Monotone Measures Defined by Pan-Integral

Given a pan-space ( ) , , , , , X R μ + ⊕ ⊗  and a nonnegative measurable function f on measurable space ( ) , X  , the pan-integral of f with respect to monotone measure μ and pan-operation ( ) , ⊕ ⊗ determines a new monotone measure ( ) , f λ ⊕ ⊗ on ( ) , X  . Such the new monotone measure ( ) , f λ ⊕ ⊗ is absolutely continuous with respect to the monotone measure μ. We show that the new monotone measure preserves some important structural characteristics of the original monotone measures, such as continuity from below, subadditivity, null-additivity, weak null-additivity and (S) property. Since the pan-integral based on a pair of pan-operations ( ) , ⊕ ⊗ covers the Sugeno integral (based on ( ) , ∨ ∧ ) and the Shilkret integral (based on ( ) , ∨ ⋅ ), therefore, the previous related results for the Sugeno integral are covered by the results presented here, in the meantime, some special results related the Shilkret integral are also obtained.


Introduction
In generalized measure and integral theory, there are several kinds of important nonlinear integrals, the Choquet integral [1], the Sugeno integral [2], the pan-integral [3] and the concave integral introduced by Lehrer [4], etc. (see also [5]).The pan-integral with respect to monotone measure μ relates to a commutative isotonic semiring ( ) , , R + ⊕ ⊗ , where ⊕ is a pan-addition and ⊗ is a pan-multiplication related by the distributivity property (see also [6]).This integral generalizes the Lebesgue integral and the Sugeno integral.When consi-, , R + + ⋅ , the Lebesgue integral coincides with the pan-integral; when considering a monotone measure μ and the commutative isotonic semiring ( ) , , R + ∨ ∧ , the Suge- no integral is recovered by the pan-integral with respect to ( ) , ∨ ∧ .The more details on the pan-integrals can be also found in [6]- [12].
In classical measure theory [13], given measure space ( ) , , X m  and a nonnegative measurable function f on ( ) , X  , the set function f ν , defined by the Lebesgue integral of f, ( ) ∫  is a measure, i.e., f ν is σ-additive and ( ) 0 f ν ∅ = , and f ν is absolutely con- tinuous with respect to m.
For the Sugeno integral and the Choquet integral, the issue above has also been discussed respectively in [14] and [15].Given a measurable space ( ) , X  , a monotone measure μ, and a nonnegative measurable function f on ( ) , X  , the Sugeno integral (or the Choquet integral) with respect to μ determines a new monotone measure ( )   Su f λ (or ( )   Ch f λ ).In general the monotone measures lose additivity, and they appear to be much looser than the classical measures.This new construction raises an important question: Do structural features of the original monotone measure μ still hold for these new monotone measures?It has been shown that the new monotone measures inherit almost all desirable structural characteristics of μ, such as, subadditivity, continuity from below, continuity from above, null-additivity, weak null-additivity and autocontinuity (see [14] [15] [16]).
In this paper we discuss the above-mentioned issue for the pan-integrals.

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and a nonnegative measurable function f on ( ) , X  , the pan-integral of f with respect to μ and pan-operation ( ) determines a new monotone measure ( )   , f λ ⊕ ⊗ on ( ) , X  .Such the monotone measure ( )   , f λ ⊕ ⊗ is absolutely continuous with respect to μ.We are going to show that ( )   , f λ ⊕ ⊗ reserves some important structural characteristics of μ, which is similar to the cases of the Choquet integral and the Sugeno integral.We are also going to show that if monotone measure is subadditive, then the pan-integral with respect to the monotone measure is a monotone superadditive functional.Since the pan-integral covers the Sugeno integral, while the Shilkret integral, the previous related results for the Sugeno integral become special cases of the results presented here, in the meantime, some special results related the Shilkret integral are shown.
In this paper we restrict our discussion on fixed measurable space ( ) the set of all monotone measures defined on ( ) is said to be: continuous from below [20], if ( ) ( ) whenever n P P  ; continuous from above [20], if ( ) ( ) continuous, if it is continuous both from below and from above; subadditive [20], if ( ) ( ) ( ) whenever , P Q∈  ; superadditive, if ( ) ( ) ( ) Obviously, the subadditivity of μ implies the null-additivity, and the later implies weak null-additivity.The converse implications may not be true.
. We say that 1) μ is absolutely continuous of Type I with respect to ν, denoted by 2) μ is absolutely continuous of Type VI with respect to ν, denoted by Obviously,  .The inverse statement may not be true.

Pan-Operation and Integrals
In [3] Yang introduced the concept of pan-integral (see also [6] and [12]), which is a type of nonlinear integral with respect to fuzzy measure.These integrals involve two binary operations, the pan-addition ⊕ and pan-multiplication ⊗ of real numbers.We recall a commutative isotonic semiring [6] (see also [8] [10] [11] [17] [23]).Definition 3. A binary operation ⊕ on R + is called a pseudo-addition on R + if and only if it satisfies the following requirements: (PA1) r s s r From associativity (PA2), we may write  lim Definition 4. A binary operation ⊗ on R + is called a pseudo-multiplication (with respect to pseudo-addition ⊕ ) on R + if and only if it fulfills the following conditions: (PM6) there exists e R + ∈ , which is called the unit element, such that e r r ⊗ = for any r R + ∈ ; (PM7) When ⊕ is a pseudo-addition on R + and ⊕ is a pseudo-multiplication (with respect to ⊕ ) on R + , the triple ( ) and ( ) Next we recall the concept of pan-integral.
( ) , A∈  , the pan-integral of f over A with respect to μ, is defined by where  is the set of all finite partitions of X.
The pan-integral of f over A with respect to μ can be expressed by pan-simple functions, as follows: ) and ˆA  is the set of all finite partitions of A.
Note.A related concept of generalized Lebesgue integral based on a generalized ring ( ) , , R + ⊕ ⊗ was proposed and discussed (see [12]).
The following is some basic properties of the pan-integrals (see [6] [12]).Proposition 6.Let A A a a The following is the Monotone Convergence Theorem of the pan-integral ([6] [12]).
. We have the following: , from Proposition 6 3), we can get 1). 2).Let , then ( ) = and for any subset B of A we have Thus, for any ( ) . In a similar way, we have , and thus 2) holds.□ The following is three kinds of important nonlinear integrals [1] [2] [24], see also [5].Consider a nonnegative real-valued measurable function The Choquet integral [1] of f on X with respect to μ, denoted by ( ) where the right side integral is Riemann integral.
The Sugeno integral [2] of f on X with respect to μ, denoted by ( ) The Shilkret integral [24] of f on X with respect to μ, is defined by ), we say that f is Choquet integrable (resp.Sugeno integrable).
Note that in the case of commutative isotonic semiring ( ) , , R + ∨ ∧ , the Suge- no integral is recovered, while for ( )

Monotone Measures Defined by Pan-Integral
Given a pan-space ( ) , , , , , X R From Definition 5 and Proposition 6 (v), we know that ( ) The new monotone measure ( )   , f λ ⊕ ⊗ is said to be monotone measure defined by pan-integral.

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. We denote ( ) Proposition 9. ( [6]) We have the following: Proposition 10.We have the following: Proof. 1) From definition of the pan-integral, the (3.2) is obvious. Let , and noting that 0 The proof of 2) is similar.□ The following is an alternative statement of Proposition 6 2).
Proposition 11.Let μ be continuous from below and A∈  .Then 0 f = a.e. on A if and only if ( ) ( ) = , by Proposition 10 1), we have a.e. on X, from Proposition 6 3) and 8 .
Similarly, we can get the following result.Proposition 13.Let 0 Proof.From Proposition 7, i.e., the Monotone Convergence Theorem of the pan-integral, and Proposition 6 3), we can obtain the desired conclusion.□ We consider the Sugeno integral and Shilkret integral.
Proposition 16.We have the following: 1) If μ is ∨-additive (it is also called to be fuzzy additive, i.e., ( ) ( ) ( ) for any , P Q∈  ), then so is ( ) 2) If μ is fuzzy multiplicative ( ( ) ( ) ( ) Similar to Proposition 16 we can get the following result for the Shilkret integral.
Proof.For any , A B ∈  , we assume that A B = ∅  without loss of generality.From Definition 5, we notice that μ is sub-⊕-additive, thus x The proof is complete.□ We also have the following results (see [12]).