Pan-integral on credibility space and its properties

Abstract

Pan-integral is an important kind of nonlinear integral, and credibility measure is a monotone measure with self-duality. In this paper, we define the pan-integral on credibility space, a new type of integral construction, whose basic properties are discussed in detail. The relationship between the pan-integral and H-fuzzy integral on a special pan-credibility space is established as well. Moreover, some monotone convergence theorems for a sequence of nonnegative measurable functions are investigated.

Keywords

Pan-integral credibility measure pan-credibility space monotone convergence theorem

1. Introduction

The fuzzy set theory was originally proposed by Zadeh via membership function in 1965 [28]. In order to measure a fuzzy event, Zadeh further introduced the concepts of possibility measure and necessary measure, which are proved to be normal, nonnegative and monotone [29, 30]. However, both the possibility measure and necessary measure do not obey the law of truth conservation and are inconsistent with laws of exclude middle and contradiction. This is because they do not satisfy the self-duality property which is intuitive and important in both theory and practice. To address this issue, Liu and Liu presented the concept of credibility measure [8], a self-dual measure. Credibility theory, founded by Liu in 2004 [9] and refined by Liu in 2007 [11], is a new branch of mathematics for studying the behavior of fuzzy phenomena. Since then, the credibility theory has been developed rapidly and applied widely [1 , 21].

It is well-known that the development of measure and integral theory goes with the demand of mathematics and its applications. For example, the fuzzy integral (also known as the Sugeno integral) was built based on a normalized monotone measure [18 , 24]. Pan-integral was introduced and its properties discussed by Yang [26], and further investigations of pan-integral on pan-additive monotone measure spaces were pursued by Wang et al., Yang and Song [22 , 27]. It deserves mentioning that the fuzzy integral involves two binary operations, maximum and minimum of real numbers, while pan-integral involves two pan-operations, pan-addition and pan-multiplication. The relationships among pan-integral and other types of integrals were extensively examined recently [2 , 31].

Considering that minimum is a special t-norm (in fact, the largest t-norm), ⊤-fuzzy integral, a natural extension of the classical fuzzy integral, was introduced by scholars in fuzzy set community [25, 32]. Subsequently, the research was further developed by employing many other operations [13 , 19]. On the other hand, taking into account that credibility measure is a specific normalized monotone measure, Hu et al. brought out the concept of H-fuzzy integral, a combination of the ⊤-fuzzy integral and credibility theory [3]. Yet, the pan-integral on a credibility space, which is the main focus of the present work, has not been studied so far.

The other parts of this paper are organized as follows. In Section 2, we recall briefly basic concepts of credibility measure and pan-operations. In Section 3, we introduce the pan-integral on credibility space, and some elementary properties are then developed. The transformation theorem for pan-integral on a special pan-credibility space is given as well. In Section 4, monotone convergence theorems for sequence of nonnegative measurable functions are investigated. The paper concludes with a summary in Section 5.

2. Preliminaries

In this section, the credibility theory and pan-operations are reviewed. Moreover, a brief and effective proof of the credibility subadditivity theorem is given, and also the corrected proof of the credibility semicontinuity theorem is presented.

Throughout the present paper, let Θ be a nonempty set and $P$ the power set of Θ (i.e., the largest σ-algebra over Θ). Each element in $P$ is called an event. The following concepts and results on credibility measure can be found in [11, 12].

Definition 2.1. The set function $Cr : P \to [0, 1]$ is called a credibility measure if it satisfies the following four axioms:

[Axiom 1.] (Normality) Cr {Θ} =1.

[Axiom 2.] (Monotonicity) Cr {A} ≤ Cr {B} whenever A ⊆ B.

[Axiom 3.] (Self-Duality) Cr {A} + Cr {A^c} =1 for any event A.

[Axiom 4.] (Maximality) $Cr {⋃_{i} A_{i}} = sup_{i} Cr {A_{i}}$ for any events {A_i} with $sup_{i} Cr {A_{i}} < 0.5$ .

It is easy to see from Definition 2.1 that Cr {∅} =0 and 0 ≤ Cr {A} ≤1 for all $A \in P$ .

Definition 2.2. Let Θ be a nonempty set and $P$ the power set of Θ, and Cr a credibility measure. Then the triplet $(Θ, P, Cr)$ is called a credibility space.

Lemma 2.3. Let $(Θ, P, Cr)$ be a credibility space. For any $A, B \in P$ , if Cr{A}+ Cr{B}<1, then $Cr {A \cup B} = Cr {A} \lor Cr {B} .$ Theorem 2.4. (Credibility Subadditivity Theorem). Let $(Θ, P, Cr)$ be a credibility space. Then Cr is subadditive, that is, $Cr {A \cup B} \leq Cr {A} + Cr {B}$ for any events A and B.

Proof. The argument breaks down into two cases.

Case 1: If Cr {A} + Cr {B} <1, it follows from Lemma 2.3 that Cr {A ∪ B} = Cr {A} ∨ Cr {B} ≤ Cr {A} + Cr {B}.

Case 2: If Cr {A} + Cr {B} ≥1, obviously, we have Cr {A ∪ B} ≤ Cr {A} + Cr {B}.

Theorem 2.5. (Credibility Semicontinuity Theorem). Let $(Θ, P, Cr)$ be a credibility space. For any events ${A_{i}}_{i = 1}^{\infty}$ , we have $lim_{n \to \infty} Cr {A_{n}} = Cr {lim_{n \to \infty} A_{n}}$ if one of the following conditions is satisfied:

Cr{A}≤ 0.5 and A_n ↗ A;

$lim_{n \to \infty} Cr {A_{n}} < 0.5$ and A_n ↘ A;

$lim_{n \to \infty} Cr {A_{n}} > 0.5$ and A_n ↘ A.

Proof. Since Cr {A} ≤0.5 and A_n ↗ A, we have Cr {A_n} ≤0.5 for each n by the monotonicity axiom. Thus, we have $\sup_{n} Cr {A_{n}} \leq 0.5$ .

If $\sup_{n} Cr {A_{n}} < 0.5$ , then it follows from the maximality axiom that $\begin{matrix} Cr {A} & = Cr {⋃_{n = 0}^{\infty} A_{n}} \\ = sup_{n} Cr {A_{n}} \\ = lim_{n \to \infty} Cr {A_{n}} . \end{matrix}$

If ${sup}_{n} Cr {A_{n}} = 0.5$ , then under the supposition Cr {A} <0.5, there exists n₀ such that Cr {A_{n
₀}} > Cr {A}, which contradicts with the monotonicity axiom. Thus, we have $Cr {A} = 0.5 = {sup}_{n} Cr {A_{n}}$ .

Other items can be easily derived by the self-duality and maximality axioms. □

In the sequel, for the sake of convenience, we denote $R_{+} = [0, \infty)$ , $\bar{R_{+}} = [0, \infty]$ , B₊ the Borel field on $R_{+}$ . The readers can refer to the quoted papers, e.g. [22, 23], for more details.

Definition 2.6. Let ⊕ be a binary operation on $\bar{R_{+}}$ . The pair $(\bar{R_{+}}, \oplus)$ is called a commutative isotonic semigroup and ⊕ is called a pan-addition on $\bar{R_{+}}$ iff ⊕ is commutative, associative, and such that a ≤ b implies a ⊕ c ≤ b ⊕ c for every c, a ⊕ 0 = a, and the existence of $lim_{n \to \infty} a_{n}$ and $lim_{n \to \infty} b_{n}$ implies the existence of $lim_{n \to \infty} (a_{n} \oplus b_{n})$ , and $lim_{n \to \infty} (a_{n} \oplus b_{n}) = lim_{n \to \infty} a_{n} \oplus lim_{n \to \infty} b_{n} .$

Definition 2.7. Let ⊗ be a binary operation on $\bar{R_{+}}$ . The triple $(\bar{R_{+}}, \oplus, \otimes)$ , where ⊕ is a pan-addition on $\bar{R_{+}}$ , is called a commutative isotonic semiring with respect to ⊕ and ⊗ iff ⊗ is commutative, associative, and distributive with respect to ⊕, and such that a ≤ b implies a ⊗ c ≤ b ⊗ c for every c, a ≠ 0 and b ≠ 0 iff a ⊗ b ≠ 0, there exist I such that I ⊗ a = a for every $a \in \bar{R_{+}}$ , and the existence of a finite $lim_{n \to \infty} a_{n}$ and a finite $lim_{n \to \infty} b_{n}$ implies the equality $lim_{n \to \infty} (a_{n} \otimes b_{n}) = lim_{n \to \infty} a_{n} \otimes lim_{n \to \infty} b_{n} .$

The operation ⊗ is called a pan-multiplication on $\bar{R_{+}}$ , and the number I is called the unit element of $(\bar{R_{+}}, \oplus, \otimes)$ .

Obviously, $(\bar{R_{+}}, +, \cdot)$ is a commutative isotonic semiring. Moreover, $\bar{R_{+}}$ with maximum and minimum (the common multiplication, respectively) of real numbers is a commutative isotonic semiring, and it is denoted by $(\bar{R_{+}}, \lor, \land)$ ( $(\bar{R_{+}}, \lor, \cdot)$ , respectively) and its unit element is ∞ (1, respectively).

3. Pan-integral on credibility space

In this section, the pan-integral on credibility space is first introduced, and its fundamental properties are then investigated.

Definition 3.1. Let $(Θ, P, Cr)$ be a credibility space and $(\bar{R_{+}}, \oplus, \otimes)$ a commutative isotonic semiring. Then the 6-ary tuple $(Θ, P, Cr, \bar{R_{+}}, \oplus, \otimes)$ is called a pan-credibility space.

Definition 3.2. Let $(Θ, P, Cr, \bar{R_{+}}, \oplus, \otimes)$ be a pan-credibility space. The function defined on Θ given by $χ_{E} (x) = {\begin{matrix} I, & x \in E, \\ 0, & otherwise, \end{matrix}$ is called the pan-characteristic function of E ⊆ Θ, where I is the unit element of $(\bar{R_{+}}, \oplus, \otimes)$ .

Definition 3.3. A real-valued function $f : Θ \to R_{+}$ is measurable iff $f^{- 1} (B) = {x \in Θ : f (x) \in B} \in P$ for any Borel set B ∈ B₊. The set of all finite nonnegative measurable function is denoted by G. Given f, g ∈ G, we write f ≤ g if f (x) ≤ g (x) for all x ∈ Θ.

For a given f ∈ G, denote f_α = {x ∈ Θ : f (x) ≥ α}, f_{α
⁺} = {x ∈ Θ : f (x) > α}, where $α \in \bar{R_{+}}$ , and f_α and f_{α
⁺} are referred to as the α-level set and strict α-level set of f, respectively.

Definition 3.4. Let $(Θ, P, Cr, \bar{R_{+}}, \oplus, \otimes)$ be a pan-credibility space. The function on Θ given by $s (x) = \oplus_{i = 1}^{n} [a_{i} \otimes χ_{E_{i}} (x)]$ is called pan-simple measurable, where $a_{i} \in R_{+}$ , i = 1, 2, …, n, and E = {E_i : i = 1, 2, …, n} is a measurable partition of Θ.

The set of all pan-simple measurable function is denoted by $S$ .

For any given nonnegative measurable function $f : Θ \to R_{+}$ , there exists a pan-simple measurable function s such that s ≤ f. For example, we can take $s (x) = {\begin{matrix} \oplus_{i = 1}^{n \cdot 2^{n}} \frac{i - 1}{2^{n}} χ_{E_{ni}} (x), & f (x) < n, \\ n, & f (x) \geq n, \end{matrix}$ where $E_{ni} = {x \in Θ : \frac{i - 1}{2^{n}} \leq f (x) < \frac{i}{2^{n}}}$ , i = 1, 2, …, n · 2ⁿ.

In a pan-credibility space $(Θ, P, Cr, \bar{R_{+}}, \oplus, \otimes)$ , for any pan-simple measurable function $s (x) = \oplus_{i = 1}^{n} [a_{i} \otimes χ_{E_{i}} (x)]$ , we write $P (s | A) = \oplus_{i = 1}^{n} [a_{i} \otimes Cr {A \cap E_{i}}],$ where A ⊆ Θ. When A = Θ, P (s|Θ) will be denoted by P (s) for short.

Definition 3.5. Let f ∈ G and $A \in P$ . The pan-integral of f on A with respect to Cr, which is denoted by (p) ∫_Af d Cr, is given by $(p) \int_{A} f d Cr = sup_{0 \leq s \leq f, s \in S} P (s | A) .$ When A = Θ, we simply write (p) ∫f d Cr instead of (p) ∫_Θf dCr.

Theorem 3.6. Let f, g ∈ G and f = g on A a . e . (i.e., Cr{B}=0, where B = {x ∈ A : f (x) ≠ g (x)}). Then $(p) \int_{A} f d Cr = (p) \int_{A} g d Cr .$

Proof. For any $s \in S$ such that s ≤ f, denote $s^{'} (x) = {\begin{matrix} s (x), & if f (x) = g (x), \\ max {s (x), g (x)}, & otherwise . \end{matrix}$

Certainly, we have $s^{'} \in S$ and s ≤ s′ ≤ g. Thus, it follows that P (s|A) ≤ P (s′|A). Hence, (p) ∫_Af d Cr ≤ (p) ∫_Ag d Cr.

Similarly, we have (p) ∫_Ag d Cr ≤ (p) ∫_Af d Cr.

Summarizing above, the conclusion holds. □

Theorem 3.7. Let f ∈ G, $A \in P$ and $\hat{P}$ denote the set of all measurable partitions of Θ. Then, $\begin{matrix} (p) \int_{A} f d Cr \\ = sup_{E \in \hat{P}} {\oplus_{E_{i} \in E} [(inf_{x \in E_{i}} f (x)) \otimes Cr {A \cap E_{i}}]} . \end{matrix}$

Proof. On the one hand, for any given $E \in \hat{P}$ and E = {E_i : i = 1, 2, …, n}, we take $s (x) = \oplus_{i = 1}^{n} [(inf_{x \in E_{i}} f (x)) \otimes χ_{E_{i}} (x)]$ . Then, $s \in S$ and s ≤ f. Thus, $\oplus_{i = 1}^{n} [(inf_{x \in E_{i}} f (x)) \otimes Cr {A \cap E_{i}}] \leq (p) \int_{A} f d Cr$ . Therefore, $\begin{matrix} sup_{E \in \hat{P}} {\oplus_{E_{i} \in E} [(inf_{x \in E_{i}} f (x)) \otimes Cr {A \cap E_{i}}]} \\ \leq (p) \int_{A} f d Cr . \end{matrix}$

On the other hand, take $s (x) = \oplus_{i = 1}^{n} [a_{i} \otimes χ_{E_{i}} (x)] \in S$ , where $E = {E_{i} : i = 1, 2, \dots, n} \in \hat{P}$ . Moreover, if s ≤ f, then $a_{i} \leq inf_{x \in E_{i}} f (x)$ . Thus, we have $\begin{matrix} P (s | A) & \leq \oplus_{i = 1}^{n} [(inf_{x \in E_{i}} f (x)) \otimes Cr {A \cap E_{i}}] \\ \leq sup_{E \in \hat{P}} {\oplus_{E_{i} \in E} [(inf_{x \in E_{i}} f (x)) \otimes Cr {A \cap E_{i}}]} . \end{matrix}$ Hence, it follows that $\begin{matrix} (p) \int_{A} f d Cr \\ \leq sup_{E \in \hat{P}} {\underset{E_{i} \in E}{\oplus} [(inf_{x \in E_{i}} f (x)) \otimes Cr {A \cap E_{i}}]} . \end{matrix}$

Consequently, the proof is complete. □

The following example is employed to substantiate the conceptual argument.

Example 3.8. Let Θ = {a, b, c}, Cr be a credibility measure on $(Θ, P)$ with Cr {a} =0.35, Cr {b} =0.15, Cr {c} =0.65. For this case, there are eight events: ∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, Θ with their credibilities being 0, 0.35, 0.15, 0.65, 0.35, 0.85, 0.65, 1, respectively. The credibility of A = {a, b}, for example, is obtained by the self-duality axiom (Cr {A} =1 - Cr {c} =0.35) or by Lemma 2.3 (since Cr {a} + Cr {b} <1, Cr {A} = Cr {a} ∨ Cr {b} =0.35). And $\begin{matrix} \hat{P} = { & {{a}, {b}, {c}}, {{a, b}, {c}}, \\ {{b}, {a, c}}, {{a}, {b, c}}, {a, b, c}} . \end{matrix}$ For the pan-credibility space $(Θ, P, Cr, \bar{R_{+}}, +, \cdot)$ , take $f (x) = {\begin{matrix} 2, & x = a, \\ 3, & x = b, \\ 1, & x = c . \end{matrix}$

By Theorem 3.7, we have $\begin{matrix} (p) \int_{\emptyset} f d Cr = 0, \\ (p) \int_{{a}} f d Cr = 2 Cr {a} = 0.7, \\ (p) \int_{{b}} f d Cr = 3 Cr {b} = 0.45, \\ (p) \int_{{c}} f d Cr = Cr {c} = 0.65, \end{matrix}$ $\begin{matrix} (p) \int_{{a, b}} f d Cr \\ = max {2 Cr {a} + 3 Cr {b}, 2 Cr {a, b}} \\ = 1.15, \\ (p) \int_{{a, c}} f d Cr \\ = max {2 Cr {a} + Cr {c}, Cr {a, c}} \\ = 1.35, \\ (p) \int_{{b, c}} f d Cr \\ = max {3 Cr {b} + Cr {c}, Cr {b, c}} \\ = 1.1, \end{matrix}$ $\begin{matrix} (p) \int_{Θ} f d Cr = max { & 2 Cr {a} + 3 Cr {b} + Cr {c}, \\ 2 Cr {a, b} + Cr {c}, \\ Cr {a, c} + 3 Cr {b}, \\ Cr {b, c} + 2 Cr {a}, Cr {Θ}} \\ = 1.8 . \end{matrix}$

Theorem 3.9. Let $(\bar{R_{+}}, \lor, ⊤)$ be a commutative isotonic semiring. Then, we have $(p) \int_{A} f d Cr = (H) \int_{A} f d Cr$ for any f ∈ G and $A \in P$ . That is, the pan-integral and the H-fuzzy integral coincide with respect to $(\bar{R_{+}}, \lor, ⊤)$ .

Proof. Recall that [3] $(H) \int_{A} f d Cr = sup_{E \subseteq Θ} {(inf_{x \in E} f (x)) ⊤ Cr {A \cap E}} .$

And for any E ⊆ Θ, ${E, E^{c}} \in \hat{P}$ . Thus, we have (H) ∫_Af d Cr (p) ≤ ∫_Af d Cr.

Conversely, for any given ɛ > 0 and any $E \in \hat{P}$ , there exists E₀ ∈ E such that $\begin{matrix} sup_{E_{i} \in E} {(inf_{x \in E_{i}} f (x)) ⊤ Cr {A \cap E_{i}}} \\ = sup_{E_{i} \subseteq Θ} {(inf_{x \in E_{i}} f (x)) ⊤ Cr {A \cap E_{i}}} \\ \leq (inf_{x \in E_{0}} f (x)) ⊤ Cr {A \cap E_{0}} + ɛ \\ \leq (H) \int_{A} f d Cr + ɛ . \end{matrix}$

Thus, it follows that (p) ∫_Af d Cr ≤ (H) ∫_Af d Cr from the fact that ɛ is arbitrary.□

Theorem 3.10. For any f ∈ G and $A \in P$ , we have $(p) \int_{A} f d Cr = (p) \int f \otimes χ_{A} d Cr .$ Proof. Taking s′ = s ⊗ χ_A for any given $s (x) = \oplus_{i = 1}^{n} [a_{i} \otimes χ_{E_{i}} (x)]$ satisfying s ≤ f, we have $s^{'} \in S$ and s′ ≤ f ⊗ χ_A. Thus, it follows that $\begin{matrix} s^{'} & = \oplus_{i = 1}^{n} [a_{i} \otimes χ_{E_{i}} \otimes χ_{A}] \\ = \oplus_{i = 1}^{n} [a_{i} \otimes χ_{A \cap E_{i}}] . \end{matrix}$

Then $P (s^{'}) = \oplus_{i = 1}^{n} [a_{i} \otimes Cr {A \cap E_{i}}]$ . Hence, we have P (s|A) = P (s′) ≤ (p) ∫f ⊗ χ_A d Cr. Therefore, (p) ∫_Af d Cr ≤ (p) ∫f ⊗ χ_A d Cr.

Conversely, for any given $s (x) = \oplus_{i = 1}^{n} [a_{i} \otimes χ_{E_{i}} (x)]$ satisfying s ≤ f ⊗ χ_A, we omit, without any loss of generality, those terms in which a_i = 0 and we may assume that a_i > 0, i = 1, 2, …, n. From s ≤ f ⊗ χ_A, we can deduce E_i ⊆ A, i = 1, 2, …, n. Thus, $\begin{matrix} P (s) & = \oplus_{i = 1}^{n} [a_{i} \otimes Cr {E_{i}}] \\ = \oplus_{i = 1}^{n} [a_{i} \otimes Cr {A \cap E_{i}}] = P (s | A) \\ \leq (p) \int_{A} f d Cr . \end{matrix}$ So we have (p) ∫f ⊗ χ_A d Cr ≤ (p) ∫_Af d Cr. This completes the proof. □

Corollary 3.11. Let f ∈ G, $A, B \in P$ . If Cr{B}=0, then $(p) \int_{A \ B} f d Cr = (p) \int_{A \cup B} f d Cr = (p) \int_{A} f d Cr .$ Proof. By Theorem 3.10, we have $\begin{matrix} (p) \int_{A \ B} f d Cr = (p) \int f \otimes χ_{A \ B} d Cr, \\ (p) \int_{A \cup B} f d Cr = (p) \int f \otimes χ_{A \cup B} d Cr, \\ (p) \int_{A} f d Cr = (p) \int f \otimes χ_{A} d Cr, \end{matrix}$ and f ⊗ χ_A\B = f ⊗ χ_A, a.e. and f ⊗ χ_A∪B = f ⊗ χ_A, a.e. hold by Cr {B} =0. Then by Theorem 3.6, we obtain the conclusion. □

Proposition 3.12. Assume that f, g ∈ G and $A, B \in P$ , then we have the following properties.

If Cr(A)=0, then (p) ∫_Af d Cr = 0.

If f = 0 on A a.e., then (p) ∫_Af d Cr = 0.

If A ⊆ B, then (p) ∫_Af d Cr ≤ (p) ∫_Bf d Cr.

If f ≤ g on A, then (p) ∫_Af d Cr ≤ (p) ∫_Ag dCr.

(p) ∫_Aa d Cr ≥ a ⊗ Cr {A}.

The equality in Proposition 3.12 (5) may not hold, and a counterexample is given as follows.

Example 3.13. (Continued from Example 3.8). For the pan-credibility space $(Θ, P, Cr, \bar{R_{+}}, +, \cdot)$ , take A = {a, b}, then $\begin{matrix} (p) \int_{A} d Cr & = max {Cr {a} + Cr {b}, Cr {a, b}} \\ = max {0.5, 0.35} = 0.5 \\ > Cr {A} . \end{matrix}$

Corollary 3.14. Let f, g ∈ G and $A, B \in P$ . Then we have the following inequalities.

(p) ∫_A∪Bf d Cr ≥ (p) ∫_Af d Cr ∨ (p) ∫_Bf d Cr.

(p) ∫_A∩Bf d Cr ≤ (p) ∫_Af d Cr ∧ (p) ∫_Bf d Cr.

(p) ∫_A (f ∨ g) d Cr ≥ (p) ∫_Af d Cr ∨ (p) ∫_Af d Cr.

(p) ∫_A (f ∧ g) d Cr ≤ (p) ∫_Af d Cr ∧ (p) ∫_Af d Cr.

Theorem 3.15. Suppose that f ∈ G and $A \in P$ , if (p) ∫_Af d Cr = 0, then Cr{A∩ f₀+}=0.

Proof. Denoting B = A ∩ f_0⁺, $B_{n} = A \cap f_{\frac{1}{n}}$ , then we have $B_{n} ↗ ⋃_{n = 1}^{\infty} B_{n} = B$ . Thus, $\begin{matrix} 0 & = (p) \int_{A} f d Cr \\ \geq (p) \int_{B_{n}} f d Cr & by Proposition 3.12 (3) \\ \geq (p) \int_{B_{n}} \frac{1}{n} d Cr & by Proposition 3.12 (4) \\ \geq \frac{1}{n} \otimes Cr {B_{n}} & by Proposition 3.12 (5) \\ \geq 0 . \end{matrix}$ Hence, we have Cr {B_n} =0, n = 1, 2, … . It follows from the credibility semicontinuity theorem that Cr {B} =0. □

Theorem 3.16. (Transformation Theorem). On a pan-credibility space $(Θ, P, Cr, \bar{R_{+}}, +, \cdot)$ , we have $(p) \int_{0}^{\infty} Cr {A \cap f_{α}} d α \leq (p) \int_{A} f d Cr,$ where f ∈ G, $A \in P$ and α is the Lebesgue measure on $\bar{R_{+}}$ .

Proof. There is no loss generality in assuming that A = Θ.

For any finite subclass of any partition of $\bar{R_{+}}$ , {B_i : i = 1, 2, …, n}, we denote β_i = inf B_i and β₀ =∞, and we may assume that β₀ ≥ β₁ ≥ β_n (otherwise, we just need to rearrange the order of B_i).

Accordingly, we have α (B_i) = β_i-1 - β_i, i ≥ 2. Let E_i = f_{β
_i} \ f_{β
_i-1}, i = 1, 2, …, n. Then E_i = {x ∈ Θ : f (x) ∈ [β_i, β_i-1)}. Thus {E_i : i = 1, 2, …, n} is a partition of Θ.

Since f_{β
_i} = ⋃ _j≤iE_j and Cr is subadditive, it follows that $inf_{γ \in B_{i}} Cr {f_{γ}} \leq Cr {f_{β_{i}}} \leq \sum_{j \leq i} Cr {E_{j}}$ and $\sum_{j \geq i} α (B_{j}) = β_{i} \leq inf_{x \in f_{β_{i}}} f (x) \leq inf_{x \in E_{i}} f (x)$ , i = 1, 2, …, n.

And it is evident that the equality $\sum_{i = 1}^{n} (\sum_{j \leq i} a_{j}) b_{i} = \sum_{i = 1}^{n} a_{i} (\sum_{j \geq i} b_{j})$ holds for any sequence {a_i} , {b_i}, i = 1, 2, …, n. Thus, we have $\begin{matrix} \sum_{i = 1}^{n} (inf_{γ \in B_{i}} Cr {f_{γ}} \cdot α (B_{i})) \\ \leq \sum_{i = 1}^{n} (\sum_{j \leq i} Cr {E_{j}} \cdot α (B_{i})) \\ = \sum_{i = 1}^{n} (Cr {E_{i}} \cdot \sum_{j \geq i} α (B_{j})) \\ \leq \sum_{i = 1}^{n} (inf_{x \in E_{i}} f (x) \cdot Cr {E_{i}}) \\ \leq (p) \int f d Cr . \end{matrix}$ Hence, indeed, the conclusion is valid. □

4. Monotone convergence theorems

In this section, some monotone convergence theorems for sequence of nonnegative measurable functions are discussed.

Theorem 4.1. Let $(Θ, P, Cr, \bar{R_{+}}, \oplus, \otimes)$ be a pan-credibility space, $A \in P$ and f_n, f ∈ G. If Cr{A}≤ 0.5 and f_n ↗ f on A, then $lim_{n \to \infty} (p) \int_{A} f_{n} d Cr = (p) \int_{A} f d Cr .$

Proof. It follows from the hypothesis f_n ≤ f that (p) ∫_Af_n d Cr ≤ (p) ∫_Af d Cr for each n, and thus $lim_{n \to \infty} (p) \int_{A} f_{n} d Cr \leq (p) \int_{A} f d Cr .$

Now we have merely to verify that the opposite inequality also holds.

For any fixed real number c ∈ (0, 1), denote B_n = {x ∈ A : f_n (x) ≥ cf (x)}, n = 1, 2, …. Then we have B_n ↗ A. Thus,

$\begin{matrix} \underset{E_{i} \in E}{\oplus} {inf_{x \in E_{i}} [χ_{B_{n}} \otimes cf (x)] \otimes Cr {B_{n} \cap E_{i}}} \\ \leq \underset{E_{i} \in E}{\oplus} {inf_{x \in E_{i}} [χ_{B_{n}} \otimes f_{n} (x)] \otimes Cr {B_{n} \cap E_{i}}} \\ \leq (p) \int_{B_{n}} χ_{B_{n}} \otimes f_{n} (x) d Cr \\ \leq (p) \int_{A} f_{n} d Cr . \end{matrix}$ So,

$\begin{matrix} lim_{n \to \infty} \oplus_{E_{i} \in E} {inf_{x \in E_{i}} [χ_{B_{n}} \otimes cf (x)] \otimes Cr {B_{n} \cap E_{i}}} \\ \leq lim_{n \to \infty} (p) \int_{A} f_{n} d Cr, \end{matrix}$ i.e., $\begin{matrix} \underset{E_{i} \in E}{\oplus} {[inf_{x \in E_{i}} χ_{A} \otimes cf (x)] \otimes Cr {A \cap E_{i}}} \\ \leq lim_{n \to \infty} (p) \int_{A} f_{n} d Cr . \end{matrix}$ Moreover, since c ∈ (0, 1), letting c → 1, we have $\begin{matrix} \underset{E_{i} \in E}{\oplus} {[inf_{x \in E_{i}} χ_{A} \otimes f (x)] \otimes Cr {A \cap E_{i}}} \\ \leq lim_{n \to \infty} (p) \int_{A} f_{n} d Cr . \end{matrix}$ Thus, $(p) \int_{A} f d Cr \leq lim_{n \to \infty} (p) \int_{A} f_{n} d Cr$ . □

Theorem 4.2. Let $(Θ, P, Cr, \bar{R_{+}}, \oplus, \otimes)$ be a pan-credibility space, $A \in P$ , Cr{A}≤ 0.5, and {B_n} be a decreasing sequence of events with Cr{B_n}→0 as n→ ∞. Then, for all f ∈ G, we have $lim_{n \to \infty} (p) \int_{A \ B_{n}} f d Cr = (p) \int_{A} f d Cr .$

Proof. By Theorem 3.10, we have $\begin{matrix} (p) \int_{A \ B_{n}} f d Cr = (p) \int_{A} f \otimes χ_{A \ B_{n}} d Cr, \\ (p) \int_{A} f d Cr = (p) \int_{A} f \otimes χ_{A} d Cr, \end{matrix}$ and f ⊗ χ_{A\B_n} ↗ f ⊗ χ_A due to the facts that {B_n} is a decreasing sequence and $lim_{n \to \infty} Cr {B_{n}} = 0$ . Then by Theorem 4.1, we have $lim_{n \to \infty} (p) \int_{A} f \otimes χ_{A \ B_{n}} d Cr = (p) \int_{A} f \otimes χ_{A} f d Cr .$ So we have $lim_{n \to \infty} (p) \int_{A \ B_{n}} f d Cr = (p) \int_{A} f d Cr$ . □

A result similar to Fatou’s Lemma in classical measure theory can be obtained as follows.

Theorem 4.3. Let $(Θ, P, Cr, \bar{R_{+}}, \oplus, \otimes)$ be a pan-credibility space, $A \in P$ and f_n ∈ G. If Cr{A}≤ 0.5, then $(p) \int_{A} \underset{n \to \infty}{lim inf} f_{n} d Cr \leq \underset{n \to \infty}{lim inf} (p) \int_{A} f_{n} d Cr .$

Proof. Denote that $g_{n} (x) = inf_{i \geq n} f_{i} (x)$ , n = 1, 2, …, then g_n ≤ f_n, and $g_{n} (x) ↗ \underset{n \to \infty}{lim inf} f_{n} (x)$ . So (p)∫_Ag_n d Cr ≤ (p) ∫_Af_n d Cr, n = 1, 2, …. By Theorem 4.1, we obtain $lim_{n \to \infty} (p) \int_{A} g_{n} d Cr = (p) \int_{A} \underset{n \to \infty}{lim inf} f_{n} (x) d Cr .$ Thus we complete the proof. □

Definition 4.4. Let $(Θ, P, Cr, \bar{R_{+}}, \oplus, \otimes)$ be a pan-credibility space, f_n, f ∈ G. Then f is referred to as the qusi-control function of f_n, if for all k > 1 there exists a positive integer N (k) such that f_n ≤ kf, whenever n ≥ N (k).

Theorem 4.5. Let $(Θ, P, Cr, \bar{R_{+}}, \oplus, \otimes)$ be a pan-credibility space, f_n, f ∈ G. If, for $A \in P$ , f_n ↘ f on A and f is the qusi-control function of f_n, then $lim_{n \to \infty} (p) \int_{A} f_{n} d Cr = (p) \int_{A} f d Cr .$

Proof. Since f_n ↘ f, we have $(p) \int_{A} f d Cr \leq lim_{n \to \infty} (p) \int_{A} f_{n} d Cr .$ Because f is the qusi-control function of f_n, for any chosen real number k > 1, there exists a positive integer N (k) such that f_n ≤ kf, whenever n ≥ N (k). Therefore, (p) ∫_Af_n d Cr ≤ (p) ∫_Akf d Cr, n ≥ N (k). So $lim_{n \to \infty} (p) \int_{A} f_{n} d Cr \leq (p) \int_{A} kf d Cr$ holds for any k > 1.

Let k → 1, we have $lim_{n \to \infty} (p) \int_{A} f_{n} d Cr \leq (p) \int_{A} f d Cr,$ which concludes the proof. □

Theorem 4.6. Let $(Θ, P, Cr, \bar{R_{+}}, \oplus, \otimes)$ be a pan-credibility space, $A \in P$ , and f_n, f ∈ G. If Cr{A}≤ 0.5, f_n → f on A and f is the qusi-control function of f_n, then $lim_{n \to \infty} (p) \int_{A} f_{n} d Cr = (p) \int_{A} f d Cr .$

Proof. Denote $g_{n} (x) = sup_{i \geq n} f_{i} (x)$ , n = 1, 2, …. Then, g_n ≥ f_n, n = 1, 2, … and g_n ↘ f. Thus, we have $\begin{matrix} (p) \int_{A} f d Cr & = (p) \int_{A} \underset{n \to \infty}{lim inf} f_{n} d Cr \\ \leq \underset{n \to \infty}{lim inf} (p) \int_{A} f_{n} d Cr \\ \leq lim_{n \to \infty} (p) \int_{A} g_{n} d Cr \\ = (p) \int_{A} f d Cr . \end{matrix}$ Hence, $lim_{n \to \infty} (p) \int_{A} f_{n} d Cr = (p) \int_{A} f d Cr$ . □

5. Conclusions

In this paper, we define the pan-integral on credibility space and discuss some of its elementary properties. We also examine the relationship between pan-integral and H-fuzzy integral on credibility space, and it is shown that the pan-integral and the H-fuzzy integral coincide with respect to a commutative isotonic semiring $(\bar{R_{+}}, \lor, ⊤)$ . Finally, several monotone convergence theorems are investigated, and concretely, it is proved that the symbols of the limit and the integral can be exchanged under some given conditions.

Our further work will focus on other types of integrals, Choquet integral, for example, on credibility space. And the relationships among them are deserved to be investigated.

Footnotes

Acknowledgments

This research was supported by the National Natural Science Foundation of China (Grant no. 61806182), the Scientific Research Fund for Young Teachers of Zhengzhou University (Grant no. 32220326) and the Training Project for Young Backbone Teachers of Colleges and Universities of Henan Province.

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