A Strong Law of Large Numbers for Random Sets in Fuzzy Banach Space

. The main purpose of this paper is to consider the strong law of large numbers for random sets in fuzzy metric space. Since many years ago, limited theorems have been expressed and proved for fuzzy random variables, but despite the uncertainty in fuzzy discussions, the nonfuzzy metric space has been used. Given that the fuzzy random variable is deﬁned on the basis of random sets, in this paper, we generalize the strong law of large numbers for random sets in the fuzzy metric space. The embedded theorem for compact convex sets in the fuzzy normed space is the most important tool to prove this generalization. Also, as a result and by application, we use the strong law of large numbers for random sets in the fuzzy metric space for the bootstrap mean.


Introduction
e study of the theory of random sets started with Robbins [1,2]. Kendall [3], Matheron [4], and Fortet and Kambouzia [5] were among others who studied this field. e motivation for studying random sets is both theoretical and practical. eoretically, they generalized random variables, random vectors, and fuzzy random variables. Also, practically, they depicted geometrical objects in certain models of growth [6]. e strong law of large numbers (SLLN) for random sets and fuzzy random variables in the Pompeiu-Hausdorff metric and the generalized Pompeiu-Hausdorff metric has been studied since 1982.
ese studies are based on the embedding theorems, namely, Rådström and Harmender theorems. Note that because the compact subset in the Banach space is not a vector space (with respect to Minkowski addition) in the Pompeiu-Hausdorff metric, proving the SLLN is not easy in this space (see [6]). e studies in this field began with the Puri and Ralescu [6] in 1983 for random sets in Banach space (Artstein et al. [7] in 1975 and Cressie [8] in 1978 also conducted studies on the SLLN in the Euclidean p-dimensional space). Puri and Ralescu [9] in 1986 provided a definition of a fuzzy random variable. In the same year, Kelement et al. [10] established the SLLN for the fuzzy random variable. Random sets and fuzzy random variables in the Pompeiu-Hausdorff metric and generalized Pompeiu-Hausdorff metric (d ∞ ) are not separable (see [9]). On the other hand, one of the essential conditions in the SLLN is separability. Also, the previous studies show that the convergence in the metric d ∞ is stronger than the convergence in the metric d 1 . In 2002, Proske et al. [11] studied the SLLN in the d ∞ metric.
López et al. [12] introduced simple convex random sets. At this time, a new approach was begun to express and prove the SLLN. In this approach, the embedding theorems were not used. Also, since the metric space for compact sets in d ∞ is not separable, to solve this problem, simple random sets were used. Colubi et al. [13] derived the SLLN for independent identically distributed (i.i.d) fuzzy random sets by the approximation method that was a result of López's studies ( [12]). Also, Molchanov [14] in the same year demonstrated the SLLN for the upper semicontinuous functions with a simpler approach. Li and Ogura [15] presented the SLLN for independent (not necessarily identically distributed) fuzzy set-valued random variables whose base space is a separable Banach space or a Euclidean space, in the sense of the extended Pompeiu-Hausdorff metric.
Fu and Zhang [16] obtained some SLLN for arrays of row-wise independent (not necessarily identically distributed) random compact sets and fuzzy random sets whose underlying spaces are separable Banach spaces. Kim et al. established [17] two types of the SLLN for fuzzy random variables taking values on the space of normal and upper semicontinuous fuzzy sets with compact support in a separable Banach space.
Probabilistic metric spaces were introduced by Menger [18] who generalized the theory of metric spaces. In Menger's theory, the concept of distance is considered to be statistical or probabilistic; i.e., he proposed associating a distribution function with every pair of elements x, y instead of associating a number. Many research studies have been done on probabilistic metric spaces in recent years. e motivation of introducing the probabilistic metric space is the fact that in many situations the distance between two points is inexact rather than a single real number. But when the uncertainty is due to fuzziness rather than randomness, as sometimes in the measurement of an ordinary length, it seems that the concept of a fuzzy metric space is more suitable. e concept of fuzzy metric space introduced by Kramosil and Michalek [19] and George and Veeramani [20] modified this concept.
Puri and Ralescu [9] provided a definition of a fuzzy random variable based on random sets. Now, if we want to use the concept of fuzzy metric space and the SLLN for fuzzy random variables, it is necessary to consider this concept and theorem for random sets. To reach this purpose, we prove an embedding theorem for random sets in fuzzy metric space. Also, we will introduce the generalization of Lebesgue dominated convergence theorem, in the case of random sets in fuzzy metric space.
Efron's bootstrapping [21] is a resampling scheme that is used on a variety of estimation problems. Considering the importance of the SLLN in the bootstrap method, much has been done in this area by various researchers (see Athreya [22], Athreya et al. [23]). In this paper, we generalize the SLLN in the fuzzy metric space for the bootstrap mean.
In Section 2, some preliminaries and lemmas will be presented. In Section 3, the generalized Rådström embedding theorem is expressed. In the next section, the Lebesgue dominated convergence theorem will be generalized. e SLLN in fuzzy metric space is stated in Section 5. In Section 6, we use the SLLN in fuzzy metric space for the bootstrap mean. e final section is the conclusion.

Preliminaries
In this section, at first, we define the t-norm, the new generalized Hukuhara difference (T-difference), fuzzy metric space, and fuzzy normed space and give several lemmas and theorems in this space that will be used in the next section.
Triangular norms (t-norms for short), introduced by Schweizer and Sklar [24], play a key role in the theory of fuzzy metric spaces. Also, a fuzzy subset of X (fuzzy set) is a function of u: X ⟶ [0, 1]. In the following, the definitions of t-norm, fuzzy metric space, and fuzzy normed space and its properties are presented.
Definition 1 (see [25] Let (X, ‖.‖) be a separable normed space. Denote by K(X) and K c (X) the collection of nonempty compact and compact convex subsets of X. e Minkowski addition and scalar multiplication are defined in K c (X): for A, B ∈ K c (X) and λ ∈ R. Note that K c (X) is not a vector space but it becomes a complete metric space when endowed with the Pompeiu-Hausdorff distance. We know that the Pompeiu-Hausdorff distance is defined as where ‖.‖ denotes the norm in X and A, B ∈ K c (X) [6]. We use the notation where A ∈ K c (X). From here on throughout the article, d H is the Pompeiu-Hausdorff metric.
To partially overcome this situation, Hukuhara in [26] introduced the following H-difference: and From an algebraic point of view, the difference of two sets A and B may be interpreted both in terms of addition as in (4) or in terms of negative addition; i.e., where (− 1)C is the opposite set of C. Conditions (4) and (5) are compatible with each other; that is why Stefanini [27] suggested a generalization of the Hukuhara difference as follows.
Definition 2 (see [27]). Let A, B ∈ K c (X); we define the generalized Hukuhara difference (gH-difference) of A and B as the set C ∈ K c (X) such that 2 Advances in Fuzzy Systems Sometimes, the gH-difference in (6) of A, B ∈ K c (R n ) does not exist (see [27]).
Stefanini and Bede [28] defined a generalized difference for compact convex sets, even if the gH-difference A⊖ g B does not exist. is difference is called the total gH-difference of A, B (T-difference for short). In the following, we introduce some preliminary concepts for compact convex sets which are required to express the T-difference.
where (R n , 〈., .〉) is a (real) Hilbert space with internal product 〈., .〉 and associated norm 〈x, x〉 1/2 . e gH-difference of A, B ∈ K c (R n ) can be expressed by the use of the support functions [28]. Consider A, B ∈ K c (R n ) with C � A⊖ g B as defined in (6); let s A , s B , s C , and s − C be the support functions of A, B, C, and If A ∈ K c (R n ), then A can be associated with a family of compact intervals that characterize it. For x ∈ R n , the support function s A : R n ⟶ R is defined by As a dual for the support function, l A : R n ⟶ R is defined by Also, Stefanini and Bede [28] defined for each P ∈ R n the compact intervals e following gH-differences for intervals are well defined for all P ∈ S n− 1 : where Definition 3 (see [28]). Let A, B ∈ K c (R n ) and consider the following family of sets: where I A,B and I C are the interval-valued functions defined in (11) and (12), respectively. e set D(A, B) will be called the (generic) difference set of the pair (A, B). It is immediate that D(B, A) � − D (A, B); i.e., C ∈ D (A, B) if and only if − C ∈ D(B, A). e new generalized difference will be defined as an element of the family D (A, B), by requiring appropriate additional conditions. Definition 4 (see [28]). C ∈ D(A, B) is called minimal with respect to set magnitude (norm-minimal for short) if no e set of all elements of D (A, B) with the normminimality property will be denoted by D norm (A, B). It is immediate that D norm (B, A) � − D norm (A, B). Furthermore, there exists a real number α(A, B) ≥ 0, depending only on A and B, such that Clearly, Definition 5 (see [28]). Let A, B ∈ K c (R n ) be given. e following convex set always exists and is unique where cl is the closure of D norm (A, B) with respect to convex unions of its elements. A⊖ T B ∈ K c (R n ) has the following basic properties: e set A⊖ T B ∈ K c (R n ) will be called the total gHdifference of A and B (T-difference for short).
Definition 6 (see [20]). Let X be an arbitrary nonempty set and * is a continuous t-norm. e 3-tuple (X, M, * ) is said to be a fuzzy metric space if M is a fuzzy set on X × X × (0, ∞) satisfying the following conditions for all x, y, z ∈ X and t, s > 0:  Example 1 (see [20]). Let (X, d) be a metric space. Define a * b � ab or a * b � min a, b { } and ∀x, y ∈ X, M(x, y, t) � kt n kt n + md(x, y) , k, m, n ∈ R + .
In this case, (X, M, * ) is a fuzzy metric space. In particular, if k � n � m � 1, then which is called the standard fuzzy metric induced by metric d.
Definition 7 (see [20]). A sequence x n in a fuzzy metric space (X, M, * ) is a Cauchy sequence if and only if for each 0 < ε < 1 and t > 0 there exist n 0 ∈ N such that for all n, m ≥ n 0 A fuzzy metric space is said to be complete if and only if every Cauchy sequence is convergent.
Definition 8 (see [29]). e 3-tuple (X, N, * ) is said to be a fuzzy normed space if X is a vector space, * is a continuous t-norm, and N is a fuzzy set on X × (0, ∞) satisfying the following conditions for every x, y ∈ X and t, s > 0: It is immediate that (X, N d H , * ) is a fuzzy normed space where * is a continuous t-norm.
Example 2 (see [29]). Let (X, ‖.‖) be a normed space. Suppose that a * b � ab or a * b � min a, b { }, and ∀A ∈ K c (X), where ‖A‖ is defined in (3). In this case, (X, N d H , * ) is a fuzzy normed space.

en, M d H is a fuzzy metric on X, which is called the fuzzy metric induced by the fuzzy norm N d H .
Proof. According to Definitions 3 and 4 and Lemma 3 in [29], it is easy to show that Lemma 1 is established.

Lemma 2. A fuzzy metric M d H , which is induced by a fuzzy norm N d H , has the following properties for all
A, B, C ∈ K c (X) and every scalar λ ≠ 0: Proof. By Lemma 1, Definition 9, and Lemma 4 [29], it is easy to show that the result is established; for example, for (2), we have

Generalized Rådström Embedding Theorem
As mentioned in Section 1, the compact subset in the Banach space is not a vector space (with respect to Minkowski addition) in the Pompeiu-Hausdorff metric (see [6]). e Rådström embedding theorem states that the collection of nonempty closed bounded and convex subsets of a Banach space can be embedded in a normed space. is theorem enables us to prove the SLLN.
In the following, first, we present the properties of the fuzzy metric space K c (X) in the fuzzy metric M d H and then generalize the Rådström embedding theorem.

Theorem 1. Let (X, N d H , * ) be a fuzzy normed space and M d H is fuzzy metric induced by N d H . Suppose that K c (X) is the collection of nonempty compact convex subsets of X; then,
Proof. By using Definition 9 and T-difference, we show that for all A, B, C ∈ K c (X), the conditions of Definition 6 are established as follows: Advances in Fuzzy Systems Rådström in [30] showed that K c (X), a class of compact convex sets, can be embedded isometrically into normed space. In the following, we will show that this property is established also into a fuzzy normed space. e space K c (X) plays an important role since it can be embedded isometrically into a fuzzy normed space. Actually, this theorem generalizes the Rådström embedding theorem [30] from K c (X) into a fuzzy normed space. □ Theorem 2. Let X be a separable normed space. ere exist a fuzzy normed space χ and a function j: K c (X) ⟶ χ with the following properties: Note that fuzzy normed space χ is not complete in general, but one can always take the completion of χ, and thus, K c (X) is embedded into a fuzzy normed space by j(·).
Proof. Since K c (X) is a class of compact convex sets in normed space, for all A, B, C ∈ K c (X) except 3 and 8-10, all the conditions of the Rådström embedding theorem in [30] are established. Now, we prove 3 and 8-10 conditions (with M d H instead of d H ).
In eorem 1, we showed that K c (X) with M d H is metrizable. Furthermore, from Lemma 2 in [30], condition 3 is confirmed.
Also, given that M d H is a decreasing function with respect to d H , from eorem 2 in [30]

Generalized Lebesgue Convergence Theorem
An important tool in Section 5, which is used to prove the SLLN for random sets in fuzzy metric space, is the Lebesgue convergence theorem. In this section, after defining a random set, we will generalize this theorem for random sets in the fuzzy metric space. Suppose that (Ω, A, P) is a probability space. e following definitions describe the concept of the random closed set, random compact, and random compact convex set.
Definition 10 (see [31]). Let C(X) be the family of closed subsets of X. A map X: Ω ⟶ C(X) is called a random closed set if, for every K ∈ K(X), ω: Definition 11 (see [31]). A random closed set X with almost everywhere compact values (so that X ∈ K(X)a.e.) is called a random compact set. e X-valued random set (i.e., random sets whose values are compact subsets of X ) is a Borel measurable function X: Ω ⟶ K(X).
For more information about this concept, see [31]. A random closed set X in the separable normed space X is called integrably bounded if has a finite expectation [31]. In other words, EX < ∞. e expected value of the random set was defined by Aumann [32] and later by Debreu [33]. ese definitions were shown to be equivalent to Byrne [34]. If X is a random compact set, then EX is defined as Here, f: Ω ⟶ X is a selection of X and Ef denotes the classical expectation (via the Bochner integral). In general, EX may be empty, but if EX < ∞, then EX ∈ K(X) (Aumann [32], Debreu [33]).
Note that X is a random compact convex set, whenever X ∈ K c (X). In the following, the random compact convex set is called a random set for brevity.
In the following, we will introduce the generalized Lebesgue dominated convergence type theorem in the case of random sets in fuzzy metric space, which is used in the next section. e almost everywhere convergence of random sets is usually defined with respect to the Pompeiu-Hausdorff metric as d H (X n , X) ⟶ 0 [31]. It can equally be said X n ⟶ X a.e. In fuzzy metric M d H whenever, M d H (X n , X, t) ⟶ 1 a.e. Theorem 3. Let X k | k ≥ 1 and X be random sets with values in K c (X) such that E‖X k ‖ < ∞ and E‖X‖ < ∞. Assume that X k in the fuzzy metric M d H is convergent a.e. to X and d H ( Proof. Using the inequality in Debreu [33], p.366-367, we have We know that d H (X k , X) is a random variable, so, Ed H (X k , X) will be real. erefore, according to Definition 11 and Lemma 1, we have On the other hand, from the hypothesis, since According to the integrability h and X ∈ K c (X), d H (X k , X) is integrable. Now, due to the classical Lebesgue dominated convergence theorem therefore M Ed H X k , X, t ⟶ 1.

Strong Law of Large Numbers in Fuzzy Metric Space
In this section, by using the Rådström embedding theorem, we establish the SLLN for random sets in fuzzy Banach space. In the following, we first express the definition of the fuzzy Banach space and then will show that (K c (F), M d H , * ) is a separable fuzzy metric space. In the last step, the SLLN will be expressed and proved in the fuzzy Banach space.
Definition 13 (see [29]). e fuzzy normed space (F, N, * ) is said to be a fuzzy Banach space whenever F is complete with respect to the fuzzy metric induced by a fuzzy norm.
As we can see in Figure 1, when m(n) ⟶ ∞ and t � 5, the expectation of the random set tends to the sample mean by using the bootstrap method in the fuzzy metric. In other words, Also, the effect of value t on the behavior of M d H and convergence rate can be seen in Figure 2. In fact, using a low value of t reduces the value of M d H to 0 too quickly as the Pompeiu-Hausdorff distance increases. On the other hand, if a high value is used, the value of M d H decreases too slowly (Figure 2). e expert's opinion is important in determining the appropriate value of t.

Conclusion
When the uncertainty is fuzziness, as sometimes in the measurement of an ordinary length, the concept of a fuzzy metric space is more suitable. Since the fuzzy random variable is defined on the basis of a random set, the SLLN for random sets in fuzzy metric space assists us in expressing this theorem for fuzzy random variables in a fuzzy metric space. erefore, we have presented a new theorem for the study of the SLLN for random sets in fuzzy metric spaces in the sense of George and Veeramani [20]. Also, we generalized the Rådström embedding theorem and Lebesgue dominated convergence that are important tools to prove these theorems. In fact, this article can provide the conditions for the expression of limit theorems for fuzzy random variables in fuzzy metric space. As an application of the SLLN for random sets in fuzzy metric space, we show that 1/m(n) m(n) i�1 Y n,i tends to E(X 1 ) in fuzzy metric space for bootstrap method. In fact, when an expert's opinion is important in determining distance, the value of t in fuzzy metric becomes significant.

Data Availability
is research is a theoretical study, and no data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.