#### Abstract

Let be the set of all fuzzy complex numbers. In this paper some classical and measure-theoretical notions are extended to the case of complex fuzzy sets. They are fuzzy complex number-valued distance on , fuzzy complex number-valued measure on , and some related notions, such as null-additivity, pseudo-null-additivity, null-subtraction, pseudo-null-subtraction, autocontionuous from above, autocontionuous from below, and autocontinuity of the defined fuzzy complex number-valued measures. Properties of fuzzy complex number-valued measures are studied in detail.

#### 1. Introduction

It is well known that additivity of a classical measure primly depicted measure problems under error-free condition. But when measure error was unavoidable, additivity could not fully depict the measure problems under certain condition. To overcome such difficulties, fuzzy measure has been developed. Research on fuzzy measures was very deep in those aspects: research based on a certain number of subsets of a classic set and a real value nonaddable measure (such as Choquet’s content theory [1], Sugeno’s measure theory [2]), research based on fuzzy sets and a real value measure (e.g., Zadeh’s addable measure [3]), and especially the research on fuzzy value measures which generalizes the set value measure theory.

Being a newly developing theory developed in the later 1960s, set value measure had been applied in many fields [4–6]. After the appearance of fuzzy numbers, people naturally thought of related measure and integral. In 1986 Zhang [7] defined a kind of fuzzy set measure on , in 1998 Wu et al. generalized the codomain of fuzzy measure to fuzzy real number field and defined the Sugeno integral of fuzzy number fuzzy measure [8], and Guo et al. also defined the fuzzy value measure integral of fuzzy value function [9] which generalized the Sugeno integral about fuzzy value fuzzy measure to fuzzy set [10]. In 1989, Buckley presented the concept of fuzzy complex number [11] which inspired that people needed to consider the measure and integral problem about fuzzy complex number.

At the beginning of the 90s, Guang-Quan [12–21] introduced fuzzy real distance and discussed the fuzzy real measure based on fuzzy sets and then gave the fuzzy real value fuzzy integral and established fuzzy real valued measure theory on fuzzy set space. During 1991-1992, Buckley and Qu [22, 23] studied the problems of fuzzy complex analysis: fuzzy complex function differential and fuzzy complex function integral. During 1996–2001, Qiu et al. studied serially basic problems of fuzzy complex analysis theory, including the continuity of fuzzy complex numbers and fuzzy complex valued series [24], fuzzy complex valued functions and their differentiability [25], and fuzzy complex valued measure and fuzzy complex valued integral function [26, 27]. Wang and Li [28] gave the fuzzy complex valued measure based on the fuzzy complex number concept of Buckley, studied Lebesgue integral of fuzzy complex valued function, and obtained some important results.

As for applications of fuzzy complex number theory, Ramot et al. [29, 30] studied complex fuzzy sets and complex fuzzy logic, Dick [31] studied fuzzy complex logic more profoundly, Ha et al. [32] applied fuzzy complex set in statistical learning theory and obtained a key theorem of statistical learning theory, Fu and Shen [33] studied modeling problems of fuzzy complex number, and Fu and Shen [34] applied fuzzy complex in pattern recognition and classification and obtained important results. Please see [35–37] for other applications.

This paper will extend the classical measure to fuzzy complex number-valued measure, which can better express the interactions among the attributes (cf. [32, 34–37]) and, thus, is expected to have extensive applications in information fusion technology, classification technology, machine learning, pattern recognition, and other fields. Section 2 is some preliminary notions (including fuzzy complex number, real fuzzy distance between two fuzzy real numbers, and two fuzzy complex numbers) and some basic operations and order relation of fuzzy complex. Section 3 is prepared for the next section. We defined, based on Ha’s work [38], the concepts of fuzzy complex distance and complex fuzzy set value complex fuzzy measure (an extension of fuzzy measure) on fuzzy complex number field. We also present, based on Zhang’s work [21], the concepts of null-additivity, pseudo-null-additivity, null-subtraction, pseudo-null-subtraction, autocontinuous from above, autocontinuous from below, and autocontinuity of fuzzy complex value fuzzy complex measure on complex fuzzy number set (this measure has the properties PGP and SA/SB). In Section 4, we deduced some important properties on complex fuzzy set value complex fuzzy measure which are generalizations of the corresponding results in measure theory; we also obtain some results on related integral theory.

#### 2. Preliminaries

##### 2.1. Fuzzy Complex Numbers

In this paper, is the set of all real numbers set, is the set of all complex numbers, is an ordinary set, is the set of all real fuzzy numbers on , is the set of all interval numbers, is a measurable space (thus is a -algbra), and is the set of all fuzzy complex numbers on . Let and let .

*Definition 1 (see [12]). *Let . Then the mapping defined by is called a fuzzy complex number, where is called the real part of (written as Re) and is called the imaginary part (written as ), and . One will identify with and, thus, think fuzzy complex numbers are an extension of fuzzy real numbers. The set of all fuzzy complex numbers on is denoted by .

For any subsets of , write . The operation is described as follows: (1) for any ;(2) for any and (). is said to be a fuzzy infinity [21] (written as ) if one of the supports of and is an unbound set. For any , one makes the following appointments:if and only if and ; if and only if and ; if and only if and or ; if and only if , .One uses to denote a family (which is obviously nonempty) of subsets of that satisfies the following conditions:(1)for each , if has upper bound, then ;(2)for each , if has lower bound, then .

##### 2.2. Fuzzy Distance of Fuzzy Numbers

*Definition 2 (see cf. [21]). *A mapping satisfying the following conditions is called a fuzzy metric or a fuzzy distance on :(1)for any , , and if and only if ;(2)for any , ;(3)for any , . is called the fuzzy distance of fuzzy real numbers and .

*Example 3. *The mapping defined by
is a fuzzy distance on .

*Remark 4. *Analogously, a mapping satisfying the following conditions is called a fuzzy metric or a fuzzy distance on :(1)for any , , and if and only if ;(2)for any , ;(3)for any , . is called the fuzzy distance of fuzzy complex numbers and .

*Example 5. *Let be a fuzzy distance on . Then the mapping defined by
is a fuzzy distance on .

*Definition 6 (see cf. [12]). *Let and let . is said to converge to according to a fuzzy metric on (written as ) if, for each , there exists a positive integer such that for all .

#### 3. Complex Fuzzy Set-Valued Complex Fuzzy Measures

The notion of complex fuzzy measure on family of classical sets was given in [26].

*Definition 7 (see [26]). *Let and let . A fuzzy measure on a -algebra composed of subsets of is a mapping which satisfies the following conditions:(1);(2)if , then ;(3)if , then ;(4)if and for some , then .

In this paper we need an expansion of this notion. First we defined the concept of fuzzy complex value distance.

*Definition 8. *A mapping satisfying the following conditions is called a fuzzy complex value metric or a fuzzy complex value distance on :(1)for any , , and if and only if ;(2)for any , ;(3)for any , . is called the fuzzy complex value distance of fuzzy complex numbers and .

*Remark 9. *It can be easily seen that a mapping is a fuzzy complex value metric on if and only if for some two fuzzy metrics and on .

*Definition 10. *Let , , and be a fuzzy complex value distance, and let (). If for each , there exists a positive integer such that and for all hold, then is said to be convergent to according to distance , denoted by .

*Definition 11. *Let be a nonempty complex number set, let be the set of all complex fuzzy sets on , and let be a fuzzy complex value metric on . A complex fuzzy set-value complex fuzzy measure is a mapping (where , ) which satisfies the following conditions:(1);(2)for any satisfying , (i.e., and );(3)(lower semicontinuous) if with , , then ;(4)(upper semicontinuous) if with , and for some , then .

Apparently, a complex fuzzy set-value complex fuzzy measure is also a kind of special generalized fuzzy measures.

*Definition 12. *A mapping is said to be(1)0-add if for any satisfying and ;(2)null-additive (briefly, 0-sub) if for any satisfying and ;(3)autocontinuous from above (briefly, autoc. ) if for any for all and satisfying , and ;(4)autocontinuous from below (briefly, autoc. ) if for any for all and satisfying , and ;(5)autocontinuous if it is both autoc. and autoc. .

*Definition 13. *A complex fuzzy set-value complex fuzzy measure on is said to be pseudo-null-additive (briefly, P.0-add**/**, where with ) if it satisfies for all and all with . It is said to be pseudo-null-subtraction (briefly, P.0-sud**/**, where with ) if it satisfies for all and all with .

*Definition 14. *A complex fuzzy set-value complex fuzzy measure on fuzzy -algebra is said to have property (PGP) if, for each , there exists a such that whenever . It is said to have property (S/A) if, for any with , there exists a subsequence of such that . It is said to have property (S/B) if for any .

#### 4. Main Results

Let be a set and the set of all fuzzy sets on . Then a subfamily is addable if and only if it satisfies the following conditions (see [21]):(1);(2)if , then , ,

where and (for all ).

We first have the following result.

Theorem 15. *Every fuzzy complex measure on a addable class is a complex fuzzy value fuzzy complex measure on .*

*Proof. *We only prove the upper continuity and lower continuity of . Suppose , , and for some . By monotonicity of , we have and for any . Since , we have
Thereby
It follows that . Similarly, we can prove , which means that is upper continuous.

Assume and . Then is a monotonic decrease sequence and , so
Similarly we can prove ; thus is also lower continuous. In summary, is a complex fuzzy set-value complex fuzzy measure.

Theorem 16. *Every complex fuzzy set-value complex fuzzy measure on a fuzzy -algebra is exhaustive.*

*Proof. *Suppose is a disjoin sequence; then
Assume ; then for some or for some ; that is, for all or for all . Without loss of generality, we assume the first. Then there are two distinct indexes and such that and (and, thus, ), which conflicts with the fact that is a disjoin sequence. Therefore holds. Thus
Then we get .

On the other hand, since is complex fuzzy set-value complex fuzzy measure on the fuzzy -algebra , , so .

Theorem 17. *If , then
*

*Proof. *Since about , and .

As and for all , and . Therefore
which means .

Similarly, .

From properties of upper limits and lower limits we can see the following theorem holds.

Theorem 18. *Let and . If is 0-addable and upper continuous on , then, for any satisfying , and , .*

*Proof. *Let . Since is upper continuous,
As and is upper continuous and 0-addable, we have
So holds.

Similar to Theorem 18, we have the following.

Theorem 19. *Let and . If is 0-subtractable and continuous on , then, for any satisfying and .*

Theorem 20. *Assume that , is a complex fuzzy set-value complex fuzzy measure on which is pseudo-zero addable about , and . If for any satisfying , then for any .*

*Proof. *Let . As is lower continuous, we have
Therefore . As , for any . By upper continuity and P.0-add of , we have
Hence .

Similarly, we have the following.

Theorem 21. *Suppose that , is a complex fuzzy set-value complex fuzzy measure on which is pseudo-zero subtractable about , and . If for any satisfying , then for any .*

Theorem 22. *Let be a complex fuzzy set-value complex fuzzy measure on a fuzzy -algebra and . If is , then possesses (PGP) property.*

*Proof. *Suppose that does not possess (P.G.P) property; then there exists an (where are positive real numbers) such that, for any natural numbers , there exist such that
Thus and, thus, and . From upper autocontinuity of , we have
Therefore for some . This conflicts with the hypothesis.

Theorem 23. *Suppose that possesses (P.G.P) property. If and , then there exists a sequence of real numbers satisfying (for all n) and and a subsequence of such that (for all ). Furthermore, possesses (SA) property.*

*Proof. *For any real numbers , let , , and . Since possesses (P.G.P) property, there exists a such that and whenever and . Since , there exists an such that , whenever and . Therefore . Since possesses (P.G.P) property, there exists a such that , , , and whenever and . As , there exists an such that and . Hence , , and, thus, and .

For and there exists , , and such that ,
For , , since , there exists , such that and . Therefore , , and .

Generally we can get , and , such that , .

Let and ; then , .

Hence ; that is, possesses (SA) property.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This work was supported by the International Science and Technology Cooperation Foundation of China (Grant no. 2012DFA11270) and the National Natural Science Foundation of China (Grant no. 11071151).