Generalization of inclusion–exclusion integral for nondiscrete monotone measure space

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Generalization of inclusion–exclusion integral for nondiscrete monotone measure space Aoi Honda a,∗ , Yoshiaki Okazaki b a Kyushu Institute of Technology, 680-4 Kawazu, Iizuka, Fukuoka, 820-8502, Japan b Fuzzy Logic Systems Institute, 680-41 Kawazu, Iizuka, Fukuoka, 820-0067, Japan

Received 10 July 2017; received in revised form 3 June 2018; accepted 4 June 2018

Abstract We generalize the inclusion–exclusion integral for nondiscrete monotone measure spaces. The inclusion–exclusion integral for a monotone measure space on a finite space is discussed in previous papers. This is an integral with respect to a nonadditive monotone measure, which includes the Lebesgue and Choquet integrals. We are concerned with the monotonicity of the integral, and a sufficient condition for the monotonicity of the t-norm based inclusion–exclusion integral is given by introducing the complete monotonicity of the t-norm based interaction operator. Moreover concrete examples of the interaction operators that satisfy the complete monotonicity are shown. © 2018 Elsevier B.V. All rights reserved. Keywords: Monotone measure; Nonadditive measure; Interaction operator; t-Norm; Measurable partition; Inclusion–exclusion integral

1. Introduction In non-additive measure theory, several nonlinear integrals with respect to a monotone measure, including the Choquet and Sugeno integrals, have been defined and discussed (e.g. [1–5]). We previously proposed an integral with respect to a monotone measure that is called the inclusion–exclusion integral and introduced the concept of the interaction operator [6,7]. This integral is for nonnegative functions on finite space and includes the Lebesgue and Choquet integrals as special cases. In this paper, we generalize the inclusion–exclusion integral to infinite space. Here, the integrands are allowed to be nonnegative bounded functions on a general infinite space. The organization of this paper is as follows. In Section 2, we recall the inclusion–exclusion integral on finite space and introduce the t-norm-based interaction operator. We show several properties of the integral with the t-norm-based interaction operator. In particular, quasi-surjectivity and a kind of monotonicity of t-norm used as interaction operators are sufficient conditions for monotonicity of the inclusion–exclusion integral. In Section 3, we generalize the inclusion–exclusion integral to the nondiscrete case, that is, the integral on infinite space. We give a sufficient condition for the interaction operator * Corresponding author.

E-mail address: [email protected] (A. Honda). https://doi.org/10.1016/j.fss.2018.06.005 0165-0114/© 2018 Elsevier B.V. All rights reserved.

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that provides the inclusion–exclusion integral appropriate properties as the integral and we show that the inclusion– exclusion integral coincides with the Choquet integral if the interaction operator is based on the minimum product t-norm. Moreover, in Section 4 we show how to construct the interaction operator using t-norms and show the product t-norm, the minimum product and Dubois and Prade’s t-norm satisfy quasi-surjectivity and completely monotonicity. The inclusion–exclusion integral with interaction operators based on these t-norms satisfy monotonicity. In the Appendix, some proofs that are not included in the main sections are shown. 2. Inclusion–exclusion integral for the finite monotone measure space In this section, the inclusion–exclusion integral for functions on a finite space is introduced. A finite n-point set is denoted by  := {1, . . . , n}, P() denotes the power set of , and the cardinal number of a subset A of a whole set  is denoted by |A|. If A ∩ B = ∅, then the union of sets A and B is denoted by A  B instead of A ∪ B when emphasizing disjoint union. A set function μ : P() → [0, +∞] is a monotone measure if μ satisfies (MM1) μ(∅) = 0, and (MM2) For any A, B ∈ P(), A ⊂ B implies μ(A) ≤ μ(B). For K ∈ (0, ∞), F[0,K] () denotes the set of all [0, K]-valued functions f :  → [0, K], which is represented by f = (f1 , f2 , . . . , fn ) ∈ [0, K]n , where fi := f (i), i ∈ . Definition 1 (interaction operator [7]). I : F[0,K] () × P() → [0, K] is an interaction operator if I satisfies the following conditions: (I1) For any f ∈ F[0,K] (), I (f | ∅) = K. (I2) For any f = (f1 , f2 , . . . , fn ) ∈ F[0,K] () and any i ∈ , I (f | {i}) = fi . (I3) For any f, g ∈ F[0,K] () and any A ∈ P(), f ≤ g implies I (f | A) ≤ I (g | A), where f ≤ g means fi ≤ gi for any i ∈ . (I4) For any f ∈ F[0,K] () and any A, B ∈ P(), A ⊂ B implies I (f | A) ≥ I (f | B). We call (, P(), μ, I, [0, K]) a finite interactive monotone measure space. The inclusion–exclusion integral of f ∈ F[0,K] () is defined with the interaction operator as follows. Definition 2 (discrete inclusion–exclusion integral [7]). Let (, P(), μ, I, [0, K]) be a finite interactive monotone measure space. Then, the inclusion–exclusion integral of f ∈ F[0,K] () with respect to μ and I is defined by   I (I ) f dμ := M (f | A) μ(A), A∈P ()\{∅}



where I M (f | A) :=



(−1)|B\A| I (f | B).

B∈P (), B⊃A I (f | A) to be non-negative for any f ∈ F We give the condition needed for M [0,K] () and any A ∈ P(A).

Definition 3 (disjoint meet supermodularity of order k). Let k be a natural number. We say a set function ϕ : P() → [0, K] is disjointly “meet” supermodular of order k if it satisfies for any m ≤ k and any S, S1 , . . . , Sm ∈ P() satisfying Si  S, i = 1, . . . , m and Si ∩ Sj = S for any i, j , 1 ≤ i < j ≤ m, m       |A|+1 ϕ Si ≥ (−1) ϕ Si . i=1

A⊂{1,...,m}, A=∅

i∈A

If ϕ satisfies this condition for all natural numbers, ϕ is said to be totally disjointly meet supermodular.

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Proposition 4. Let (, P(), μ, I, [0, K]) be a finite interactive monotone measure space. If I is disjointly meet n I supermodular  of order || − 1 (= n − 1), then, for any x ∈ [0, K] and any A ∈ P(), M (x | A) ≥ 0 holds, which implies (I )  f dμ ≥ 0 for any f ∈ F[0,K] (). Proof. Let f ∈ F[0,K] () and let A be a nonempty subset of . Case 1. |A| = n − 1. By (I4), we have M I (f | A) = I (f | A) − I (f | ) ≥ 0. Case 2. |A| < n − 1. Put m := | \ A|, then we have m ∈ {2, . . . , n − 1}. We may assume A = {m + 1, m + 2, . . . , n} without loss of generality. Put Ai := A  {i}, i = 1, . . . , m. Then by the disjoint meet supermodularity of I ,  M I (f | A) = I (f | A) + (−1)|B\A| I (f | B) B∈P (), BA

= I (f | A) +



(−1)|B| I (f | A  B)

B⊂{1,...,m},B=∅

 m   =I f Ai − i=1

which implies  (I ) f dμ =





A∈P ()\{∅}

 B⊂{1,...,m},B=∅

   (−1)|B|+1 I f Ai ≥ 0, i∈B

M I (f | A)μ(A) ≥ 0

by the nonnegativity of μ. 2 Remark 5. For order 1, disjointly meet supermodularity does not make sense. For order 2, I (f | A) + I (f | A  B  C) ≥ I (f | A  B) + I (f | A  C) corresponds to the supermodularity. For comparison supermodularity can also be generalized to the “join” supermodularity of order m, which directly corresponds to the inclusion–exclusion principle. Join supermodularity is discussed in several papers (for example [8]). The assumption of A1 , . . . , Am ⊂  such that Ai ∩ Aj = A is not needed for join supermodularity, in contrast to the “meet” submodularity. Next, we construct concretely the interaction operators using a t-norm. First, we generalize the classical t-norm [9] (see also [10]), which is defined on [0, 1], to a t-norm on [0, K], K ∈ (0, ∞). Definition 6 (generalized) t-norm. A binary operator ⊗ : [0, K]2 → [0, K], K ∈ (0, +∞) is called a t-norm on [0, K] if it satisfies the following conditions for any x, y, z ∈ [0, K]: (GT1) (GT2) (GT3) (GT4)

x ⊗ 0 = 0, x ⊗ K = x for x ∈ [0, K]. x ≤ y implies x ⊗ z ≤ y ⊗ z. x ⊗ y = y ⊗ x. (x ⊗ y) ⊗ z = x ⊗ (y ⊗ z).

By (GT4), the t-norm can be extended to multivariable functions |A| = k, A = {i1 , i2 , . . . , ik } as follows:

K, A = ∅, xi := xi1 ⊗ xi2 ⊗ · · · ⊗ xik , A = {i1 , i2 , . . . , ik } i∈A

for {xi | i ∈ A} ∈ [0, K]k , A = {i1 , i2 , . . . , ik }.



i∈A xi

:

∞

k k=0 [0, K]

→ [0, K], where A ⊂ N,

(1)

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Remark 7. The classical t-norms ⊗ on [0, 1] can be generalized to ⊗G on [0, K], K ∈ (0, +∞), by x y x ⊗G y := ⊗ K K K for x, y ∈ [0, K]. The following t-norms on [0, K] are generalized from the classical t-norms, minimum product, product t-norm, Łukasiewicz t-norm, Hamacher product [11], Dubois–Prade t-norm [12], and Sugeno–Weber t-norm [13]. For x, y ∈ [0, K], x ∧ y := min(x, y), xy , x ⊗P y := K x ⊗L y := max(x + y − K, 0), xy , x ⊗H y := K(x + y) − xy xy , 0 ≤ λ ≤ K, x ⊗DP λ y := max(x, y, λ)   x + y − K + λxy/K x ⊗SW y := max , 0 , λ > −1. λ 1+λ Proposition 8. Let ⊗ be a t-norm on [0, K]. For f ∈ F[0,K] () and A ∈ P(), put I : F[0,K] () × P() → [0, K] as I (f | A) := fi . (2) i∈A

Then, I becomes an interaction operator. Proof. Conditions (I1), (I2), (I3), and (I4) can be easily checked by (GT2), (1), and (2). 2 The interaction operator defined by (2) is called the t -norm-based or ⊗-based interaction operator. Alternatively, ⊗ is directly called the interaction operator. Denote by (, P(), μ, ⊗, [0, K]) in the case where I is the ⊗-based interaction operator. Moreover, if an interaction operator I is based on ⊗, then the inclusion–exclusion integral is I ⊗ called  the t-norm-based or ⊗-based inclusion–exclusion integral, and M and (I ) f dμ are also denoted by M and ⊗ f dμ, respectively. Definition 9 (monotone t-norm of order k, completely monotone t-norm). Let ⊗ be a t-norm on [0, K] and k be a natural number. We say ⊗ is a monotone t-norm of order k if ⊗ satisfies, for any natural number m satisfying m ≤ k, any (x1 , . . . , xm ) ∈ [0, K]m and any x0 ∈ [0, K],    |A| (−1) x0 ⊗ xi ≥ 0. A⊂{1,...,m}

i∈A

If ⊗ satisfies the condition for all natural numbers k, ⊗ is called a completely monotone t-norm (cf. “completely monotone function” and “Bernstein function”). Definition 10 (quasi-surjectivity). Let ⊗ be a t-norm on [0, K]. If for any x ∈ (0, K], a ∈ [0, x], there exists y ∈ [0, K] such that x ⊗ y = a, then we say ⊗ is quasi-surjective. Proposition 11. Let (, P(), μ, ⊗, [0, K]) be a finite interactive monotone measure space. If ⊗ is a monotone t-norm of order k, then ⊗-based interaction operator defined by (2) in Proposition 8 is disjointly meet supermodular of order k.

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Proof. Putting I (f | S) := x0 , I (f | Si \ S) := xi , we have m       I Si − (−1)|A|+1 I f Si i=1

A⊂{1,...,m}, A=∅

i∈A



= I (f | S) −

(−1)

|A|+1

i∈A

A⊂{1,...,m}, A=∅



= x0 −

(−1)

=

(−1)



|A|+1

x0 ⊗



 xi

i∈A

A⊂{1,...,m}, A=∅



    I f S Si \ S



|A|

x0 ⊗



 xi ≥ 0. 2

i∈A

A⊂{1,...,m}

Corollary 12. If ⊗ is a completely monotone t-norm, then the t-norm-based interaction operator is totally disjoint meet supermodular. The product t-norm, the minimum product and Dubois and Prade’s t-norm are completely monotone (cf. Proposition 26). Proposition 13. Let (, P(), μ, ⊗, [0, K]) be a finite interactive monotone measure space. If the t-norm ⊗ is quasisurjective and monotone of order n, then the inclusion–exclusion integral with respect to μ and ⊗ is monotone, that is, for any f, g ∈ F[0,K] () satisfying f ≤ g, it holds that   ⊗ f dμ ≤ ⊗ gdμ. 



 Proof. If I is a ⊗-based interaction operator, then ⊗  f dμ is invariant for any permutations of , so we have only to prove the assertion for f = (f1 , . . . , fn ), g = (f1 . . . , fn−1 , gn ), fn < gn . We generally have for (I )  f dμ,   (I ) g dμ − (I ) f dμ 

=





A∈P ()\{∅}

=





M I (g | A) μ(A) − 

M I (f | A) μ(A)

A∈P ()\{∅}

 M I (g | A) μ(A) − M I (f | A) μ(A)

A⊂\{n}, A=∅

  + M I (g | A  {n}) μ(A  {n}) − M I (f | A  {n}) μ(A  {n})   + M I (g | {n}) μ({n}) − M I (f | {n}) μ({n}) ⎞ ⎛    ⎝ ⎠ (−1)|B\A| {I (g | B) − I (f | B)} μ(A) = + A⊂\{n}, A=∅

B⊃A,B n /

+

 B⊃A{n}

B⊃A,Bn

 (−1)|B\A|−1 {I (g | B) − I (f | B)} μ(A  {n})

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+



(−1)|B\{n}|−1 {I (g | B) − I (f | B)} μ({n})

B⊃{n}



=

A⊂\{n}, A=∅

−





(−1)|B\A| {I (f | B) − I (f | B)} μ(A)

B⊃A,B n /



(−1)|B\A|−1 {I (g | B) − I (f | B)} μ(A)

B⊃A{n}



+



(−1)

|B\A|−1

{I (g | B) − I (f | B)} μ(A  {n})

B⊃A{n}

+



(−1)|B\{n}|−1 {I (g | B) − I (f | B)} μ({n})

B⊃{n}





=

(−1)|B\A|−1 {I (g | B) − I (f | B)} {μ(A  {n}) − μ(A)}

A⊂\{n}, B⊃A{n} A=∅

+



(−1)|B\{n}|−1 {I (g | B) − I (f | B)} μ({n})

B⊃{n}

⎛



=



⎝{μ(A  {n}) − μ(A)}

⎞ (−1)|B\A|−1 {I (g | B) − I (f | B)}⎠ .

B⊃A{n}

A⊂\{n}

Hence we have only to show that for any A ⊂  \ {n},  (−1)|B\A|−1 {I (g | B) − I (f | B)} ≥ 0 B⊃A{n}

holds. We may assume  \ (A  {n}) = {1, 2, . . . , m} without loss of generality. Then m takes a value from 0 for A =  \ {n}, to n − 1 for A = ∅. Case 1. A =  \ {n}.  (−1)|B\A|−1 {I (g | B) − I (f | B)} = I (g | ) − I (f | ) ≥ 0 B⊃A{n}

by (I3). Case 2. A   \ {n}.  (−1)|B\A|−1 {I (g | B) − I (f | B)} B⊃A{n}

=



(−1)|(A{n}B)\A|−1 {I (g | A  {n}  B) − I (f | A  {n}  B)}

B⊂\(A{n})

=



(−1)|B| {I (g | A  {n}  B) − I (f | A  {n}  B)}

B⊂{1,...,m}

Put Sj := S  {n}  {j }, j = 0, 1, . . . , m. On the other hand, by the monotonicity of order n, for any x0 ∈ [0, K], x := {x1 , . . . , xm } ∈ [0, K]m , we obtain    0≤ (−1)|B| x0 ⊗ xi i∈B

B⊂{1,...,m,m+1}

=



(−1)|B| x0 ⊗

B⊂{1,...,m+1},B {m+1} /



i∈B

 xi

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+

(−1)

|B|

B⊂{1,...,m+1},B{m+1}





=

(−1)

|B|

x0 ⊗



+

(−1)

|B|+1

B⊂{1,...,m}

Putting x0 := x0 ⊗





0≤

xi

xi 



 xi .

i∈B



, we have     |B|  (−1) x0 ⊗ xi ⊗ xi i∈A xi

i∈A





− (−1)|B| x0 B⊂{1,...,m} 



(−1)|B|



⊗





(−1)|B|

xi ⊗ xm+1 ⊗

xi



 xi



 xi

i∈B

i∈B

  xi ⊗ (x0 ⊗ xm+1 ). ⊗

i∈A

B⊂{1,...,m}



  xi ⊗ x0 ⊗

i∈A



i∈B



i∈A



B⊂{1,...,m}

−



(x0 ⊗ xm+1 ) ⊗

B⊂{1,...,m}

=



i∈B

i∈B

B⊂{1,...,m}



x0 ⊗



7

i∈B

x0

We replace xi := fi , i = 1, . . . , m and := gn , and note that there exists y ∈ [0, K] such that x0 ⊗ y = gn ⊗ y = fn by the quasi-surjectivity of ⊗. Then, by replacing xm+1 := y, we obtain           |B| 0≤ (−1) fi ⊗ fi ⊗ gn − fi ⊗ f i ⊗ fn i∈A

B⊂{1,...,m}

=



(−1)|B\A|−1

B⊃A{n}

=





B⊃A{n}

i∈B



fi ⊗

i∈A

 (−1)|B\A|−1





gi −

i∈B





fi ⊗ gn −

i∈B





i∈A



fi .

i∈A



i∈B

fi ⊗







f i ⊗ fn

i∈B

2

i∈B

Remark 14. The concave integral for discrete monotone measure space (, P(), μ) is defined as follows [2]. ⎧ ⎫  ⎨   ⎬ f dμ := max aA μ(A) aA χA = f, aA ≥ 0 . ⎩ ⎭ A∈P ()

cav

#

A∈P ()

I gives a special decomposition of f , so that the Putting aA := | A) we have A∈P () aA χA = f , that is, M concave integral and our inclusion–exclusion integral are based on a similar concept. Incidentally, if μ is superadditive, then the concave integral corresponds with the Choquet integral. If μ is subadditive, then the concave integral corresponds with the Lebesgue integral. I (f M

3. Generalization of the inclusion–exclusion integral for nondiscrete monotone measure space In this section, we assume the whole space is X, not finite. B denotes a σ -algebra over X. A set function μ : B → [0, +∞] is a monotone measure if μ satisfies (GMM1) μ(∅) = 0, and (GMM2) For any A, B ∈ B, A ⊂ B implies μ(A) ≤ μ(B).

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The interaction operator has to be slightly generalized to the nondiscrete case. Let F[0,K](X) denote the set of all [0, K]-valued function f : X → [0, K]. Note that the functions do not necessarily need to be B-measurable. Definition 15 (interaction operator on a general space). I : F[0,K] (X) × B → [0, K] is an interaction operator if I satisfies the following conditions: For any f ∈ F[0,K] (X), I (f | ∅) = K. For any a ≥ 0 and any S ∈ B, there exists A ∈ B such that I (a χS | A) = a. For any f, g ∈ F[0,K] (X) and A ∈ B, f ≤ g implies I (f | A) ≤ I (g | A). For any f ∈ F[0,K] (X) and A, B ∈ B, A ⊂ B implies I (f | A) ≥ I (f | B).

(GI1) (GI2) (GI3) (GI4)

We call (X, B, μ, I, [0, K]) an infinite interactive monotone measure space. We define an integral for nonnegative functions on a general space X using the inclusion–exclusion integral for discrete$ functions. D = {D1 , D2 , . . . , Dn }, Di ∈ B denotes a partition of X, that is, Di ∩ Dj = ∅, 1 ≤ i < j ≤ n and ni=1 Di = X and let (X) be the set of all partitions D of X. A non-negative simple# function fD ≥ 0 associated with the partition D = {D1 , D2 , . . . , Dn }, Di ∈ B is the function of the form fD (x) = ni=1 ai χDi (x), ai ≥ 0, i = 1, 2, . . . , n. Denote by σ (D), the sub-σ -algebra of B generated by the partition D, that is, σ (D) is the family of all finite unions of elements of D. For A ∈ σ (D), |A|D denotes the number of elements Di ∈ D included in A. Definition 16 (inclusion–exclusion integral on general space). Let (X, B, μ, I, [0, K]) be an infinite interactive monotone measure space. Then, the inclusion–exclusion integral of f ∈ F[0,K] (X) with respect to μ and I is defined by ⎞ ⎛   ⎜  ⎟ |B\A|D ⎟ μ(A). ⎜ (I ) f dμ := sup (−1) I (f | B) D ⎠ ⎝ D ∈(X), A∈σ (D ), fD ≤f A=∅

Let I MD (fD | A) :=

and

 fD dμ :=

(I ) D

then we can express  (I ) f dμ =



B∈σ (D ), B⊃A

(−1)|B\A|D I (fD | B)

B∈σ (D ),B⊃A

 A∈σ (D ),A=∅

I MD (fD | A)μ(A),

 sup

D ∈(X),fD ≤f



(I )

fD dμ. D

(I ) D fD dμ corresponds to the inclusion–exclusion integral for the finite case. Indeed for a partition D = {D1 , D2 , . . . , DN }, let ∼ be the equivalence relation on X defined by D, that is, x ∼ y means that there exists i ∈ {1, 2, . . . , N } such that x, y ∈ Di . Then the quotient space X/∼ is the finite set of N -points. Denote by D = X/∼ = {d1 , d2 , . . . , dN } and ϕ : X → D be the quotient map defined by ϕ(x) := di for x ∈ Di , i = 1, 2, . . . , N . Remark that ϕ : (X, σ (D)) → (D , P(D )) is measurable. We shall introduce the finite interactive monotone measure space (D , P(D ), μD , ID , [0, K]) by μD (U ) := μ(ϕ −1 (U )) for U ∈ P(D ) and ID (FD | U ) := I (FD ◦ ϕ | ϕ −1 (U )) for U ∈ P(D ), FD : D → [0, K]. Then it holds that for any A ∈ σ (D) there exists U ⊂ D such that

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A = ϕ −1 (U ) =



9

Di

i:di ∈U

for any σ (D)-measurable function fD : X → [0, K], there exists P(D )-measurable function FD : D → [0, K] such that fD = FD ◦ ϕ, and I (fD | A) = I (FD ◦ ϕ | |ϕ −1 (U )) = ID (FD | U ). So that in the above definition, we have   (I ) fD dμ = (ID ) FD dμD D

D

where the right hand side is the inclusion–exclusion integral on the finite interactive monotone measure space (D , P(D ), μD , ID , [0, K]). We show that under the condition that M I is nonnegative, a finer D leads to a larger inclusion–exclusion integral on D. Proposition 17. Let (X, B, μ, I, [0, K]) be an infinite interactive monotone measure space, D ∈ (X), and let fD n  I (f | A) ≥ 0 for any be a nonnegative simple function on X satisfying fD := ai χDi , ai ≥ 0. If I satisfies MD D D ∈ (X), any fD and any A ∈ σ (D), then it holds that   (I ) fD dμ ≤ (I ) fD dμ

i=1

D

D

for any D ∈ (X) which are refinements of D. Lemma 18. Let (X, B, μ, I, [0, K]) be an infinite interactive monotone measure space, D = {D1 , . . . , Dn } ∈ (X), n  ai χDi , ai ≥ 0. For any refinement D ∈ (X) and fD be a nonnegative simple function on X such that fD := of D, any i ∈ {1, . . . , n} and any A ∈ σ (D \ {Di }), A = ∅, we have

i=1

I I I  I  MD (fD | A) = MD  (fD | A) + MD  (fD | A  Di ) + MD  (fD | A  Di ).

Proof. We have only to prove the assertion for the refinement of the form D = {D1 , . . . , Di−1 , Di  , Di  , Di+1 , . . . , Dn } ∈ (X), Di   Di  = Di . For any A ∈ σ (D \ {Di }), A = ∅, we have  I MD (−1)|B\A|D I (fD | B)  (fD | A) = B∈σ (D  ), B⊃A

⎛ ⎜ ⎜ =⎜ ⎝ ⎛ ⎜ ⎜ =⎜ ⎝



+

B∈σ (D  ),B⊃A, B∩(Di Di )=∅

 B⊂σ (D  ),B⊃A, B∩(Di Di )=∅

⎞ 

+

B∈σ (D  ),B⊃ADi , B∩Di =∅

−

 B∈σ (D  ), B⊃A(Di Di )

+



+

B∈σ (D  ),B⊃ADi , B∩Di =∅

 B∈σ (D  ),B⊃(ADi ), B∩Di =∅

+

 B∈σ (D  ), B⊃A(Di Di )

 B∈σ (D  ), B⊃A(Di Di )

⎟ ⎟ ⎟ (−1)|B\A|D I (fD | B) ⎠

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⎞ 

+

B∈σ (D  ),B⊃ADi , B∩Di =∅

⎞

⎛ ⎜ ⎜ =⎜ ⎝

 B∈σ (D  ),B⊃A, B∩(Di Di )=∅

B∈σ (D  ), B⊃A(Di Di )

⎛

⎜ ⎜ −⎜ ⎝ ⎛ ⎜ ⎜ −⎜ ⎝



+





⎟ ⎟ ⎟ (−1)|B\A|D I (fD | B) ⎠

⎟ ⎟ ⎟ (−1)|B\A|D I (fD | B) ⎠ ⎞

B∈σ (D  ), B⊃A(Di Di )

⎟  ⎟ ⎟ (−1)|B\(ADi )|D I (fD | B) ⎠



+

B∈σ (D  ),B⊃ADi , B∩Di =∅

B∈σ (D  ), B⊃A(Di Di )



+

B∈σ (D  ),B⊃ADi , B∩Di =∅



+

B∈σ (D  ), B⊃A(Di Di )

⎞ ⎟  ⎟ ⎟ (−1)|B\(ADi )|D I (fD | B) ⎠

I I  I  (fD | A) − MD = MD  (fD | A  Di ) − MD  (fD | A  Di ).

2

Proof of Proposition 17. 

A∈σ (D  )\{∅}

D

⎛ ⎜ ⎜ =⎜ ⎝ =

 A∈σ (D  ), A⊃Di



A∈σ (D  ), A⊃Di Di

+

A∈σ (D  ), A⊃Di Di

A=Di ,Di



+

A∈σ (D  ),A=Di ,Di , A∩Di =Di or Di



A=Di ,Di

 A∈σ (D  )\{∅}, A∩Di =∅

⎟⎟ ⎟⎟ I ⎟⎟ MD (fD | A) μ(A) ⎠⎠

I MD  (fD | A) μ(A)

I I   MD  (fD | A) μ(A) + MD  (fD | A  Di ) μ(A  Di )

I MD (fD | A) μ(A) +





⎜ ⎜ +⎜ ⎝ 



A∈σ (D  )\{∅}, A∩(Di Di )=∅

=

⎞⎞

I MD (fD | A) μ(A) +







+

A∈σ (D  )\{∅}, A∩(Di Di )=∅

≥

I MD  (fD | A) μ(A)

⎛

A∈σ (D  ), A⊃Di Di

+



fD dμ =

(I )

 A=Di ,Di

 I   +MD  (fD | A  Di ) μ(A  Di ) I MD  (fD | A) μ(A)

  I I  I  MD  (fD | A) + MD  (fD | A  Di ) + MD  (fD | A  Di ) μ(A)

I MD (fD | A) μ(A) +

 A=Di ,Di

I MD  (fD | A) μ(A) +

 A∈σ (D  )\{∅}, A∩(Di Di )=∅

I MD (fD | A) μ(A)

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=

A∈σ (D )\{∅}



≥

A∈σ (D )\{∅}

I MD (fD | A) μ(A) +

 A=Di ,Di

11

I MD  (fD | A) μ(A)

I MD (fD | A) μ(A)



= (I )

fD dμ D

The third equality from the bottom is obtained by Lemma 18.

2

At the end of this section, we show the relation between the inclusion–exclusion integral and the Choquet integral. Denote by (X, B, μ, ∧, [0, K]) in the case where I is the interaction operator given by the minimum product ∧ as ' I (f | A) = f (x) = inf f (x). x∈A

x∈A

If (, P(), μ, ∧, [0, K]) is a “finite” interactive monotone measure space, then the inclusion–exclusion integral on  with respect to μ and ∧ coincides with the Choquet integral [7]. The same is true of the nondiscrete case (X, B, μ, ∧, [0, K]). The definition of the Choquet integral in the nondiscrete case is as follows. Definition 19 (Choquet integral [1,14]). Let (X, B, μ) be a monotone measure space and f ∈ F[0,K] (X) be a B-measurable function. The Choquet integral of f with respect to μ is defined by ∞

 (C)

f dμ :=

μ({x | f (x) ≥ t}dt, 0

where the integral on the right-hand side is the Lebesgue integral. Proposition 20. Let (X, B, μ, ∧, [0, K]) be the infinite interactive monotone measure space defined by the minimum product and f ∈ F[0,K] (X, B) be a B-measurable function. Then, the inclusion–exclusion integral of f with respect to μ and ∧ coincides with the Choquet integral of f . Proof. The Choquet integral is represented as the Šipoš integral by  n  (C) f dμ = lim (ai − ai−1 ) μ(f ≥ ai ) P ∈+

= sup

i=1 n 

P ∈+ i=1

(ai − ai−1 ) μ(f ≥ ai ),

where + is the set of all partitions P = {a0 , a1 , . . . , an } of [0, +∞) satisfying a0 = 0 ≤ a1 < a2 < · · · < an < +∞ # [15,16]. Remark that the sum ni=1 (ai −ai−1 ) μ(f ≥ ai−1 ) is the lower Darboux sum of the non-increasing monotone  ∞ function μ(f ≥ t). Since μ(f ≥ t) is monotone, the Lebesgue integral 0 μ(f ≥ t) dt (= (C) f dμ) coincides with the Riemann integral. So that we have the above equality. Let P = {a0 , a1 , . . . , an } and an > K without loss of generality. Putting Ei (P , f ) := {x ∈ X | ai−1 ≤ f (x) < ai }, we have E1 (P , f )  E2 (P , f )  · · ·  En (P , f ) = X because an > K. Define fP (x) :=

n  i=1

ai−1 χEi (P ,f ) (x),

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then fP (x) ≤ f (x). The Choquet integral of fp (x) is  n   n   (C) fp dμ = (ai − ai−1 ) μ Ek (P , f ) k=i

i=1

n  = (ai − ai−1 ) μ(f ≥ ai ). i=1

Define a partition D(P , f ) ∈ (X) by D(P , f ) := {Ei (P , f )}ni=1 . Then we have by [7, Example 2],   ∧ fP dμ = (C) fP dμ. D (P ,f )





∧

f dμ ≥ sup ∧ D ∈(X)

 fD dμ ≥ sup ∧

D

 ≥ sup (C) P ∈+

P ∈+

fDn dμ

Dn

 n  fP dμ = sup (ai − ai−1 ) μ(f ≥ ai ) = (C) f dμ. P ∈+ i=1

Conversely, by f ≥ fD    (C) f dμ ≥ (C) fD dμ ≥ ∧ fD dμ; D

hence, (C)



 f dμ ≥ sup ∧ D ∈(X)

 fD dμ = ∧

f dμ

D

holds. Consequently, we obtain   ∧ f dμ = (C) f dμ. 2 4. Example of interaction operator on general space In this section, we show a way of constructing an interaction operator on a general space (X, B, μ) using generalized t-norms. Lemma 21. Let ⊗ be a t-norm on [0, K]. In the case of |A| = ∞, we have xi := inf xj ∈ [0, K]. i∈A

B⊂A,|B|<∞

j ∈B

Proof. For any finite sets B and C, B ⊂ C ⊂ A, we have

i∈A xi ∈ [0, K]. 2



i∈B xi

≥



i∈C xi

≥ 0 by (GT1) and (GT2). Hence

Proposition 22. Let ⊗ be a generalized t-norm, D = (D1 , . . . , Dn ) ∈ (X), d > 0 and m be a monotone measure on (X, B). Put I : F[0,K] (X) × B → [0, K] as ⎧ K, A = ∅, ⎪ ⎪ ⎪ ⎨  m(Di ) +1 d ai , A = Di , i = 1, . . . , n, (3) I (f | A) := ⊗ ⎪ I (f | D ), otherwise, ⎪ i ⎪ ⎩ Di ⊂A

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13

where   ai := sup a ≥ 0 | a χDi ≤ f = inf f (x), x∈Di

x is the largest integer less than or equal to x and ⊗1 x := x, ⊗n+1 := x ⊗ (⊗n x). Then, I is an interaction operator. Proof. Conditions (GI1), (GI2), (GI3) and (GI4) can be easily checked by (3). 2 The monotone measure m for defining I can be the same μ of the monotone measure space, (X, B, μ), and also can be another monotone measure or additive measure on (X, B). For example, if X = R, then m can be the Lebesgue measure. Denote by (X, B, μ, (⊗, d), [0, K]) the infinite interactive monotone measure space in the case where I is the interaction operator given in Proposition 22. We call the inclusion–exclusion integral defined on this space, the t-norm based or ⊗-based inclusion–exclusion integral and denote it by ⊗ f dμ as well as for the finite case in Section 2. If ⊗ is quasi-surjective and completely monotone, then the generalized inclusion–exclusion integral preserves basic properties similar to discrete integral, as shown in Proposition 25. Lemma 23. Let (X, B, μ, (⊗, d), [0, K]) be the infinite interactive monotone measure space. If ⊗ is quasi-surjective and completely monotone, then   ⊗ f dμ = sup ⊗ fD∗ dμ, D ∈(X)

D

f∗

where D is the lower approximation of f , that is,   n  ∗ fD := sup fD := ai χDi fD ≤ f , ai ≥ 0 . i=1

Proof. By Proposition 13, we have the assertion. 2 Remark 24. If ⊗ is the minimum product ∧, then d is not needed. In fact, ∧ Therefore, in this case, I becomes  K, if there is no Di satisfying Di ⊂ A, I (f | A) := min ai , otherwise,

m(Di ) d +1

ai equals ai for any d > 0.

i:Di ⊂A

where ai is given in Proposition 22. Proposition 25. Let (X, B, μ, (⊗, d), [0, K]) be the infinite interactive monotone measure space. If ⊗ is quasisurjective and completely monotone, then the inclusion–exclusion integral satisfies the following properties. For any f, g ∈ F[0,K] (X, B), (i) If μ is additive, then   ⊗ f dμ = f dμ, where the right-hand side is the Legesgue integral. (ii)  ⊗ f dμ ≥ 0. (iii) Let χAK (x) :=

K, 0,

x ∈ A, x∈ / A.

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14

Then,

 ⊗

χAK dμ = Kμ(A).

(iv) f ≤ g implies   ⊗ f dμ ≤ ⊗ gdμ. Proof. (i)     fD∗ dμ = f dμ ⊗ f dμ = sup ⊗ fD∗ dμ = sup D ∈(X)

D ∈(X)

D

by Lemma 23 and [7, Theorem 4]. (ii)   ⊗ f dμ = sup ⊗ fD∗ dμ ≥ sup 0 = 0 D ∈(X)

D ∈(X)

D

by Corollary 12 and Proposition 4. (iii)   ⊗ χAK dμ = sup ⊗ (χAK )∗D dμ = sup Kμ(A) = Kμ(A) D ∈(X)

D

D ∈(X)

by [7, Theorem 6]. (iv) By Lemma 23,     ∗ ⊗ f dμ = sup ⊗ fD∗ dμ ≤ sup ⊗ gD dμ = ⊗ gdμ, D ∈(X)

f∗

because D

≤ g∗

D.

D

D ∈(X)

D

2

Proposition 26. (i) The product t-norm is quasi-surjective and completely monotone. (ii) The minimum product is quasi-surjective and completely monotone. (iii) Dubois and Prade’s t-norm is quasi-surjective and completely monotone. Remark 27. The order sum t-norm, which is proposed by Ling [17], is also generalized to a t-norm on [0, K]. Let {(⊗i , [αi , βi ])}i∈I be a collection of pairs of t-norm and closed interval with an index set I , where [αi , βi ] ∩ [αj , βj ] = ∅, i = j . Then, the ordered sum t-norm on [0, K] generated by {(⊗i , [αi , βi ])}i∈I is defined by ⎧ ⎨ α + (β − α ) x − αi ⊗ y − αi , (x, y) ∈ [α , β ]2 , i i i i i i x ⊗ y := βi − αi βi − αi ⎩ x ∧ y, otherwise. Dubois and Prade’s t-norm on [0, K] is the ordered sum t-norm whose generator is (⊗P , [0, λ]). 5. Conclusion We have proposed the inclusion–exclusion integral for a nondiscrete monotone measure space, which is a generalization of one for the discrete monotone measure space. We have given several properties of this integral as an integral and concrete examples of the interaction operators. The t-norms could play an important role as an interaction operator for the integral. In the future work, precise conditions of interaction operators or t-norms used in definition of the integral should be clarified. Convergence properties of the integral is also the subject of further study.

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Acknowledgements This work was supported by Japan Society for the Promotion of Science KAKENHI Grant Number 15K05003. Appendix A. Proof of Proposition 26 (i) Putting y := aK/x for any x ∈ [0, K], a ∈ [0, x], we obtain that x ⊗P y = a and ⊗P is quasi-surjective. Next, we show the complete monotonicity. The n-array product t-norm is represented by ) xi P xi := i∈A . |A|−1 K i∈A

We have for any k ∈ N, 

(−1)

|A|



x0 ⊗

P

P

i∈A

A⊂{1,...,k}



= x0





(−1)

xi =



|A|

(−1)

|A|

A⊂{1,...,k}

* xi i∈A

A⊂{1,...,k}

 

= x0

K

1 x0 K

)

i∈A xi K |A|−1



k  * xi  ≥ 0. 1− K i=1

(ii) Putting y := a for any x ∈ [0, K], a ∈ [0, x], we obtain that x ∧ y = a and ∧ is quasi-surjective. Next, we show the complete monotonicity. Let k be an arbitrary natural number. Without loss of generality, we may assume x1 ≤ · · · ≤ x < x0 ≤ x +1 ≤ · · · ≤ xk . We have ⎛ ⎞   '  ⎟  ' ⎜  ⎟ (−1)|A| (−1)|A| x0 ∧ xi = ⎜ + (x0 ∧ xi ) ⎝ ⎠ i∈A

A⊂{1,...,k}

i∈A

A⊂{1,...,k}, A=∅,{k}

A=∅,{k}

⎧ ⎫  k−1 ⎨ k−i  ⎬   k−i = {x0 − (x0 ∧ xk )} + (−1)j (x0 ∧ xi ) ⎩ ⎭ j j =0

i=1

≥

k−1 

(x0 ∧ xi )(1 − 1)k−1 ≥ 0.

i=1

(iii) Putting y :=

λa/x, x ≤ λ, for any x ∈ [0, K], a ∈ [0, x], we obtain that x ⊗DP y = a and ⊗DP is quasia, x>λ

surjective. Next, we show the complete monotonicity. The n-array Dubois and Prade’s t-norm is represented by ⎧ K, A = ∅, ⎪ ' ⎪ ⎪ ⎪ xi , xi > λ, i ∈ A ⎨ DP xi = i∈A * xi ∧ λ ⎪ ⎪ i∈A ⎪λ ⎪ , otherwise ⎩ λ i∈A

Let k be an arbitrary natural number. Without loss of generality, we can assume x1 ≤ x2 ≤ · · · ≤ x ≤ λ < x +1 ≤ · · · ≤ xk . Then, we have    DP (−1)|A| x0 ⊗DP xi A⊂{1,...,k}

⎛

=⎝

i∈A

 A⊂{ +1,...,k}

+

 A⊂{ +1,...,k}

⎞ ⎠ (−1)

 |A|

x0 ⊗

DP



DP

i∈A

 xi

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=

(−1)

|A|

x0 ⊗

DP



'

xi

i∈A

A⊂{ +1,...,k}



+

(−1)



|A|

x0 ⊗

DP

λ

A⊂{ +1,...,k}

Case 1. For x0 > λ, 

(−1)

 |A|

x0 ⊗

DP



=

(−1)

A⊂{ +1,...,k}

|A|

DP

x0 ∧

⎛ ≥ λ⎝

−

A⊂{1,...,m}

=λ )



'



k  *

1−

i=1

(−1)

|A|

A⊂{ +1,...,k}



−

⎞



⎠ (−1)

|A|

A⊂{ +1,...,m}

* xi ∧ λ i∈A

λ

|A|



* xi ∧ λ i∈A



⎠ (−1)

A⊂{ +1,...,m}

⎞







xi +

A⊂{1,...,m}



.

xi



= {x0 − (x0 ∧ xk )} + ⎝

λ

i∈A

i∈A

⎛





i∈A

A⊂{1,...,k}

* xi ∧ λ

λ

λ

* xi ∧ λ i∈A





λ

λ

   k  k  * * xi ∧ λ λ xi ∧ λ −λ =λ ≥ 0. 1− 1− λ λ λ i= +1

i=1

Note that i∈∅ xi = 1 and that the first term of the second formula deformation is obtained in a similar way to that of (ii). Case 2. For x0 ≤ λ,    DP |A| DP (−1) x0 ⊗ xi i∈A

A⊂{1,...,k}

  * xi ∧ λ x 0 (−1)|A| x0 + (−1)|A| λ = λ λ i∈A A⊂{ +1,...,k} A⊂{ +1,...,k} ⎛ ⎞   * xi ∧ λ   k−

|A| ⎠ (−1) = x0 (1 − 1) + x0 ⎝ − λ 



A⊂{1,...,k}

= x0

k  *

1−

i=1



xi ∧ λ − x0 λ

k * i= +1

i∈A

A⊂{ +1,...,k}

   k  * λ xi ∧ λ = x0 ≥ 0. 1− 1− λ λ i=1

References [1] [2] [3] [4]

G. Choquet, Theory of capacities, Ann. Inst. Fourier 5 (1953) 131–295. E. Lehrer, A new integral for capacities, Econ. Theory 39 (2009) 157–176. N. Shilkret, Maxitive measure and integration, Indag. Math. 33 (1971) 109–116. M. Sugeno, Fuzzy measures and fuzzy integrals—a survey, in: M. Gupta, G. Saridis, B. Gaines (Eds.), Fuzzy Automata and Decision Processes, North Holland, Amsterdam, 1977, pp. 89–102. [5] Q. Yang, The pan-integral on the fuzzy measure space, Fuzzy Math. 3 (1985) 107–114 (in Chinese). [6] A. Honda, J. Okamoto, Inclusion–exclusion integral and its application to subjective video quality estimation, in: Information Processing and Management of Uncertainty in Knowledge-Based Systems. Theory and Methods, in: Communications in Computer and Information Science, vol. 80, 2010, pp. 480–489. [7] A. Honda, Y. Okazaki, Theory of inclusion–exclusion integral, Inf. Sci. 376 (2017) 136–147, https://doi.org/10.1016/j.ins.2016.09.063.

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[8] A. Chateauneuf, J.-Y. Jaffray, Some characterizations of lower probabilities and order monotone capacities through the use of Möbius inversion, Math. Soc. Sci. 17 (1989) 263–283. [9] B. Schweizer, A. Sklar, Statistical metric spaces, Pac. J. Math. 10 (1960) 313–334. [10] K. Menger, Statistical metrics, Proc. Natl. Acad. Sci. USA 28 (1942) 535–537. [11] H. Hamacher, Über logische aggregationen nicht-binär expliziter Entscheidungskriterien, Dissertation, R.G. Fischer Verlag, Frankfurt/Main, 1978. [12] D. Dubois, H. Prade, New results about properties and semantics of fuzzy set-theoretic operators, in: P.P. Wang, et al. (Eds.), Fuzzy Sets, Plenum Press, 1980, pp. 59–75. [13] S. Weber, A general concept of fuzzy connectives, negation and implications based on t-norms and t-conorms, Fuzzy Sets Syst. 11 (1983) 115–134. [14] J. Kawabe, The Choquet integral in Riesz space, Fuzzy Sets Syst. 159 (2008) 629–645. [15] J. Šipoš, Integral with respect to a pre-measure, Math. Slovaca 29 (1979) 141–155. [16] J. Šipoš, Non linear integrals, Math. Slovaca 29 (1979) 257–270. [17] C.-H. Ling, Representation of associative functions, Publ. Math. (Debr.) 12 (1965) 189–212.