Coincidences of the concave integral and the pan-integral

: In this note, we discuss when the concave integral coincides with the pan- integral with 1 respect to the standard arithmetic operations + and · . The subadditivity of the underlying monotone 2 measure is one sufﬁcient condition for this equality. We show also another sufﬁcient condition, which, 3 in the case of ﬁnite spaces, is necessary, too. Some convergence results concerning pan-integrals are 4 also included.

We pointed out that in the above-mentioned study we have only considered the case that the 48 underlying space is finite. But our approach based on minimal atoms does not apply to infinite spaces, 49 see [25]. 50 This paper will focus on the relationship between the concave integrals and pan-integrals on 51 general spaces (not necessarily finite). We shall show that if the underlying monotone measure µ is 52 subadditive, then the concave integral coincides with the pan-integral w.r.t. the usual addition + and 53 usual multiplication ·.  When µ is a monotone measure, the triple (X, A, µ) is called a monotone measure space ([15,26, 64 34]). In some literature, such a monotone measure µ constrained by the boundary condition µ(X) = 1 65 is also called a capacity or a fuzzy measure or a nonadditive probability, etc..
In our discussions we concern three types of nonlinear integrals, the Choquet integral, the concave 74 integral and the pan-integral. We recall their definitions. 75 We consider a given monotone measure space (X, A, µ), and let f ∈ F + , χ A denote the indicator 76 function of measurable set A.
The Choquet integral [3] (see also [4,26]) of f on X with respect to µ, is defined by where the right side integral is the Riemann integral. 78 Lehrer [13] introduced a new integral known as concave integral (see also [12,31]), as follows:

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The concave integral of f on X is defined by The pan-integral of f on X w.r.t. the usual addition + and usual multiplication · (in short, 85 pan-integral), is given by All these integrals are covered by a resent concept of decomposition integrals by Even and Lehrer

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Note that the pan-integral is related to finite partitions of X, the concave integral to any finite 89 set systems of measurable subsets of X. The Choquet integral is based on chains of sets, it can be 90 expressed in the following form: Comparing above three definitions, it is obvious that for each f ∈ F + , In general, cav f dµ = pan f dµ, cav f dµ = Cho f dµ.

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Example 2. Let X = N (the set of all positive integers). The monotone measure µ : 2 N → [0, 1] is defined by We take Observe that the Choquet integral and the pan-integral are not comparable.

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The above examples indicate that any two of the three integrals do not coincide, in general. They

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We consider a given measurable space (X, A), and let M be the class of all monotone measures 101 defined on (X, A).

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For the convenience of our discussion, we denote Ch µ ( f ) = Cho f dµ, Cav µ ( f ) = cav f dµ and In [13] (see also [1,12,14]) the relationship between the the concave integral and the Choquet 105 integral was discussed, as follows: Proof. Observe that Ch µ (χ E ) = µ(E) for any E ⊆ X and, thus for any A, B ⊆ X, A ∩ B = ∅, we i.e., µ is superadditive. Proof. It suffices to prove that pan f dµ ≥ cav f dµ holds for any f ∈ F + . To prove this fact, it suffices to prove that for any {A i } N i=1 ⊂ A and λ i ≥ 0, i = 1, 2, . . . , N, there is a sequence of pairwise disjoint subsets B j M j=1 ⊂ A and a sequence of nonnegative numbers l j , j = 1, 2, . . . , M such that For N = 2, observe that If we let l 1 = λ 1 , l 2 = λ 2 , l 3 = λ 1 + λ 2 and Moreover, thanks to the subadditivity of µ, we have Now suppose that (3.3) and (3.4) hold for N = k, we need to verify that they are also true for N = k + 1. 120 If we let 122 B j = C j − (C j ∩ A k+1 ), j = 1, 2, . . . , N B N +j = C j ∩ A k+1 , j = 1, 2, . . . , N , The following example shows that the subadditivity in Theorem 9 is not a necessary condition.

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The next theorem gives another sufficient condition ensuring the coincidence of the pan-integral 127 and concave integral, now covering Example 10, too.

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Theorem 11. Let µ be a monotone measure on (X, A). If there is a countable partition {E t | t ∈ T} ⊂ A of X, so that e t = µ(E t ), t ∈ T, and then the concave integral coincides with the pan-integral with respect to the usual arithmetic operation " + " 129 and " · ".

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Proof. It is not difficult to check that under the above constraints on µ, for any f ∈ F + it holds Observe that if X is a finite space, then the constraints on µ given in Theorem 11 are also necessary, see [25]. Moreover, consider a lower probability µ on a finite set X = {1, 2, . . . , n} in the sense of de Finetti [19], i.e., there is partition {E 1 , E 2 , · · · , E r } of X such that µ(E 1 ) = e 1 , µ(E 2 ) = e 2 , · · · , µ(E r ) = e r , e 1 + e 2 , · · · , e r = 1, and for any E ⊂ X it holds Note that µ is then a belief measure [34] which is k-additive [7]. Clearly, µ satisfies the constraints of Theorem 11, and thus Cav µ = Pan µ . Moreover, both these integrals coincide in this case also with the Choquet integral, i.e., Ch µ = Cav µ = Pan µ . Note that the case when µ is σ-additive (i.e., a discrete probability measure on X) is a particular subcase of the mentioned class of lower probabilities related to the finest partition of X into the singletons, i.e., when E 1 = {1}, E 2 = {2}, · · · , E n = {n}. Another particular subclass of de Finetti's lower probabilities, known from the game theory, is formed by the unanimity games. In that case, for a non-empty subset E of X, we define a monotone measure µ E on X as and then for all three considered integrals their equal output is min{ f (i) | i ∈ E}.

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In further research, we shall investigate the relationships among these four integrals on a fixed 147 generalized ring (R + , ⊕, ⊗).