Some Characterizations of the Choquet Integral with Respect to a Monotone IntervalValued Set Function
 Author: Jang LeeChae
 Organization: Jang LeeChae
 Publish: International Journal of Fuzzy Logic and Intelligent Systems Volume 13, Issue1, p83~90, 25 March 2013

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

KEYWORD
Intervalvalued 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 CamposBolanos [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 decisionmaking 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 multivalued functions [15], in the integrals of setvalued functions [16], in the Choquet integrals of intervalvalued (or closed setvalued) functions [1722], and for intervalvalued capacity functions [23]. CousoMontesGil [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 intervalvalued set function of measurable functions. Based on the results of CousoMotesGil [24], we shall provide characterizations of monotone intervalvalued set functions as well as applications of the Choquet integral regarding a monotone intervalvalued 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 intervalvalued set function and the Choquet integral with respect to a monotone intervalvalued 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 intervalvalued 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 nonempty set, and let denote aA σ algebra of subsets ofX . Letand
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 mappingis said to be a set function if
μ (Ø)=0.(2) A set function
μ is said to be monotone if(3) A set function
μ is said to be continuous from below (or lower semicontinuous) if for any sequence {A_{n} } ⊂ andA A ∈ such thatA (4) A set function
μ is said to be continuous from above (or upper semicontinuous) if for any sequence {A_{n} } ⊂ andA A ∈ such thatA (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 ifA ,B ∈ and A ∩ B = Ø, thenA (7) A set function
μ is said to be submodular ifA ,B ∈ , thenA (8) A set function
μ is said to be nulladditive ifDefinition 2..2. [2,3,5,24] Letμ be a monotone set function on . (1) IfA f :X → ？^{+} is a nonnegative measurable function, then the Choquet integral off with respect toμ is defined bywhere
for all
α ∈ 2 ？^{+} and the integral on the righthand side is the Lebesgue integral ofμ_{f} .(2) If
f :X → ？ is a realvalued measurable function, then the Choquet integral off with respect toμ is defined bywhere
f ^{+} = max{f ,0},f ^{？}=max{？f ,0},A^{c} is the complementary set ofA , andμ ^{*} is the conjugate ofμ , that is,(3) A measurable function
f is said to beμ integrable if the Choquet integral off onX exists.We note that
for all
α ∈ ？^{+} andfor all
α ∈ ？^{？} = (？∞, 0). Thus, we haveWe introduce almost everywhere convergence, convergence in
μ mean, and uniformμ integrability as follows:Definition 2..3. Letμ be a monotone set function on a measurable space (X , ), {A f_{n} } a sequence of measurable functions fromX to ？, andf a measurable function fromX to ？.(1) A sequence {
f_{n} } almost everywhere converges tof if there exists a measurable and null subsetN ∈ ,A μ (N ) = 0 such that(2) A sequence {
f_{n} } converges inμ mean tof ifwhere  ？  is the absolute value on ？.
Definition 2..4. [24] Letμ be a monotone set function on andA I ⊂ ？ an index set. A class of realvalued measurable functions {f_{n} }_{n∈I} is said to be uniformμ integrable ifNow, 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 , ) be a measurable space. If a monotone set functionA μ : → ？^{+} is submodular andA f ,g :X → ？ are realvalued measurable functions, then we haveTheorem 2..6. Let (X , ) be a measurable space. If a monotone set functionA μ : → ？^{+} is subadditive andA f ,g :X → ？ are measurable functions with disjoint support, that is, {x ∈X f (x ) > 0}∩{x ∈X g (x ) > 0} = Ø, then we have3. Main Results
In this section, we consider intervals, intervalvalued functions, and the Aumann integral of measurable intervalvalued functions. Let
I (？) be the class of all bounded and closed intervals (intervals, for short) in ？ as follows:For any
a ∈ ？, we definea = [a ,a ]. Obviously,a ∈I (？) [1821].Recall that if (？,
？ ,m ) is the Lebesgue measure space andC (？) is the set of all closed subsets of ？, then the Aumann integral of a closed setvalued functionF : ？ →C (？) is defined bywhere
S (F ) is the set of all integrable selections ofF , that is,where
m ？a.e. means almost everywhere in the Lebesgue measurem , and g  is the absolute value ofg [15,16]. In [13,23], we can see that (A )？ Fdm is a nonempty bounded and closed interval in ？ wheneverF is an intervalvalued function as in the following theorem.Theorem 3..1. If an intervalvalued functionF = [g_{l} ,g_{r} ] : ？ →I (？) is measurable and integrably bounded, theng_{l} ,g_{r} ∈S (F ) andwhere the two integrals on the righthand side are the Lebesgue integral with respect to
m .Note that we write
for all bounded continuous function
g . LetC (？) be the class of all closed ？. We recall that the Hausdorff metricd_{H} :C (？) ×C (？) → ？^{+} is defined byfor all
A ,B ∈C (？). It is wellknown that for all？ = [a_{l} ,a_{r} ],Next, we shall define monotone intervalvalued set functions and discuss their characterization.
Definition 3..2. (1) A mappingis said to be an intervalvalued set function if
(2) An intervalvalued set function
is said to be monotone if
(3) An intervalvalued set function
is said to be continuous from below if for any sequence {
A_{n} }⊂ andA A ∈ such thatA A_{n} ↑A , thenthat is,
(4) An intervalvalued set function
is said to be continuous from above if for any sequence {
A_{n} }⊂ andA A ∈ such thatA is a bounded interval and
A_{n} ↓A , then(5) An intervalvalued set function
is said to be continuous if it is both continuous from above and continuous from below.
(6) An intervalvalued set function
is said to be subadditive if
A ,B ∈ , thenA (7) An intervalvalued set function
is said to be submodular if
A ,B ∈ , thenA (8) An intervalvalued set function
is said to be nulladditive if
From Definition 3.2 and Eq. (25), we can directly derive the following theorem [23,25].
Theorem 3..3. (1) A mappingis an intervalvalued set function if only only if
μ_{l} andμ_{r} are set functions, andμ_{l} ≤μ_{r} .(2) An intervalvalued set function
is monotone if only only if the set functions
μ_{l} andμ_{r} are monotone.(3) An intervalvalued set function
is continuous from below if only only if the set functions
μ_{l} andμ_{r} are continuous from below, andμ_{l} ≤μ_{r} .(4) An intervalvalued set function
is continuous from above if only only if the set functions
μ_{l} andμ_{r} are continuous from above, andμ_{l} ≤μ_{r} .(5) An intervalvalued set function
is subadditive if and only if the set functions
μ_{l} andμ_{r} are subadditive, andμ_{l} ≤μ_{r} .(6) An intervalvalued set function
is submodular if and only if the set functions
μ_{l} andμ_{r} are submodular, andμ_{l} ≤μ_{r} .(7) An intervalvalued set function
is nulladditive if and only if the set functions
μ_{l} andμ_{r} are nulladditive, andμ_{l} ≤μ_{r} .By using Definition 2.2 and Theorem 3.3, we define the Choquet integral of a nonnegative 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 nonnegative measurable functionf :X → ？^{+}, with respect to a monotone intervalvalued set functionis defined by
where
m is the Lebesgue measure on ？ and the integral on the righthand side is the Aumann integral with respect tom of(2) The Choquet integral of a realvalued measurable function
f :X → ？, with respect to a monotone intervalvalued set functionis defined by
where
f ^{+} = max{f ,0} andf ^{？}=max{？f ,0}, andis the conjugate of
that is,
(3) A measurable function
f is said to beintegrable if
We note that Eq. (36) implies
where
for all
α ∈ ？^{+}. By the definition ofwe easily get the following theorem.
Theorem 3..5. (1) A monotone intervalvalued set functionis continuous from below (resp. from above) if and only if
is continuous from above (resp. from below).
(2) If
is a monotone intervalvalued set function and
μ_{l} (X ) =μ_{r} (X ), thenwhere
In [21], we can find the theorem below. This gives a useful and interesting tool for the application of the Choquet integral of a nonnegative measurable function
f , with respect to a monotone intervalvalued set functionTheorem 3..6. ([23, Lemma 2.5 (i) and (v)]) Letf be a nonnegative measurable function anda monotone intervalvalued function. If
is continuous from above and we take
for all
α ∈ ？^{+}, then we have(1)
F is continuous from above, and(2)
where
m is the Lebesgue measure andμ_{lf} (α )=μ_{l} ({x ∈X f (x ) >α }) andμ_{rf} (α )=μ_{r} ({x ∈X f (x ) >α }).Note that Theorem 3.6(2) implies the following equation (36) under the same condition of
f andBy 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 intervalvalued set function.Theorem 3..7. Letf be a measurable function anda monotone intervalvalued set function. If
is continuous and
μ_{l} (X ) =μ_{r} (X ), then we haveProof. Letf ^{+} = max{f , 0} andf ^{？} = max{？f , 0}. Sinceis continuous from above, by (40), we have
Since
is continuous from below and
μ_{l} (X ) =μ_{r} (X ), by Theorem 3.5(2),is continuous from above. Thus, by (36), we have
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 intervalvalued set function.
Theorem 3..8. Letbe a monotone intervalvalued set function and let
A ∈ . IfA is continuous from above, then we have
Proof. Ifa ≥ 0, then by Eq. (36), we haveIf
a < 0, then by Eq. (36), we haveTheorem 3..9. Let a monotone intervalvalued set functionbe continuous from above and let
f a nonnegativeintegrable function. If
is continuous from above and
A ,B ∈ withA A ⊂B , then we haveProof. Sinceis a monotone intervalvalued set function, by Theorem 3.3 (1) and (2),
μ_{l} andμ_{r} are monotone intervalvalued set functions. Thus,By Eq. (36) and Eq. (42), we have the result.
We remark that if we take a
integrable function
f which isf ^{+} = 0 andf ^{？} > 0, thenis not monotone, that is, for each pair
A ,B ⊂ withA A ∈B ,Theorem 3..10. Letbe a monotone intervalvalued set function which is continuous from above, and let
A ∈ . IfA f andg are nonnegativeintegrable functions with
f ≤g , then we haveProof. The proof is similar to the proof of Theorem 3.10.Theorem 3..11. Letbe monotone intervalvalued set functions,
f a nonnegativeintegrable and
integrable function, and
A ∈ .A (1) If
then we have
(2) If
then we have
Proof. (1) Note thatif and only if
μ _{1l} ≤μ _{2l} andμ _{lr} ≤μ _{2r}. Thus, we haveBy (36) and (47), we have the result.
(2) Note that
if and only if
μ _{2l} ≤μ _{1l} andμ _{1r} ≤μ _{2r}. Thus, we haveBy Eq. (36) and eq. (48), we have the result.
Finally, we investigate the subadditivity of the Choquet integral under some conditions for the monotone intervalvalued set functions.
Theorem 3..12. Let (X , ) be a measurable space. If a continuous monotone intervalvalued set functionA with
μ_{l} (X ) =μ_{r} (X ), is submodular andare measurable functions, then we have
Proof. Sinceis a submodular monotone intervalvalued set function, by Theorem 3.3(6),
μ_{l} andμ_{r} are submodular monotone set functions.By Theorem 2.5, we have
and
By Eq. (36), eq. (50), and eq. (51), we have the result.
Theorem 3..13. Let (X , ) be a measurable space. If a continuous monotone intervalvalued set functionA with
μ_{l} (X ) =μ_{r} (X ), is subadditive, andare measurable functions with disjoint support, then
Proof. Sinceis a subadditive monotone intervalvalued set function, by Theorem 3.3(5),
μ_{l} andμ_{r} are subadditive monotone set functions. By Theorem 2.6, we haveand
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 intervalvalued set function of measurable functions.
From Theorem 3.2, Definition 3.3(3), and the condition that
μ_{l} (X ) =μ_{r} (X ) of a continuous monotone set function, we can deal with the new concept of the Choquet integral of a monotone intervalvalued set functionof 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 intervalvalued set function on the space of nonnegativeintegrable 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 biChoquet integral with respect to a monotone intervalvalued set function
> Conflict of Interest
No potential conflict of interest relevant to this article was reported.