Lacunary Statistical Convergence in Measure for Double Sequences of Fuzzy Valued Functions

Based on the concept of lacunary statistical convergence of sequences of fuzzy numbers, the lacunary statistical convergence, uniformly lacunary statistical convergence, and equi-lacunary statistical convergence of double sequences of fuzzy-valued functions are defined and investigated in this paper. ,e relationship among lacunary statistical convergence, uniformly lacunary statistical convergence, equi-lacunary statistical convergence of double sequences of fuzzy-valued functions, and their representations of sequences of α-level cuts are discussed. In addition, we obtain the lacunary statistical form of Egorov’s theorem for double sequences of fuzzy-valued measurable functions in a finite measurable space. Finally, the lacunary statistical convergence in measure for double sequences of fuzzy-valued measurable functions is examined, and it is proved that the inner and outer lacunary statistical convergence in measure are equivalent in a finite measure set for a double sequence of fuzzy-valued measurable functions.


Introduction
Fast [1] initiated statistical convergence for a real sequence. After the work of Fridy [2] andŠalát [3], it became a noteworthy topic in summability theory. Mursaleen and Edely [4] examined the statistical convergence via double sequences.
Various types of convergence for sequences of functions, such as pointwise, equi-statistical (or ideal), and uniform convergence, were originated by Balcerzak et al. [5]. Pointwise and uniform statistical convergence of double sequences was studied by Gökhan et al. [6].
Additionally, Duman and Orhan [7] studied convergence in μ-density and μ-statistical convergence of sequences of functions and presented the notions of μ-statistical pointwise convergence and μ-statistical uniform convergence.
Zadeh [10] initiated the notion of fuzzy sets. e publication of the paper affected deeply all the scientific fields.
In recent times, Gong et al. studied statistical convergence, equi-statistical convergence, and uniformly statistical convergence for sequences of fuzzy-valued functions. ese concepts were extended to the double sequences by Hazarika et al. Kişi and Dündar examined lacunary statistical convergence in measure for sequences of fuzzy-valued functions and established noteworthy results.
According to Zadeh [10], a fuzzy subset of T is a nonempty (i) w is convex, i.e., w(t) ≥ w(s)Λw(r) � min w(s), { w(r)}, where s < t < r (ii) w is normal, i.e., there exists an t 0 ∈ R such that w(t 0 ) � 1 (iii) w is upper semicontinuous, i.e., for every ε > 0, We indicate the set of all fuzzy numbers by F(R). e set R of real numbers can be included in F(R) if we take r ∈ F(R) by which is a closed bounded interval of R. As in [13], the Hausdorff distance between two fuzzy numbers w and q is denoted by D: where d is the Hausdorff metric. For K ⊂ N and j ∈ N, δ j (K) is named jth partial density of K if exists, it is named the natural density of K. Ψ � K ⊂ N: A sequence of fuzzy numbers (x n ) is called to be statistically convergent to a fuzzy number x 0 if for every ε > 0, where I r � (k r− 1 , k r ]. e number δ θ (A) is denoted as the lacunary density exists. Also, Λ � A ⊂ N: δ θ (A) � 0 is called to be zero density set.
Here, h is named the lacunary statistical limit function of (h m ).
Some definitions and significant results about lacunary statistical convergence in measure for sequences of fuzzyvalued functions were given in.
A double sequence θ 2 � (k r , j u ) is named lacunary sequence if there exist two increasing sequences of integers (k u ) and (j u ) such that We use the following symbols in the sequel: In the paper, by θ 2 � (k r , j u ) , we will indicate a double lacunary sequence of positive real numbers.
In this article, we proposed the concepts of lacunary statistical convergence, uniformly lacunary statistical convergence, and equi-lacunary statistical convergence for double sequences of fuzzy-valued functions and proved some classical results in this new setting and their representations of sequences of α-level cuts. We proved lacunary statistical form of Egorov's theorem for double sequences of fuzzy-valued measurable functions defined on a finite measure space (Ω, M, μ). Finally, we define the notion of lacunary statistical convergence in measure for double sequences of fuzzy-valued measurable functions and prove some interesting results. Our results were emphasized with examples.

Main Results
In the section, we presume that h: [a, b] ⟶ F(R) and h mn : [a, b] ⟶ F(R) are the fuzzy-valued function and a double sequence of fuzzy-valued functions for all m, n ∈ N. We indicate FVF and DSFVF in place of fuzzy-valued function and double sequence of fuzzy-valued functions.
e number δ θ 2 (T) is named the lacunary density or exists. Additionally, is named to be zero density set.
if for every z ∈ [a, b] and for all ε > 0 there exists T z ∈ Λ 2 such that, for all (m, n) ∈ N × N∖T z , we have It is obvious that h mn ⟶ p S θ 2 h if for every z ∈ [a, b] and for each ε > 0 Definition 3. We call that a DSFVF (h mn ) is uniformly which provides for all z ∈ [a, b]. It is obvious that h mn ⇉ uS θ 2 h if, for all ε > 0, for all z ∈ [a, b].
Proof. Assume that sup z∈[a,b] D(h mn (z), h(z)) ⟶ pS θ 2 0. en, for each z ∈ [a, b] and for every ε > 0, we have as ε is arbitrary. is gives that (m, n) ∈ N × N∖A 2 . erefore, we obtain which gives Proof. First, we demonstrate that h mn is continuous. Let ε > 0 and z 0 ∈ [a, b]. By the equi-continuity of h mn , then there exists c > 0 such that So, there exists (m, n) ∈ N × N such that We obtain

Journal of Mathematics
erefore, we proved the continuity of h.
At the moment, we will show that h mn Using the finite covering theorem, select finite open coverings Note that, by monotonicity of δ ru θ 2 , we can also utilize β � ε. Remark 3. It is obvious that h mn ⟶ p S θ 2 h iff for every z ∈ Y and for every ε, β > 0 ∃k, l ∈ N, for all r ≥ k, u ≥ l, In this instance, we may take β � ε. Obviously, Proof. Proving the above result, we can consider the examples as given below.
for all (m, n) ∈ N × N∖M, z ∈ [a, b], and any α ∈ [0, 1]. As a result, we have that is, is concludes the proof.

Journal of Mathematics
Proof. h mn ⟶ e S θ h shows that, for any ε > 0 and σ > 0, there exists k, l ∈ N, for all r ≥ k, u ≥ l and any z ∈ [a, b] such that So, for any α ∈ [0, 1], we obtain erefore, for any α ∈ [0, 1], we acquire en, emphasize that Conversely, let ε > 0 and σ > 0, and there exists k 1 , l 1 ∈ N such that for all r ≥ k 1 , u ≥ l 1 and any z ∈ [a, b] and for any α ∈ [0, 1]. Similarly, there exists k 2 , l 2 ∈ N such that δ ru for all r ≥ k 2 and u ≥ l 2 and any z ∈ [a, b] and for any α ∈ [0, 1]. en, select k � max k 1 , k 2 and l � max l 1 , l 2 . We obtain for all r ≥ k and u ≥ l and any z ∈ [a, b] and for any α ∈ [0, 1]. us, we have that is, is concludes the proof. e next result is the lacunary statistical form of Egorov's theorem for the DSFVF. en, for every ε > 0, there exists A ⊂ M such that μ(Ω\A) < ε and h mn|A ⟶ e S θ 2 h |A on A.

Journal of Mathematics
Hence, for all ∀r ≥ r 0 and u ≥ u 0 , one acquires (78) To acquire relation (73), it is enough to demonstrate that For the set S * η ⊂ Ω × P and for every fix r, u ∈ N, we can utilize the Fubini theorem for the characteristic function of S * η of the finite measure μ × δ ru θ 2 . Actually, where μ S η (m, n) < 1(∀(m, n) ∈ P), us, Suppose r 0 and u 0 ∈ N such that r ≥ r 0 and u ≥ u 0 , one obtains which gives that strict inequality (80) is valid. (ii) Presume that μ(Ω) < ∞. Fix η > 0. To verify our result, we need to denote that ∀ε, q > 0, ∃r 0 , u 0 ∈ N, ∀r Let ε > 0 and q > 0 be given. Since h mn ⟶ μ , δ θ 2 h, one can find r 0 , u 0 ∈ N such that, for all r ≥ r 0 and u ≥ u 0 , we have By considering the Fubini theorem for the function of Supposing r 0 and u 0 such that, for all r ≥ r 0 and u ≥ u 0 , we have is concludes the proof.
□ eorem 6 indicates that both kinds of convergence (in Definition 5) in measure are equivalent if Ω is a finite measurable set. By thinking the set Ω, we give the following definition.
Proof. We presume that h mn ⇉ uS θ 2 h. Take q > 0. In this way, there is a set W ∈ Λ 2 such that us, we obtain is indicates that h mn (z) ⟶ μ S θ 2 h(z). □ Theorem 7. Let h be a FVMF such that, for each z ∈ Ω, h mn (z) ⟶ p S θ 2 h(z). en, given ε > 0 and δ > 0, there is a measurable set W ⊂ Ω with μ(W) < δ and integers m 0 and n 0 such that Proof. Let and set We have that Ω m 0 +1,n 0 +1 ⊂ Ω m 0 ,n 0 , and for each z ∈ Ω, there is some Ω m 0 ,n 0 to which z does not belong, since h mn (z) ⟶ h(z).
(101) erefore, one obtains Proof. Presume that h mn (z) ⟶ μ S θ 2 h(z), so any subsequence (h m k n l ) of (h mn ) also lacunary statistically convergent in measure to h.
both lacunary statistically convergent measure to zero for each q > 0. Notice that this concept is given by the following formula: In that case, we can write η � q or q � (1/r), r ∈ N.

Conclusion 1
In this study, we propose the notions of pointwise lacunary statistical convergence, uniformly lacunary statistical convergence, and equi-lacunary statistical convergence of the double sequence of fuzzy-valued functions and discuss the reationship between various kinds of lacunary statistical convergence for the double sequence of fuzzyvalued functions and the sequence of α-level cut functions. In addition, we obtain the Egorov theorem for the double sequence of fuzzy-valued functions and investigate the lacunary statistical convergence in measure for the double sequence of fuzzy-valued functions. Note that, in contrast to the case for single sequences, a convergent double sequence need not be bounded, so the studies on lacunary statistical convergence of double sequences have a rapid growth and an emerging area in mathematical research. We conclude that our results are more general than the one proved earlier for single sequences by Kişi and Dündar. As an application, researchers who linked two theories such as the theory of approximation and the theory of lacunary statistical summability may prove fuzzy analogue of Korovkin's type approximation theorem for several test functions by using our convergence methods.

Data Availability
No data were used to support this study.

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