Monotone Measures Defined by Pan Integral

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ISSN Online: 2160-0384 ISSN Print: 2160-0368

DOI: 10.4236/apm.2018.86031 Jun. 20, 2018 535 Advances in Pure Mathematics

Monotone Measures Defined by Pan-Integral

Peipei Wang

1

_{, Minhao Yu}

2

_{, Jun Li}

2*

1_{Property and Finance Department, National Museum of China, Beijing, China} 2_{School of Sciences, Communication University of China, Beijing, China}

Abstract

Given a pan-space

(

X, , , , ,

µ

R+ ⊕ ⊗

)

and a nonnegative measurable

func-tion f on measurable space

(

X,

)

, the pan-integral of f with respect to mo-notone 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 characteris-tics of the original monotone measures, such as continuity from below, sub-additivity, null-sub-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}

( )

∨ ⋅, _),

there-fore, 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.

Keywords

Monotone Measure, Pan-Integral, Sugeno Integral, Choquet Integral, Shilkret Integral

1. 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 commut-ative 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-How to cite this paper: Wang, P.P., Yu, M.H. and Li, J. (2018) Monotone Measures Defined by Pan-Integral. Advances in Pure Mathematics, 8, 535-547.

https://doi.org/10.4236/apm.2018.86031

Received: April 29, 2018 Accepted: June 17, 2018 Published: June 20, 2018

http://creativecommons.org/licenses/by/4.0/

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DOI: 10.4236/apm.2018.86031 536 Advances in Pure Mathematics

dering a σ-additive measure m and the commutative isotonic semiring

(

R+, ,+ ⋅

)

,

the Lebesgue integral coincides with the pan-integral; when considering a mo-notone 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 non-negative measurable function f on

(

X,

)

, the set function νf, defined by the

Lebesgue integral of f,

( )

d , .

f A _Af m A

ν

=

_∫

∀ ∈

is a measure, i.e., νf is σ-additive and νf

( )

∅ =0, and νf 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 struc-tural characteristics of μ, such as, subadditivity, continuity from below, continu-ity from above, null-additivcontinu-ity, weak null-additivcontinu-ity and autocontinucontinu-ity (see [14] [15][16]).

In this paper we discuss the above-mentioned issue for the pan-integrals. Given a pan-space

(

X, , , , ,

µ

R+ ⊕ ⊗

)

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.

2. Preliminaries

Let X be a non-empty set,  a σ-algebra of subsets of X. As usual the pair

(

X,

)

denotes a measurable space. Denotes R+=

[

0,+∞

)

, R+=

[

0,+∞

]

.

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called a monotone measure defined on

(

X,

)

, if μ satisfies the following two conditions:

1) µ

( )

∅ =0_and µ

( )

X >0_{; (vanishing at}∅₎

2) ∀P Q, ∈, P Q⊂ implies µ

( )

P ≤µ

( )

Q _{. (monotonicity)}

When μ is a monotone measure, the triple

(

X, , µ

)

is called a monotone measure space ([10]).

In many literature, a monotone measure is also known as “non-additive measure”, “capacity”, “fuzzy measure”, “non-additive measure” or “non-additive probability”, etc. (see [1][6][17][18][19]).

In this paper we restrict our discussion on fixed measurable space

(

X,

)

. Let _(X,)_{the set of all monotone measures defined on}

(

_X_,_

)

_.

A monotone measure µ:A→R₊ is said to be: continuous from below[20], if limn→∞µ

( )

Pn =µ

( )

P whenever PnP; continuous from above [20], if

( )

limn→∞µ Qn =µ Q whenever QnQ and µ

( )

Q1 < ∞; continuous, if it is continuous both from below and from above; subadditive[20], if

(

P Q

)

( )

P

( )

Q

µ  ≤µ +µ whenever P Q, ∈; superadditive, if

(

P Q

)

( )

P

( )

Q

µ _ ≥µ +µ _whenever P Q, ∈, P Q = ∅; null-additive ([6]), if µ

(

A N_

)

=µ

( )

A _{for any}_A_whenever µ

( )

N =0_;_{weakly null-additive}_[6]_,

if µ

(

R S_

)

=0_{, for any} R S A, ∈ _{such that} µ

_{( )}

R =µ

_{( )}

S =0_.

Obviously, the subadditivity of μ implies the null-additivity, and the later im-plies weak null-additivity. The converse implications may not be true.

Definition 2 ([21][22]) Let

_{µ ν}

_, _∈_(X,)_._{We say that}

1) μ is absolutely continuous of Type I with respect to ν, denoted by µIν,

iff µ

( )

A = 0_wheneverν

_{( )}

A =0_;

2) μ is absolutely continuous of Type VI with respect to ν, denoted by

VI

µ ν, iff µ

( )

A_n →0

(

n→ ∞

)

_wheneverν

( )

A_n →0

(

n→ ∞

)

_.

Obviously, µVIν implies µIν. The inverse statement may not be

true.

3. 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 in-volve 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⊕ = ⊕ _;

(PA2)

(

r s⊕ ⊕ = ⊕ ⊕

)

t r

(

s t

)

_;

(PA3) r s≤ ⇒ ⊕ ≤ ⊕r t s t_{for any}_t_;

(PA4) r⊕ =0 r_;

(PA5) rn→r and sn→s ⇒ rn⊕ → ⊕sn r s.

From associativity (PA2), we may write r r1⊕ ⊕ ⊕2  rn as

1

n i i= r

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1 lim 1

n

i _n i

i r i r

∞

→∞

= =

=

⊕

.

Definition 4. A binary operation ⊗ _on R+ is called a

pseu-do-multiplication (with respect to pseudo-addition ⊕_{) on} R+ if and only if it

fulfills the following conditions: (PM1) r s s r⊗ = ⊗ _;

(PM2)

(

r s⊗ ⊗ = ⊗ ⊗

)

t r

(

s t

)

_;

(PM3) r⊗ ⊕ =

(

s t

) (

r s⊗ ⊕ ⊗

) (

r t

)

_;

(PM4) r s≤ ⇒ ⊗ ≤ ⊗r t s t_{for any}_t_;

(PM5) r s⊗ = ⇔ =0 r 0 or s=0;

(PM6) there exists e R∈ +, which is called the unit element, such that

e r r⊗ = _{for any} r R∈ +;

(PM7) r s, ∈

(

0,+∞

)

_, r_n→r_and r_n→r ⇒ r_n⊗ → ⊗s_n r s_.

When ⊕_{is a pseudo-addition on}R+ and ⊕ is a pseudo-multiplication

(with respect to ⊕_{) on} R+, the triple

(

R+, ,⊕ ⊗

)

is called a commutative iso-tonic semiring on R+ ([6]), and

(

X, , , , ,

µ

R+ ⊕ ⊗

)

is called a pan-space

([6]).

Next we recall the concept of pan-integral. +(X,) denotes the set of all finite

nonnegative -measurable functions on X. A finite partition of A∈ is a finite disjoint system of sets

{ }

A_{i i}n₌₁⊂ such that A Ai j = ∅ for i j≠ and

1 n

i i= A A=



.

Definition 5. ([3] [6]) Let

(

X, , , , ,

µ

R+ ⊕ ⊗

)

be a pan-space. For any

(X, )

f∈+  , A∈, the pan-integral of f over A with respect to μ, is defined by ( )

( )

(

)

(

)

,

ˆ

d sup inf ,

A f µ _E x A E f x µ A E

⊕ ⊗

∈ ∈ ∈

  

=  __ ⊗ __



⊕



∫

_ 

   (3.1)

where ˆ is the set of all finite partitions of X. When A X= _,

_∫

_X(⊕ ⊗, )fdµ is written as

_∫

(⊕ ⊗, )fdµ.

The pan-integral of f over A with respect to μ can be expressed by pan-simple functions, as follows:

( )

(

)

(

)

{ }

,

1

1 1

ˆ d sup n _i _i : n _i _A_i , _i 0, _i n_i _A ,

A f µ _i λ µ A _i λ χ f λ A

⊕ ⊗

=

= =

 

=  ⊗ ⊗ ≤ ≥ ∈ 



⊕



∫



where χA is the pan-characteristic function of A, i.e.,

( )

_{0 otherwise,}if A

e x A

x

χ _{= } ∈



( )

(

)

1 n

i i i=

λ

⊗

µ

A

⊕

is pan-simple function ([6]) and ˆA is the set of all finite

partitions of A.

Note. A related concept of generalized Lebesgue integral based on a genera-lized ring

(

R+, ,⊕ ⊗

)

was proposed and discussed (see [12]).

The following is some basic properties of the pan-integrals (see [6][12]). Proposition 6. Let _{f g}, (X, )

+

∈_  _and _{a R}

+

∈ _{. Then we have the following:}

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2) if f =0_a.e.

[ ]

µ _on_A_,_i_._e_.,

µ

(

{

f ≠0

}

_A

)

=0_{, then}

_∫

_A(⊕ ⊗, )fdµ=0. Further, if μ is continuous from below, then

_∫

_A(⊕ ⊗, )fdµ=0 if and only if f =0 a.e.

[ ]

µ _on_A_;

3) _A( , )fdµ ( , )f χ µAd

⊕ ⊗ ⊕ ⊗

= ⊗

∫

;

4) if f g≤ _{, then}

_∫

_A(⊕ ⊗, )fdµ≤

_∫

_A(⊕ ⊗, )gdµ; 5) if A B⊂ , then

_∫

_A(⊕ ⊗, )fdµ≤

_∫

_B(⊕ ⊗, )fdµ

6)

_∫

(⊕ ⊗, )a⊗fdµ≥ ⊗a

_∫

(⊕ ⊗, )fdµ. In particular, for A∈,

( , ) ( , )

d d .

A A

a χ µ a χ µ

⊕ ⊗ ⊕ ⊗

⊗ ≥ ⊗

∫

The following is the Monotone Convergence Theorem of the pan-integral ([6] [12]).

Proposition 7. Let μ be continuous from below, A∈,

( , )

, X , 1,2,

n

f f∈+  n= . If fn f n

(

→ ∞

)

, then

( , ) ( , )

d lim nd .

A f µ n A f µ

⊕ ⊗ ⊕ ⊗

→∞

=

∫

Proposition 8. Let _{f g}_, (X, )

+

∈_  _._{We have the following}_:

1) ( , ) _{( ,_}) 0

d _f d

f µ f µ

⊕ ⊗ ⊕ ⊗

>

=

∫

, where

{

f >0

}

= ∈

{

x X f x:

( )

>0

}

;

2) If μ is null-additive and f g= _a.e.

[ ]

µ _on_A_,_i_._e_.,

µ

(

{

f g≠

}

_A

)

=0_,

then

( , ) ( , )

d d .

A f µ A g µ

⊕ ⊗ ⊕ ⊗

=

∫

Proof. Noting that f = ⊗f

χ

{f>0}, from Proposition 6 3), we can get 1). 2). Let A1=

{

x A f x∈ ,

( )

≠g x

( )

}

, then µ

( )

A1 =0 and for any subset B of A we have µ

( )

B =µ

(

B A\ 1

)

. Thus, for any 1 i

( )

n

i= 

λ

i

χ

B f x

χ

A

⊕ _ ⊗ _≤ ⊗ _,

( )

1

1 i\

n

i= 

λ

i

χ

B A g x

χ

A

⊕ _ ⊗ _≤ ⊗ . Moreover,

( )

(

)

( )

,

1

1 1

d n _i _i \ n _i _i ,

A g µ _i λ µ B A _i λ µ B

⊕ ⊗

= =

   

≥

_⊕

_ ⊗ _=

_⊕

_ ⊗ _

∫

which implies that

_∫

_A(⊕ ⊗, )gdµ≥

_∫

_A(⊕ ⊗, )fdµ. In a similar way, we have

( , ) ( , )

d d

A f µ A g µ

⊕ ⊗ ⊕ ⊗

≥

∫

, 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 _f (X, )

+

∈_ 

and

µ

∈_(X,)_,_i_._e_.,

₍

_µ

_,_f

₎

(X, ) (X, )

+

∈_  ×_ 

.

The Choquet integral[1] of f on X with respect to μ, denoted by

( )

Ch f

_∫

d

µ

, is defined by

( )

Ch fdµ 0µ

(

{

x f x:

( )

t

}

)

dt

∞

= ≥

∫

where the right side integral is Riemann integral.

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( )

(

{

( )

}

)

0

d sup :

t

S u f µ t µ x f x t

≤ <+∞

 

= _ ∧ ≥ _

∫

The Shilkret integral[24] of f on X with respect to μ, is defined by

( )

(

{

( )

}

)

0

d sup :

t

Sh f µ t µ x f x t

≤ <+∞

 

= _ ⋅ ≥ _

∫

For f∈+(X,), if

( )

Ch f

∫

d

µ

< ∞ (resp.

( )

Su f

∫

d

µ

< ∞), 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

(

R+, ,∨ ⋅

)

, the Shilkret integral is covered by

the pan-integral in Definition 5, i.e.,

( )

Su f

_∫

d

µ

=

_∫

fd

µ

_(3.2)

and

( )

Sh f

_∫

d

µ

=

_∫

fd .

µ

_(3.3)

4. Monotone Measures Defined by Pan-Integral

Given a pan-space

(

X, , , , ,

µ

R+ ⊕ ⊗

)

and ( )

, X

f∈+  , we define a new set

function ( , ) f

λ

⊕ ⊗

by

( , )

_{( )}

( , ) _{d ,} _.

f A _A f A

λ⊕ ⊗ ₌ ⊕ ⊗ µ _{∀ ∈}

∫

 (4.1) From Definition 5 and Proposition 6 (v), we know that ( , )

f

λ

⊕ ⊗

is a monotone measure on

(

X,

)

, too, i.e., ( , ) (X, )

f

λ

⊕ ⊗ _∈_  _.

The new monotone measure ( , ) f

λ

⊕ ⊗

is said to be monotone measure defined by pan-integral.

In the following we discuss some properties of ( , ) f

λ

⊕ ⊗

on the given pan-space

(

X, , , , ,

µ

R+ ⊕ ⊗

)

. We denote infx X∈ f x

( )

by αf.

Proposition 9. ([6]) We have the following: ∀A B, ∈, _{f g}, (X, )

+

∈_  _,

1) ( , )

(

)

( , )

( )

( , )

( )

f A B f A f B

λ

⊕ ⊗ _≥

λ

⊕ ⊗ _∨

λ

⊕ ⊗

 .

2) ( , )

(

)

( , )

( )

( , )

( )

f A B f A f B

λ

⊕ ⊗ _≤

λ

⊕ ⊗ _∧

λ

⊕ ⊗

 .

3) ( , )

( )

( , )

( )

( , )

( )

f g A f A g A

λ

⊕ ⊗

λ

⊕ ⊗

λ

⊕ ⊗

∨ ≥ ∨ .

4) ( , )

( )

( , )

( )

( , )

( )

f g A f A g A

λ

⊕ ⊗

λ

⊕ ⊗

λ

⊕ ⊗

∧ ≤ ∧ .

Proposition 10. We have the following: 1) ( , )

f I

λ

⊕ ⊗

µ

 , i.e., ∀ ∈A ,

( )

_{0 implies} ( , )

_{( )}

_0.

f

A A

µ

₌

λ

⊕ ⊗ ₌

(4.2) Furthermore, if αf >0, i.e., infx X∈ f x

( )

>0, then for any A∈,

( )

_{0 if and only if} ( , )

_{( )}

_0.

f

A A

µ

₌

λ

⊕ ⊗ ₌

2) If α_f >0, then ( , ) VI f

µ

λ

⊕ ⊗

 , i.e., ∀

{ }

An ⊂, as n→ ∞,

( , )

_{( )}

_{0 implies}

_{( )}

_0.

f An An

λ

⊕ ⊗ _→

µ

_→

Proof. 1) From definition of the pan-integral, the (3.2) is obvious. Let α_f >0_{, then} ∀ ∈A , if ( , )

( )

₀

f A

λ

⊕ ⊗ ₌

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( _, )

_{( )}

( , ) ( , )

_{( )}

d d .

f A _A f _A f f A

λ⊕ ⊗ ₌ ⊕ ⊗ µ_≥ ⊕ ⊗α µ α_≥ _⊗µ

∫

Therefore αf ⊗µ

( )

A =0, and noting that αf >0, we get µ

( )

A =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 f =0

a.e. on A if and only if ( , )

( )

₀ f A

λ

⊕ ⊗ ₌

.

Proposition 12. Let α_f >0. If μ is null-additive, then so is ( , ) f

λ

⊕ ⊗

.

Proof. For any A B, ∈ and

λ

(f⊕ ⊗, )

( )

B =0, by Proposition 10 1), we have

( )

B 0

µ = . Noting that fχA B = fχA a.e. on X, from Proposition 6 3) and 8 2)

we have

( )

₍

₎

( ) ( ) ( )

( ) ₍ ₎

( )

, , ,

,

, _,

d d d

d .

f _{A B} A B A

f A

A B f f f

f A

λ µ χ µ χ µ

µ λ

⊕ ⊗ ⊕ ⊗ ⊕ ⊗

⊕ ⊗

⊕ ⊗ _{⊕ ⊗}

= = =

= =

∫

 



□

Similarly, we can get the following result.

Proposition 13. Let α_f >0_{. If}_μ_{is weakly null-additive, then so is}

λ

(_f⊕ ⊗, )

. Proposition 14. If μ is continuous from below, then so is ( , )

f

λ

⊕ ⊗

.

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 15. 1) If μ is sub-additive, then so is ( ),

f

λ

∨ ∧

, that is, ∀A B, ∈, ( , )

₍

₎

( , )

_{( )}

( , )

_{( )}

_,

f A B f A f B

λ

∨ ∧ _≤

λ

∨ ∧ ₊

λ

∨ ∧



i.e.,

( )

Su

_∫

_{A B}fd

µ

≤

( )

Su

_∫

_Afd

µ

+

( )

Su

_∫

_Bfd .

µ

 (4.3)

2) If μ is sub-additive, then so is ( ), f

λ

∨ ⋅

, that is, ∀A B, ∈, ( ),

₍

₎

( ),

_{( )}

( ),

_{( )}

_, f A B f A f B

λ

∨ ⋅ _≤

λ

∨ ⋅ ₊

λ

∨ ⋅



i.e.,

( )

Sh

_∫

_{A B}fd

µ

≤

( )

Sh

_∫

_Afd

µ

+

( )

Sh

_∫

_Bfd .

µ

 (4.4)

The following result was obtained in [14]. Proposition 16. We have the following:

1) If μ is ∨_{-additive (it is also called to be fuzzy additive,}_i_._e_.,

(

P Q

)

( )

P

( )

Q

µ _ =µ ∨µ _{for any}P Q, ∈ ), then so is ( ), f

λ

∨ ∧

, that is,

,

A B

∀ ∈,

( , )

₍

₎

( , )

_{( )}

( , )

_{( )}

_,

f A B f A f B

λ

∨ ∧ ₌

λ

∨ ∧ _∨

λ

∨ ∧



i.e.,

( )

Su

_∫

_{A B}fd

µ

=

( )

Su

_∫

_Afd

µ

∨

( )

Su

_∫

_Bfd .

µ



(8)

then so is

λ

( )f ,

∨ ∧

, that is, ∀A B, ∈,

( , )

₍

₎

( , )

_{( )}

( , )

_{( )}

_,

f A B f A f B

λ

∨ ∧ ₌

λ

∨ ∧ _∧

λ

∨ ∧



i.e.,

( )

Su

_∫

_{A B}fd

µ

=

( )

Su

_∫

_Afd

µ

∧

( )

Su

_∫

_Bfd .

µ



Similar to Proposition 16 we can get the following result for the Shilkret integral.

Proposition 17. We have the following: 1) If μ is ∨_{-additive, then so is} ( ),

f

λ

∨ ⋅

, that is, ∀A B, ∈, ( ),

₍

₎

( ),

_{( )}

( ),

_{( )}

_, f A B f A f B

λ

∨ ⋅ ₌

λ

∨ ⋅ _∨

λ

∨ ⋅



i.e.,

( )

Sh

_∫

_{A B}fd

µ

=

( )

Sh

_∫

_Afd

µ

∨

( )

Sh

_∫

_Bfd

µ



2) If μ is fuzzy multiplicative, then so is ( ), f

λ

∨ ⋅

, that is, ∀A B, ∈, ( , )

₍

₎

( ),

_{( )}

( ),

_{( )}

_,

f A B f A f B

λ

∨ ∧ ₌

λ

∨ ⋅ _∧

λ

∨ ⋅



i.e.,

( )

Sh

_∫

_{A B}fd

µ

=

( )

Sh

_∫

_Afd

µ

∧

( )

Sh

_∫

_Bfd .

µ



Definition 18. Let

µ

∈_(X,)_{, and}

(

_X_{, , , , ,}

_µ

_R

)

+ ⊕ ⊗

 be a pan-space. The monotone measure μ is said to be

1) sub-⊕_-additive_{, if} µ

(

A B_

)

≤µ

( )

A ⊕µ

( )

B _whenever A B, ∈; 2) super-⊕_-additive_{, if} µ

(

_{A B}

)

≥µ

( )

_A ⊕µ

( )

_B whenever A B, ∈,

A B_ = ∅_;

3) ⊕_-additive_{, if} µ

(

_{A B}

)

=µ

( )

_A ⊕µ

( )

_B for any A B, ∈, A B = ∅; 4) σ-⊕_-additive_{, if for any disjoint sequence of sets}

{ }

A_n ⊂,

( )

1 1 . n n n

n A A

µ ∞ ∞ µ

= =

 ₌

 







⊕

Proposition 19. Given the pan-space

(

X, , , , ,

µ

R+ ⊕ ⊗

)

and ( )

, X f∈+  . If μ is sub-⊕_-_{additive, then so is} ( , )

f

λ

⊕ ⊗

.

Proof. For any A B, ∈, we assume that A B = ∅ without loss of general-ity. From Definition 5, we notice that μ is sub-⊕_{-additive, thus}

( )

₍

₎

( ) ( )

( )

(

)

( )

(

)

(

)

( )

(

)

(

)

(

( )

)

(

)

, , ˆ ˆ ˆ

d sup inf

sup inf

sup inf inf

su

f _{A B} _{x A B E}

E

x A B E E

x A E x B E

E E

A B f f x A B E

f x A E B E

f x A E f x B E

λ µ µ

µ µ µ µ ⊕ ⊗ ⊕ ⊗ ∈ ∈ ∈ ∈ ∈ ∈ ∈ ∈ ∈ ∈ ∈     = = _ __ _⊗ __           ≤ _ __ _⊗ ⊕ __        _ _  _ _ ≤  __ ⊗ __ ⊕ __ ⊗ __       ≤

⊕

∫

⊕

                       

( )

(

)

(

)

(

( )

)

(

)

( ) ( ) ₍ ₎

( )

_{( )}

ˆ ˆ , , _, _,

p inf sup inf

d d .

x A E x B E

E E

A B f f

f x A E f x B E

f f A B

µ µ

χ µ χ µ λ λ

(9)

The proof is complete. □

We also have the following results (see [12]).

(

X, , , , ,

µ

R+ ⊕ ⊗

)

and ( )

, X f∈+  .

1) If μ is ⊕_{-additive, then so is} ( , ) f

λ

⊕ ⊗

. 2) If μ is σ-⊕_{-additive, then so is} ( , )

f

λ

⊕ ⊗

.

Given the pan-space

(

X, , , , ,

µ

R+ ⊕ ⊗

)

, the pan-integral ( _, )

_{( )}

( , ) ( _, )

d , X

F⊕ ⊗ f ⊕ ⊗ f µ f

+

=

_∫

∀ ∈_  _(4.5)

determines a monotone functional on +(X,).

In the following we show that if μ is sub-⊕_{-additive, then the monotone}

func-tional _F(⊕ ⊗, )_{is super-}_⊕_-additive.

(

X, , , , ,

µ

R+ ⊕ ⊗

)

. If μ is

sub-⊕_{-additive, then for any} _{f g}, (X, )

+

∈_  _{, it holds}

( , )

₍

₎

( , )

_{( )}

( , )

_{( )}

_,

F ⊕ ⊗ f _⊕g _≥F⊕ ⊗ f _⊕F⊕ ⊗ g

(4.6)

i.e.,

( , )

₍

₎

( , ) ( , )

d d d .

f g µ f µ g µ

⊕ ⊗ ⊕ ⊗ ⊕ ⊗

⊕ ≥ ⊕

∫

(4.7)

Proof: Let _{f g}, (X, )

+

∈_  _{. For any}

(

)

(

)

1 i and 1 j ,

r k

i A j B

i f j g

λ χ δ χ

= =

⊗ ≤ ⊗ ≤

⊕

where _i 0, _j 0, 1,2, , , 1,2, , ,

{ }

_i r_i₁ ˆ,

{ }

_j k₁ ˆ

j

i r j k A B

λ ≥ δ ≥ = _ = _ ₌ ∈ ₌ ∈, we have

(

)

(

)

1 i 1 j .

r k

i A j B

i j f g

λ

χ

δ

χ

= =    _⊗ _⊕ _⊗ _{≤ ⊕}     

⊕

 

⊕



On the other hand,

(

)

(

)

(

)

₍

₎

1 1 i j 1 i 1 j .

r k r k

i j A B i A j B

i j i j

λ δ

χ

λ

χ

δ

χ

= = = =

 _⊕ _⊗ ₌ _⊗ _⊕ _⊗ 

  _ _  

   

⊕ ⊕



⊕

Noting that

{

A B ii j: 1,2, , ,=  r j=1,2, , k

}

∈ˆ, it follows

(

) (

)

(

)

(

)

1 1 d .

r k

i j i j

i j A B f g

λ δ µ µ

= =

⊕ ⊗ ≤ ⊕

⊕ ⊕



∫

Since μ is sub-⊕_{-additive, we have}

( )

(

)

(

( )

)

(

)

(

)

(

) (

)

(

)

( )

(

)

1 1

1 1 1 1

1 1

,

d .

r k

i i j j

i j

r k k r

i i j j i j

i j j i

r k

i j i j i j

A B

A B A B

A B

f g

λ µ λ µ

λ µ δ µ

λ δ µ

µ = = = = = = = = ⊕ ⊗    _⊗ _⊕ _⊗               _ _   _ _ ≤_ _ ⊗ _ ___{ }⊕ _ ⊗ _ ___          ≤ ⊕ ⊗ ≤ ⊕

⊕

⊕ ⊕

∫

   Therefore, ( , ) ( , ) ( , )

₍

₎

d d d .

f µ g µ f g µ

⊕ ⊗ ⊕ ⊗ ⊕ ⊗

⊕ ≤ ⊕

(10)

The proof is complete. □

In the special case of the usual addition + and multiplication, the following results have been obtained in [25]:

(

X, , , , ,

µ

R+ + ⋅

)

and ( )

, X

f∈+  . If μ

is subadditive, then so is

λ

(f , )

⊕ ⊗

, i.e., any A B, ∈,

( ), ( ), ( ),

d d d .

A Bf µ A f µ B f µ

+ ⋅ + ⋅ + ⋅

= +

∫



∫

(

X, , , , ,

µ

R+ + ⋅

)

If μ is subadditive,

then for any _{f g}, (X, )

+

∈_  _{, it holds}

( ),

₍

₎

( ),

_{( )}

( ),

_{( )}

_, F+ ⋅ f g₊ ₌F + ⋅ f ₊F+ ⋅ g

(4.8)

i.e.,

( ),

₍

₎

( ), ( ),

d d d .

f g µ f µ g µ

+ ⋅ + ⋅ + ⋅

+ = +

∫

(4.9)

5. Conclusions

We have discussed the monotone measures defined by the pan-integral (with respect to monotone measure μ and operations

(

⊕ ⊗,

)

_{). As what has been}

dis-cussed above, this construction inherited almost all desirable structural charac-teristics of the original monotone measure μ. They are continuity from below,

⊕_-additivity,⊕_{-subadditivity, null-additivity, weak null-additivity, etc. We have}

also shown that the pan-integral with respect to the subadditive monotone measure is monotone superadditive functional.

In general, monotone measures lose additivity, they appear to be much looser than the classical measures. Thus, it is quite difficult to develop a general theory of monotone measures and nonlinear integral without any additional condition. The concepts mentioned above replace additivity play important roles in genera-lized measure theory (see [6] [18] [19] [26] [27] [28]). So, the results we ob-tained here are significant when applying monotone measure theory to expert systems, decision making, pattern recognition, image processing, risk analysis, and other areas.

The concave integral is a type of important nonlinear integral (with respect to capacity) [4] [29]. In further research, we will investigate properties of mono-tone measures defined by concave integral.

We point out that there are three important structural characteristics of mo-notone set function: strong order continuity [26] (see also [30]), the property (S) and the condition (E) [27]. It has been shown that Lebesgue’s theorem, Riesz’s theorem and Egoroff’s theorem in classical measures theory ([20] [31]) hold in the case of monotone measures if and only if the monotone measures possess strong order continuity [26][27][28] (see also [30]). We state these three con-cepts in the following:

Definition 24. ([26]) A monotone measure µ:→

[

0,+∞

]

is said to be

strongly order continuous, if limn→∞µ

( )

Pn =0 whenever PnN with

( )

N 0

(11)

Definition 25. ([28] A fuzzy measure μ is said to have property) (S), if for any

{ }

A_{n n} with lim_n_→∞µ

( )

A_n =0_{, there exists a subsequence}

{ }

A_{n i}_i of

{ }

A_{n n}

such that

1

0.

i n k i kA

µ ∞ ∞

= =

 ₌

 



 



Definition 26. ([27]) A fuzzy measure μ is said to fulfill the condition (E), if for every double sequence

{

( )k | 1, 1

}

n

A n≥ k≥ ⊂ satisfying the conditions: for fixed n=1,2,,

( )

₍

₎

1

and 0,

k

n n n

n

A D k µ ∞ D

=

 

→ ∞ _ _=

 





there exist increasing sequences

{ }

k_{i i}_∈ and

{ }

n_{i i}_∈ of natural numbers, such that

( )

lim i 0.

i k n j

µ

_{i j}A

∞

→+∞ ₌

 

=

 







We don’t know whether the monotone measure ( , ) f

λ

⊕ ⊗

defined by the pan-integral with respect to monotone measure μ and operations

(

⊕ ⊗,

)

_can

inherit these three important structural characteristics of monotone measures (even in special cases of

(

∨ ∧,

)

_or

( )

∨ ⋅, _{, it is not clear). We will further}

inves-tigate these problems.

Acknowledgements

This research was partially supported by the National Natural Science Founda-tion of China (Grant No. 11571106).

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