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

Authors: Ömer Kişi
Journal: Journal of Mathematics
 (2021)

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## Abstract

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. The 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 ofα-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.


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## Body

## 1. Introduction

Fast [1] initiated statistical convergence for a real sequence. After the work of Fridy [2] and Šalát [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.Lacunary statistical convergence was firstly studied by Fridy and Orhan [8]. In [9], lacunary statistical convergence via double sequence was investigated.Zadeh [10] initiated the notion of fuzzy sets. The publication of the paper affected deeply all the scientific fields. This notion is significant for real-life conditions, but has no adequate solution to some problems. Such problems lead to new quests. Matloka [11] identified ordinary convergence of a sequence of fuzzy numbers. Nanda [12] examined the seqeuences with fuzzy numbers. Negoita [13] gave the Hausdorff distance between two fuzzy numbers. Statistical convergence by utilizing fuzzy numbers was given by Nuray and Savaş [14]. Savaş and Mursaleen [15] investigated statistical convergent double sequences via fuzzy numbers. By using fuzzy numbers, Aytar and Pehlivan [16] defined the statistical convergence of sequences. Pointwise statistical convergence sequences of fuzzy mappings studied by Altin et al. [17]. Some beneficial results on this topic can be found in [18–32].In recent times, Gong et al. studied statistical convergence, equi-statistical convergence, and uniformly statistical convergence for sequences of fuzzy-valued functions. These 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 subset t,w¯t:t∈T of T×J=0,1 for some function w¯:T⟶J=0,1. A function w¯:ℝ⟶J=0,1 is called a fuzzy number if the function w¯ holds the following properties:(i)
w¯ is convex, i.e., w¯t≥w¯sΛw¯r=minw¯s,w¯r, where s<t<r(ii)
w¯ is normal, i.e., there exists an t0∈ℝ such that w¯t0=1(iii)
w¯ is upper semicontinuous, i.e., for every ε>0, w¯−10,a+ε, for all a∈0,1 is open in the usual topology of ℝ(iv)
w¯0=clt∈ℝ:w¯t≥0 is compact, where cl is the closure operatorWe indicate the set of all fuzzy numbers byFℝ. The set ℝ of real numbers can be included in Fℝ if we take r¯∈Fℝ by(1)r¯t=1,ift=r,0,ift≠r.For0<α≤1, α-cut of w¯ is given by w¯α=t∈ℝ:w¯t≥α=w¯α−,w¯α+ 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:Fℝ×Fℝ⟶0,∞:(2)Dw¯,q¯=supα∈0,1dw¯α,q¯α=supα∈0,1maxw¯α−−w¯α+,q¯α−−q¯α+,where d is the Hausdorff metric.ForK⊂ℕ and j∈ℕ, δjK is named jth partial density of K if(3)δjK=K∩1,2,…,jj.If(4)δK=limn⟶∞1nk≤n:k∈Ki.e.,δK=limj⟶∞δjK,exists, it is named the natural density of K. Ψ=K⊂ℕ:δK=0 is denoted the zero density set.A sequence of fuzzy numbersx¯n is called to be statistically convergent to a fuzzy number x¯0 if for every ε>0, δn∈ℕ:Dx¯n,x¯0≥ε=0, i.e., n∈ℕ:Dx¯n,x¯0≥ε∈Ψ. It is demonstrated by st−limx¯n=x¯0 or x¯n⟶stx¯0, n⟶∞.LetA⊂ℕ and r∈ℕ. δθrA is named the rth partial lacunary density of A if(5)δθrA=A∩Irhr,where Ir=kr−1,kr.The numberδθA is denoted as the lacunary density (θ-density) of A if(6)δθA=limr⟶∞1hrk∈Ir:k∈Ai.e.,δθA=limr⟶∞δθrA,exists. Also, Λ=A⊂ℕ:δθA=0 is called to be zero density set.A SFVFh¯m is pointwise lacunary statistically convergent to FVF h¯ on a,b, denoted by pSθ−limh¯mz=h¯z or h¯m⟶pSθh¯; if for each z∈a,b and all ε>0 there exists Tz∈Λ such that, for all m∈ℕ∖Tz, we get Dh¯mz,h¯z<ε. It is obvious that h¯m⟶pSθh¯ if for every z∈a,b and all ε>0, δθm∈ℕ:Dh¯mz,h¯z≥ε=0. 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 fuzzy-valued functions were given in.A double sequenceθ2=kr,ju is named lacunary sequence if there exist two increasing sequences of integers ku and ju such that(7)k0=0,hr=kr−kr−1⟶∞,j0=0,h¯u=ju−ju−1⟶∞,r,u⟶∞.We use the following symbols in the sequel:(8)kru=krju,hru=hrh¯u,Iru=k,j:kr−1<k≤krandju−1<j≤ju,qr=krkr−1,q¯u=juju−1,qur=quq¯r.In the paper, byθ2=kr,ju, 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 ofα-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.


## 2. Main Results

In the section, we presume thath¯:a,b⟶Fℝ and h¯mn:a,b⟶Fℝ 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.Definition 1.
LetT⊂ℕ×ℕ and r,u∈ℕ. δθ2ruT is called the ruth partial lacunary density of T if(9)δθ2ruT=T∩Iruhru.
The numberδθ2T is named the lacunary density or θ2-density of T if(10)δθ2T=limr,u⟶∞1hruk,j∈Iru:k,j∈Ti.e.,δθ2T=limr,u⟶∞δθ2ruTexists. Additionally,(11)Λ2=T⊂ℕ×ℕ:δθ2T=0is named to be zero density set.Definition 2.
We call that a DSFVFh¯mn is pointwise lacunary statistically convergent to FVF h¯ on a,b, indicated by(12)pSθ2−limh¯mnz=h¯zorh¯mn⟶pSθ2h¯,if for every z∈a,b and for all ε>0 there exists Tz∈Λ2 such that, for all m,n∈ℕ×ℕ∖Tz, we have(13)Dh¯mnz,h¯z<ε.
It is obvious thath¯mn⟶pSθ2h¯ if for every z∈a,b and for each ε>0(14)δθ2m,n∈ℕ×ℕ:Dh¯mnz,h¯z≥ε=0.Definition 3.
We call that a DSFVFh¯mn is uniformly lacunary statistically convergent to FVF h¯ on a,b, denoted by uSθ2−limh¯mnz=h¯z on a,b, or h¯mn⇉uSθ2h¯, if for each ε>0 there exists T∈Λ2 such that, for all m,n∈ℕ×ℕ∖T, we obtain(15)Dh¯mnz,h¯z<ε,which provides for all z∈a,b. It is obvious that h¯mn⇉uSθ2h¯ if, for all ε>0,(16)δθ2m,n∈ℕ×ℕ:Dh¯mnz,h¯z≥ε=0,for all z∈a,b.Remark 1.
Ifh¯mn⇉uSθ2h¯, then h¯mn⟶pSθ2h¯.Remark 2.
h¯mn⇉uSθ2h¯ iff supz∈a,bDh¯mnz,h¯z⟶pSθ20.Proof.
Assume thatsupz∈a,bDh¯mnz,h¯z⟶pSθ20. Then, for each z∈a,b and for every ε>0, we have Dh¯mnz,h¯z≤supz∈a,bDh¯mnz,h¯z so that(17)m,n∈ℕ×ℕ:Dh¯mnz,h¯z≥ε,∀z∈a,b⊆m,n∈ℕ×ℕ:supz∈a,bDh¯mnz,h¯z≥ε.
Hence,h¯mn⇉uSθ2h¯.
Next, suppose thath¯mn⇉uSθ2h¯ on a,b. For every ε>0, we write(18)A1=m,n∈ℕ×ℕ:Dh¯mnz,h¯z≥ε,∀z∈a,b,A2=m,n∈ℕ×ℕ:supz∈a,bDh¯mnz,h¯z≥ε.
Ifm,n∈ℕ×ℕ∖A1, then Dh¯mnz,h¯z<ε, ∀z∈a,b which implies(19)supz∈a,bDh¯mnz,h¯z<ε,as ε is arbitrary. This gives that m,n∈ℕ×ℕ∖A2. Therefore, we obtain(20)ℕ×ℕ∖A1⊆ℕ×ℕ∖A2⇒A2⊆A1,which gives δθ2A2≤δθ2A1=0. Hence, supz∈a,bDh¯mnz,h¯z⟶pSθ20.Theorem 1.
Presume that the sequence of FVFh¯mn⟶pSθ2h¯ on a,b, where h¯mn is equi-continuous on a,b; then, h¯ is continuous and h¯mn⇉uSθ2h¯ on a,b.Proof.
First, we demonstrate thath¯mn is continuous. Let ε>0 and z0∈a,b. By the equi-continuity of h¯mn, then there exists γ>0 such that(21)Dh¯mnz,h¯mnz0<ε3,for any m,n∈ℕ×ℕ and z∈z0−γ,z0+γ. For some z∈z0−γ,z0+γ, since h¯mn⟶pSθ2h¯, the set(22)m,n∈ℕ×ℕ:Dh¯mnz0,h¯z0≥ε3∪m,n∈ℕ×ℕ:Dh¯mnz,h¯z≥ε3∈Λ2.
So, there existsm,n∈ℕ×ℕ such that(23)Dh¯mnz0,h¯z0<ε3,Dh¯mnz,h¯z<ε3.
We obtain(24)Dh¯z0,h¯z≤Dh¯z0,h¯mnz0+Dh¯mnz0,h¯mnz+Dh¯mnz,h¯z<ε3+ε3+ε3=ε.
Therefore, we proved the continuity ofh¯.
At the moment, we will show thath¯mn⇉uSθ2h¯ on a,b. Let ε>0. Since h¯ is continuous on a,b, it provides that h¯ is uniformly continuous and h¯mn is equi-uniformly continuous on a,b. So, pick γ>0 such that, for any z,z′∈a,b and z−z′<γ, we have(25)Dh¯mnz,h¯mnz′<ε3,Dh¯z,h¯z′<ε3.
Using the finite covering theorem, select finite open coverings(26)z1−γ,z1+γ,z2−γ,z2+γ,…,zr−γ,zr+γ,from the cover of a,b. Using h¯mn⟶pSθ2h¯, there exists a set Mzi∈Λ2 such that(27)Dh¯mnzi,h¯zi<ε3,for all m,n∉Mzi and i∈1,1,2,2…,r,u. Let m,n∉Mzi and z∈a,b. Hence, z∈zi−γ,zi+γ for some i∈1,1,2,2…,r,u. As a result(28)Dh¯mnz,h¯z≤Dh¯mnz,h¯mnzi+Dh¯mnzi,h¯zi+Dh¯zi,h¯z<ε3+ε3+ε3=ε,which gives that h¯mn⇉uSθ2h¯ on a,b.Definition 4.
We call that a DSFVFh¯mn is equi-lacunary statistically convergent to FVF h¯, denoted by h¯mn⟶eSθ2h¯, if, for given ε>0,(29)Gru,ε=δθ2rum,n∈ℕ×ℕ:Dh¯mnz,h¯z≥ε,with regards to z∈a,b is uniformly convergent to zero function. Thus, h¯mn⟶eSθ2h¯ iff for all ε,β>0, ∃k,l∈ℕ, for each r≥k,u≥l, for all z∈a,b,(30)δθ2rum,n∈ℕ×ℕ:Dh¯mnz,h¯z≥ε<β.
Note that, by monotonicity ofδθ2ru, we can also utilize β=ε.Remark 3.
It is obvious thath¯mn⟶pSθ2h¯ iff for every z∈Y and for every ε,β>0∃k,l∈ℕ, for all r≥k,u≥l,(31)δθ2rum,n∈ℕ×ℕ:Dh¯mnz,h¯z≥ε<β.
In this instance, we may takeβ=ε. Obviously, h¯mn⟶eSθ2h¯ implies h¯mn⟶pSθ2h¯. Furthermore, we see that h¯mn⇉uSθ2h¯ indicates h¯mn⟶eSθ2h¯.Corollary 1.
Leth¯mn be a DSFVF and h¯ be a FVF on a,b. Then, h¯mn⇉uSθ2h¯⇒h¯mn⟶eSθ2h¯⇒h¯mn⟶pSθ2h¯. The converse implications do not hold in general.Proof.
Proving the above result, we can consider the examples as given below.Example 1.
Consider DSFVFh¯mn is defined by h¯mnz=e−mnz¯ for z∈0,1. Then, we get Dh¯mnz,0¯=e−mnz for each m,n∈ℕ×ℕ and z∈0,1. Therefore, h¯mn converges pointwise to h¯=0¯ and so h¯mn⟶pSθ20¯. However, for each k,l∈ℕ×ℕ, we consider k,l∈mn,2mn−1. Therefore, for all z∈0,1/2mn−1, we have(32)Dh¯klz,0¯=e−mnz≥e−2mn−1z≥1easz∈0,12mn−1≥13.
Thus, for allz∈0,1/2mn−1, one obtains(33)Amnz=δθ2k,l∈mn,2mn−1:Dh¯klz,0¯≥13⟶1≠0.
Hence,h¯mn is not equi-lacunary statistically convergent to 0¯ on 0,1.Example 2.
Let us define a DSFVFh¯mn by(34)h¯mnz=m2n2z1+m3n3z2¯,forz∈1mn+2,1mn+1,0¯,otherwise.
Then, for everyz∈0,1, we have(35)δθ2m,n∈ℕ×ℕ:h¯mnz≠0¯=0.
Therefore, for everyε>0,(36)1hrum,n∈Iru:Dh¯mnz,0¯≥ε≤1hrum,n∈Iru:h¯mnz≠0¯∩1,1,2,2,…,m,n≤1hru⟶0.
This yields thath¯mn is pointwise lacunary statistically convergent to h¯=0¯. However,(37)supz∈0,1Dh¯mnz,0¯=supz∈0,1m2n2z1+m3n3z2=supz∈0,111/m2n2z+mnz≠0,for all m,n∈ℕ and h¯mn is not uniformly lacunary statistically convergent to h¯=0¯ on 0,1.Theorem 2.
A of SFVMF h¯mn is uniformly lacunary statistically convergent to FVMF h¯ iff h¯mnzα is uniformly lacunary statistically convergent to h¯zα uniformly with regards to α and z.Proof.
Selectε>0. Given h¯mn⇉uSθ2h¯, there exists M∈Λ2 such that Dh¯mnz,h¯z<ε, for any m,n∈ℕ×ℕ∖M and z∈a,b, i.e.,(38)supα∈0,1maxh¯mnzα−−h¯α−z,h¯mnzα+−h¯α+z<ε.
That is, there are(39)h¯mnzα−−h¯α−z<ε,h¯mnzα+−h¯α+z<ε,for any m,n∈ℕ×ℕ∖M and z∈a,b. Moreover,(40)h¯mnzα=h¯mnzα−,h¯mnzα+,h¯zα=h¯α−z,h¯α+z.
Consequently, we geth¯mnzα is uniformly lacunary statistically convergent to h¯zα uniformly with regards to α and z.
Conversely, for anyα∈0,1 and for any z∈a,b, h¯mnzα is uniformly lacunary statistically convergent to h¯zα with regards to α and z. Thus, for given ε>0, there exists M1∈Λ2 such that(41)h¯mnzα−−h¯α−z<ε,for all m,n∈ℕ×ℕ∖M1, z∈a,b, and any α∈0,1. Additionally, we see that, for given ε>0, there exists M2∈Λ2 such that(42)h¯mnzα+−h¯α+z<ε,for all m,n∈ℕ×ℕ∖M2, z∈a,b and any α∈0,1. Take M=M1∪M2∈Λ2. We get(43)h¯mnzα−−h¯α−z<ε,h¯mzα+−h¯α+z<ε,for all m,n∈ℕ×ℕ∖M, z∈a,b, and any α∈0,1. As a result, we have(44)supα∈0,1maxh¯mnzα−−h¯α−z,h¯mnzα+−h¯α+z<ε,that is,(45)Dh¯mnz,h¯z<ε.
This concludes the proof.Theorem 3.
Leth¯ FVMF and h¯mn SFVMF. Fix z0∈a,b if h¯mn⟶eSθ2h¯ on a,b and all h¯mn are continuous on z0, then h¯ is continuous on z0.Proof.
Letε>0. h¯mn⟶eSθ2h¯, and we can identify numbers r,u∈ℕ such that, for all z∈a,b,(46)δθ2rum,n∈ℕ×ℕ:Dh¯mnz,h¯z≥ε3<12.
LetKz=m,n∈ℕ×ℕ:Dh¯mnz,h¯z<ε/3, z∈a,b. Therefore, δθ2ruKz>1/2 for all z∈a,b. Via the continuity of h¯11,h¯22,…,h¯ru at z0, there is a neighborhood z0−ζ,z0+ζ of z0 such that(47)Dh¯iz,h¯iz0<ε3,for all i=1,1,2,2…,r,u and z∈z0−ζ,z0+ζ. Fix z∈z0−ζ,z0+ζ. Since δθ2ruKz>1/2 and δθ2ruKz0>1/2, we find p∈Kz∩Kz0. Thus,(48)Dh¯z,h¯z0≤Dh¯z,h¯pz+Dh¯pz,h¯pz0+Dh¯pz0,h¯z0<ε3+ε3+ε3=ε.
Thus, we obtain(49)Dh¯z,h¯z0<ε,for all z∈Uz0,ζ, i.e., h¯ is continuous on z0.Theorem 4.
A of SFVMF h¯mn is equi-lacunary statistically convergent to FVMF h¯ iff h¯mnzα is equi-lacunary statistically convergent to h¯zα uniformly for any α∈0,1 and any z∈a,b.Proof.
h¯mn⟶eSθh¯ 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(50)δθ2rum,n∈ℕ×ℕ:Dh¯mnz,h¯z≥ε<σ.
So, for anyα∈0,1, we obtain(51)δθ2rum,n∈ℕ×ℕ:supα∈0,1maxh¯mnzα−−h¯α−z,h¯mnzα+−h¯α+z≥ε<σ.
Therefore, for anyα∈0,1, we acquire(52)δθ2rum,n∈ℕ×ℕ:h¯mnzα−−h¯α−z≥ε<σ,δθ2rum,n∈ℕ×ℕ:h¯mnzα+−h¯α+z≥ε<σ.
Then, emphasize that(53)h¯mnzα=h¯mnzα−,h¯mnzα+,h¯zα=h¯α−z,h¯α+z.
Hence,h¯mnzα is uniformly lacunary statistically convergent to h¯zα for any α∈0,1 and any z∈a,b.
Conversely, letε>0 and σ>0, and there exists k1,l1∈ℕ such that(54)δθ2rum,n∈ℕ×ℕ:h¯mnzα−−h¯α−z≥ε<σ,for all r≥k1, u≥l1 and any z∈a,b and for any α∈0,1. Similarly, there exists k2,l2∈ℕ such that(55)δθ2rum,n∈ℕ×ℕ:h¯mnzα+−h¯α+z≥ε<σ,for all r≥k2 and u≥l2 and any z∈a,b and for any α∈0,1. Then, select k=maxk1,k2 and l=maxl1,l2. We obtain(56)δθ2rum,n∈ℕ×ℕ:h¯mnzα−−h¯α−z≥ε<σ,δθ2rum,n∈ℕ×ℕ:h¯mnzα+−h¯α+z≥ε<σ,for all r≥k and u≥l and any z∈a,b and for any α∈0,1. Thus, we have(57)δθ2rum,n∈ℕ×ℕ:supα∈0,1maxh¯mnzα−−h¯α−z,h¯mnzα+−h¯α+z≥ε<σ,that is,(58)δθ2rum,n∈ℕ×ℕ:Dh¯mnz,h¯z≥ε<σ.
This concludes the proof.
The next result is the lacunary statistical form of Egorov’s theorem for the DSFVF.Theorem 5.
Take measurable spaceΩ,ℳ,μ. Presume that h¯ and h¯mn are measurable and identified almost everywhere on Ω. Suppose also that h¯mn⟶pSθ2h¯ almost everywhere on Ω. Then, for every ε>0, there exists A⊂ℳ such that μΩ\A<ε and h¯mn|A⟶eSθ2h¯|A on A.Proof.
We presume thath¯ and h¯mn are defined everywhere on Ω and also suppose that h¯mnz⟶pSθ2h¯z for all z∈Ω. Now, for any fix σ,r,u∈ℕ, consider that the set(59)P=z∈Ω:δθ2rum,n∈ℕ×ℕ:Dh¯mnz,h¯z≥1σ<1σ,is measurable. Then, the function Φmnz=Dh¯mnz,h¯z, z∈Ω, is measurable. Let Hmn=Φmn−11/σ,∞. For every z∈Ω, we have z∈P iff(60)1hru∑m=1r∑n=1uχHmnz<1σ.
Since the function(61)h=1hru∑m=1r∑n=1uχHmnzis measurable, so we have P=h−1−∞,1/σ. For k,l∈ℕ, one writes(62)Ψσ,k,l=z∈Ω:∀r≥k,u≥l,δθ2rum,n∈ℕ×ℕ:Dh¯mnz,h¯z≥1σ<1σ.
From the previous observation, we conclude thatΨσ,k,l is measurable. Also, we obtain(63)Ψσ,k,l⊂Ψσ,k+1,l+1,∀k,l∈ℕ,Ω=∪k=1∞∪l=1∞Ψσ,k,l.
As a result,μΩ=limm,n⟶∞μΨσ,k,l. Take ε>0. For each k,l∈ℕ, select kσ,lσ∈ℕ such that μΩ∖Ψσ,kσ,lσ<ε/2σ. Set(64)T0=∪σ=1∞Ω∖Ψσ,kσ,lσ.
Then, we have(65)μT0≤∑σ=1∞μΩ∖Ψσ,kσ,lσ<ε.
Let(66)T=Ω∖T0=∩σ=1∞Ψσ,kσ,lσ.
Thus,μΩ∖T=μT0<ε. Hence, we get ∀σ∈ℕ,∀r≥kσ,u≥lσ,∀z∈T,(67)δθ2rum,n∈ℕ×ℕ:Dh¯mnz,h¯z≥1σ<1σ.
This gives thath¯mn|A⟶eSθ2h¯|A on A.Now, we give the concepts of inner and outer lacunary statistical convergence in measure of DSFVF and demonstrate the equivalence of these concepts. For our convenience, we use FVMF and DSFVMF in place of fuzzy-valued function and double sequence of fuzzy-valued measurable function, respectively.Definition 5.
Take measurable spaceΩ,ℳ,μ. Presume that the set ℒ0 of all FVMF defined almost everywhere on Ω. Take h¯mn and h¯ in ℒ0. The outer lacunary statistical convergence in measure of a DSFVMF h¯mn to a FVMF h¯ is defined by(68)δθ2rum,n∈ℕ×ℕ:μz∈Ω:Dh¯mnz,h¯z≥η≥ζ⟶0,ifr,u⟶∞,for η, ζ>0. It is denoted by h¯mn⟶δθ2,μh¯. If we take (68) as follows, we get the inner statistical convergence in measure of a DSFVMF h¯mn to a FVMF h¯:(69)μz∈Ω:δθ2rum,n∈ℕ×ℕ:Dh¯mnz,h¯z≥η≥ζ⟶0,ifr,u⟶∞.
It is demonstrated byh¯mn⟶μ,δθ2h¯.Theorem 6.
Take measurable spaceΩ,ℳ,μ. Presume h¯mn and h¯ in ℒ0:(i)
Ifh¯mn⟶δθ2,μh¯, then h¯mn⟶μ,δθ2h¯(ii)
Ifh¯mn⟶μ,δθ2h¯, then h¯mn⟶δθ2,μh¯, provided μΩ<∞Proof.
Sinceδθ2ru:P1⟶0,1r,u∈ℕ is a probability measure, we can think the product measure μ×δθ2ru on the product algebra ℳ⊗P1 of subsets of Ω×ℕ×ℕ. For fix η>0, we obtain(70)Sη=z,m,n∈Ω×ℕ×ℕ:Dh¯mnz,h¯z≥η.
Define a functionΦ:Ω×ℕ×ℕ⟶ℝ as(71)Φz,m,n=Dh¯mnz,h¯z,z,m,n∈Ω×ℕ×ℕ,is ℳ⊗P1-measurable. Therefore, we get Sη∈ℳ⊗P1. For any K⊂Ω×ℕ×ℕ, one writes(72)Kz=m,n∈ℕ×ℕ:z,m,n∈K,ifz∈Ω,Km,n=z∈Ω:z,m,n∈K,ifm,n∈ℕ×ℕ.(i)
In order to acquire this, we have to show that(73)∀ε,q>0,∃r0,u0∈ℕ,∀r≥r0,u≥u0,μz∈Ω:δθ2ruSηz≥q<ε.Fixq>0 and ε>0. Since h¯mn⟶δθ2,μh¯, one can find r0,u0∈ℕ such that r≥r0,u≥u0, one get the following:(74)δθ2rum,n∈ℕ×ℕ:μSηm,n≥1<q2,(75)δθ2rum,n∈ℕ×ℕ:μSηm,n≥qε4<qε4.Assume that(76)P=m,n∈ℕ×ℕ:μSηm,n<1.Then, we have from situation (74) that(77)δθ2ruℕ×ℕ∖P<q2∀r≥r0,u≥u0.Hence, for all∀r≥r0andu≥u0, one acquires(78)μz∈Ω:δθ2ruSηz≥q≤μz∈Ω:δθ2ruSηz∩P≥q2+μz∈Ω:δθ2ruSηz∖P≥q2≤μz∈Ω:δθ2ruSηz∩P≥q2.TakeSη∗=Sη∩Ω×P. Therefore, we have(79)Sη∗z=Sηz∩Pz∈Ω,Sη∗m,n=Sηm,nm,n∈P.To acquire relation (73), it is enough to demonstrate that(80)∀r≥r0,u≥u0,μz∈Ω:δθ2ruSη∗z≥q2<ε.For the setSη∗⊂Ω×P and for every fix r,u∈ℕ, we can utilize the Fubini theorem for the characteristic function of Sη∗ of the finite measure μ×δθ2ru. Actually,(81)Sη∗=∪m,n∈Pm,n×Sηm,n,where(82)μSηm,n<1∀m,n∈P,δθ2rum,n=0∀m>r,n>u.Thus,(83)∬PμSη∗m,ndmdn=μ×δθ2ruSη∗=∫Ωδθ2ruSη∗zdz.Supposer0 and u0∈ℕ such that r≥r0andu≥u0, one obtains(84)qε2>qε4+δθ2rum,n∈ℕ×ℕ:μSηm,n≥qε4≥∬m,n∈P:μSηm,n<qε/4μSqm,ndmdn+∬m,n∈P:μSηm,n≥qε/41⋅dmdn≥∬PμSηm,ndmdn=∫∫PμSη∗m,ndmdn=∫Ωδθ2ruSη∗zdz≥∫z∈Ω:μSη∗z≥q/2δθ2ruSη∗zdz≥q2μz∈Ω:δθ2ruSη∗z≥q2,which gives that strict inequality (80) is valid.(ii)
Presume thatμΩ<∞. Fix η>0. To verify our result, we need to denote that(85)∀ε,q>0,∃r0,u0∈ℕ,∀r≥r0,u≥u0,δθ2rum,n∈ℕ×ℕ:μSηm,n≥q<ε.
Letε>0 and q>0 be given. Since h¯mn⟶μ,δθ2h¯, one can find r0,u0∈ℕ such that, for all r≥r0 and u≥u0, we have(86)μz∈Ω:δθ2ruSηz≥qε2μΩ<qε2.
By considering the Fubini theorem for the function ofSη⊂Ω×ℕ×ℕ, we obtain(87)∫Ωδθ2ruSηzdz=μ×δθ2ruSη=∬ℕ×ℕμSηm,ndmdn.
Supposingr0 and u0 such that, for all r≥r0andu≥u0, we have(88)qε>qεμΩ2μΩ+μz∈Ω:δθ2ruSηz≥qε2μΩ≥∫z∈Ω:δθ2ruSηz<qε/2μΩδθ2ruSηzdz+∫z∈Ω:δθ2ruSηz>qε/2μΩ1dz≥∫Ωδθ2ruSηzdz=∬ℕ×ℕμSηm,ndmdn≥∬m,n∈ℕ×ℕ:μSηm,n≥qμSηm,ndmdn≥qδθ2rum,n∈ℕ×ℕ:μSηm,n≥q.
This concludes the proof.Theorem6 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.Definition 6.
DSFVMFh¯mn is called to be lacunary statistical convergent in measure (shortly, LSCM) to a FVMF h¯, in symbol, h¯mn⟶μSθ2h¯, if μz∈Ω:Dh¯mnz,h¯z≥q is lacunary statistically convergent to zero for any q>0 and all m,n∈ℕ×ℕ. This concept can be denoted by the following formula:(89)∀η>0,∀q>0,m,n∈ℕ×ℕ:μz∈Ω:Dh¯mnz,h¯z≥q>η∈Λ2.
Here, we can takeη=q or q=1/r, r∈ℕ.Proposition 1.
Take measurable spaceΩ,ℳ,μ. Suppose that h¯mn and h¯ in ℒ0. Then, h¯mn⇉uSθ2h¯⇒h¯mn⟶μSθ2h¯.Proof.
We presume thath¯mn⇉uSθ2h¯. Take q>0. In this way, there is a set W∈Λ2 such that(90)Dh¯mnz,h¯z<q,∀m,n∉W,z∈Ω.
Thus, we obtain(91)m,n∈ℕ×ℕ:μz∈Ω:Dh¯mnz,h¯z≥q>q⊂m,n∈ℕ×ℕ:μz∈Ω:Dh¯mnz,h¯z≥q≠∅⊂W∈Λ2.
This indicates thath¯mnz⟶μSθ2h¯z.Theorem 7.
Leth¯ be a FVMF such that, for each z∈Ω, h¯mnz⟶pSθ2h¯z. Then, given ε>0 and δ>0, there is a measurable set W⊂Ω with μW<δ and integers m0andn0 such that(92)Dh¯mnz,h¯z<ε,∀z∉Wandm≥m0,n≥n0.Proof.
Let(93)Zmn=z∈Ω:Dh¯mnz,h¯z≥ε,and set(94)Ωm0,n0=∪m=m0∞∪n=n0∞Zmn=z∈Ω:Dh¯mnz,h¯z≥ε,for somem≥m0,n≥n0.
We have thatΩm0+1,n0+1⊂Ωm0,n0, and for each z∈Ω, there is some Ωm0,n0 to which z does not belong, since h¯mnz⟶h¯z. Thus, we get ∩m0∩n0Ωm0,n0=∅, so we have limμΩm0,n0=0. Hence, given δ>0, ∃m0,n0∈ℕ so that μΩm0,n0<δ, i.e.,(95)μz∈Ω:Dh¯mnz,h¯z≥ε,for somem≥m0,n≥n0<δ.
If we writeW for this Ωm0,n0, then μW<δ and(96)W=z∈Ω:Dh¯mnz,h¯z≥ε,for somem≥m0,n≥n0.Theorem 8.
If DSFVMFh¯mn pointwise lacunary statistically convergent to a FVMF h¯ almost everywhere on Ω, then h¯mn⟶μSθ2h¯.Proof.
Assume thath¯mnz⟶pSθ2h¯z almost everywhere on Ω. We have from Theorem 6 that h¯mn⟶μ,δθ2h¯ is the same as h¯mnz⟶μSθ2h¯z. So, to demonstrate our result, we have to show that h¯mnz⟶μ,δθ2h¯z. Presume that ε>0 and q>0. It is clear from Theorem 5 that A⊂ℳ such that h¯mn|A⟶eSθ2h¯|A and μΩ\A<ε. Select indexes k and l such that(97)δθ2rum,n∈ℕ×ℕ:Dh¯mnz,h¯z≥q<q,for every r≥k,u≥l, and y∈A. Thus, for all r≥k and u≥l, we obtain(98)z∈Ω:δθ2rum,n∈ℕ×ℕ:Dh¯mnz,h¯z≥q≥q⊂Ω\A.
Therefore, for everyr≥k and u≥l,(99)μz∈Ω:δθ2rum,n∈ℕ×ℕ:Dh¯mnz,h¯z≥q≥q<ε,as desired.Theorem 9.
Ifh¯mnz⟶pSθ2h¯z almost everywhere on Ω, then h¯mnz⟶μSθ2h¯z.Proof.
Takeq,ε>0. By considering Theorem 5, there is an A⊂ℳ such that h¯mn|A⟶eSθ2h¯|A and μΩ\A<ε. Consider k,l∈ℕ such that(100)δθ2rum,n∈ℕ×ℕ:Dh¯mnz,h¯z≥q<q∀r≥k,u≥landz∈A,which yields(101)z∈Ω:δθ2rum,n∈ℕ×ℕ:Dh¯mnz,h¯z≥q≥q⊂Ω\A∀r≥k,u≥l.
Therefore, one obtains(102)μz∈Ω:δθ2rum,n∈ℕ×ℕ:Dh¯mnz,h¯z≥q≥q<ε.Corollary 2.
Ifh¯mnz⟶μSθ2h¯z, then ∃ a subsequence h¯mknl of h¯mn such that h¯mknlz⟶pSθ2h¯z almost everywhere on Ω.Proof.
Presume thath¯mnz⟶μSθ2h¯z, so any subsequence h¯mknl of h¯mn also lacunary statistically convergent in measure to h¯. In this way, h¯mn possesses a subsequence that lacunary statistically convergent in measure to h¯ almost everywhere on Ω. This gives that h¯mknlz⟶pSθ2h¯z almost everywhere on Ω.Definition 7.
Take finite measurable spaceΩ,ℳ,μ. Assume that h¯mnα+, h¯mnα−, h¯α+, and h¯α−∈ℒ0. The double sequence h¯mnzα is uniformly lacunary statistically convergent in measure (briefly, we can write ULSCM) to h¯zα with regards to α if(103)μz∈Ω:supα∈0,1h¯mnzα+−h¯zα+≥q,μz∈Ω:supα∈0,1h¯mnzα−−h¯zα−≥q,both lacunary statistically convergent measure to zero for each q>0. Notice that this concept is given by the following formula:(104)∀η>0,∀q>0,m,n∈ℕ×ℕ:μz∈Ω:supα∈0,1h¯mnzα+−h¯zα+≥q≥η∈Λ2,∀η>0,∀q>0,m,n∈ℕ×ℕ:μz∈Ω:supα∈0,1h¯mnzα−−h¯zα−≥q≥η∈Λ2.
In that case, we can writeη=q or q=1/r, r∈ℕ.Theorem 10.
DSFVMFh¯mn is LSCM to FVMF h¯ iff h¯mnzα is ULSCM to h¯zα with regards to α.Proof.
Suppose thath¯mn is LSCM to h¯. Then,(105)μz∈Ω:Dh¯mnz,h¯z≥q,is lacunary statistically convergent to zero for each q>0, i.e.,(106)m,n∈ℕ×ℕ:μz∈Ω:Dh¯mnz,h¯z≥q≥q∈Λ2.
Thus,(107)m,n∈ℕ×ℕ:μz∈Ω:supα∈0,1maxh¯mnzα−−h¯zα−,h¯mnzα+−h¯zα+≥q≥q∈Λ2.
For everyq>0, one obtains(108)m,n∈ℕ×ℕ:μz∈Ω:supα∈0,1h¯mnzα−−h¯zα−≥q≥q∈Λ2,m,n∈ℕ×ℕ:μz∈Ω:supα∈0,1h¯mnzα+−h¯zα+≥q≥q∈Λ2,which gives that(109)μz∈Ω:supα∈0,1h¯mnzα−−h¯zα−≥q⟶μSθ20,μz∈Ω:supα∈0,1h¯mnzα+−h¯zα+≥q⟶μSθ20.
Hence,h¯mnzα is ULSCM to h¯zα with regards to α.
Next, we presume thath¯mnzα is ULSCM to h¯zα with regards to α. Then, for every q>0, one has(110)μz∈Ω:supα∈0,1h¯mnzα−−h¯zα−≥q⟶μSθ20,μz∈Ω:supα∈0,1h¯mnzα+−h¯zα+≥q⟶μSθ20.
Thus,(111)m,n∈ℕ×ℕ:μz∈Ω:supα∈0,1h¯mnzα−−h¯zα−≥q≥q∈Λ2,m,n∈ℕ×ℕ:μz∈Ω:supα∈0,1h¯mnzα+−h¯zα+≥q≥q∈Λ2.
From the last two relations, we obtain(112)m,n∈ℕ×ℕ:μz∈Ω:supα∈0,1maxh¯mnzα−−h¯zα−,h¯mnzα+−h¯zα+≥q≥q∈Λ2,which gives that(113)m,n∈ℕ×ℕ:μz∈Ω:Dh¯mnz,h¯z≥q≥q∈Λ2.
It means thath¯mn is LSCM to FVMF h¯.


## 3. 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 fuzzy-valued functions and the sequence ofα-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.



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