lager Image
is continuous from above and A , B A with A B , then we have
PPT Slide
Lager Image
Proof. Since
PPT Slide
Lager 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,
PPT Slide
Lager Image
By Eq. (36) and Eq. (42), we have the result.
We remark that if we take a
PPT Slide
Lager Image
-integrable function f which is f + = 0 and f > 0, then
PPT Slide
Lager Image
is not monotone, that is, for each pair A , B A with A B ,
PPT Slide
Lager Image
Theorem 3..10. Let
PPT Slide
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be a monotone intervalvalued set function which is continuous from above, and let A A . If f and g are non-negative
PPT Slide
Lager Image
-integrable functions with f g , then we have
PPT Slide
Lager Image
Proof. The proof is similar to the proof of Theorem 3.10.
Theorem 3..11. Let
PPT Slide
Lager Image
be monotone interval-valued set functions, f a non-negative
PPT Slide
Lager Image
-integrable and
PPT Slide
Lager Image
-integrable function, and A A .
(1) If
PPT Slide
Lager Image
then we have
PPT Slide
Lager Image
(2) If
PPT Slide
Lager Image
then we have
PPT Slide
Lager Image
Proof. (1) Note that
PPT Slide
Lager Image
if and only if μ 1l μ 2l and μ lr μ 2r . Thus, we have
PPT Slide
Lager Image
By (36) and (47), we have the result.
(2) Note that
PPT Slide
Lager Image
if and only if μ 2l μ 1l and μ 1r μ 2r . Thus, we have
PPT Slide
Lager 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
PPT Slide
Lager Image
with μl ( X ) = μr ( X ), is submodular and
PPT Slide
Lager Image
are measurable functions, then we have
PPT Slide
Lager Image
Proof. Since
PPT Slide
Lager 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
PPT Slide
Lager Image
and
PPT Slide
Lager 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
PPT Slide
Lager Image
with μl ( X ) = μr ( X ), is subadditive, and
PPT Slide
Lager Image
are measurable functions with disjoint support, then
PPT Slide
Lager Image
Proof. Since
PPT Slide
Lager 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
PPT Slide
Lager Image
and
PPT Slide
Lager 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
PPT Slide
Lager Image
of measurable functions f : X → ℝ. 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
PPT Slide
Lager 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.
Acknowledgements
This work was supported by Konkuk University in 2013.
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