The theory of metric space is an important topic in topology. The methods of constructing a fuzzy metric have been extensively studied [1–4]. It is worth noting that George and Veeramani  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 [5–11]. It also has been applied to color image filtering to improve image quality (see  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 . 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 σ-algebras and develop a measure theory of fuzzy sets [13–15]. So far, there are many different classes of fuzzy measures, such as possibility measure [16, 17], decomposable measure [18–20], pseudo-additive measure [21, 22], and T-measure [23–26] etc. A systematic study of fuzzy measure theory can be found in [27–30].
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  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 σ-⊥-superdecomposable measure. The authors of  further use a topological approach to extend the t-conorm-based decomposable measures by introducing a fuzzy pseudometric structure on an algebra of sets. In  we constructed a pseudo-metric (in the sense of Pap) on the measurable sets of a given σ-⊥-decomposable measure, and then analyzed the connection between the induced pseudo-metric and the σ-⊥-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.
Definition 2.1 (Klement et al.): A function ⊤ : [0, 1]2 → [0, 1] is called triangular norm (t-norm for short) if it satisfies the following conditions for all a, b, c, d ∈ [0, 1]:
a⊤1 = a. (boundary condition) a⊤b ≤ c⊤d whenever a ≤ c and b ≤ d.(monotonicity) a⊤b = b⊤b. (commutativity) a⊤(b⊤c) = (a⊤b)⊤c. (associativity)
A t-Norm ⊤ is said to be continuous if it is a continuous function in [0, 1]2. Typical examples of continuous t-Norms are the minimum TM, the product TP and the Łukasiewicz t-norm TL, which are given by, respectively:
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 (xi)i∈ℕ of elements of [0, 1], the following limit can be considered:
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.
a⊤1 = a. (boundary condition)
a⊤b ≤ c⊤d whenever a ≤ c and b ≤ d.(monotonicity)
a⊤b = b⊤b. (commutativity)
a⊤(b⊤c) = (a⊤b)⊤c. (associativity)
Definition 2.2 (George and Veeramani ): 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 X2 × (0, ∞) satisfying the following conditions, for all x, y, z ∈ X, t, s > 0:
M(x, y, t) > 0, M(x, y, t) = 1 iff x = y, M(x, y, t) = M(x, y, t) M(x, z, t + s) ≥ M(x, y, t)⊤M(y, z, t) M(x, y, ·) : (0, ∞) → (0, 1] is continuous.
If the condition (GV2) is replaced by the condition (GVp): M(x, x, t) = 1, then (X, M, ⊤) is said to be a fuzzy pseudometric space.It was proved in  that in a fuzzy metric space X, the function M(x, y, ·) is nondecreasing for all x, y ∈ X. A sequence (xi)i∈ℕ in a fuzzy metric space (X, M, ⊤) is said to converge  to x if limi→∞ M(xi, x, t) = 1 for all t > 0; a sequence (xi)i∈ℕ in a fuzzy metric space (X, M, ⊤) is said to be Cauchy  if limi, j →∞ M(xi, x, t) = 1 for all t > 0; (X, M, ⊤) is said to be complete  if every Cauchy sequence is convergent. A mapping f from a fuzzy metric space (X, M, ⊤1) to a fuzzy metric space (X, M, ⊤2) is called uniformly continuous  if for each ε ∈ (0, 1) and each t > 0, their exist r ∈ (0, 1) and s > 0 such that N(f(x), f(y), t) > 1 – ε whenever M(x, y, s) > 1 – r.
M(x, y, t) > 0,
M(x, y, t) = 1 iff x = y,
M(x, y, t) = M(x, y, t)
M(x, z, t + s) ≥ M(x, y, t)⊤M(y, z, t)
M(x, y, ·) : (0, ∞) → (0, 1] is continuous.
Definition 2.3: Let X be a nonempty set, I the unit interval [0, 1]. A subset of IX is a fuzzy σ-algebra iff
0, 1 ∈ , A ∈ implies 1 — A ∈ , if is a sequence in , then = sup Ai ∈ .
0, 1 ∈ ,
A ∈ implies 1 — A ∈ ,
if is a sequence in , then = sup Ai ∈ .
Definition 2.4: A finite fuzzy measure (or F-measure) on a fuzzy σ-algebra is a function μ : → [0, ∞) satisfying:
μ(0) = 0, for A ∈ , μ(1 – A) = μ(1) – μ(A), for A, B ∈ , μ(A ∨ B) + μ (A ∧ B) = μ(A) + μ(B), if is a sequence in such that Ai ↗ A, A ∈ , then μ(A) sup = μ(Ai).
We call (X, , μ) an F-measure space, elements of are referred as fuzzy measurable sets.
μ(0) = 0,
for A ∈ , μ(1 – A) = μ(1) – μ(A),
for A, B ∈ , μ(A ∨ B) + μ (A ∧ B) = μ(A) + μ(B),
if is a sequence in such that Ai ↗ A, A ∈ , then μ(A) sup = μ(Ai).
3 Constructing fuzzy metric based on F-measure
In this follow-up, ⊤ stands for the minimum t-norm TM. The following result is the natural fuzzy metric structure on fuzzy measurable sets.
Theorem 3.1: Let (X, , μ) be an F-measure space. If we define the fuzzy set M on 2 × (0, ∞) by where A, B ∈ . Then M is a fuzzy pseudometric on .
Proof: From the definition of M, it is obvious that for any A, B ∈ , t > 0, we have (i) M(A, B, t) > 0; (ii) M(A, A, t) = 1 and M(A, B, t) = M(B, A, t); (iii) M(x, y, ·) : (0, ∞) → [0, 1] is continuous. The only thing that we need to prove is the triangular inequality. For any A, B, C ∈ and t, s > 0, we have and . We are going to verify M(A, B, C, t + s) ≥ M(A, B, t) ⊤ M(B, C, s). For each A, B, C, ∈ , we have and so we can distinguish two cases: M(B, C, s) ≥ M(A, B, s) or M(A, B, t) ≥ M(B, C, s).Case (i). If M(B, C, s) ≥ M(A, B, t), or equivlently, and hence Consequently, This implies that and hence Case (ii). Similar to (i). Thus, M is a fuzzy pesudometric on .
Remark 3.2: Based on Theorem 3.1, M is a fuzzy pseudometric on under any left-continuous t-norms since TM is the biggest left-continuous t-norm.
Generally speaking, the fuzzy pseudometric space ( , M, ⊤) is usually not a fuzzy metric space. But we can construct a fuzzy metric space from a fuzzy pseudometric metric space ( , M, ⊤) and, at the same time, keep the general characteristics of the fuzzy pseudometric metric space.
Lemma 3.3: Let (X, , μ) be an F-measure space. For each A, B ∈, we define the relation “~” on : A ~ B if and only if μ(A ∨ B) = μ(A ∧ B). Then ~ is an equivalence relation on .
Theorem 3.4: Let (X, , μ) be an F-measure space. Let /μ be the set of all equivalence classes for the relation “∼”. The fuzzy pseudometric M has a natural extension to where [A] ([B]) denote the equivalence class of A (B). Then M is a fuzzy metric on /μ.
The following Lemma shows that the collection of equivalence class /μ forms a fuzzy σ-algebra.
Lemma 3.5: Let (X μ) be an F -measure space. Then for Ai, Bi ∈ , Ai ∼ Bi, i ∈ ℕ, we have
1 - Ai ∼ 1 - Bi A1 ∨ A2 ~ B1 ∨ B2 and A1 ∨ A2 ~ B1 ∨ B2 .
1 - Ai ∼ 1 - Bi
A1 ∨ A2 ~ B1 ∨ B2 and A1 ∨ A2 ~ B1 ∨ B2
Lemma 3.6: Let (X, μ) be an F-measure space. If, for Ai, Bi ∈ , Ai ∼ Bi, i ∈ ℕ, and then A ∈ B.
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 , [Ai] ∈ /μ denote the equivalence class of Ai in . Hence, /μ is a fuzzy σ algbra. We therefore properly define μ on /μ by setting
The pair (/μ, μ) is a said to be an F-measure algebra.
For convenience and simplicity, we denote members [A] of /μ by A, and functions μ: /μ → [0, ∞) by μ: /μ → [0, ∞)
4 Properties of the fuzzy metric space( , M, ⊤)
In this section, we study some properties of the fuzzy metric space ( , M, ⊤) based on F -measure µ.
Theorem 4.1: Let (X, , µ) be an F -measure space and M be the fuzzy metric on defined in Theorem 3.4. Then the maps (i) (A, B) ↦ A ∨ B and (ii) (A, B) ↦ A ∧ B are uniformly continuous from × to .
Now, we shall consider the completeness of the fuzzy metric space , M, ⊤.
Lemma 4.2: Let (X, , µ) be an F -measure space and Ai (i = 1, 2, ...) be elements of . Then the fuzzy metric M as defined in Theorem 3.4 satisfies the following properties:
for each t > 0; ), for all k ∈ ℕ, ti > 0.
for each t > 0;
), for all k ∈ ℕ, ti > 0.
Lemma 4.3: Let (xn)n∈ℕ be a Cauchy sequence in fuzzy metric space (x, M, ⊤). If there is a subsequence (xk(n)n∈ℕ of (xn)n∈ℕ that converges to x in X, then the Cauchy sequence (xn)n∈ℕ converges to x.
Theorem 4.4: The fuzzy metric space (, M, ⊤) based on F -measure µ is complete.
5 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 ( , µ).
Definition 5.1: If, for two distinct elements A, B ⊂ X, there exists t, s > 0 and an element P ⊂ X, different from both A and B such that M(A, B, t + s) = M(A, P, t)⊤M(P, B, s), then fuzzy metric space (X, M, ⊤) is called convex..
Definition 5.2: Let ( , µ) be an F-measure algebra. Then the measure µ is called nonatom if for A, B ∈ , A ≤ B and µ(A) < µ(B), there exists P ∈ , A ≤ P ≤ B such that µ(A) < µ(P) <(B).
Theorem 5.3: The F-measure algebra ( , µ) is nonatom if and only if fuzzy metric space ( , M, ⊤) is convex.
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].
The authors are enormously grateful to the editors and the anonymous reviewers for their professional comments and valuable suggestions. This work is supported in part by the National Natural Science Foundation of China (No. 11371130, 61103052, 11401195), the Natural Science Foundation of Fujian Province (No. 2014H0034, 2016J01022), the projects of Education Department of Fujian Province (No. JA15280) and Li Shangda Discipline Construction Fund of Jimei University.
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Published Online: 2016-09-20
Published in Print: 2016-01-01
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