Egoroff’s theorem and Lusin’s theorem for complex uncertain sequences

1 Introduction

Uncertainty is an extremely important feature of the real world. In order to model uncertainty, an uncertainty theory was founded by Liu [5] in 2007 and refined by Liu [11] in 2011 which based on an uncertain measure that satisfies normality, duality, subadditivity, and product axioms. Thereafter, a concept of uncertain variable was proposed to represent the uncertain quantity and a concept of uncertain distribution to describe the uncertain variable. Up to now, uncertainty theory has successfully been applied to uncertain finance (see, e.g., Peng and Yao [14], Yu [22], Sun and Chen [17]), uncertain differential equation (see, e.g., Liu [6], Chen and Liu [1], Gao [3], Liu [13], Yao [19], Yao and Chen [20]), uncertain programming (see, e,g., Liu [8], Liu and Chen [12]), uncertain risk analysis and uncertain reliability analysis (see, e.g., Liu [10]), uncertain control (see, e.g., Liu [9], Gao [4]), uncertain calculus (see, e.g., Liu [7]), and uncertain statistics (see, e.g., Tripathy and Nath [18]), etc.

In practical application, uncertainty exists in complex quantities as well as in real uncertainties. In 2012, for modeling complex uncertain quantities, Peng [15] put forward the concepts of complex uncertain variables that are measurable functions from uncertainty spaces to the set of complex numbers.

Since the convergence of sequences play a crucial role in the fundamental theory of mathematics, there are many researchers who have studied these in the field of uncertain measure. Liu [5] first introduced convergence in measure, convergence in mean, convergence almost surely, and convergence in distribution in 2007. In 2009, You [21] introduced another type of convergence named convergence uniformly almost surely and discussed the relationships among those convergence concepts. Building in this development, the convergence of complex uncertain sequences was first studied by Chen, Ning and Wang [2] in 2016. After that, in 2017, Tripathy and Nath [18] introduced the statistical convergence concepts of complex uncertain sequences. In addition, decomposition theorems and the relationships among them were discussed in [18]. Inspired by these, we study Egoroff’s theorem and Lusin’s theorem for complex uncertain sequences.

In this paper, our main results are Egoroff’s theorem and Lusin’s theorem for complex uncertain sequences. For studying Egoroff’s theorem and Lusin’s theorem in the framework of uncertain measure, we introduce the concepts of strongly order continuous and regular. The remainder of this paper is organized as follows: In section 2, some notations, notions and theorems which are used in this paper are introduced. In section 3, we give Egoroff’s theorem and Lusin’s theorem for complex uncertain sequences including the proofs.

2 Preliminaries

In this section, some fundamental concepts and theorems in uncertainty theory are introduced, which are used in this paper.

Definition 2.1. ([5]) Let L be a σ-algebra on a non-empty set Γ. A set function M is called an uncertain measure if it satisfies the following axioms:

Axioms 1. (Normality Axiom) M { Γ } = 1 ;

Axioms 2. (Duality Axiom) M { Λ } + M { Λ c } = 1 for any Λ ∈ L ;

Axioms 3. (Monotonicity Axiom) M { Λ 1 } ≤ M { Λ 2 } for any Λ 1 , Λ 2 ∈ L and Λ₁ ⊆ Λ₂;

Axioms 4. (Subadditivity Axiom) For every countable sequence of { Λ j } ⊂ L , we have

M { ⋃ j = 1 ∞ Λ j } ≤ ∑ j = 1 ∞ M { Λ j } .(1)

The triplet ( Γ , L , M ) is called an uncertainty space, and each element Λ in L is called an event. In order to obtain an uncertain measure of compound events, a product uncertain measure is defined by Liu [7] as follows:

Axioms 5. (Product Axiom) Let ( Γ k , L k , M k ) be uncertainty space for k = 1, 2, ·· ·. The product uncertain measure M is a measure satisfying

M { ∏ k = 1 ∞ Λ k } = ⋀ k = 1 ∞ M k { Λ k }(2)

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

Definition 2.2. ([5]) An uncertain variable ξ is a measurable function from an uncertainty space ( Γ , L , M ) to the set of real numbers, i.e. for any Borel set of B of real numbers, the set

{ ξ ∈ B } = { γ ∈ Γ | ξ ( γ ) ∈ B }(3) is an event.

Definition 2.3. ([5]) The uncertainty distribution Φ of an uncertain variable ξ is defined by

Φ ( x ) = M { ξ ≤ x } , ∀ x ∈ R .(4)

Definition 2.4. ([7]) The uncertain variables ξ₁, ξ₂, ·· · , ξ _n are said to be independent if

M { ⋂ j = 1 n ( ξ j ∈ B j ) } = ⋀ j = 1 n M { ξ j ∈ B j }(5) for any Borel sets B₁, B₂, ·· · , B _n of real numbers.

Definition 2.5. ([5]) Let ξ be an uncertain variable. The expected value of ξ is defined by

E [ ξ ] = ∫ 0 + ∞ M { ξ ≥ r } dr - ∫ - ∞ 0 M { ξ ≤ r } dr(6) provided that at least one of the above two integrals is finite.

  Considering the crucial role of sequence convergence in mathematics, some convergence concepts of uncertain sequences, e.g., convergence almost surely (a.s.) and convergence uniformly almost surely were introduced in [5, 21] as follows. Throughout this paper, "a.s" is the abbreviation of "almost surely".

Definition 2.6. ([5]) The uncertain sequence {ξ _n } is said to be convergent almost surely (a.s.) to ξ if there exists an event Λ with M { Λ } = 1 such that

lim n → ∞ ∣ ξ n ( γ ) - ξ ( γ ) ∣ = 0(7)

for every γ ∈ Λ. In that case we write ξ _n  → ξ, a.s.

Definition 2.7. ([21]) The sequence {ξ _n } is said to be convergent uniformly almost surely to ξ if there exists {E _k }, M { E k } → 0 such that {ξ _n } converges uniformly to ξ in E k c : = Γ - E k , for any fixed k ∈ N.

  Now we introduce some concepts and theorems of complex uncertain variable, which were first proposed by Peng [15] in 2012. As a complex function on uncertainty space, complex uncertain variable is mainly used to model a complex uncertain quantity.

Definition 2.8. ([15]) A complex uncertain variable is a measurable function ζ from an uncertainty space ( Γ , L , M ) to the set of complex numbers, i.e., for any Borel set of B of complex numbers, the set

{ ζ ∈ B } = { γ ∈ Γ ∣ ζ ( γ ) ∈ B }(8) is an event.

Theorem 2.1. ([15]) A variable ζ from an uncertainty space ( Γ , L , M ) to the set of complex numbers is a complex uncertain variable if and only if Reζ and Imζ are uncertain variables where Reζ and Imζ represent the real and the imaginary part of ζ, respectively.

Definition 2.9. ([15]) The complex uncertainty distribution Φ (x) of a complex uncertain variable ζ is a function from ℂ to [0, 1] defined by

Φ ( c ) = M { Re ( ζ ) ≤ Re ( c ) , Im ( ζ ) ≤ Im ( c ) }(9) for any complex c.

In the following, we introduce two convergence concepts of complex uncertain sequence: convergence almost surely (a.s.) and convergence uniformly almost surely.

Definition 2.10. ([2]) The complex uncertain sequence {ζ _n } is said to be convergent almost surely (a.s.) to ζ if there exists an event Λ with M { Λ } = 1 such that

lim n → ∞ ∥ ζ n ( γ ) - ζ ( γ ) ∥ = 0(10) for every γ ∈ Λ.

Definition 2.11. ([2]) The complex uncertain sequence {ζ _n } is said to be convergent uniformly almost surely to ζ if there exists {E _k }, M { E k } → 0 such that {ζ _n } converges uniformly to ζ in E k c : = Γ - E k , for any fixed k ∈ N.

For more details of convergence concepts of complex uncertain sequence, one can refer to Chen, Ning, Wang [2], Tripathy and Nath [18], etc.

From now on, let ( Γ , L , M ) be an uncertainty space, Λ _n and Λ are both events in L . Then, we introduce two concepts of uncertain measure M .

Definition 2.12. M is called strongly order continuous if it satisfies that lim n → ∞ M { Λ n } =0 whenever Λ _n  ↘ Λ and M { Λ } = 0 .

Definition 2.13. M is called strongly continuous at Γ if it satisfies that lim n → ∞ M { Λ n } = 1 whenever Λ _n  ↗ Λ and M { Λ } = 1 .

Remark 2.1. For an uncertain measure M , the strongly order-continuity is equivalent to the strong continuity at Γ.

Proof. "⇒" From Definition 2.12, M is strongly order continuous, i.e., lim n → ∞ M { Λ n ′ } =0 whenever Λ n ′ ↘ Λ ′ and M { Λ ′ } = 0 .

Suppose that Λ _n  ↗ Λ and M { Λ } = 1 . Let Λ n ′ : = Γ - Λ n , for n = 1, 2, ·· ·, then Λ n ′ ↘ Λ ′ : = Γ - Λ and M { Λ ′ } = 0 . By the fact that M is strongly order continuous, we have

0 = lim n → + ∞ M { Λ n ′ } ∥ = lim n → + ∞ M { Γ - Λ n } ∥ = 1 - lim n → + ∞ M { Λ n } .(11) Hence lim n → + ∞ M { Λ n } = 1 , i.e., M is strongly continuous at Γ.

"⇐" From Definition 2.13, M is strongly continuous at Γ, i.e., lim n → ∞ M { Λ n ′ } = 1 whenever Λ n ′ ↗ Λ ′ and M { Λ ′ } = 1 .

Suppose that Λ _n  ↘ Λ and M { Λ } = 0 . Let Λ n ′ : = Γ - Λ n , for n = 1, 2, ·· ·, then Λ n ′ ↗ Λ ′ : = Γ - Λ and M { Λ ′ } = 1 . By the fact that M is strongly continuous at Γ, we have

1 = lim n → + ∞ M { Λ n ′ } ∥ = lim n → + ∞ M { Γ - Λ n } ∥ = 1 - lim n → + ∞ M { Λ n } .(12) Hence lim n → + ∞ M { Λ n } = 0 , i.e., M is strongly order continuous.

Theorem 3.1 (Egoroff’s Theorem). Let {ζ _n } be a complex uncertain sequence and ζ be a complex uncertain variable in ( Γ , L , M ) , which satisfy the following condition that {ζ _n } converges almost surely (a.s.) to ζ. Then {ζ _n } converges uniformly almost surely to ζ if and only if M is strongly order continuous.

Proof. "⇐" Let D be the set of these points γ ∈ Γ at which {ζ _n } does not converge to ζ. Then

D = ⋃ m = 1 ∞ ⋂ i = 1 ∞ ⋃ n = i ∞ { ∥ ζ n - ζ ∥ ≥ 1 m } .(13)

Since {ζ _n } converges almost surely (a.s.) to ζ, we have M { D } = 0 . Thus, for any fixed positive integer m,

M ( ⋂ i = 1 ∞ ⋃ n = i ∞ { ∥ ζ n - ζ ∥ ≥ 1 m } ) = 0 .(14)

Noting the fact that

⋃ n = i ∞ { ∥ ζ n - ζ ∥ ≥ 1 m } ↘ ⋂ i = 1 ∞ ⋃ n = i ∞ { ∥ ζ n - ζ ∥ ≥ 1 m } ,(15)

and by the condition that M is strongly order continuous, we have

lim n → ∞ M ( ⋃ n = i ∞ { ∥ ζ n - ζ ∥ ≥ 1 m } ) ∥ = M ( ⋂ i = 1 ∞ ⋃ n = i ∞ { ∥ ζ n - ζ ∥ ≥ 1 m } ) ∥(16) = 0 .

Hence, for any k, m ∈ N, there exists a positive integer m _k such that

M ( ⋃ n = m k ∞ { ∥ ζ n - ζ ∥ ≥ 1 m } ) ≤ 1 2 m · 1 k .(17)

Let E k = ⋃ m = 1 ∞ ⋃ n = m k ∞ { ∥ ζ n - ζ ∥ ≥ 1 m } , then

M { E k } ≤ ∑ m = 1 ∞ M ( ⋃ n = m k ∞ { ∥ ζ n - ζ ∥ ≥ 1 m } ) ∥ ≤ ∑ m = 1 ∞ 1 2 m · 1 k ∥(18) = 1 k → 0 .

And, for any m ∈ N, there exists a positive integer m _k such that for any n ≥ m _k ,

sup γ ∈ Γ - E k ∥ ζ n - ζ ∥ ≤ 1 m .(19) Thus, {ζ _n } converges uniformly almost surely to ζ.

"⇒" Let Λ _n  ↘ Λ and M { Λ } = 0 . Define

ζ n ( γ ) = { 1 , γ ∈ Λ n 0 , γ ∈ Γ - Λ n(20)

for n = 1, 2, ·· ·. It is easy to verify that {ζ _n } converges almost surely (a.s.) to 0, i.e., lim n → ∞ | ζ n ( γ ) | = 0 for every γ ∈ Γ - Λ and M { Γ - Λ } = 1 . From the hypothesis, we know that {ζ _n } converges uniformly almost surely to 0, i.e., there exists {E _k }, M { E k } → 0 such that {ζ _n } converges uniformly to 0 in Γ - E _k , for any fixed k ∈ N.

For any k ∈ N, there exists n _k  ∈ N such that for any γ ∈ Γ - E _k , we have | ζ i ( γ ) | < 1 k whenever i ≥ n _k . Hence, for any k ∈ N, we have

Γ - E k ⊂ ⋂ i = n k + ∞ { | ζ i ( γ ) | < 1 k } = Γ - Λ n k .(21) So Λ _{n k}  ⊂ E _k . By the fact that M { E k } → 0 , we have M { Λ n k } → 0 , and hence

lim n → + ∞ M { Λ n } = 0 .(22)

This shows that M is strongly order continuous. The proof is completed.

Example 3.1. Consider the uncertainty space ( Γ , L , M ) to be {γ₁, γ₂, ·· ·} with

M { Λ } = { sup γ n ∈ Λ 1 ( n + 1 ) , if sup γ n ∈ Λ 1 ( n + 1 ) < 0.5 1 - sup γ n ∈ Λ c 1 ( n + 1 ) , if sup γ n ∈ Λ c 1 ( n + 1 ) < 0.5 1 , if Λ = Γ 0 , if Λ = ∅ .(23)

And the complex uncertain variables are defined by

ζ n ( γ ) = { ( n + 1 ) i , if γ = γ n 0 , otherwise(24)

for n = 1, 2, ·· · and ζ ≡ 0. Then {ζ _n } converges uniformly almost surely to ζ.

Proof. Obviously, {ζ _n } converges almost surely (a.s.) to ζ. By Theorem 3.1, we just need to prove that M is strongly order continuous.

Suppose that Λ _n  ↘ Λ and M { Λ } = 0 . We prove the fact that lim n → ∞ M { Λ n } = 0 by contradiction.

Assume that

lim n → ∞ M { Λ n } = a > 0 ,(25) then, for a given δ > 0 satisfying that a - δ > 0, there exists a positive integer N such that n ≥ N,

M { Λ n } ≥ a - δ > 0 .(26) Hence, for any n ≥ N, there is at least one γ ∈ Λ _n .

Noting that Λ _n  ↘ Λ, we have γ ∈ Λ, and hence

M { Λ } > 0 .(27)

That contradicts to the fact that

M { Λ } = 0 .(28)

Thus, we know that M is strongly order continuous. The proof is completed.

Example 3.2. Consider the uncertainty space ( Γ , L , M ) to be {γ₁, γ₂, ·· ·} with

M { Λ } = ∑ γ n ∈ Λ 1 2 n .(29)

The complex uncertain variables are defined by

ζ n ( γ ) = { i 2 n , if γ = γ n 0 , otherwise(30)

for n = 1, 2, ·· · and ζ ≡ 0. Then {ζ _n } converges uniformly almost surely to ζ.

The proof is similar to that of Example 3.1, so it is omitted.

For study Lusin’theorem, we suppose that (Γ, ρ) is a metric space, and O and C are the classes of all open and closed sets in (Γ, ρ), respectively, and L is the Borel σ-algebra on Γ, i.e. it is the smallest σ-algebra containing O .

Definition 3.1. An uncertain measure M is called regular, if for any Λ ∈ L and δ > 0, there exists a closed set F _δ and an open set G _δ of Γ, such that F δ ⊂ Λ ⊂ G δ(31) and M { G δ - F δ } < δ .(32)

Lemma 3.1. Suppose that M is strongly order continuous, then M is regular.

Proof. Let A be the class of all set Λ ∈ L such that for any δ > 0, there exists a closed set F _δ and an open set G _δ of Γ satisfying

F δ ⊂ Λ ⊂ G δ and M { G δ - F δ } < δ .(33)

To prove this lemma, it is sufficient to show that L ⊂ A .

Firstly, we prove that A is an algebra. It is easy to find out that Γ ∈ A . Suppose Λ 1 , Λ 2 ∈ A , then for any δ > 0, there exists closed sets F_1,δ, F_2,δ of Γ and open sets G_1,δ, G_2,δ of Γ such that

F 1 , δ ⊂ Λ 1 ⊂ G 1 , δ , ∥ M { G 1 , δ - F 1 , δ } < δ ; ∥ F 2 , δ ⊂ Λ 2 ⊂ G 2 , δ , ∥ M { G 2 , δ - F 2 , δ } < δ .(34)

So we have F 1 , δ G 2 , δ c ⊂ Λ 1 Λ 2 c ⊂ G 1 , δ F 2 , δ c ,(35)

where F 1 , δ G 2 , δ c is a closed set of Γ, G 1 , δ F 2 , δ c is an open set of Γ, and

M { G 1 , δ F 2 , δ c - F 1 , δ G 2 , δ c } ∥ = M { G 1 , δ F 2 , δ c F 1 , δ c G 2 , δ } ∥ = M { ( G 1 , δ - F 1 , δ ) ⋂ ( G 2 , δ - F 2 , δ ) } ∥ ≤ min { M ( G 1 , δ - F 1 , δ ) , M ( G 2 , δ - F 2 , δ ) } ∥ < δ .(36)

That is, Λ 1 Λ 2 c ∈ A . So A is an algebra of Γ.

Next, we prove that A is closed under the formation of pairwise disjoint countable unions. Let { Λ n } ⊂ A be the sequence of pairwise disjoint set and δ > 0 be given. From the definition of A and Λ n ∈ A , we know that for each given n, there exists an open set G _n and a closed set F _n of Γ such that

F n ⊂ Λ n ⊂ G n , ∥ M { G n - F n } < δ 2 n + 1 . ∥(37)

Noting the fact that

⋃ n = 1 ∞ F n - ⋃ n = 1 k F n ↘ ∅ ,(38)

and by the condition that M is strongly order continuous, we have

lim k → ∞ M { ⋃ n = 1 ∞ F n - ⋃ n = 1 k F n } = 0 .(39)

Hence, there exists a positive integer k₀, such that

M { ⋃ n = 1 ∞ F n - ⋃ n = 1 k 0 F n } < δ 2 .(40)

Denote G δ : = ⋃ n = 1 ∞ G n and F δ : = ⋃ n = 1 k 0 F n , then G _δ is an open set of Γ, F _δ is a closed set of Γ, and

F δ ⊂ ⋃ n = 1 ∞ Λ n ⊂ G δ .(41)

It follows from the subadditivity axiom of M that

M { G δ - F δ } ∥ = M { ⋃ n = 1 ∞ G n - ⋃ n = 1 k 0 F n } ∥ ≤ M { ( ⋃ n = 1 ∞ ( G n - F n ) ) ⋃ ( ⋃ n = 1 ∞ F n - ⋃ n = 1 k 0 F n ) } ∥ ≤ M { ⋃ n = 1 ∞ ( G n - F n ) } + M { ⋃ n = 1 ∞ F n - ⋃ n = 1 k 0 F n } ∥ ≤ ∑ n = 1 ∞ M { G n - F n } + M { ⋃ n = 1 ∞ F n - ⋃ n = 1 k 0 F n } ∥ < δ .(42)

That is, ⋃ n = 1 ∞ Λ n ∈ A .(43)

Thus, we proved that A is a σ-algebra of Γ.

In real analysis theory, we know that for any closed set F of Γ, there exists a sequence of open sets {E _n } of Γ such that

E n - F ↘ ∅ .(44)

Hence, by the condition that M is strongly order continuous, we have lim n → ∞ M { E n - F } = 0 . So C ⊂ A . Since A is closed under the formation of complements, we have O ⊂ A . This shows that A is a σ-algebra containing O . Thus L ⊂ A . The proof is completed.

Remark 3.1. Let M be an uncertain measure satisfying the condition of strongly order continuous. For the complex uncertain sequence {ζ _n } and complex uncertain variable ζ, suppose that {ζ _n } converges almost surely (a.s.) to ζ, then there exists a sequence of closed sets {F _k } of Γ, M { Γ - F k } → 0 such that {ζ _n } converges uniformly to ζ in F _k , for any fixed k ∈ N.

Proof. By Theorem 3.1, there exists {E _k } of Γ, M { Γ - E k } → 0 such that {ζ _n } converges uniformly to ζ in E _k , for any fixed k ∈ N.

By Lemma 3.1, we know that M is regular. Hence, for any fixed k ∈ N, there exists a closed set F _k of Γ and an open set G _k of Γ such that F k ⊂ E k ⊂ G k , and M { E k - F k } ≤ M { G k - F k } < 1 k → 0 .(45) Thus, we have

M { Γ - F k } ∥ ≤ M { Γ - E k } + M { E k - F k } ∥ → 0 ,(46) and {ζ _n } converges uniformly to ζ in F _k for any fixed k ∈ N. The proof is completed.

Remark 3.2. Let M be an uncertain measure satisfying the condition of strongly order continuous. For the complex uncertain sequence {ζ _n } and complex uncertain variable ζ, suppose that {ζ _n } converges almost surely (a.s.) to ζ, then there exists an increasing sequence of closed sets {H _k } of Γ,

M { Γ - ⋃ k = 1 ∞ H k } = 0(47)

such that {ζ _n } converges uniformly to ζ in H _k , for any fixed k ∈ N.

Proof. By Remark 3.1, there exists a sequence of closed sets {F _k } of Γ, M { Γ - F k } → 0 such that {ζ _n } converges uniformly to ζ in F _k , for any fixed k ∈ N.

If {F _k } is not an increasing sequence, let

H k : = ⋃ i = 1 k F i ,(48) then {H _k } is an increasing sequence of closed sets of Γ and

M { Γ - H k } ∥ ≤ M { Γ - F k } ∥ → 0 .(49)

Noting that

Γ - H k ↘ Γ - ⋃ k = 1 ∞ H k ,(50) then

M { Γ - ⋃ k = 1 ∞ H k } ∥ ≤ M { Γ - H k } ∥ → 0 .(51)

Hence, we have

M { Γ - ⋃ k = 1 ∞ H k } = 0 .(52)

It is easy to verify that {ζ _n } converges uniformly to ζ in H _k for any fixed k ∈ N. The proof is completed.

In the following, we present Lusin’s Theorem for complex uncertain sequences.

Theorem 3.2 (Lusin’s Theorem). Let M be an uncertain measure satisfying the condition of strongly order continuous and ζ be a complex uncertain variable in Γ. Then, for each δ > 0, there exists a closed set F _δ of Γ such that M { Γ - F δ } < δ and ζ is continuous in F _δ .

Proof. We prove this theorem stepwise in the following two situations.

(a) Suppose that ζ is a simple function, i.e. ζ = ∑ k = 1 m a k X A k + i · ∑ j = 1 n b k X B j , where X A k and X B j are characteristic functions of A _k and B _j , respectively. Obviously, Γ = ⋃ k = 1 m ⋃ j = 1 n A k B j . For Reζ, by Lemma 3.1, for any δ > 0, we know that for each k, there exists a closed set C _k of Γ, such that C _k  ⊂ A _k and

M { A k - C k } < δ m .(53)

In the similar manner of the above, for Imζ, for each j, there exists a closed set D _j of Γ, such that D _j  ⊂ B _j and

M { B j - D j } < δ n .(54)

Let

F δ : = ⋃ k = 1 m ⋃ j = 1 n C k D j ,(55) then ζ is continuous in F _δ and

M { Γ - F δ } ∥ = M { ⋃ k = 1 m ⋃ j = 1 n A k B j - ⋃ k = 1 m ⋃ j = 1 n C k D j } ≤ M { ⋃ k = 1 m ⋃ j = 1 n ( A k B j - C k D j ) } ∥ = M { ⋃ k = 1 m ⋃ j = 1 n A k C k c B j D j c } ∥ = M { ⋃ k = 1 m A k C k c } ⋂ { ⋃ j = 1 n B j D j c } ≤ min { ∑ k = 1 m M ( A k - C k ) , ∑ j = 1 n M ( B j - D j ) } ∥ < δ .(56)

(b) Let ζ be a complex uncertain variable. Then, there exists a sequence {φ _n } of simple functions such that φ n → ζ in Γ. With the help of Remark 3.2, we know that there exists an increasing sequence of closed sets {H _k } of Γ,

M { Γ - ⋃ k = 1 ∞ H k } = 0(57) such that {φ _n } converges uniformly to ζ on H _k for any fixed k ∈ N.

Applying(a), we can obtain that for any fixed n ∈ N, there exists a closed set Q k ( n ) of Γ satisfying that

M { Γ - Q k ( n ) } < 1 2 n · 1 k ,(58)

for k = 1, 2, ·· · and φ _n is continuous in Q k ( n ) . Let

Q k : = ⋂ n = 1 ∞ Q k ( n ) .(59)

Then, we have

M { Γ - Q k } ∥ = M { ⋃ n = 1 ∞ ( Γ - Q k ( n ) ) } < 1 k · ∑ n = 1 ∞ 1 2 n = 1 k .(60)

Without loss of generality, we can choose {Q _k } to be increasing. Then

Γ - Q k ↘ Γ - ⋃ k = 1 ∞ Q k(61) and

M { Γ - ⋃ k = 1 ∞ Q k } ∥ ≤ M { Γ - Q k } ∥ < 1 k → 0 .(62) Hence M { Γ - ⋃ k = 1 ∞ Q k } = 0 .(63)

It follows from the subadditivity axiom of M that

M { ( Γ - ⋃ k = 1 ∞ H k ) ⋃ ( Γ - ⋃ k = 1 ∞ Q k ) } = 0 .(64)

Noting the fact that

( Γ - H k ) ⋃ ( Γ - Q k ) ↘ ( Γ - ⋃ k = 1 ∞ H k ) ⋃ ( Γ - ⋃ k = 1 ∞ Q k )(65) and by the condition that M is strongly order continuous, we have

lim k → ∞ M { ( Γ - H k ) ⋃ ( Γ - Q k ) } = 0 .(66) Then for any given δ > 0, there exists k₀ ∈ N,

M { ( Γ - H k 0 ) ⋃ ( Γ - Q k 0 ) } = M { Γ - ( H k 0 ⋂ Q k 0 ) } < δ .(67)

Let F δ = H k 0 ⋂ Q k 0 .(68)

Since {φ _n } is continuous in Q _k and converges uniformly to ζ in H _k for any fixed k ∈ N, then {φ _n } is continuous and converges uniformly to ζ in F _δ .

At last, we show that ζ is continuous in F _δ . For any ɛ > 0 and any γ, γ₀ ∈ F _δ , there exists a positive integer n₀ and a positive constant α such that

∥ φ n 0 ( γ ) - ζ ( γ ) ∥ ≤ ɛ 3 , ∥ φ n 0 ( γ ) - φ n 0 ( γ 0 ) ∥ ≤ ɛ 3 ,(69)

whenever ρ (γ, γ₀) < α. Thus, we have

∥ ζ ( γ ) - ζ ( γ 0 ) ∥ ∥ = ∥ ζ ( γ ) - φ n 0 ( γ ) + φ n 0 ( γ ) - φ n 0 ( γ 0 ) + φ n 0 ( γ 0 ) - ζ ( γ 0 ) ∥ ∥ ≤ ∥ ζ ( γ ) - φ n 0 ( γ ) ∥ + ∥ φ n 0 ( γ ) - φ n 0 ( γ 0 ) ∥ + ∥ φ n 0 ( γ 0 ) - ζ ( γ 0 ) ∥ ∥ < ɛ .(70)

So ζ is continuous in F _δ .

Remark 3.3. Let M be an uncertain measure satisfying the condition of strongly order continuous and ζ be a complex uncertain variable in Γ. By Theorem 3.2, we know that for any fixed n ∈ N, there exists a sequence of closed sets {F _n } of Γ such that ζ is continuous in F _n and

M { Γ - F n } < 1 n .(71)

Theorem 3.3 (Continuous Function Approximation Theorem). Let M be an uncertain measure satisfying the condition of strongly order continuous and ζ be a complex uncertain variable in Γ. Then, there exists a complex uncertain sequence {ζ _n } in Γ such that {ζ _n } converges uniformly almost surely to ζ. Furthermore, if ∥ζ ∥ ≤ M, then ∥ζ _n  ∥ ≤ M (n = 1, 2, ·· ·), where M is a positive constant.

Proof. By Remark 3.3, we know that for any k ∈ N, there exists a closed set F _k of Γ such that ζ is continuous on F _k and M { Γ - F k } < 1 k . By Tietze’s extension theorem (see, e.g., [16]), for each k = 1, 2, ·· · , there exists a continuous complex uncertain variable ψ _k in Γ such that ψ _k  (γ) = ζ (γ), for γ ∈ F _k . And if ∥ζ ∥ ≤ M, then ∥ψ _k  ∥ ≤ M. Therefore, for any ɛ > 0, we have

{ ∥ ψ k ( γ ) - ζ ( γ ) ∥ ≥ ɛ } ⊂ Γ - F k .(72)

Hence, for each k = 1, 2, ·· ·,

M { ∥ ψ k ( γ ) - ζ ( γ ) ∥ ≥ ɛ } ≤ M { Γ - B k } < 1 k .(73)

So we have

lim k → ∞ M { ∥ ψ k ( γ ) - ζ ( γ ) ∥ ≥ ɛ } = 0 .(74)

From the above fact, we can choose a subsequence {ψ _{k n} } of {ψ _k } such that

M { ∥ ψ k n ( γ ) - ζ ( γ ) ∥ ≥ 1 2 n } < 1 2 n .(75)

Let

Λ n : = { ∥ ψ k n ( γ ) - ζ ( γ ) ∥ ≥ 1 2 n } .(76)

Then

∑ n = 1 ∞ M { Λ n } < ∞ .(77)

Next, we prove

M ( ⋂ n = 1 ∞ ⋃ v = 1 ∞ { ∥ ψ k n + v ( γ ) - ζ ( γ ) ∥ ≥ ɛ } ) = 0 .(78)

Indeed, for any ɛ > 0, there exists a positive integer n₀ such that for any n ≥ n₀, 1 2 n < ɛ and M ( ⋂ n = 1 ∞ ⋃ v = 1 ∞ { ∥ ψ k n + v ( γ ) - ζ ( γ ) ∥ ≥ ɛ } ) ∥ ≤ ∑ m = n ∞ M ( { ∥ ψ k m ( γ ) - ζ ( γ ) ∥ ≥ ɛ } ) ∥ ≤ ∑ m = n ∞ M { Λ m } → 0 .(79)

Take ζ _n  = ψ _{k n} , n = 1, 2, ·· ·. Then, there exists an event Λ : = Γ - ⋂ n = 1 ∞ ⋃ v = 1 ∞ { ∥ ζ n + v ( γ ) - ζ ( γ ) ∥ ≥ ɛ } with M { Λ } = 1 such that

lim n → ∞ ∥ ζ n ( γ ) - ζ ( γ ) ∥ = 0(80) for every γ ∈ Λ. Hence, {ζ _n } converges almost surely (a.s.) to ζ. By Theorem 3.1, we have {ζ _n } converges uniformly almost surely to ζ.