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

Abstract

Complex uncertain variables are measurable functions from uncertainty spaces to the set of complex numbers and are used to model complex uncertain quantities. In this paper, we investigate Egoroff’s theorem and Lusin’s theorem for complex uncertain sequences. For studying these theorems, we introduce two concepts: strongly order continuous and regular. And as far as we know, our results are new.

Keywords

Complex uncertain variables Egoroff’s theorem Lusin’s theorem

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 $Λ \in L$ ;

Axioms 3. (Monotonicity Axiom) $M {Λ_{1}} \leq M {Λ_{2}}$ for any $Λ_{1}, Λ_{2} \in L$ and Λ₁ ⊆ Λ₂;

Axioms 4. (Subadditivity Axiom) For every countable sequence of ${Λ_{j}} \subset L$ , we have

$\begin{matrix} M {⋃_{j = 1}^{\infty} Λ_{j}} \leq \sum_{j = 1}^{\infty} M {Λ_{j}} . \end{matrix}$ (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

$\begin{matrix} M {\prod_{k = 1}^{\infty} Λ_{k}} = ⋀_{k = 1}^{\infty} M_{k} {Λ_{k}} \end{matrix}$ (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

$\begin{matrix} {ξ \in B} = {γ \in Γ | ξ (γ) \in B} \end{matrix}$ (3) is an event.

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

$\begin{matrix} Φ (x) = M {ξ \leq x}, \forall x \in R . \end{matrix}$ (4)

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

$\begin{matrix} M {⋂_{j = 1}^{n} (ξ_{j} \in B_{j})} = ⋀_{j = 1}^{n} M {ξ_{j} \in B_{j}} \end{matrix}$ (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

$\begin{matrix} E [ξ] = \int_{0}^{+ \infty} M {ξ \geq r} dr - \int_{- \infty}^{0} M {ξ \leq r} dr \end{matrix}$ (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

$\begin{matrix} lim_{n \to \infty} ∣ ξ_{n} (γ) - ξ (γ) ∣ = 0 \end{matrix}$ (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}, $M {E_{k}} \to 0$ 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

$\begin{matrix} {ζ \in B} = {γ \in Γ ∣ ζ (γ) \in B} \end{matrix}$ (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

$\begin{matrix} Φ (c) = M {Re (ζ) \leq Re (c), Im (ζ) \leq Im (c)} \end{matrix}$ (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

$\begin{matrix} lim_{n \to \infty} ∥ ζ_{n} (γ) - ζ (γ) ∥ = 0 \end{matrix}$ (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}, $M {E_{k}} \to 0$ 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 $lim_{n \to \infty} M {Λ_{n}}$ =0 whenever Λ_n ↘ Λ and $M {Λ} = 0$ .

Definition 2.13. $M$ is called strongly continuous at Γ if it satisfies that $lim_{n \to \infty} M {Λ_{n}} = 1$ 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., $lim_{n \to \infty} M {Λ_{n}^{'}}$ =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

$\begin{matrix} 0 = lim_{n \to + \infty} M {Λ_{n}^{'}} ∥ \\ = lim_{n \to + \infty} M {Γ - Λ_{n}} ∥ \\ = 1 - lim_{n \to + \infty} M {Λ_{n}} . \end{matrix}$ (11) Hence $lim_{n \to + \infty} M {Λ_{n}} = 1,$ i.e., $M$ is strongly continuous at Γ.

"⇐" From Definition 2.13, $M$ is strongly continuous at Γ, i.e., $lim_{n \to \infty} M {Λ_{n}^{'}} = 1$ 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

$\begin{matrix} 1 = lim_{n \to + \infty} M {Λ_{n}^{'}} ∥ \\ = lim_{n \to + \infty} M {Γ - Λ_{n}} ∥ \\ = 1 - lim_{n \to + \infty} M {Λ_{n}} . \end{matrix}$ (12) Hence $lim_{n \to + \infty} M {Λ_{n}} = 0,$ i.e., $M$ is strongly order continuous.

3 Main results

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

$D = ⋃_{m = 1}^{\infty} ⋂_{i = 1}^{\infty} ⋃_{n = i}^{\infty} {∥ ζ_{n} - ζ ∥ \geq \frac{1}{m}} .$ (13)

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

$M (⋂_{i = 1}^{\infty} ⋃_{n = i}^{\infty} {∥ ζ_{n} - ζ ∥ \geq \frac{1}{m}}) = 0 .$ (14)

Noting the fact that

$⋃_{n = i}^{\infty} {∥ ζ_{n} - ζ ∥ \geq \frac{1}{m}} ↘ ⋂_{i = 1}^{\infty} ⋃_{n = i}^{\infty} {∥ ζ_{n} - ζ ∥ \geq \frac{1}{m}},$ (15)

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

$\begin{matrix} lim_{n \to \infty} M (⋃_{n = i}^{\infty} {∥ ζ_{n} - ζ ∥ \geq \frac{1}{m}}) ∥ \\ = M (⋂_{i = 1}^{\infty} ⋃_{n = i}^{\infty} {∥ ζ_{n} - ζ ∥ \geq \frac{1}{m}}) ∥ \end{matrix}$ (16) $= 0 .$

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

$M (⋃_{n = m_{k}}^{\infty} {∥ ζ_{n} - ζ ∥ \geq \frac{1}{m}}) \leq \frac{1}{2^{m}} \cdot \frac{1}{k} .$ (17)

Let $E_{k} = ⋃_{m = 1}^{\infty} ⋃_{n = m_{k}}^{\infty} {∥ ζ_{n} - ζ ∥ \geq \frac{1}{m}}$ , then

$\begin{matrix} M {E_{k}} \leq \sum_{m = 1}^{\infty} M (⋃_{n = m_{k}}^{\infty} {∥ ζ_{n} - ζ ∥ \geq \frac{1}{m}}) ∥ \\ \leq \sum_{m = 1}^{\infty} \frac{1}{2^{m}} \cdot \frac{1}{k} ∥ \end{matrix}$ (18) $= \frac{1}{k} \to 0 .$

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

$\sup_{γ \in Γ - E_{k}} ∥ ζ_{n} - ζ ∥ \leq \frac{1}{m} .$ (19) Thus, {ζ_n} converges uniformly almost surely to ζ.

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

$ζ_{n} (γ) = {\begin{matrix} 1, & γ \in Λ_{n} \\ 0, & γ \in Γ - Λ_{n} \end{matrix}$ (20)

for n = 1, 2, ·· ·. It is easy to verify that {ζ_n} converges almost surely (a.s.) to 0, i.e., $lim_{n \to \infty} | ζ_{n} (γ) | = 0$ 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}, $M {E_{k}} \to 0$ 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 $| ζ_{i} (γ) | < \frac{1}{k}$ whenever i ≥ n_k. Hence, for any k ∈ N, we have

$Γ - E_{k} \subset ⋂_{i = n_{k}}^{+ \infty} {| ζ_{i} (γ) | < \frac{1}{k}} = Γ - Λ_{n_{k}} .$ (21) So Λ_{n
_k} ⊂ E_k. By the fact that $M {E_{k}} \to 0$ , we have $M {Λ_{n_{k}}} \to 0$ , and hence

$lim_{n \to + \infty} M {Λ_{n}} = 0 .$ (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

$M {Λ} = {\begin{matrix} sup_{γ_{n} \in Λ} \frac{1}{(n + 1)}, & if sup_{γ_{n} \in Λ} \frac{1}{(n + 1)} < 0.5 \\ 1 - sup_{γ_{n} \in Λ^{c}} \frac{1}{(n + 1)}, & if sup_{γ_{n} \in Λ^{c}} \frac{1}{(n + 1)} < 0.5 \\ 1, & if Λ = Γ \\ 0, & if Λ = \emptyset . \end{matrix}$ (23)

And the complex uncertain variables are defined by

$ζ_{n} (γ) = {\begin{matrix} (n + 1) i, & if γ = γ_{n} \\ 0, & otherwise \end{matrix}$ (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 $lim_{n \to \infty} M {Λ_{n}} = 0$ by contradiction.

Assume that

$\begin{matrix} lim_{n \to \infty} M {Λ_{n}} = a > 0, \end{matrix}$ (25) then, for a given δ > 0 satisfying that a - δ > 0, there exists a positive integer N such that n ≥ N,

$\begin{matrix} M {Λ_{n}} \geq a - δ > 0 . \end{matrix}$ (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

$\begin{matrix} M {Λ} > 0 . \end{matrix}$ (27)

That contradicts to the fact that

$\begin{matrix} M {Λ} = 0 . \end{matrix}$ (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

$\begin{matrix} M {Λ} = \sum_{γ_{n} \in Λ} \frac{1}{2^{n}} . \end{matrix}$ (29)

The complex uncertain variables are defined by

$ζ_{n} (γ) = {\begin{matrix} i 2^{n}, & if γ = γ_{n} \\ 0, & otherwise \end{matrix}$ (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 $Λ \in L$ and δ > 0, there exists a closed set F_δ and an open set G_δ of Γ, such that $F_{δ} \subset Λ \subset G_{δ}$ (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 $Λ \in L$ such that for any δ > 0, there exists a closed set F_δ and an open set G_δ of Γ satisfying

$F_{δ} \subset Λ \subset G_{δ} and M {G_{δ} - F_{δ}} < δ .$ (33)

To prove this lemma, it is sufficient to show that $L \subset A$ .

Firstly, we prove that $A$ is an algebra. It is easy to find out that $Γ \in A$ . Suppose $Λ_{1}, Λ_{2} \in A$ , 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

$\begin{matrix} F_{1, δ} \subset Λ_{1} \subset G_{1, δ}, ∥ \\ M {G_{1, δ} - F_{1, δ}} < δ; ∥ \\ F_{2, δ} \subset Λ_{2} \subset G_{2, δ}, ∥ \\ M {G_{2, δ} - F_{2, δ}} < δ . \end{matrix}$ (34)

So we have $F_{1, δ} G_{2, δ}^{c} \subset Λ_{1} Λ_{2}^{c} \subset G_{1, δ} F_{2, δ}^{c},$ (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

$\begin{matrix} M {G_{1, δ} F_{2, δ}^{c} - F_{1, δ} G_{2, δ}^{c}} ∥ \\ = M {G_{1, δ} F_{2, δ}^{c} F_{1, δ}^{c} G_{2, δ}} ∥ \\ = M {(G_{1, δ} - F_{1, δ}) ⋂ (G_{2, δ} - F_{2, δ})} ∥ \\ \leq \min {M (G_{1, δ} - F_{1, δ}), M (G_{2, δ} - F_{2, δ})} ∥ \\ < δ . \end{matrix}$ (36)

That is, $Λ_{1} Λ_{2}^{c} \in A$ . So $A$ is an algebra of Γ.

Next, we prove that $A$ is closed under the formation of pairwise disjoint countable unions. Let ${Λ_{n}} \subset A$ be the sequence of pairwise disjoint set and δ > 0 be given. From the definition of $A$ and $Λ_{n} \in A$ , we know that for each given n, there exists an open set G_n and a closed set F_n of Γ such that

$\begin{matrix} F_{n} \subset Λ_{n} \subset G_{n}, ∥ \\ M {G_{n} - F_{n}} < \frac{δ}{2^{n + 1}} . ∥ \end{matrix}$ (37)

Noting the fact that

$⋃_{n = 1}^{\infty} F_{n} - ⋃_{n = 1}^{k} F_{n} ↘ \emptyset,$ (38)

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

$lim_{k \to \infty} M {⋃_{n = 1}^{\infty} F_{n} - ⋃_{n = 1}^{k} F_{n}} = 0 .$ (39)

Hence, there exists a positive integer k₀, such that

$M {⋃_{n = 1}^{\infty} F_{n} - ⋃_{n = 1}^{k_{0}} F_{n}} < \frac{δ}{2} .$ (40)

Denote $G_{δ} : = ⋃_{n = 1}^{\infty} G_{n}$ 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

$F_{δ} \subset ⋃_{n = 1}^{\infty} Λ_{n} \subset G_{δ} .$ (41)

It follows from the subadditivity axiom of $M$ that

$\begin{matrix} M {G_{δ} - F_{δ}} ∥ \\ = M {⋃_{n = 1}^{\infty} G_{n} - ⋃_{n = 1}^{k_{0}} F_{n}} ∥ \\ \leq M {(⋃_{n = 1}^{\infty} (G_{n} - F_{n})) ⋃ (⋃_{n = 1}^{\infty} F_{n} - ⋃_{n = 1}^{k_{0}} F_{n})} ∥ \\ \leq M {⋃_{n = 1}^{\infty} (G_{n} - F_{n})} + M {⋃_{n = 1}^{\infty} F_{n} - ⋃_{n = 1}^{k_{0}} F_{n}} ∥ \\ \leq \sum_{n = 1}^{\infty} M {G_{n} - F_{n}} + M {⋃_{n = 1}^{\infty} F_{n} - ⋃_{n = 1}^{k_{0}} F_{n}} ∥ \\ < δ . \end{matrix}$ (42)

That is, $⋃_{n = 1}^{\infty} Λ_{n} \in A .$ (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

$E_{n} - F ↘ \emptyset .$ (44)

Hence, by the condition that $M$ is strongly order continuous, we have $lim_{n \to \infty} M {E_{n} - F} = 0 .$ So $C \subset A$ . Since $A$ is closed under the formation of complements, we have $O \subset A$ . This shows that $A$ is a σ-algebra containing $O$ . Thus $L \subset A$ . 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, $M {Γ - F_{k}} \to 0$ 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, $M {Γ - E_{k}} \to 0$ 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 $F_{k} \subset E_{k} \subset G_{k},$ and $\begin{matrix} M {E_{k} - F_{k}} \leq M {G_{k} - F_{k}} < \frac{1}{k} \to 0 . \end{matrix}$ (45) Thus, we have

$\begin{matrix} M {Γ - F_{k}} ∥ \\ \leq M {Γ - E_{k}} + M {E_{k} - F_{k}} ∥ \\ \to 0, \end{matrix}$ (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,

$M {Γ - ⋃_{k = 1}^{\infty} H_{k}} = 0$ (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, $M {Γ - F_{k}} \to 0$ 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

$\begin{matrix} H_{k} : = ⋃_{i = 1}^{k} F_{i}, \end{matrix}$ (48) then {H_k} is an increasing sequence of closed sets of Γ and

$\begin{matrix} M {Γ - H_{k}} ∥ \\ \leq M {Γ - F_{k}} ∥ \\ \to 0 . \end{matrix}$ (49)

Noting that

$\begin{matrix} Γ - H_{k} ↘ Γ - ⋃_{k = 1}^{\infty} H_{k}, \end{matrix}$ (50) then

$\begin{matrix} M {Γ - ⋃_{k = 1}^{\infty} H_{k}} ∥ \\ \leq M {Γ - H_{k}} ∥ \\ \to 0 . \end{matrix}$ (51)

Hence, we have

$M {Γ - ⋃_{k = 1}^{\infty} H_{k}} = 0 .$ (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. $ζ = \sum_{k = 1}^{m} a_{k} X_{A_{k}} + i \cdot \sum_{j = 1}^{n} b_{k} X_{B_{j}}$ , 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

$M {A_{k} - C_{k}} < \frac{δ}{m} .$ (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

$M {B_{j} - D_{j}} < \frac{δ}{n} .$ (54)

Let

$F_{δ} : = ⋃_{k = 1}^{m} ⋃_{j = 1}^{n} C_{k} D_{j},$ (55) then ζ is continuous in F_δ and

$\begin{matrix} M {Γ - F_{δ}} ∥ \\ = M {⋃_{k = 1}^{m} ⋃_{j = 1}^{n} A_{k} B_{j} - ⋃_{k = 1}^{m} ⋃_{j = 1}^{n} C_{k} D_{j}} \\ \leq M {⋃_{k = 1}^{m} ⋃_{j = 1}^{n} (A_{k} B_{j} - C_{k} D_{j})} ∥ \\ = M {⋃_{k = 1}^{m} ⋃_{j = 1}^{n} A_{k} C_{k}^{c} B_{j} D_{j}^{c}} ∥ \\ = M {⋃_{k = 1}^{m} A_{k} C_{k}^{c}} ⋂ {⋃_{j = 1}^{n} B_{j} D_{j}^{c}} \\ \leq min {\sum_{k = 1}^{m} M (A_{k} - C_{k}), \sum_{j = 1}^{n} M (B_{j} - D_{j})} ∥ \\ < δ . \end{matrix}$ (56)

(b) Let ζ be a complex uncertain variable. Then, there exists a sequence {φ_n} of simple functions such that $φ_{n} \to ζ$ in Γ. With the help of Remark 3.2, we know that there exists an increasing sequence of closed sets {H_k} of Γ,

$M {Γ - ⋃_{k = 1}^{\infty} H_{k}} = 0$ (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

$M {Γ - Q_{k}^{(n)}} < \frac{1}{2^{n}} \cdot \frac{1}{k},$ (58)

for k = 1, 2, ·· · and φ_n is continuous in $Q_{k}^{(n)}$ . Let

$Q_{k} : = ⋂_{n = 1}^{\infty} Q_{k}^{(n)} .$ (59)

Then, we have

$\begin{matrix} M {Γ - Q_{k}} ∥ \\ = M {⋃_{n = 1}^{\infty} (Γ - Q_{k}^{(n)})} \\ < \frac{1}{k} \cdot \sum_{n = 1}^{\infty} \frac{1}{2^{n}} \\ = \frac{1}{k} . \end{matrix}$ (60)

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

$\begin{matrix} Γ - Q_{k} ↘ Γ - ⋃_{k = 1}^{\infty} Q_{k} \end{matrix}$ (61) and

$\begin{matrix} M {Γ - ⋃_{k = 1}^{\infty} Q_{k}} ∥ \\ \leq M {Γ - Q_{k}} ∥ \\ < \frac{1}{k} \to 0 . \end{matrix}$ (62) Hence $\begin{matrix} M {Γ - ⋃_{k = 1}^{\infty} Q_{k}} = 0 . \end{matrix}$ (63)

It follows from the subadditivity axiom of $M$ that

$\begin{matrix} M {(Γ - ⋃_{k = 1}^{\infty} H_{k}) ⋃ (Γ - ⋃_{k = 1}^{\infty} Q_{k})} = 0 . \end{matrix}$ (64)

Noting the fact that

$\begin{matrix} (Γ - H_{k}) ⋃ (Γ - Q_{k}) ↘ (Γ - ⋃_{k = 1}^{\infty} H_{k}) ⋃ (Γ - ⋃_{k = 1}^{\infty} Q_{k}) \end{matrix}$ (65) and by the condition that $M$ is strongly order continuous, we have

$lim_{k \to \infty} M {(Γ - H_{k}) ⋃ (Γ - Q_{k})} = 0 .$ (66) Then for any given δ > 0, there exists k₀ ∈ N,

$\begin{matrix} M {(Γ - H_{k_{0}}) ⋃ (Γ - Q_{k_{0}})} = M {Γ - (H_{k_{0}} ⋂ Q_{k_{0}})} < δ . \end{matrix}$ (67)

Let $\begin{matrix} F_{δ} = H_{k_{0}} ⋂ Q_{k_{0}} . \end{matrix}$ (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

$\begin{matrix} ∥ φ_{n_{0}} (γ) - ζ (γ) ∥ \leq \frac{ɛ}{3}, \\ ∥ φ_{n_{0}} (γ) - φ_{n_{0}} (γ_{0}) ∥ \leq \frac{ɛ}{3}, \end{matrix}$ (69)

whenever ρ (γ, γ₀) < α. Thus, we have

$\begin{matrix} ∥ ζ (γ) - ζ (γ_{0}) ∥ ∥ \\ = ∥ ζ (γ) - φ_{n_{0}} (γ) + φ_{n_{0}} (γ) - φ_{n_{0}} (γ_{0}) + φ_{n_{0}} (γ_{0}) - ζ (γ_{0}) ∥ ∥ \\ \leq ∥ ζ (γ) - φ_{n_{0}} (γ) ∥ + ∥ φ_{n_{0}} (γ) - φ_{n_{0}} (γ_{0}) ∥ + ∥ φ_{n_{0}} (γ_{0}) - ζ (γ_{0}) ∥ ∥ \\ < ɛ . \end{matrix}$ (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

$\begin{matrix} M {Γ - F_{n}} < \frac{1}{n} . \end{matrix}$ (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 $M {Γ - F_{k}} < \frac{1}{k}$ . 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

$\begin{matrix} {∥ ψ_{k} (γ) - ζ (γ) ∥ \geq ɛ} \subset Γ - F_{k} . \end{matrix}$ (72)

Hence, for each k = 1, 2, ·· ·,

$\begin{matrix} M {∥ ψ_{k} (γ) - ζ (γ) ∥ \geq ɛ} \leq M {Γ - B_{k}} < \frac{1}{k} . \end{matrix}$ (73)

So we have

$\begin{matrix} lim_{k \to \infty} M {∥ ψ_{k} (γ) - ζ (γ) ∥ \geq ɛ} = 0 . \end{matrix}$ (74)

From the above fact, we can choose a subsequence {ψ_{k
_n}} of {ψ_k} such that

$\begin{matrix} M {∥ ψ_{k_{n}} (γ) - ζ (γ) ∥ \geq \frac{1}{2^{n}}} < \frac{1}{2^{n}} . \end{matrix}$ (75)

Let

$\begin{matrix} Λ_{n} : = {∥ ψ_{k_{n}} (γ) - ζ (γ) ∥ \geq \frac{1}{2^{n}}} . \end{matrix}$ (76)

Then

$\begin{matrix} \sum_{n = 1}^{\infty} M {Λ_{n}} < \infty . \end{matrix}$ (77)

Next, we prove

$\begin{matrix} M (⋂_{n = 1}^{\infty} ⋃_{v = 1}^{\infty} {∥ ψ_{k_{n + v}} (γ) - ζ (γ) ∥ \geq ɛ}) = 0 . \end{matrix}$ (78)

Indeed, for any ɛ > 0, there exists a positive integer n₀ such that for any n ≥ n₀, $\frac{1}{2^{n}} < ɛ$ and $\begin{matrix} M (⋂_{n = 1}^{\infty} ⋃_{v = 1}^{\infty} {∥ ψ_{k_{n + v}} (γ) - ζ (γ) ∥ \geq ɛ}) ∥ \\ \leq \sum_{m = n}^{\infty} M ({∥ ψ_{k_{m}} (γ) - ζ (γ) ∥ \geq ɛ}) ∥ \\ \leq \sum_{m = n}^{\infty} M {Λ_{m}} \to 0 . \end{matrix}$ (79)

Take ζ_n = ψ_{k
_n}, n = 1, 2, ·· ·. Then, there exists an event $Λ : = Γ - ⋂_{n = 1}^{\infty} ⋃_{v = 1}^{\infty} {∥ ζ_{n + v} (γ) - ζ (γ) ∥ \geq ɛ}$ with $M {Λ} = 1$ such that

$\begin{matrix} lim_{n \to \infty} ∥ ζ_{n} (γ) - ζ (γ) ∥ = 0 \end{matrix}$ (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.

Footnotes

Acknowledgments

The authors would like to thank the Associate Editor and the anonymous referees for their constructive suggestions and valuable comments that greatly improved this paper. This work is supported partly by the National Natural Science Foundation of China (11801307) and the Natural Science Foundation of Shandong Province of China (ZR2017MA012, ZR2021MA009).

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