The relationships between choquet integral, pan-integral, upper integral and lower integral

The Relationships between ChoquetIntegral, Pan-integral, Upper Integraland Lower IntegralGang Li∗

School of Mathematics and Physical Sciences, Shandong Institute of LightIndustry, Ji’nan, Shandong 250353, People’s Republic of China

Various types of integrals with respect to fuzzy measure on finite sets in the literature (Grabischet al., Fuzzy Measures and Integrals. Theory and Applications, Heidelberg, Germany, 2000; Wanget al., Int J Intell Syst 2006; 21:1073–1092; Wang and Klir, Int J Uncertainty, Fuzziness knowl-based Syst 1997;5(2):163–175; Wang and Klir, Fuzzy Measures Theory, New York; Plenum,1992) have been proposed, including Choquet integral, pan-integral, the upper integral, and thelower integral. In this paper, we present the relationships between these different types of integrals.The sufficient and necessary conditions for the equality between these integrals are discussed. C©2011 Wiley Periodicals, Inc.

1. INTRODUCTION

Integrals are the most important aggregation tools in information fusion. Theclassical integral including the weighted average method is taken with respect to anadditive set function and is linear functional on the class of real-valued measurablefunctions. However, in many real problems, the classical integral fails due to the in-teraction among attributes. Nonadditive set functions including fuzzy measures,1−4

imprecise probabilities5 have been used to describe the interaction. A new type ofintegral such as the Choquet integral2,3 with respect to them as new aggregationtools have been used in information fusion. The Choquet integral is nonlinear andpossess some properties. Furthermore, other types of nonlinear integrals, such aspan-integral,4,6 the lower integral, and the upper integral have been introduced inRef. 7. So, it is necessary to discuss the relationships between them.

This article is arranged as follows. Section 2 provides a basic knowledge onnonadditive set function and relevant nonlinear integrals, including Choquet integral,pan-integral, the lower integral, and the upper integral. In Section 3, the order of

∗Author to whom all correspondence should be addressed: e-mail: [email protected].

INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, VOL. 26, 464–473 (2011)C© 2011 Wiley Periodicals, Inc.

View this article online at wileyonlinelibrary.com. • DOI 10.1002/int.20477

RELATIONSHIPS BETWEEN CHOQUET AND NONLINEAR INTEGRALS 465

these integrals is given. Furthermore, the sufficient and necessary conditions for theequality between these nonlinear integrals are provided.

2. BASIC DEFINITIONS

DEFINITION 1. 1 Let X = {x1, x2, . . . , xn} be a finite set and μ a set function

μ : ℘(X) → [0, 1]

where ℘(X) is the power set of X. We will say μ is a fuzzy measure on X if itsatisfied:

(i)μ(ø) = 0; μ(X) = 1.

(ii)∀A, B ⊆ X, if A ⊆ B

then

μ(A) ≤ μ(B).

A specially interesting class of fuzzy measures is the capacities of order 2,because it covers a great numbers of fuzzy measures, and at the same time, capacitiesof order 2 possess enough mathematical properties.

DEFINITION 2. 1,3 Let a fuzzy measure μ is supermodular if and only of

∀A, B ⊆ X, μ(A ∪ B) + μ(A ∩ B) ≥ μ(A) + μ(B).

a fuzzy measure μ is submodular if and only if

∀A, B ⊆ X, μ(A ∪ B) + μ(A ∩ B) ≤ μ(A) + μ(B).

Some often used classes of fuzzy measures, such as belief measures, necessitymeasures, λ-measures (λ > 0 Sugeno4) are supermodular; plausibility measures,possibility measures, λ-measures (λ < 0 Sugeno4) are submodular.

DEFINITION 3. 2,3 Let μ be a fuzzy measure on X and f : X → R+0 a nonnegative

measurable function.The Choquet integral of f with respect to μ is defined as

(C)∫

f dμ =∫ +∞

0μ(Fα) dα

with Fα = {x ∈ X|f (x) ≥ α}.International Journal of Intelligent Systems DOI 10.1002/int

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Some of the important properties of the Choquet integral3 are

(1) (C)∫

IAdμ = μ(A), where IA is the indicator function of A.(2) (C)

∫f dμ ≤ (C)

∫g dμ, whenever f (x) ≤ g(x),∀x ∈ X.

(3) (C)∫

(f + g) dμ = (C)∫

f dμ + (C)∫

gdμ, in general.If f and g are commonotone (i.e., ∀x1, x2 ∈ X, f (x1) < f (x2), ⇒ g(x1) ≤ g(x2)),

then (C)∫

(f + g)dμ = (C)∫

f dμ + (C)∫

gdμ.(4) (C)

∫(f1 + f2) dμ ≥ (C)

∫f1 dμ + (C)

∫f2dμ ⇔ μ is supermodular;

(C)∫

(f1 + f2) dμ ≤ (C)∫

f1 dμ + (C)∫

f2dμ ⇔ μ is submodular.

DEFINITION 4. 6 s = ∑kj=1 aj IEj

is called a nonnegative simple function, where{E1, E2, . . . , Ek} is a partition of X and aj , j = 1, 2, . . . , k, are nonnegative con-stants. The set of all nonnegative simple function is denoted by S+. A value, insymbol Q(s), is assigned to function s by

Qs = sup

⎧⎨⎩

k∑j=1

ajμEj|s =

k∑j=1

aj IEj

⎫⎬⎭

in which μEj= μ(Ej ) for j = 1, 2, . . . , k. The pan-integral from below

(abbreviated as PFB-integral) of f with respect to μ, in symbol (pf b)∫

f dμ,is defined by

(pf b)∫

f dμ = sups≤f,s∈S+

Qs

When f is bounded, the pan-integral from above (abbreviated as PFA-integral) off with respect to μ, in symbol (pf a)

∫f dμ, is defined by

(pf a)∫

f dμ = infs≥f,s∈S+

Qs

When f is unbounded, the pan-integral from above (abbreviated as PFA-integral)of f with respect to μ,in symbol (pf a)

∫f dμ, is defined by

(pf a)∫

f dμ = limn→∞(pf a)

∫fn dμ

where

fn(x) ={f (x) f (x) ≤ n

n f (x) > n

for every x ∈ X.

When f is bounded, we may use f to replace fn in the definition directly.

International Journal of Intelligent Systems DOI 10.1002/int


The relationship between (pf b)∫

f dμ and (pf a)∫

f dμ is

PROPOSITION 1. 6 (pf b)∫

f dμ ≥ (pf a)∫

f dμ.

DEFINITION 5. 7 The upper integral of f with respect to μ denoted by (U )∫

f dμ

is defined by

(U )∫

f dμ = sup

⎧⎨⎩

2n−1∑j=1

ajμAj

∣∣∣2n−1∑j=1

aj IAj= f

⎫⎬⎭

where aj ≥ 0 and Aj = ⋃i:ji=1{xi} if j is binary digits as jnjn−1 . . . j1 for every

j = 1, 2, . . . 2n − 1.

The value of (U )∫

f dμ is the solution of the following linear programmingproblem, where a1, a2, . . . , a2n−1 are unknown parameters.

max z = ∑2n−1j=1 ajμj

s.t.∑2n−1

j=1 aj IAj(xi) = f (xi), i = 1, 2, . . . , n

aj ≥ 0, j = 1, 2, . . . , 2n − 1

(1)

in which μj = μAjfor j = 1, 2, . . . , 2n − 1.

DEFINITION 6. 7 The lower integral of f with respect to μ denoted by (L)∫

f dμ isdefined by

(L)∫

f dμ = inf

⎧⎨⎩

2n−1∑j=1

ajμAj

∣∣∣2n−1∑j=1

aj IAj= f

⎫⎬⎭

where aj ≥ 0 and Aj = ⋃i:ji=1{xi} if j is binary digits as jnjn−1 . . . j1 for every

j = 1, 2, . . . 2n − 1.The value of (L)

∫f dμ is the solution of the following linear programming

problem, where a1, a2, . . . , a2n−1 are unknown parameters:

min z = ∑2n−1j=1 ajμj

s.t.∑2n−1

j=1 aj IAj(xi) = f (xi), i = 1, 2, . . . , n

aj ≥ 0, j = 1, 2, . . . , 2n − 1

(2)

in which μj = μAjfor j = 1, 2, . . . , 2n − 1.


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Remark . The upper integral and lower integral of function f in Ref. 7 are alldefined with respected to signed fuzzy measure.

In this paper, we will mainly discuss the relationships between them andChoquet integral, pan-integral. So, above definitions are presented.

Furthermore, denote the class of nonnegative bounded measurable functionsby F .

3. MAIN RESULTS

In Ref. 7 Wang et.al. gave the order relation between Choquet integral andupper (lower) integral.

THEOREM 1 . Let f, g ∈ F be nonnegative functions on X, and let μ be fuzzymeasures on ℘(X), then

(L)∫

f dμ ≤ (C)∫

f dμ ≤ (U )∫

f dμ

So it is interesting to discuss the conditions under which the Choquet integralcoincides with the upper (or lower) integral.

THEOREM 2 . Let f ∈ F be any nonnegative functions on X, μ be fuzzy measureson ℘(X),

(C)∫

f d μ = (U )∫

f dμ

holds for every f ∈ F if and only if μ is supermodular.

Proof. (a) Necessary condition:If μ is not supermodular, then there are two sets A and B such that

μ(A) + μ(B) > μ(A ∪ B) + μ(A ∩ B)

So

(U )∫

(IA + IB) dμ ≥ μ(A) + μ(B) > μ(A ∪ B) + μ(A ∩ B)

= (C)∫

IA∪B dμ + (C)∫

IA∩B dμ = (C)∫

(IA∪B + IA∩B) dμ

= (C)∫

(IA + IB) dμ



where the above equalities are due to the properties (3) of the Choquetintegral.

It is a contradiction with the hypothesis.(b) Sufficient condition:If μ is supermodular, we assume (U )

∫f dμ = ∑2n−1

j=1 ajμAj, f = ∑2n−1

j=1 aj IAj

so

(U )∫

f dμ =2n−1∑j=1

ajμAj=

2n−1∑j=1

(C)∫

aj IAjdμ

≤ (C)∫ ⎛

⎝2n−1∑j=1

aj IAj

⎞⎠ dμ = (C)

∫f dμ

where the above inequality is due to the property (4) of the Choquet integral. Owingto Theorem 1,

(C)∫

f dμ = (U )∫

f dμ

With respect to the lower integral, we can get the similar conclusion. �

THEOREM 3. Let f ∈ F be any nonnegative functions on X, μ be fuzzy measureson ℘(X),

(C)∫

f dμ = (L)∫

f dμ

holds for every f ∈ F if and only if μ is submodular.

Example . Let X = {x1, x2, x3, x4}. Assume the fuzzy measure

μ(x1) = 0.4, μ(x1, x2) = 0.4, μ(x1, x3) = 0.4, μ(x1, x4) = 0.4,

μ(x2, x3) = 0.1, μ(x1, x2, x3) = 0.5, μ(x1, x2, x4) = 0.4,

μ(x1, x3, x4) = 0.7, μ(x2, x3, x4) = 0.1, μ(X) = 1,

for else sets A ⊆ X, μ(A) = 0. Let f : X → [0, 1], f (x1) = 3, f (x2) = 9,

f (x3) = 6, f (x4) = 4.So, we have (C)

∫f dμ = 3.3.


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For the upper integral (U )∫

f dμ, we can get the value is the solution offollowing linear programming:

max z = 0.4a1 + 0.4a3 + 0.4a5 + 0.4a9 + 0.1a6

+ 0.5a7 + 0.4a11 + 0.1a14 + a15

s.t. a1 + a3 + a5 + a7 + a9 + a11 + a13 = 3a2 + a3 + a6 + a7 + a10 + a11 + a14 = 9a4 + a5 + a6 + a7 + a12 + a13 + a14 = 6a8 + a9 + a10 + a11 + a12 + a13 + a14 = 4aj ≥ 0, j = 1, 2, . . . , 15

(3)

(U )∫

f dμ = 3.3.

We can find μ is belief measure, which is supermodular.If μ is replaced by its dual measure μ∗, i.e., μ∗(A) = 1 − μ(Ac), ∀A ⊆ X,then (C)

∫f dμ∗ = 5.7.

The value of the lower integral (L)∫

f dμ∗ is the solution of following linearprogramming:

max z = 0.9a1 + 0.3a2 + 0.6a4 + 0.5a8 + a3 + a5 + 0.9a9 + 0.6a6

+ 0.6a10 + 0.6a12 + a7 + a11 + a13 + 06a14 + a15

s.t. a1 + a3 + a5 + a7 + a9 + a11 + a13 = 3a2 + a3 + a6 + a7 + a10 + a11 + a14 = 9a4 + a5 + a6 + a7 + a12 + a13 + a14 = 6a8 + a9 + a10 + a11 + a12 + a13 + a14 = 4aj ≥ 0, j = 1, 2, . . . , 15

(4)

(L)∫

f dμ∗ = 5.7.

We can find that μ∗ is plausibility measure, which is submodular.

THEOREM 4. (U )∫

f dμ ≥ (pf b)∫

f dμ ≥ (pf a)∫

f dμ ≥ (L)∫

f dμ for everyf ∈ F

Proof. In the definition of (L)∫

f dμ, the class {E1, E2, . . . , Ek} may be eitherinteractive or disjoint and may or may not cover X. However, in the definition of(pf a)

∫f dμ,{E1, E2, . . . , Ek} is restricted to be a partition of X.

So

(pf a)∫

f dμ ≥ (L)∫

f dμ



Similarly,

(U )∫

f dμ ≥ (pf b)∫

f dμ

We can get the conclusion by Proposition 1. �

THEOREM 5. when μ is supermodular measure. Then

(U )∫

f dμ = (C)∫

f dμ = (pf b)∫

f dμ

for every f ∈ F if and only if μ is additive or simple supported.

Proof. (a) Sufficient condition:There exist ai ≥ 0, Ej ∈ ℘(X) such that

f =n∑

i=1

aiIEi

and

(U )∫

f dμ =n∑

i=1

ajμEj

If μ is additive, the three integrals all coincide with the Lebesgue integral. If μ issimple support measure with support set A, i.e.,

μ(E) ={

1 A ∈ E

0 otherwise

Denote

s(x) ={∑

i:Ei⊃A ai x ∈ A

0 otherwise

It is clear that s ∈ S+, s ≤ f , and

Qs ≥∑

i:Ei⊃A

ai ≥n∑

i=1

ajμEj


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Thus

(pf b)∫

f dμ ≥n∑

i=1

ajμEj= (U )

∫f dμ

We can get the conclusion by the Theorem 2,4.(b) Necessary condition:If μ is neither an additive measure nor a simple supported measure.

Let m be the mobius transform8 of supermodular measure μ. We know that m

focuses only on singletons when μ is an additive measure and m focuses on just oneset when μ is a simple support measure. Hence, if μ is neither an additive measurenor a simple supported measure, then there exist sets A and B, where A consists of atleast two points, such that m(A) > 0 and m(B) > 0. Without any loss of generality,we can assume that B do not contain A. Taking x ∈ A − B, E = (A ∪ B) − {x} andF = {x}, we have μ(E) ≥ μ(B) > 0, E ∩ F = ∅ and

μ(E ∪ F ) − μ(E) − μ(F ) ≥ m(A) > 0

This means that

μ(E ∪ F ) > μ(E) + μ(F )

If μ(E) = a > 0, μ(F ) = b, μ(E ∪ F ) = c > a + b, we take

f (x) =

⎧⎪⎨⎪⎩

1 x ∈ Ea

c − bx ∈ F

0 otherwise

Thus

(U )∫

f dμ = (C)∫

f dμ = ac

c − b+ (1 − a

c − b)a >

ac

c − b= (pf b)

∫f dμ

This is a contradiction with

(U )∫

f dμ = (C)∫

f dμ = (pf b)∫

f dμ

for every f ∈ F . �

THEOREM 6 . (U )∫

f dμ = (C)∫

f dμ = (pf b)∫

f dμ = (pf a)∫

f μ = (L)∫

f dμ for every f ∈ F if and only if μ is additive.



Proof. If μ is additive, they all coincide with the Lebesgue integral. If all theintegrals are equal, then μ is supermodular and submodular (i.e., additive) byTheorem 2,3. �

4. CONCLUSION

We have given the sufficient and necessary conditions for the coincidence ofChoquet integral (or pan-integral) and the upper (or lower) integral on finite set,respectively. The relationships between these nonlinear integral on general set X,which is not necessarily finite, are still open problem.

References

1. Campos LM, Bolanos MJ. Representaion of fuzzy measures through probabilities. FuzzySet Syst 1989;31:23–36.

2. Choquet G. Theory of capacities. Ann l’Inst Fourier 1954;5:131–295.3. Grabisch M, Murogushi T, Sugeno M. Fuzzy measures and integrals. Theory and applica-

tions. Heidelberg, Germany: Physica-Verlag; 2000.4. Wang Z, Klir GJ. Fuzzy measures theory. New York: Plenum; 1992.5. Walley P. Statistical reasoning with imprecise probabilities. London: Chapman and Hall;

1991.6. Wang Z, Klir GJ. PFB-integral and PFA-integrals with respect to monotone set functions,

Int J Uncertainty, Fuzziness Knowl-Based Syst 1997;5(2):163–175.7. Wang Z, Leung KS, Klir GJ. Integration on finite sets. Int J Intell Syst 2006;21:1073–1092.8. Chateauneuf A, Jaffray JY. Some characterization of lower probabilities and other monotone

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The relationships between choquet integral, pan-integral, upper integral and lower integral