Open Access

Subadditivity of chance measure

Journal of Uncertainty Analysis and Applications20142:14

https://doi.org/10.1186/2195-5468-2-14

Received: 22 April 2014

Accepted: 1 May 2014

Published: 3 June 2014

Abstract

Chance theory is a mathematical methodology for dealing with indeterminacy phenomena involving uncertainty and randomness. In this paper, some properties of chance space are investigated. Based on this, the subadditivity theorem, null-additivity theorem, and asymptotic theorem of chance measure are proved.

Keywords

Uncertainty theoryChance theoryChance measureSubadditivity

Introduction

Uncertainty theory founded by Liu [1] in 2007 is a branch of axiomatic mathematics based on normality, duality, subadditivity, and product axioms. After that, many researchers widely studied the uncertainty theory and made significative progress. Liu [1] presented the concept of uncertain variable and uncertainty distribution. Then, a sufficient and necessary condition of uncertainty distribution was proved by Peng and Iwamura [2] in 2010. In addition, a measure inversion theorem was proposed by Liu [3] from which the uncertain measures of some events can be calculated via the uncertainty distribution. After proposing the concept of independence [4], Liu [3] presented the operational law of uncertain variables. In order to sort uncertain variables, Liu [3] proposed the concept of expected value of uncertain variable. A useful formula was presented by Liu and Ha [5] to calculate the expected values of monotone functions of uncertain variables. Based on the expected value, Liu [1] presented the concepts of variance, moments, and distance of uncertain variables. In order to characterize the uncertainty of uncertain variables, Liu [4] proposed the concept of entropy in 2009. Dai and Chen [6] verified the positive linearity of entropy and presented some formulas for calculating the entropy of monotone function of uncertain variables. Chen and Dai [7] discussed the maximum entropy principle for selecting the uncertainty distribution that has maximum entropy and satisfies the prescribed constraints. In order to make an extension of entropy, Chen et al. [8] proposed a concept of cross-entropy for comparing an uncertainty distribution against a reference uncertainty distribution. Liu [9] introduced a paradox of stochastic finance theory based on uncertainty theory and uncertain differential equation. In addition, an uncertain integral was proposed by Chen and Ralescu [10] presented with respect to the general Liu process.

In 2013, Liu [11] proposed chance theory by giving the concepts of uncertain random variable and chance measure in order to describe the situation that uncertainty and randomness appear in a system. Some related concepts of uncertain random variables such as chance distribution, expected value, and variance were also presented by Liu [11]. As an important contribution to chance theory, Liu [12] presented an operational law of uncertain random variables. After that, uncertain random variables were discussed widely. Yao and Gao [13] provided a law of large numbers for uncertain random variables. Gao and Yao [14] gave some concepts and theorems of uncertain random process. In addition, Yao and Gao [13] proposed an uncertain random process as a generalization of both stochastic process and uncertain process. As applications of chance theory, Liu [12] proposed uncertain random programming. Uncertain random risk analysis was presented by Liu and Ralescu [15]. Besides, chance theory was applied into many fields, and many achievements were obtained, such as uncertain random reliability analysis [16], uncertain random logic [17], uncertain random graph [18], and uncertain random network [18].

In this paper, some properties of chance space are investigated. Based on this, the subadditivity theorem, null-additivity theorem, and asymptotic theorem of chance measure are proposed.

Preliminary

As a branch of axiomatic mathematics, uncertainty theory aims to deal with human uncertainty. In this section, we will provide a brief introduction to uncertain variables and uncertain random variables, which will be used throughout this paper.

Uncertain variables

Definition 1

(Liu [1]) Let Γ be a non-empty set and be a σ-algebra on Γ. Each element in is called an event. A set function from to [0,1] is called an uncertain measure if it satisfies the following axioms:

Axiom 1. (Normality Axiom) {Γ}=1 for the universal set Γ.

Axiom 2. (Duality Axiom) {Λ}+{Λ c }=1 for any event Λ.

Axiom 3. (Subadditivity Axiom) For every countable sequence of events Λ1, Λ2,,we have
i = 1 Λ i i = 1 { Λ i } .
The triplet In 2009, Liu [4] definedproduct uncertain measure via the fourth axiom of uncertainty theory.
( Γ , , )

is called an uncertainty space.

Axiom 4. (Product Axiom) Let ( Γ k , k , k ) be uncertainty spaces for k=1,2, Then, the product uncertain measure is an uncertain measure satisfying
k = 1 Λ k = 43; k = 1 k { Λ k }

where Λ k are arbitrarily chosen events from k for k=1,2,, respectively.

An uncertain variable is a real-valued function on an uncertainty space, which is defined as follows.

Definition 2.

(Liu [1]) Let ( Γ , , ) be an uncertainty space. An uncertain variable is a measurable function from an uncertainty space ( Γ , , ) to the set of real numbers, i.e., for any Borel set B of real numbers, the set ξ−1(B)={γΓ|ξ(γ)B} is an event.

In order to describe uncertain variables, a concept of uncertainty distribution was introduced by Liu [1].

Definition 3.

(Liu [1]) The uncertainty distribution Φ of an uncertain variable ξ is defined by
Φ ( x ) = { ξ x }

for any real number x.

Definition 4.

(Liu [4]) The uncertain variables ξ1, ξ2,, ξ n are said to be independent if
i = 1 n ( ξ i B i ) = i = 1 n { ξ i B i }

for any Borel sets B1, B2,, B n of real numbers.

Theorem 1

(Liu [1]) Assume that ξ1, ξ2,, ξ n are independent uncertain variables with regular uncertainty distributions Φ1, Φ2,, Φ n , respectively. If f(x1, x2,, x n ) is strictly increasing with respect to x1, x2,, x m and strictly decreasing with respect to xm+1, xm+2,, x n , then ξ=f(ξ1, ξ2,, ξ n ) is an uncertain variable with inverse uncertainty distribution
Ψ 1 ( α ) = f ( Φ 1 1 ( α ) , , Φ m 1 ( α ) , Φ m + 1 1 ( 1 α ) , , Φ n 1 ( 1 α ) ) .

To represent the average value of an uncertain variable in the sense of uncertain measure, the expected value is defined as follows.

Definition 5.

(Liu [1]) Let ξ be an uncertain variable. Then, the expected value of ξ is defined by
E [ ξ ] = 0 + { ξ r } d r 0 { ξ r } d r

provided that at least one of the two integrals is finite.

Definition 6.

(Liu [1]) Let ξ be an uncertain variable with uncertainty distribution Φ. If the expected value exists, then
E [ ξ ] = 0 + ( 1 Φ ( x ) ) d x 0 Φ ( x ) d x.
(1)

For calculating the expected value by inverse uncertainty distribution, Liu and Ha [5] proved the following theorem.

Theorem 2.

(Liu and Ha [5]) Assume that ξ1, ξ2,, ξ n are independent uncertain variables with regular uncertainty distributions Φ1, Φ2,, Φ n , respectively. If f(x1, x2,, x n ) is strictly increasing with respect to x1, x2,, x m and strictly decreasing with respect to xm+1, xm+2,, x n , then the uncertain variable ξ=f(ξ1, ξ2,, ξ n ) has an expected value
E [ ξ ] = 0 1 f ( Φ 1 1 ( α ) , , Φ m 1 ( α ) , Φ m + 1 1 ( 1 α ) , , Φ n 1 ( 1 α ) ) d α.
(2)

Uncertain random variables

In 2013, Liu [11] first proposed chance theory, which is a mathematical methodology for modeling complex systems with both uncertainty and randomness, including chance measure, uncertain random variable, chance distribution, operational law, expected value, and so on. The chance space is referred to the product ( Γ , , ) × ( Ω , A , Pr ) , in which ( Γ , , ) is an uncertainty space and (Ω,A,Pr) is a probability space.

Definition 7.

(Liu [11]) Let ( Γ , , ) × ( Ω , A , Pr ) be a chance space, and let Θ × A be an event. Then, the chance measure of Θ is defined as
Ch { Θ } = 0 1 Pr { ω Ω { γ Γ | ( γ , ω ) Θ } r } d r.
Notation: For a real number r, the set Θ r ={ωΩ{γΓ|(γ, ω)Θ}≥r} is a subset of Ω but not necessarily an event in A. In this case, Pr{Θ r } is assigned by
Pr { Θ r } = inf A A , A Θ r Pr { A } , if inf A A , A Θ r Pr { A } < 0.5 sup A A , A Θ r Pr { A } , if sup A A , A Θ r Pr { A } > 0.5 0.5 , otherwise
(3)
Liu [11] proved that a chance measure satisfies normality, duality, and monotonicity properties, that is
  1. (a)

    Ch{Γ×Ω}=1, Ch{}=0;

     
  2. (b)

    Ch{Θ}+Ch{Θ c }=1 for any event Θ;

     
  3. (c)

    Ch{Θ 1}≤Ch{Θ 2} for any event Θ 1Θ 2.

     

First, we give an equivalent definition of Pr{·} in (3).

Lemma 1

Let ( Γ , A , ) × ( Ω , A , Pr ) be a chance space, and let Θ × A be an event. Denote that Θ B ={ωΩ{γΓ|(γ, ω)Θ}B} for any Borel set B. Then, we have
Pr { Θ B } = inf A A , A Θ B Pr { A } sup A A , A Θ B Pr { A } 0.5
(4)
Pr { Θ B } = sup A A , A Θ B Pr { A } inf A A , A Θ B Pr { A } 0.5
(5)

Proof

The argument breaks down into three cases.

Case 1:
inf A A , A Θ B Pr { A } < 0.5
. In this case, note that sup A A , A Θ B Pr { A } 0.5 0.5 . Then, we have
inf A A , A Θ B Pr { A } sup A A , A Θ B Pr { A } 0.5 = inf A A , A Θ B Pr { A } .
Case 2:
sup A A , A Θ B Pr { A } > 0.5
. Then, we have
inf A A , A Θ B Pr { A } sup A A , A Θ B Pr { A } 0.5 = inf A A , A Θ B Pr { A } sup A A , A Θ B Pr { A } = sup A A , A Θ B Pr { A } .
Case 3: Otherwise. It means inf A A , A Θ B Pr { A } 0.5 and sup A A , A Θ B Pr { A } 0.5 . Then, we have
inf A A , A Θ B Pr { A } sup A A , A Θ B Pr { A } 0.5 = inf A A , A Θ B Pr { A } 0.5 = 0.5 .
The equality (1) is proved. Note that
inf A A , A Θ B Pr { A } sup A A , A Θ B Pr { A } 0.5 = inf A A , A Θ B Pr { A } sup A A , A Θ B Pr { A } inf A A , A Θ B Pr { A } 0.5 = sup A A , A Θ B Pr { A } inf A A , A Θ B Pr { A } 0.5 .

Hence, the equality (5) holds.

Lemma 2.

Let ( Γ , , ) × ( Ω , A , Pr ) be a chance space, and let Θ × A be an event. Denote that Θ B ={ωΩ{γΓ|(γ, ω)Θ}B} for any Borel set B. Then, we have
Pr { Θ B } + Pr { Θ B c } = 1

Proof.

According to the equivalent definition of Pr{·} in Lemma 1, we have
Pr { Θ B c } = inf A A , A Θ B c Pr { A } sup A A , A Θ B c Pr { A } 0.5 = inf A A , A c Θ B Pr { A } sup A A , A c Θ B Pr { A } 0.5 = inf A A , A Θ B Pr { A c } sup A A , A Θ B Pr { A c } 0.5 = inf A A , A Θ B 1 Pr { A } sup A A , A Θ B 1 Pr { A } 0.5 = 1 sup A A , A Θ B Pr { A } 1 inf A A , A Θ B Pr { A } 0.5 = 1 sup A A , A Θ B Pr { A } 1 inf A A , A Θ B Pr { A } 0.5 = 1 sup A A , A Θ B Pr { A } inf A A , A Θ B Pr { A } 0.5 = 1 Pr { Θ B }

The lemma is proved.

Lemma 3.

Let ( Γ , , ) × ( Ω , A , Pr ) be a chance space, and let Θ 1 , Θ 2 × A be two events satisfying Θ1Θ2. Then, we have
Pr { Θ 1 } Pr { Θ 2 } .
(6)

Proof.

Θ1Θ2, we have
inf A A , A Θ 1 Pr { A } inf A A , A Θ 2 Pr { A } , sup A A , A Θ 1 Pr { A } sup A A , A Θ 2 Pr { A } .
According to Lemma 1, we have
Pr { Θ 1 } = inf A A , A Θ 1 Pr { A } sup A A , A Θ 1 Pr { A } 0.5 inf A A , A Θ 2 Pr { A } sup A A , A Θ 2 Pr { A } 0.5 = Pr { Θ 2 } .

The lemma is proved.

Theorem 3.

(Subadditivity Theorem) The chance measure is subadditive. That is, for any countable sequence of events Θ1, Θ2,, we have
Ch i = 1 Θ i i = 1 Ch Θ i .

Proof.

For each ω, it follows from the subadditivity of uncertain measure that
γ Γ | ( γ , ω ) i = 1 Θ i i = 1 { γ Γ | ( γ , ω ) Θ i } .
Thus, for any real number r, we have
ω Ω | γ Γ | ( γ , ω ) i = 1 Θ i r ω Ω | i = 1 γ Γ | ( γ , ω ) Θ i r
According to Lemma 3, we have
Pr ω Ω | γ Γ | ( γ , ω ) i = 1 Θ i r Pr ω Ω | i = 1 γ Γ | ( γ , ω ) Θ i r
By the definition of chance measure, we get
Ch i = 1 Θ i = 0 1 Pr ω Ω | γ Γ | ( γ , ω ) i = 1 Θ i r d r 0 1 Pr ω Ω | i = 1 γ Γ | ( γ , ω ) Θ i r d r 0 + Pr ω Ω | i = 1 γ Γ | ( γ , ω ) Θ i r d r = i = 1 0 + Pr ω Ω | γ Γ | ( γ , ω ) Θ i r d r = i = 1 0 1 Pr ω Ω | γ Γ | ( γ , ω ) Θ i r d r = i = 1 Ch Θ i .

That is, the chance measure is subadditive.

Null-additivity is a direct deduction from the above theorem. In fact, a more general theorem can be proved as follows.

Theorem 4.

Let ( Γ , , ) × ( Ω , A , Pr ) be a chance space and Θ1, Θ2, be a sequence of events with Ch{Θ i }→0 as i. Then, for any event Θ, we have
lim i Ch { Θ Θ i } = lim i Ch { Θ Θ i } = Ch { Θ } .

Proof.

By using the monotonicity and subadditivity of chance measure, we have
Ch { Θ } Ch { Θ Θ i } Ch { Θ } + Ch { Θ i }
(7)
for each i. For Ch{Θ i }→0 as i, we get Ch{ΘΘ i }→Ch{Θ}. Note that ΘΘ i Θ((ΘΘ i )Θ i ). We have
Ch { Θ Θ i } Ch { Θ } Ch { Θ Θ i } + Ch { Θ i } .
(8)

Hence, lim i Ch { Θ Θ i } = Ch { Θ } .

Remark

From the above theorem, we know that the chance measure is null-additive. That means Ch{Θ1Θ2}=Ch{Θ1}+Ch{Θ2} if either Ch{Θ1}=0 or Ch{Θ2}=0.

Theorem 5.

(Asymptotic Theorem) Let ( Γ , , ) × ( Ω , A , Pr ) be a chance space. For any events Θ1, Θ2,, we have
lim i Ch { Θ i } > 0 , if Θ i ↑Γ × Ω ,
(9)
lim i Ch { Θ i } < 1 , if Θ i ↓∅.
(10)

Proof.

Assume Θ i Γ×Ω. Since Γ×Ω= i Θ i , it follows from the subadditivity of chance measure that
1 = Ch { Γ × Ω } i = 1 Ch { Θ i } .
Note that Ch{Θ i } is increasing with respect to i. We get lim i Ch { Θ i } > 0 . If Θ i , then Θ i c ↑Γ × Ω . By using inequality (9) and the duality of chance measure, we have
lim i Ch { Θ i } = 1 lim i Ch { Θ i c } < 1 .

The theorem is proved.

Declarations

Acknowledgements

This work was supported by the National Natural Science Foundation of China Grant No.61273044 and University Science Research Project of Anhui Province No. KJ2011B105.

Authors’ Affiliations

(1)
Department of Mathematical Sciences, Chaohu University

References

  1. Liu B: Uncertainty Theory, 2nd Edition. Springer, Berlin; 2007.Google Scholar
  2. Peng Z, Iwamura K: A sufficient and necessary condition of uncertainty distribution, J. Interdisciplin. Math 2010, 13(3):277–285.MathSciNetMATHGoogle Scholar
  3. Liu B: Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty. Springer, Berlin; 2010.View ArticleGoogle Scholar
  4. Liu B: Some research problems in uncertainty theory. J. Uncertain Syst 2009, 3(1):3–10.Google Scholar
  5. Liu Y, Ha M: Expected value of function of uncertain variables. J. Uncertain. Syst 2010, 4(3):181–186.Google Scholar
  6. Dai W, Chen X: Entropy of function of uncertain variables. Math. Comput. Modell 2012, 55(3–4):754–760.MathSciNetView ArticleMATHGoogle Scholar
  7. Chen X, Dai W: Maximum entropy principle for uncertain variables. Int. J. Fuzzy. Syst 2011, 13(3):232–236.MathSciNetGoogle Scholar
  8. Chen X, Kar S, Ralescu D: Cross-entropy measure of uncertain variables. Inf. Sci 2012, 201: 53–60.MathSciNetView ArticleMATHGoogle Scholar
  9. Liu B: Toward uncertain finance theory. J. Uncertain. Anal. Appl 2013., 1(1): doi:10.1186/2195–5468–1-1Google Scholar
  10. Chen X, Ralescu D: Liu process and uncertain calculus. J. Uncertain. Anal. Appl 2013., 1(3): doi:10.1186/2195–5468–1-3Google Scholar
  11. Liu Y: Uncertain random variables: a mixture of uncertainty and randomness. Soft Comp 2013, 17(4):625–634. 10.1007/s00500-012-0935-0View ArticleMATHGoogle Scholar
  12. Liu Y: Uncertain random programming with applications. Fuzzy Optim. Decis. Ma 2013, 12(2):153–169. 10.1007/s10700-012-9149-2MathSciNetView ArticleGoogle Scholar
  13. Yao K, Gao J: Law of large numbers for uncertain random variables. (2012). Accessed 1 April 2012 http://orsc.edu.cn/online/120401.pdfGoogle Scholar
  14. Gao J, Yao K: Some concepts and theorems of uncertain random process. Int. J. Intell. Syst (2014, in press)Google Scholar
  15. Liu Y, Ralescu D: Risk index in uncertain random risk analysis. Int. J. Uncertain. Fuzz (2014, in press)MATHGoogle Scholar
  16. Wen M, Kang R: Reliability analysis in uncertain random system. (2012). Accessed 19 April 2012 http://orsc.edu.cn/online/120419.pdfGoogle Scholar
  17. Liu: Uncertain random logic and uncertain random entailment. Technical Report 2013.Google Scholar
  18. Liu B: Uncertain random graph and uncertain random network. J. Uncertain Syst 2014, 8(1):3–12.Google Scholar

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© Hou; licensee Springer. 2014

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