The fuzzy metric space based on fuzzy measure

Abstract In this paper, we study the relation between a fuzzy measure and a fuzzy metric which is induced by the fuzzy measure. We also discuss some basic properties of the constructed fuzzy metric space. In particular, we show that the nonatom of fuzzy measure space can be characterized in the constructed fuzzy metric space.


Introduction
The theory of metric space is an important topic in topology. The methods of constructing a fuzzy metric have been extensively studied [1][2][3][4]. It is worth noting that George and Veeramani [5] introduced the concept of a fuzzy metric with the help of continuous t -norms. Despite being restrictive, this kind of fuzzy metric provides a more natural and intuitive way to connect with the metrizable topological spaces. This concept has been widely used in various papers devoted to fuzzy topology [5][6][7][8][9][10][11]. It also has been applied to color image filtering to improve image quality (see [9] and the references therein).
On the other hand, measure theory is one of the most important theories in mathematics and it has been extensively studied. The concept of fuzzy measure was first introduced by Sugeno [12]. It can be regarded as an extension of classical measure in which the additivity is replaced by a weaker condition, monotonicity. Klement et. al establish the axiomatic theory of fuzzy -algebras and develop a measure theory of fuzzy sets [13][14][15]. So far, there are many different classes of fuzzy measures, such as possibility measure [16,17], decomposable measure [18][19][20], pseudo-additive measure [21,22], and T -measure [23][24][25][26] etc. A systematic study of fuzzy measure theory can be found in [27][28][29][30].
Recently, the study of constructing a fuzzy metric using a fuzzy measure technique has been actively pursued. In particular, a fuzzy Prokhorov metric and ultrametric defined on the set of all probability measures in a compact fuzzy metric space have been developed in [31,32]. Cao et. al [33] introduce fuzzy analogue of the Kantorovich metric among the set of possibility distributions. In [34,35], the authors discuss the relations between the decomposable measure and the fuzzy metric. More specifically, it has been proven that, with a Hausdorff fuzzy pseudo-metric constructed on its power set, a stationary fuzzy ultrametric space can induce a -?-superdecomposable measure. sense of Pap) on the measurable sets of a given -?-decomposable measure, and then analyzed the connection between the induced pseudo-metric and the -?-decomposable measure.
In this paper we focus on the following problems: how to construct a fuzzy metric by using a fuzzy measure developed in [14,15] and what is the relation between these two? And what is the relations between these two? Specifically, by introducing the concept of an equivalence relation on fuzzy measurable sets, we construct a fuzzy metric on the associated quotient sets from a given fuzzy measure. Furthermore, we study some basic properties of the constructed fuzzy metric space such as completeness and continuity. To gain better insight into our proposed method of constructing a fuzzy metric, we study the properties of the constructed fuzzy metric which can precisely reflect those of fuzzy measure. As an illustration we obtain that the nonatom of fuzzy measure space can be characterized in the constructed fuzzy metric space.
The rest of the paper is organized as follows. In Section 2, some basic notions and results are given. Sections 3 and 4 are devoted to constructing a fuzzy metric and discussing its properties. In Section 5, we discuss the relationships between the constructed fuzzy metric and the fuzzy measure. Finally, some concluding remarks are given in Section 6.

Preliminaries
We start this section by recalling the concept of triangular norms from [20,37]. They are an important tool in extending the classical metric space to fuzzy metric space. Because of the associative property, the t -Norm > can be extended by induction to n-ary operation by setting Due to monotonicity, for each sequence .x i / i2N of elements of OE0; 1, the following limit can be considered: x i : Next we recall the concept of a fuzzy metric with the help of the continuous t -norm, which is a generalization of the concept of Menger probabilistic metric to the fuzzy setting.
Definition 2.2 (George and Veeramani [5]). The 3-tuple .X; M; >/ is said to be a fuzzy metric space if X is an arbitrary nonempty set, > is a continuous t -norm and M is a fuzzy set on X 2 .0; 1/ satisfying the following conditions, for all x; y; z 2 X; t; s > 0: x; x; t / D 1, then .X; M; >/ is said to be a fuzzy pseudometric space.
It was proved in [5] that in a fuzzy metric space X , the function M.x; y; / is nondecreasing for all x; y 2 X . A sequence .x i / i2N in a fuzzy metric space .X; M; >/ is said to converge [6] for all t > 0; .X; M; >/ is said to be complete [8] if every Cauchy sequence is convergent. A mapping f from a fuzzy metric space .X; M; > 1 / to a fuzzy metric space .Y; N; > 2 / is called uniformly continuous [7] if for each " 2 .0; 1/ and each t > 0, their exist r 2 .0; 1/ and s > 0 such that N f .x/; f .y/; t > 1 " whenever M.x; y; s/ > 1 r.
We call .X; A; / an F -measure space, elements of A are referred as fuzzy measurable sets.

Constructing fuzzy metric based on F -measure
In this follow-up, > stands for the minimum t -norm T M . The following result is the natural fuzzy metric structure on fuzzy measurable sets.
Consequently,   where OEA .OEB/ denote the equivalence class of A .B/. Then M is a fuzzy metric on A= .
Proof. We first prove that M.OEA; OEB; t / is well defined on A= . If A 1 2 OEA and B 1 2 OEB, we conclude that . The following Lemma shows that the collection of equivalence class A= forms a fuzzy -algebra. Similarly, Thus we obtain . Also, Proof. The proof is straightforward.
According to Lemma 3.5 and Lemma 3.6, by means of representatives of classes, we can introduce the operations of union, intersection and complementation on A= : where OEA i 2 A= denote the equivalence class of A i in A. Hence, A= is a fuzzy -algbra. We therefore properly define on A= by setting .OEA/ D .A/; f or al l A 2 A: The pair .A= ; / is a said to be an F -measure algebra.
For convenience and simplicity, we denote members OEA of A= by A, and functions :A= ! OE0; 1/ by :A ! OE0; 1/.

Properties of the fuzzy metric space .A; M; >/
In this section, we study some properties of the fuzzy metric space .A; M; >/ based on F -measure . Proof. (i) For any A 1 ; B 1 ; A 2 ; B 2 2 A, t; s > 0, we first prove the relation M.   (i) M.A 1 ; A 2 ; t / Ä M.A 1 ; A 1 _ A 2 ; t / for each t > 0I Proof. (i) For any A 1 ; A 2 2 A, we get and so M.A 1 ; A 2 ; t/ Ä M.A 1 ; A 1 _ A 2 ; t /, for each t > 0: (ii) By (i) and Theorem 4.1 (i), for all k 2 N, t i > 0, we have Let .x n / n2N be a Cauchy sequence in fuzzy metric space .X; M; >/. If there is a subsequence .x k.n/ / n2N of .x n / n2N that converges to x in X , then the Cauchy sequence .x n / n2N converges to x.

Proof.
Let .x k.n/ / n2N be a subsequence of .x n / n2N . Then, given r with 0 < r < 1 and t > 0, there is an n 0 2 N such that for each n n 0 , M.

The correspondence between fuzzy metric space and F -measure space
In the following, we give the characteristics of the nonatom of the F -measure algebra .A; /.

Conclusion
In this paper, by constructing a fuzzy metric on the fuzzy measurable sets, we studied the relations between these two. In particular, several satisfactory properties of the constructed fuzzy metric have been obtained. In addition, we investigated the charaterization of the nonatom of the fuzzy measure and the corresponding properties of constructed fuzzy metric space. The main results and methods presented in this paper generalize some well known results in [38,39].