A revision of sufficient and necessary condition of uncertainty distribution

1 Introduction

For the purpose of dealing with the belief degrees rationally, uncertainty theory was proposed by Liu [1] in 2007 and refined by Liu [2] in 2009. Since then, many scholars have dedicated themselves to the studies of uncertainty theory and have made significant progress.

The first essential concept in uncertainty theory is uncertain measure defined by Liu [2] based on the normality, duality, subadditivity, and product axioms. Uncertain measure has been widely studied by many scholars. As an important contribution, Gao [11] proposed the judgement conditions of continuous uncertain measure. Later, some structural characteristics of uncertain measure were investigated by Zhang [18]. Based on the product axiom, Peng and Iwamura [17] verified that the product uncertain measure is indeed an uncertain measure. After that, Liu [6] gave a polyrectangular theorem to calculate the product uncertain measure of the polyrectangle. In addition, Wang [12] constructed some inequalities about uncertain measure and gave the estimation for the uncertain measure of some uncertain events. Since then, uncertain measure has been well developed and become a momentous foundation of uncertainty theory.

The another essential concept in uncertainty theory is uncertain variable presented by Liu [1] in 2007. Meanwhile, for the sake of characterizing uncertain variables, Liu [1] proposed the concept of uncertainty distribution. In addition, Liu [3] proposed the concept of inverse uncertainty distribution, and Liu [5] verified a sufficient and necessary condition for it. Based on the independence of uncertain variables presented by Liu [2], Liu [3] proposed the operational law for the uncertainty distribution of strictly monotone function of independent uncertain variables. Furthermore, in order to measure uncertain variable, Liu [1] gave the definition of expected value of uncertain variable. After that, Liu and Ha [14] proposed an important formula to calculate the expected values of strictly monotone function of independent uncertain variables, and Yang [13] discussed the expected values of non-monotonic function of independent uncertain variables. Based on the expected value of uncertain variables, Liu [1] proposed the concepts of variance and k-th moment of uncertain variables. After that, Sheng [15] presented a formula to calculate the k-th moment of uncertain variables. Nowadays, considerable work has been done by many scholars based on uncertain variables, such as You [7], Chen et al. [9], Yao [8], etc.

Uncertainty distribution is the most important tool to characterize uncertain variable in uncertainty theory. In fact, a large number of research cases show that only getting the uncertainty distribution is enough to solve the problem. Therefore, uncertainty distribution plays a vital role in the development of uncertainty theory, and it is crucial to figure out what kind of function is an uncertainty distribution. For the sake of dealing with this problem, Peng and Iwamura [16] presented a sufficient and necessary condition of a function being uncertainty distribution. Nowadays, the sufficient and necessary condition has been widely reported by many scholars and become a theorem that must be mastered in uncertainty theory. However, the sufficient and necessary condition is not appropriate.

In this paper, we aim to illustrate the incorrectness of the existed sufficient and necessary condition and propose a revision of it. The overall structure of this paper takes the form of six sections, including this introductory section. Section 2 begins by reviewing the theorem presented by Peng and Iwamura [16], and providing a counterexample to show the incorrectness of the existed sufficient and necessary condition. Section 3 gives a useful lemma to revise the condition. Then a revision for the sufficient and necessary condition is proposed in Section 4. Section 5 gives some discussions about uncertain measure. Finally, a concise conclusion is given in Section 6.

2 A counterexample

The uncertainty distribution Φ of an uncertain variable η is defined by Φ ( x ) = M { η ≤ x } for any real number x. Up to now, uncertainty distribution has been applied in many fields of uncertainty theory. For example, uncertainty distribution can be used to calculate the expected values and variances of uncertain variables. In order to characterize the uncertainty of uncertain variables resulting from information deficiency, Liu [2] and Chen et al. [10] applied uncertainty distribution into the entropy of uncertain variables. Besides, uncertainty distribution was applied to hazard distribution by Liu [4] to calculate the residual lifetime of the uncertain system.

In order to find out the concrete form of the uncertainty distribution, Peng and Iwamura [16] presented a sufficient and necessary condition of the uncertainty distribution as follows,

Theorem 2.1. (Peng-Iwamura [16]) A real-valued function Φ (x) on R is an uncertainty distribution if and only if it is a monotone increasing function satisfying 0 ≤ Φ ( x ) ≤ 1 , Φ ( x ) ≡ 0 , Φ ( x ) ≡ 1 .(1)

However, Theorem 2.1 is not appropriate. In order to illustrate the inappropriateness of Theorem 2.1, a counterexample is given below.

OPEN IN VIEWER

Fig. 1

The Graph of Φ (x).

Example 2.1. Let Φ (x) be a function from R to [0,1] as follows, Φ ( x ) = { 0 , if x ≤ 0 7 20 x , if 0 < x ≤ 2 1 , if x > 2 .(2) As shown in Fig. 1, Φ (x) satisfies all the conditions in Theorem 2.1. Thus, according to Theorem 2.1 that Φ (x) should be an uncertainty distribution, i.e., we can design an uncertain variable η such that Φ (x) is the uncertainty distribution of η. By using measure inversion theorem, we have M { η > 2 } = 1 - Φ ( 2 ) = 0.3 and M { η > 2 + 1 k } = 1 - Φ ( 2 + 1 k ) = 0 , k = 1 , 2 , ⋯ Notice that { η > 2 } = ⋃ k = 1 ∞ { η > 2 + 1 k } . According to the subadditivity axiom, we can obtain 0.3 = M { η > 2 } ≤ ∑ k = 1 ∞ M { η > 2 + 1 k } = 0 which is a contradiction. It implies that Theorem 2.1 is not correct.

What goes wrong with Peng and Iwamura’s theorem? Assume that C is a collection consisting of all intervals of the form (- ∞ , x], (x, + ∞), ∅, R , and M is a set function on C as follows, M { ( - ∞ , x ] } = Φ ( x ) , M { ( x , + ∞ ) } = 1 - Φ ( x ) , M { R } = 1 , M { ∅ } = 0 where Φ (x) is a monotone increasing function on R satisfying (1). Then in the proof of Peng and Iwamura’s theorem ([16], page 281, line 2), without any proof, they claimed that inf B ⊂ ⋃ i = 1 ∞ A i ′ ∑ i = 1 ∞ M { A i ′ } + inf B c ⊂ ⋃ i = 1 ∞ A i ″ ∑ i = 1 ∞ M { A i ″ } ≥ 1(3) holds for any subset B of real numbers, where A i ′ , A i ″ ∈ C ,  i = 1, 2, ⋯ However, this inequality is not always valid. For example, we take the Φ (x) as the function defined in (2), let us consider the case B = (- ∞ , 2]. Then we take A 1 ′ = ( - ∞ , 2 ] , A i ′ = ∅ , i = 2 , 3 , ⋯ and A i ″ = ( 2 + 1 i , + ∞ ) , i = 1 , 2 , ⋯ It is easily seen, { A i ′ } is a cover of B and { A i ″ } is a cover of B ^c . Then we can obtain inf B ⊂ ⋃ i = 1 ∞ A i ′ ∑ i = 1 ∞ M { A i ′ } + inf B c ⊂ ⋃ i = 1 ∞ A i ″ ∑ i = 1 ∞ M { A i ″ } ≤ M { ( - ∞ , 2 ] } + ∑ i = 1 ∞ M { ( 2 + 1 i , + ∞ ) } = Φ ( 2 ) + ∑ i = 1 ∞ ( 1 - Φ ( 2 + 1 i ) ) = 0.7 < 1 which implies that the inequality (3) is not valid.

To provide a revision of the sufficient and necessary condition, we first present a useful lemma in this section.

Lemma 3.1. Let Φ (x) be a monotone increasing function on R satisfying 0 ≤ Φ ( x ) ≤ 1 , Φ ( x ) ≡ 0 , Φ ( x ) ≡ 1 , Φ ( x 0 ) = 1 if Φ ( x ) = 1 for any x > x 0 . Then there exists an uncertain measure M on R such that M { ( - ∞ , x ) } = { 0 , if lim y ↑ x Φ ( y ) = 0 μ , otherwise , M { [ x , + ∞ ) } = { 1 , if lim y ↑ x Φ ( y ) = 0 1 - μ , otherwise , M { ( - ∞ , x ] } = Φ ( x ) , M { ( x , + ∞ ) } = 1 - Φ ( x ) , M { ∅ } = 0 , M { R } = 1(4) for any real number x, where μ is an arbitrarily chosen value between lim y ↑ x Φ ( y ) and Φ (x).

Proof: First, let S be a collection consisting of all intervals of the form (- ∞ , x], (- ∞ , x), (x, + ∞), [x, + ∞), ∅ and R , and let M be a set function on S satisfy (4). Then we prove that inf B ⊂ ⋃ i = 1 2 A i ′ ∑ i = 1 2 M { A i ′ } + inf B c ⊂ ⋃ i = 1 2 A i ″ ∑ i = 1 2 M { A i ″ } ≥ 1(5) holds for any subset B of real numbers, where A i ′ , A i ″ ∈ S , i = 1, 2. For any subset B of real numbers, the union of covers of B and B ^c must be a cover of R . That is, we always have R ⊂ ⋃ i = 1 2 A i ′ ∪ ⋃ i = 1 2 A i ″ if B ⊂ ⋃ i = 1 2 A i ′ , B c ⊂ ⋃ i = 1 2 A i ″ , where A i ′ , A i ″ ∈ S , i = 1, 2. Without loss of generality, for arbitrary { A 1 ′ , A 2 ′ } which covers B and { A 1 ″ , A 2 ″ } which covers B ^c , we assume that A 1 ′ = ( - ∞ , a ′ ] or ( - ∞ , a ′ ) , A 2 ′ = [ b ′ , + ∞ ) or ( b ′ , + ∞ ) , A 1 ″ = ( - ∞ , a ″ ] or ( - ∞ , a ″ ) , A 2 ″ = [ b ″ , + ∞ ) or ( b ″ , + ∞ ) where a′, a″, b′, b″ ∈ [- ∞ , + ∞]. Here we set ( - ∞ , - ∞ ) = ( + ∞ , + ∞ ) = ∅ . Since A 1 ′ ∪ A 1 ″ ∈ S and A 2 ′ ∪ A 2 ″ ∈ S , we always have inf R ⊂ ⋃ i = 1 2 A i ∑ i = 1 2 M { A i } ≤ inf B ⊂ ⋃ i = 1 2 A i ′ , B c ⊂ ⋃ i = 1 2 A i ″ ∑ i = 1 2 M { A i ′ ∪ A i ″ } ≤ inf B ⊂ ⋃ i = 1 2 A i ′ , B c ⊂ ⋃ i = 1 2 A i ″ ∑ i = 1 2 ( M { A i ′ } + M { A i ″ } ) = inf B ⊂ ⋃ i = 1 2 A i ′ , B c ⊂ ⋃ i = 1 2 A i ″ ( ∑ i = 1 2 M { A i ′ } + ∑ i = 1 2 M { A i ″ } ) = inf B ⊂ ⋃ i = 1 2 A i ′ ∑ i = 1 2 M { A i ′ } + inf B c ⊂ ⋃ i = 1 2 A i ″ ∑ i = 1 2 M { A i ″ } . Hence, for the purpose of proving inequality (5), we only need to prove the following inequality inf R ⊂ ⋃ i = 1 2 A i ∑ i = 1 2 M { A i } ≥ 1 .(6) For an arbitrary sequence {A₁, A₂} which covers R , we write that a = sup x ∈ A 1 x , b = inf x ∈ A 2 x . Since R ⊂ ⋃ i = 1 2 A i , we have b ≤ a . When b < a, by using the monotonicity of Φ (x), we can get Φ ( b ) ≤ μ . Here we set Φ (- ∞) =0 and Φ (+ ∞) =1. Hence, we have M { A 1 } + M { A 2 } ≥ M { ( - ∞ , a ) } + M { ( b , + ∞ ) } = μ + 1 - Φ ( b ) ) ≥ 1 . Now we consider the case b = a. Let us write them by c. Since c ∈ R ⊂ ⋃ i = 1 2 A i , there is at least an interval containing c. That is, there is at least an interval that is (- ∞ , c] or [c, + ∞). For convenience, we assume that A₁ = (- ∞ , c] . Then we can get M { A 1 } + M { A 2 } ≥ M { ( - ∞ , c ] } + M { ( c , + ∞ ) } = Φ ( c ) + 1 - Φ ( c ) = 1 . It follows that ∑ i = 1 2 M { A i } ≥ 1 for any sequence {A₁, A₂} covering R . Thus inequality (6) holds and inequality (5) follows immediately.

Next we extend the set function M defined in (4) into an uncertain measure, for any subset B of real numbers, we can always take two sets A₁ and A₂ from S such that {A₁, A₂} is a cover of B, where A₁ = (- ∞ , a] or (- ∞ , a), A₂ = [b, + ∞) or (b, + ∞). Then we can define M { B } as follows, M B = inf B ⊂ ∪ i = 1 2 A i ∑ i = 1 2 M A i if inf B ⊂ ⋃ i = 1 2 A i ∑ i = 1 2 M { A i } < 0.5 , M { B } = 1 - inf B c ⊂ ⋃ i = 1 2 A i ∑ i = 1 2 M { A i } if inf B c ⊂ ⋃ i = 1 2 A i ∑ i = 1 2 M { A i } < 0.5 , and M { B } = 0.5 , otherwise, where A 1 , A 2 ∈ S . Next we will verify that the M is indeed an uncertain measure. In other words, we will verify that M satisfies the three axioms of the definition of uncertain measure.

Normality: It follows from (4) that M { R } = 1 .

Duality: We should prove that M { B } + M { B c } = 1 holds for an arbitrary subset B of real numbers. Then the argument breaks down into three cases. Case 1: Assume inf B ⊂ ⋃ i = 1 2 A i ∑ i = 1 2 M { A i } < 0.5 . Then we have inf B c ⊂ ⋃ i = 1 2 A i ∑ i = 1 2 M { A i } > 0.5 by inequality (5). By using the definition of M , we obtain M { B } + M { B c } = inf B ⊂ ⋃ i = 1 2 A i ∑ i = 1 2 M { A i } + 1 - inf B ⊂ ⋃ i = 1 2 A i ∑ i = 1 2 M { A i } = 1 . Case 2: Assume inf B c ⊂ ⋃ i = 1 2 A i ∑ i = 1 2 M { A i } < 0.5 . This case can be proved similarly. Case 3: Assume inf B ⊂ ⋃ i = 1 2 A i ∑ i = 1 2 M { A i } ≥ 0.5 and inf B c ⊂ ⋃ i = 1 2 A i ∑ i = 1 2 M { A i } ≥ 0.5 . Then we immediately have M { B } = M { B c } = 0.5 . The duality is proved.

Next, in order to verify that M satisfies the subadditivity axiom, we need to prove that M is monotone increasing. Assume that Λ₁ and Λ₂ are two subsets of R and satisfy that Λ₁ ⊂ Λ₂. If M { Λ 2 } < 0.5 , then we can get inf Λ 2 ⊂ ⋃ i = 1 2 A i ∑ i = 1 2 M { A i } = M { Λ 2 } < 0.5 . Hence we get M { Λ 1 } = inf Λ 1 ⊂ ⋃ i = 1 2 A i ∑ i = 1 2 M { A i } ≤ inf Λ 2 ⊂ ⋃ i = 1 2 A i ∑ i = 1 2 M { A i } = M { Λ 2 } < 0.5 . If M { Λ 1 } > 0.5 , we immediately have M { Λ 1 c } < 0.5 , Λ 2 c ⊂ Λ 1 c . Thus we have M { Λ 2 c } ≤ M { Λ 1 c } , that is, M { Λ 1 } ≤ M { Λ 2 } . Otherwise, we have M { Λ 1 } ≤ 0.5 , M { Λ 2 } ≥ 0.5 , it is clear that M { Λ 1 } ≤ M { Λ 2 } . Therefore, M is monotone increasing.

Subadditivity: Suppose {B _j } is a sequence in the power set over R and B = ⋃ j = 1 ∞ B j . We prove the following inequality M { B } ≤ ∑ j = 1 ∞ M { B j } . The argument breaks down into three cases. Case 1: Assume M { B j } < 0.5 , j = 1 , 2 , ⋯ By using the definition of M , we can get M { B j } = inf B j ⊂ ⋃ i = 1 2 A i j ∑ i = 1 2 M { A i j } for j = 1, 2, ⋯ Then for any given ɛ > 0 and each j, there exist sequences { A i j } in S such that B j ⊂ ⋃ i = 1 2 A i j and ∑ i = 1 2 M { A i j } ≤ M { B j } + ɛ 2 j .(7) By the inequality (5), we can get 1 - inf B c ⊂ ⋃ i = 1 2 A i ″ ∑ i = 1 2 M { A i ″ } ≤ inf B ⊂ ⋃ i = 1 2 A i ′ ∑ i = 1 2 M { A i ′ } . Hence we have M { B } ≤ inf B ⊂ ⋃ i = 1 2 A i ∑ i = 1 2 M { A i } . Moreover, we have B = ⋃ j = 1 ∞ B j ⊂ ⋃ j = 1 ∞ ⋃ i = 1 2 A i j = ( ⋃ j = 1 ∞ A 1 j ) ∪ ( ⋃ j = 1 ∞ A 2 j ) . If M { A 1 j } = 0 for j = 1, 2, ⋯, we can get M { ⋃ j = 1 ∞ A 1 j } = 0 = ∑ j = 1 ∞ M { A 1 j } . Otherwise, we notice that ⋃ j = 1 ∞ A 1 j = A 1 1 ∪ ( ⋃ j = 2 ∞ A 1 j ) . Since A 1 1 ∈ S and ⋃ j = 2 ∞ A 1 j ∈ S , we have M { ⋃ j = 1 ∞ A 1 j } ≤ M { A 1 1 } + M { ⋃ j = 2 ∞ A 1 j } . Repeating the above process, we can obtain M { ⋃ j = 1 ∞ A 1 j } ≤ ∑ j = 1 ∞ M { A 1 j } . Similarly, we have M { ⋃ j = 1 ∞ A 2 j } ≤ ∑ j = 1 ∞ M { A 2 j } . Then we can get M { B } ≤ inf B ⊂ ⋃ i = 1 2 A i ∑ i = 1 2 M { A i } ≤ M { ⋃ j = 1 ∞ A 1 j } + M { ⋃ j = 1 ∞ A 2 j } ≤ ∑ j = 1 ∞ M { A 1 j } + ∑ j = 1 ∞ M { A 2 j } = ∑ j = 1 ∞ ∑ i = 1 2 M { A i j } ≤ ∑ j = 1 ∞ ( M { B j } + ɛ 2 j ) = ∑ j = 1 ∞ M { B j } + ɛ . So that, by letting ɛ → 0, we have M { B } ≤ ∑ j = 1 ∞ M { B j } . Case 2: Assume there is only one term greater than or equal to 0.5. For simplicity, we suppose that M { B 1 } ≥ 0.5 , M { B j } < 0.5 , j = 2 , 3 , ⋯ According to the monotonicity of M , we have M { B } ≥ M { B 1 } ≥ 0.5 . If M { B } = 0.5 , then we immediately have M { B } ≤ M { B 1 } ≤ ∑ j = 1 ∞ M { B j } . Notice that we have B 1 c ⊂ B c ∪ ( ⋃ j = 2 ∞ B j ) . Otherwise, we have M { B c } = 1 - M { B } < 0.5 . By using Case 1, we can obtain M { B 1 c } ≤ M { B c ∪ ( ⋃ j = 2 ∞ B j ) } ≤ M { B c } + ∑ j = 2 ∞ M { B j } . That is, we can get 1 - M { B 1 } ≤ 1 - M { B } + ∑ j = 2 ∞ M { B j } . i.e., M { B } ≤ ∑ j = 1 ∞ M { B j } . Case 3: Assume there are at least two terms greater than or equal to 0.5, it is clear that M { B } ≤ 1 ≤ ∑ j = 1 ∞ M { B j } . Hence M is really an uncertain measure and satisfies (4) immediately. The lemma is proved.

Now we will give a revision of the sufficient and necessary condition.

Theorem 4.1. A real-valued function Φ (x) on R is an uncertainty distribution if and only if it is a monotone increasing function satisfying 0 ≤ Φ ( x ) ≤ 1 , Φ ( x ) ≡ 0 , Φ ( x ) ≡ 1 , Φ ( x 0 ) = 1 if Φ ( x ) = 1 for any x > x 0 .(8)

Proof: Assume that a real-valued function Φ (x) on R is an uncertainty distribution. According to the monotonicity theorem, we always have Φ ( a ) = M { η ≤ a } ≤ M { η ≤ b } = Φ ( b ) for any given points a < b, i.e., Φ (x) is monotone increasing. In addition, we always have 0 ≤ M { η ≤ x } ≤ 1 for an arbitrary real number x. That is, 0 ≤ Φ (x) ≤1. If Φ (x) ≡0, we can notice that Γ = { η < + ∞ } = ⋃ n = 1 ∞ { η ≤ n } . Thus, we have 1 = M { Γ } ≤ ∑ n = 1 ∞ M { η ≤ n } = ∑ n = 1 ∞ Φ ( n ) = 0 which is a contradiction. If Φ (x) ≡1, we can notice that Γ = { η > - ∞ } = ⋃ n = 1 ∞ { η > - n } . Thus, we have 1 = M { Γ } ≤ ∑ n = 1 ∞ M { η > - n } = ∑ n = 1 ∞ ( 1 - Φ ( - n ) ) = 0 which is also a contradiction. Given a point x₀, if we have Φ (x) =1 for any x > x₀, then we can get 1 - Φ ( x 0 ) = M { η > x 0 } = M { ⋃ n = 1 ∞ ( η > x 0 + 1 n ) } ≤ ∑ n = 1 ∞ M { η > x 0 + 1 n } = ∑ n = 1 ∞ ( 1 - Φ ( x 0 + 1 n ) ) = 0 . That is, Φ (x₀) =1. Therefore, the necessity of the theorem is proved.

Conversely, assume a real-valued function Φ (x) on R is a monotone increasing function satisfying (8). Next we will design an uncertain variable η such that Φ (x) is the uncertainty distribution of η. Firstly we take an uncertainty space ( ℜ , L , M ) where L is the power set over ℜ and M is the uncertain measure defined in the proof of Lemma 3.1. Then we take the uncertain variable ηdefined by η (γ) = γ. For any given point x, we can obtain M { η ≤ x } = M { γ ∈ R ∣ γ ≤ x } = M { ( - ∞ , x ] } = Φ ( x ) which means that the uncertainty distribution of η ( γ ) = γ is Φ (x). Thus, the theorem is proved.

It is clear that the counterexample in Section 2 does not satisfy the revision of the sufficient and necessary condition. It follows from the revision that all functions that are not right continuous at point xat which Φ (x)=1are not an uncertainty distribution. And the common uncertainty distribution satisfies the revision such as linear uncertainty distribution, zigzag uncertainty distribution and normal uncertainty distribution.

Now we give a method to construct an uncertain variable whose uncertainty distribution Φ is just the given function. First, for any subset Bof real numbers, we set inf B = inf x ∈ B x a n d sup B = sup x ∈ B x , and define some assistant set functions as follows, M 1 { B } = { 0 , if lim x ↑ sup B Φ ( x ) = 0 Φ ( sup B ) , otherwise , M 2 { B } = { 1 , if lim x ↑ inf B Φ ( x ) = 0 1 - Φ ( inf B ) , otherwise , M 3 { B } = inf ( a , b ] ∈ B c { Φ ( a ) + 1 - Φ ( b ) } . For convenience, let us write M * B = ∧ i = 1 3 M i B and define an uncertain measure M as follows, M { B } = { M * { B } , if M * { B } < 0.5 1 - M * { B c } , if M * { B c } < 0.5 0.5 , otherwise . Then we take an uncertainty space ( ℜ , L , M ) where L is the power set over ℜ and M is the uncertain measure defined above, and define the uncertain variable as η (γ) = γ. It’s easy to verify that the uncertain variable ηis what we need. Next we give an example to illustrate the method.

Example 4.1. Let Φ ( x ) ≡ c , 0.5 < c < 1. We can construct an uncertain variable whose uncertainty distribution is Φ. For any nonempty proper subset B of real numbers, if B is lower bounded, then M 1 { B } = Φ ( sup B ) = c or M 1 { B } = Φ ( + ∞ ) = 1 , M 2 { B } = 1 - Φ ( inf B ) = 1 - c , M 3 { B } = Φ ( - ∞ ) + 1 - c = 1 - c . Thus, we have M { B } = M * { B } = ∧ i = 1 3 M i { B } = 1 − c . If B^cis lower bounded, we have M { B c } = 1 − c , that is, M { B } = c . If B and B^care both lower unbounded, then M 1 { B } = Φ ( sup B ) = c or M 1 { B } = Φ ( + ∞ ) = 1 , M 2 { B } = 1 - Φ ( - ∞ ) = 1 , M 3 { B } = c + 1 - c = 1 . Thus we have M * { B } ≥ c > 0.5 a n d M * { B c } ≥ c > 0.5. That is, M { B } = 0.5 . Then we have M { B } = { 0 , if B = ∅ 1 - c , if B is lower bounded 0.5 , if B and B c are both lower bounded c , if B c is lower bounded 1 , if B = R , and the uncertain variable η (γ) = γ defined on the uncertainty space ( ℜ , L , M ) has the uncertainty distribution Φ ( x ) ≡ c , 0.5 < c < 1 .

It is easy for us to know the uncertain measures of {η ≤ x} and {η >x}. However, we don’t have any information about the uncertain measures of {η <x} and {η ≥ x}. Next we will discuss the uncertain measures of {η <x} and {η ≥ x}.

Theorem 5.1. For any given uncertainty distribution Φ, there exists an uncertain variable ηwhose uncertainty distribution is Φ satisfying M { η < x } = { 0 , if lim y ↑ x Φ ( y ) = 0 μ , otherwise for any given point x, where μ is an arbitrarily chosen value between lim y ↑ x Φ ( y ) and Φ (x).

Proof: We take an uncertainty space ( ℜ , L , M ) where L is the power set over ℜ and M is the uncertain measure defined in the proof of Lemma 3.1, and define an uncertain variable η (γ) = γ. According to Theorem 4.1 that Φ (x)is the uncertainty distribution of η. For any given point x, we can get M { η < x } = M { γ ∈ R ∣ γ < x } = M { ( - ∞ , x ) } = { 0 , if lim y ↑ x Φ ( y ) = 0 μ , otherwise .

Theorem 5.2. For any given uncertainty distribution Φ, there exists an uncertain variable ηwhose uncertainty distribution is Φ satisfying M { η ≥ x } = { 1 , if lim y ↑ x Φ ( y ) = 0 1 - μ , otherwise for any given point x, where μ is an arbitrarily chosen value between lim y ↑ x Φ ( y ) and Φ (x).

Proof: We take an uncertainty space ( ℜ , L , M ) where L is the power set over ℜ and M is the uncertain measure defined in the proof of Lemma 3.1, and define an uncertain variable η (γ) = γ. According to Theorem 4.1 that Φ (x)is the uncertainty distribution of η. For any given point x, we can get M { η ≥ x } = M { γ ∈ R ∣ γ ≥ x } = M { [ x , - ∞ ) } = { 1 , if lim y ↑ x Φ ( y ) = 0 1 - μ , otherwise .

6 Conclusion

The main contributions of this paper are as follows. First of all, we provided a counterexample for the sufficient and necessary condition presented by Peng and Iwamura. Then we gave a revision of the sufficient and necessary condition, and presented a method to construct an uncertain variable whose uncertainty distribution Φ is just the given function. Finally, some discussions about uncertain measure were given in this paper.