# Lacunary Statistical Limit and Cluster Points of Generalized Difference Sequences of Fuzzy Numbers

Authors: Pankaj Kumar; Vijay Kumar; S. S. Bhatia
Journal: Advances in Fuzzy Systems
 (2012)

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## Abstract

The aim of present work is to introduce and study lacunary statistical limit and lacunary statistical cluster points for generalized difference sequences of fuzzy numbers. Some inclusion relations among the sets of ordinary limit points, statistical limit points, statistical cluster points, lacunary statistical limit points, and lacunary statistical cluster points for these type of sequences are obtained.


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## Body

## 1. Introduction

The notion of statistical convergence of sequences of numbers was introduced by Fast [1] and Schoenberg [2] independently and latter discussed in [3–6], and so forth. In 1993, Fridy and Orhan [7] presented an interesting generalization of statistical convergence with the help of a lacunary sequence and called it lacunary statistical convergence or Sθ-convergence. Demirci [8] defined Sθ-limit and cluster points of number sequences and obtained some interesting results analogous to [4]. In past years, statistical convergence has also become an interesting area of research for sequences of fuzzy numbers. The credit goes to Nuray and Savaş [9] who first introduced statistical convergence of sequences of fuzzy numbers. After their pioneer work, many authors have made their contribution to study different generalizations of statistical convergence for sequences of fuzzy numbers (see [10–13], etc.).Quite recently, statistical convergence of sequences of fuzzy numbers is studied with the help of the difference operatorΔ. For instance, Bilgin [14] introduced strongly Δ-summable and Δ-statistical convergence of sequences of fuzzy numbers. Işik [15] studied some notions of generalized difference sequences of numbers. In 2006, Altin et al. [16] united lacunary sequences to introduce the concept of lacunary statistical convergence of generalized difference sequences of fuzzy numbers and obtained some interesting results. Some more work in this direction can be found in [17–19]. In present work, we continue with this study and introduce the concepts of lacunary statistical limit and cluster points of generalized difference sequences of fuzzy numbers. We obtain some relations among the sets of ordinary limit, points, lacunary statistical limit, and cluster points for these type of sequences.


## 2. Background and Preliminaries

We begin with the following terminology on fuzzy numbers. Given any intervalA, we shall denote its end points by A_,A¯ and by D the set of all closed bounded intervals on real line ℝ, that is, D={A⊂ℝ:A=[A_,A¯]}. For A,B∈D we define A≤B if and only if A_≤B_ and A¯≤B¯. Moreover, the distance function d defined by d(A,B)=max{|A_-B_|,|A¯-B¯|} is a Hausdorff metric on D and (D,d) is a complete metric space. Also ≤ is a partial order on D.A fuzzy number is a functionX from ℝ to [0,1] which is satisfying the following conditions: (i) X is normal, that is, there exists x0∈ℝ such that X(x0)=1; (ii) X is fuzzy convex, that is, for any x,y∈ℝ and λ∈[0,1], X(λx+(1-λ)y)≥min{X(x),X(y)}; (iii) X is upper semicontinuous; and (iv) the closure of the set {x∈ℝ:X(x)>0} denoted by X0 is compact.Properties (i)–(iv) imply that for eachα∈(0,1], the α-level set, Xα={x∈ℝ:X(x)≥α}=[X_α,X¯α], is a nonempty compact convex subset of ℝ. Let L(ℝ) denote the set of all fuzzy numbers. The linear structure of L(ℝ) induces an addition X+Y and a scalar multiplication λX in terms of α-level sets by
(1)[X+Y]α=[X]α+[Y]α,[λX]α=λ[X]α(X,Y∈L(R),λ∈R)
for each α∈[0,1]. Define a map d¯:L(ℝ)×L(ℝ)→ℝ by
(2)d¯(X,Y)=sup⁡α∈[0,1]⁡d(Xα,Yα).
Puri and Ralescu [20] proved that (L(ℝ),d¯) is a complete metric space. Also the ordered structure on L(ℝ) is defined as follows. For X,Y∈L(ℝ), we define X≤Y if and only if X_α≤Y_α and X¯α≤Y¯α for each α∈[0,1]. We say that X<Y if X≤Y and there exist α0∈[0,1] such that X_α0<Y_α0 or X¯α0<Y¯α0. The fuzzy numbers X and Y are said to be incomparable if neither X≤Y nor Y≤X.We next recall some definitions and results which form the base for present study. For any setK⊆ℕ, let Kn denote the set {k∈K:k≤n} and |Kn| denote the number of elements in Kn. The natural density δ of K is defined by δ(K)=limnn-1|Kn|. The natural density may not exist for each set K. But the upper density δ¯ defined by δ¯(K)=limsupnn-1|Kn| always exists for each set K. Moreover, δ(K) different from zero means δ¯(K)>0. Besides that, δ(KC)=1-δ(K) and if A⊆B, then δ¯(A)≤δ¯(B).For any sequenceX=(Xk) of fuzzy numbers, we write {Xk:k∈ℕ} to denote the range of X. If (Xk(j)) is a subsequence of X and K={k(j):j∈ℕ}, then we abbreviate (Xk(j)) by (X)K. If δ(K)=0, (X)K is called a thin subsequence, otherwise if δ(K)≠0, (X)K is called nonthin subsequence of X.Forw, the set of all sequences of fuzzy numbers, the operator Δm:w→w is defined by
(3)Δ0Xk=Xk,Δ1Xk=Xk-Xk+1,⋮=⋮ΔmXk=Δ1(Δm-1Xk).Definition 1.
A sequenceX=(Xk) of fuzzy numbers is said to be Δm-statistically convergent to a fuzzy number X0, in symbol: S(Δm)-limkXk=X0, if for each ϵ>0,
(4)lim⁡n→∞⁡1n|{k∈N:d¯(ΔmXk,X0)≥ϵ}|=0.
Let S(Δm(X)) denote the set of all Δm-statistically convergent sequences of fuzzy numbers.Definition 2.
LetX=(Xk) be a sequence of fuzzy numbers. A fuzzy number X0 is said to be a statistical limit point (s.l.p) of the generalized difference sequence (ΔmXk) of fuzzy numbers provided that there is a nonthin subsequence of X that is Δm-convergent to X0.
LetΛS(Δm(X)) denote the set of all s.l.p. of the generalized difference sequence (ΔmXk) of fuzzy numbers.Definition 3.
LetX=(Xk) be a sequence of fuzzy numbers. A fuzzy number Y0 is said to be a statistical cluster point (s.c.p) of the generalized difference sequence (ΔmXk) of fuzzy numbers provided that, for each ϵ>0,
(5)lim⁡sup⁡n→∞⁡1n|{k∈N:d¯(ΔmXk,Y0)<ϵ}|>0.
Let ΓS(Δm(X)) denote the set of all s.c.p of the generalized difference sequence (ΔmXk) of fuzzy numbers.By a lacunary sequence we mean an increasing sequenceθ=(kr) of positive integers such that k0=0 and hr=kr-kr-1→∞ as r→∞. The intervals determined by θ=(kr) will be denoted by Ir=(kr-1,kr] whereas the ratio kr/kr-1 is denoted by qr. Further, a lacunary sequence θ′=(kr′) is called a lacunary refinement of the lacunary sequence θ=(kr) if {kr}⊆{kr′}.Definition 4 (see [21]).
Letθ=(kr) be a lacunary sequence. A sequence X=(Xk) of fuzzy numbers is said to be lacunary statistical convergent to a fuzzy number X0 provided that for each ϵ>0,
(6)lim⁡r→∞⁡1hr|{k∈Ir:d¯(Xk,Y0)≥ϵ}|=0.
Let Sθ denote the set of all lacunary statistically convergent sequences of fuzzy numbers.Letθ=(kr) be a lacunary sequence and X=(Xk) a sequence of fuzzy numbers. If (X)K where K={k(j):j∈ℕ} is a subsequence of X=(Xk) such that
(7)lim⁡r→∞⁡1hr|{k(j)∈Ir:j∈N}|=0,
we call (X)K a θ-thin subsequence. On the other hand, (X)K is a θ-nonthin subsequence of X provided that
(8)lim⁡sup⁡r→∞⁡1hr|{k(j)∈Ir:j∈N}|>0.Definition 5.
Letθ=(kr) be a lacunary sequence. A sequence X=(Xk) of fuzzy numbers is said to be lacunary Δm-statistically convergent to a fuzzy number X0, in symbol: Sθ(Δm)-limkXk=X0, if for each ϵ>0,
(9)lim⁡r→∞⁡1hr|{k∈Ir:d¯(ΔmXk,X0)≥ϵ}|=0.
Let Sθ(Δm(X)) denote the set of all lacunary Δm-statistically convergent sequences of fuzzy numbers.We now consider the natural definitions of statistical limit and cluster points for generalized difference sequences of fuzzy numbers with respect to lacunary sequences.


## 3. Main Results

Definition 6.
Letθ=(kr) be a lacunary sequence and X=(Xk) a sequence of fuzzy numbers. A fuzzy number X0 is said to be a lacunary statistical limit point (l.s.l.p) of the generalized difference sequence (ΔmXk) of fuzzy numbers provided that there is a θ-nonthin subsequence of X that is Δm-convergent to X0.LetΛSθ(Δm(X)) denote the set of all l.s.l.p. of the generalized difference sequence (ΔmXk) of fuzzy numbers.Definition 7.
Letθ=(kr) be a lacunary sequence and X=(Xk) a sequence of fuzzy numbers. A fuzzy number Y0 is said to be a lacunary statistical cluster point (l.s.c.p) of the generalized difference sequence (ΔmXk) of fuzzy numbers provided that, for each ϵ>0,
(10)lim⁡sup⁡r→∞⁡1hr|{k∈Ir:d¯(ΔmXk,Y0)<ϵ}|>0.
Let ΓSθ(Δm(X)) denote the set of all l.s.c.p of the generalized difference sequence (ΔmXk) of fuzzy numbers.Example 8.
Letθ=(kr) be a lacunary sequence. We define a sequence of fuzzy numbers X=(Xk) as follows. For x∈ℝ, define
(11)Xk(x)={x-k+1,ifk-1≤x≤k-x+k+1,ifk<x≤k+10,otherwise},ifkr-[hr]+1≤k≤kr;r∈Nx-5,if5≤x≤67-x,if6<x≤70,otherwise},otherwise.
Then, we obtain
(12)[Xk]α={[k-1+α,k+1-α],ifk∈[kr-[hr]+1,kr][5+α,7-α]otherwise,[ΔXk]α={[-3+α,1-2α],ifbothk,k+1∈[kr-[hr]+1,kr][k-8+2α,k-4-2α],ifk∈[kr-[hr]+1,kr]butnotk+1,[-k+3+2α,-k+7-2α],ifk+1∈[kr-[hr]+1,kr]butnotk,[-2+2α,2-2α],otherwise.
Thus, for m=1, it is clear that the sequence ΔXk has two different subsequences which converge to μ1 and μ2, respectively, where [μ1]α=[-3+α,1-2α] and [μ2]α=[-2+2α,2-2α]. Hence, if L(Δ(X)) denotes the set of ordinary limit points of (ΔXk), then L(Δ(X))={μ1,μ2}; however, ΓSθ(Δ(X))={μ2}.Theorem 9.
Letθ=(kr) be a lacunary sequence and X=(Xk) a sequence of fuzzy numbers. Then, one has ΛSθ(Δm(X))⊆ΓSθ(Δm(X)).Proof.
SupposeX0∈ΛSθ(Δm(X)). By definition, there is a θ-nonthin subsequence (Xk(j)) of X=(Xk) which is Δm-convergent to X0, and therefore we have
(13)lim⁡sup⁡r→∞⁡1hr|{k(j)∈Ir:j∈N}|=d>0.
Since, for every ϵ>0,
(14){k∈Ir:d¯(ΔmXk,X0)<ϵ}⊇{k(j)∈Ir:d¯(ΔmXk(j),X0)<ϵ},
so we have the containment
(15){k∈Ir:d¯(ΔmXk,X0)<ϵ}⊃{k(j)∈Ir:j∈N}-{k(j)∈Ir:d¯(ΔmXk(j),X0)≥ϵ}.
Now, (Xk(j)) is Δm-convergent to X0, which implies that, for every ϵ>0, {k(j)∈Ir:d¯(ΔmXk(j),X0)≥ϵ} is finite for which we have
(16)lim⁡sup⁡r→∞⁡1hr|{k(j)∈Ir:d¯(ΔmXk(j),X0)≥ϵ}|=0.
Thus from (15), we obtain
(17)lim⁡sup⁡r→∞⁡1hr|{k∈Ir:d¯(ΔmXk,X0)<ϵ}|≥lim⁡sup⁡r→∞⁡1hr|{k(j)∈Ir:j∈N}|-lim⁡sup⁡r→∞⁡1hr|{k(j)∈Ir:d¯(ΔmXk(j),X0)≥ϵ}|≥lim⁡sup⁡r→∞⁡1hr|{k(j)∈Ir:j∈N}|=d>0,
using (13) and (16). This shows that X0∈ΓSθ(Δm(X)) and therefore the result is proved.Theorem 10.
Letθ=(kr) be a lacunary sequence. Then, for any sequence X=(Xk) of fuzzy numbers, one has ΓSθ(Δm(X))⊆L(Δm(X)).Proof.
AssumeY0∈ΓSθ(Δm(X)). By definition, for each ϵ>0 we have
(18)lim⁡sup⁡r→∞⁡1hr|{k∈Ir:d¯(ΔmXk,Y0)<ϵ}|>0.
We set (X)K a θ-nonthin subsequence of X=(Xk) such that K={k(j)∈Ir:d¯(ΔmXk,Y0)<ϵ} for ϵ>0. Since limsupr→∞(1/hr)|{K}|>0, it follows that K is an infinite set. Thus we have a subsequence (X)K of X that is Δm-convergent to Y0. This shows that Y0∈L(Δm(X)). Hence ΓSθ(Δm(X))⊆L(Δm(X)).Theorem 11.
Letθ=(kr) be a lacunary sequence. If X=(Xk) and Y=(Yk) are two sequences of fuzzy numbers such that limr→∞(1/hr)|{k∈Ir:Xk≠Yk}|=0, then ΛSθ(Δm(X))=ΛSθ(Δm(Y)) and ΓSθ(Δm(X))=ΓSθ(Δm(Y)).Proof.
We prove the theorem into two parts. In the first part we prove thatΛSθ(Δm(X))=ΛSθ(Δm(Y)); however, in the second part we shall prove ΓSθ(Δm(X))=ΓSθ(Δm(Y)).
Part (i). LetX0∈ΛSθ(Δm(Y)). By definition, there is a θ-nonthin subsequence (Y)K of Y=(Yk) that is Δm-convergent to X0. Since limr→∞(1/hr)|{k∈Ir:k∈KandXk≠Yk}|=0, it follows that limsupr→∞(1/hr)|{k∈Ir:k∈KandXk=Yk}|>0. Therefore, from the later set, we can yield a θ-nonthin subsequence (X)K′ of X=(Xk) that is Δm-convergent to X0. Hence, X0∈ΛSθ(Δm(X)), and therefore we have ΛSθ(Δm(Y))⊂ΛSθ(Δm(X)). Also by symmetry one get ΛSθ(Δm(X))⊂ΛSθ(Δm(Y)). On combining we have ΛSθ(Δm(X))=ΛSθ(Δm(Y)).
Part (ii). LetY0∈ΓSθ(Δm(X)). By definition, for each ϵ>0,
(19)lim⁡sup⁡r→∞⁡1hr|{k∈Ir:d¯(ΔmXk,Y0)<ϵ}|>0.
Since Xk=Yk for all most all k, it follows that, for each ϵ>0,
(20)lim⁡sup⁡r→∞⁡1hr|{k∈Ir:d¯(ΔmYk,Y0)<ϵ}|>0.
This shows that Y0∈ΓSθ(Δm(Y)) and therefore ΓSθ(Δm(X))⊂ΓSθ(Δm(Y)). By symmetry, we see that ΓSθ(Δm(Y))⊂ΓSθ(Δm(X)), whence ΓSθ(Δm(X))=ΓSθ(Δm(Y)).Theorem 12.
Letθ=(kr) be a lacunary sequence. If X=(Xk) is a sequence of fuzzy numbers such that Sθ(Δm)-limkXk=X0, then ΛSθ(Δm(X))=ΓSθ(Δm(X))={X0}.Proof.
We prove the theorem in two parts. In the first part, we prove thatΛSθ(Δm(X))={X0} whereas in the second part we obtain ΓSθ(Δm(X))={X0}.
Part (i). Suppose thatΛSθ(Δm(X))={X0,Y0}, where X0≠Y0, that is, Y0 is a l.s.l.p. of the generalized difference sequence (ΔmXk) different from X0. Choose ϵ>0 such that 0<ϵ<d¯(X0,Y0)/2. By definition there exist two θ-nonthin subsequences (Xk(j)) and (Xl(i)) of the sequence X=(Xk) which are Δm-convergent to X0 and Y0, respectively. Since (Xl(i)) is Δm-convergent to Y0, so for each ϵ>0, {l(i)∈In:d¯(ΔmXl(i),Y0)≥ϵ} is a finite set for which
(21)lim⁡sup⁡r→∞⁡1hr|{l(i)∈Ir:d¯(ΔmXl(i),Y0)≥ϵ}|=0.
Further, we can write
(22){l(i)∈Ir:i∈N}={l(i)∈Ir:d¯(ΔmXl(i),Y0)<ϵ}∪{l(i)∈Ir:d¯(ΔmXl(i),Y0)≥ϵ},
for which we have
(23)lim⁡sup⁡r→∞⁡1hr|{l(i)∈Ir:i∈N}|=lim⁡sup⁡r→∞⁡1hr|{l(i)∈Ir:d¯(ΔmXl(i),Y0)<ϵ}|+lim⁡sup⁡r→∞⁡1hr|{l(i)∈Ir:d¯(ΔmXl(i),Y0)≥ϵ}|.
Since (Xl(i)) is nonthin, so, by use of (21), we have
(24)lim⁡sup⁡r→∞⁡1hr|{l(i)∈Ir:d¯(ΔmXl(i),Y0)<ϵ}|>0.
Since Sθ(Δm)-limkXk=X0, so for each ϵ>0(25)lim⁡r→∞⁡1hr|{k∈Ir:d¯(ΔmXk,X0)≥ϵ}|=0,
and therefore we can write
(26)lim⁡sup⁡r→∞⁡1hr|{k∈Ir:d¯(ΔmXk,X0)<ϵ}|>0.
Furthermore for 0<2ϵ<d¯(X0,Y0),
(27){l(i)∈Ir:d¯(ΔmXl(i),Y0)<ϵ}∩{k∈Ir:d¯(ΔmXk,X0)<ϵ}=∅,
which immediately gives the containment
(28){l(i)∈Ir:d¯(ΔmXl(i),Y0)<ϵ}⊂{k∈Ir:d¯(ΔmXk,X0)≥ϵ}
for which we have
(29)lim⁡sup⁡r→∞⁡1hr|{l(i)∈Ir:d¯(ΔmXl(i),Y0)<ϵ}|≤lim⁡sup⁡r→∞⁡1hr|{k∈Ir:d¯(ΔmXk,X0)≥ϵ}|=0.
As left side of (29) cannot be negative, so we must have
(30)lim⁡sup⁡r→∞⁡1hr|{l(i)∈Ir:d¯(ΔmXl(i),Y0)<ϵ}|=0.
This contradicts (24). Hence, ΛSθ(Δm(X))={X0}.
Part (ii). LetZ0 be a l.s.c.p. of the generalized difference sequence (ΔmXk) different from X0, that is, ΓSθ(Δm(X))={X0,Z0}, where X0≠Z0. Choose ϵ such that 0<ϵ<d¯(X0,Z0)/2. Since Z0 is a l.s.c.p of (ΔmXk), so for each ϵ>0 we have
(31)lim⁡sup⁡r→∞⁡1hr|{k∈Ir:d¯(ΔmXk,Z0)<ϵ}|>0.
Since {k∈Ir:d¯(ΔmXk,X0)<ϵ}∩{k∈Ir:d¯(ΔmXk,Z0)<ϵ}=∅ for every 0<ϵ<d(X0,Z0)/2, it follows that {k∈Ir:d¯(ΔmXk,X0)≥ϵ}⊇{k∈Ir:d¯(ΔmXk,Z0)<ϵ} for which we have
(32)lim⁡sup⁡r→∞⁡1hr|{k∈Ir:d¯(ΔmXk,X0)≥ϵ}|≥lim⁡sup⁡r→∞⁡1hr|{k∈Ir:d¯(ΔmXk,Z0)<ϵ}|>0
by (31), which is impossible as by (25) limsupr→∞(1/hr)|{k∈Ir:d¯(ΔmXk,X0)≥ϵ}|=0. In this way we obtained a contradiction. Hence, ΓSθ(Δm(X))={X0}.Theorem 13.
Letθ=(kr) be a lacunary sequence and X=(Xk) a sequence of fuzzy numbers. Then one has the following:(i)
ifliminfqr>1, then ΛSθ(Δm(X))⊂ΛS(Δm(X)),(ii)
iflimsupqr<∞, then ΛS(Δm(X))⊂ΛSθ(Δm(X)),(iii)
if1<liminfqr≤limsupqr<∞, then ΛS(Δm(X))=ΛSθ(Δm(X)).Proof.
(i) Supposeliminfqr>1; there exists a δ>0 such that qr>1+δ for sufficient large r, which implies that hr/kr≥δ/(1+δ). Assume that X0∈ΛSθ(Δm(X)), then there is θ-nonthin subsequence (Xk(j)) of (Xk) that is Δm-convergent to X0 and
(33)lim⁡sup⁡r→∞⁡1hr|{k(j)∈Ir:j∈N}|=d>0.
Since
(34)1kr|{k(j)≤kr:j∈N}|≥1kr|{k(j)∈Ir:j∈N}|=hrkr1hr|{k(j)∈Ir:j∈N}|≥(δ1+δ)1hr|{k(j)∈Ir:j∈N}|,
it follows by (33) that limsupr→∞(1/kr)|{k(j)≤kr:j∈ℕ}|>0. Since (Xk(j)) is already Δm-convergent to X0, so we have X0∈ΛS(Δm(X)). Hence ΛSθ(Δm(X))⊂ΛS(Δm(X)).
(ii) Iflimsupqr<∞, then there exists a real number H such that qr<H for all r. Without loss of generality, we can assume H>1 (as otherwise kr<kr-1). Now for all r,(hr/kr-1)=(kr-kr-1)/kr-1=qr-1≤H-1. Let X0∈ΛS(Δm(X)), then there is a set K={k(j):j∈ℕ} with δ(K)≠0 and limjΔmXk(j)=X0. Let Nr=|{k∈Ir:k∈K}|=|K∩Ir| and tr=Nr/hr. For any integer n satisfying kr-1<n≤kr, we can write
(35)1n|{k≤n:k∈K}|≤1kr-1|{k≤kr:k∈K}|=1kr-1{N1+N2+⋯+Nr}=1kr-1{t1h1+t2h2+⋯trhr}=1∑i=1r-1hi∑i=1r-1‍hiti+hrkr-1tr≤1∑i=1r-1hi∑i=1r-1‍hiti+(H-1)tr.
Suppose tr→0 as r→∞. Since θ is a lacunary sequence and the first part on the right side of above expression is a regular weighted mean transform of the sequence t=(tr), therefore it too tends to zero as r→∞. Since n→∞ as r→∞, it follows that δ(K)=0 which is a contradiction as δ(K)≠0. Thus limr→∞tr≠0 and therefore X0∈ΛSθ(Δm(X)). Hence ΛS(Δm(X))⊂ΛSθ(Δm(X)).
(iii) This is an immediate consequence of (i) and (ii).Theorem 14.
Letθ=(kr) be a lacunary sequence and X=(Xk) a sequence of fuzzy numbers. Then one has the following:(i)
ifliminfqr>1, then ΓSθ(Δm(X))⊂ΓS(Δm(X)),(ii)
iflimsupqr<∞, then ΓS(Δm(X))⊂ΓSθ(Δm(X)),(iii)
if1<liminfqr≤limsupqr<∞, then ΓS(Δm(X))=ΓSθ(Δm(X)).Proof.
The proof of the theorem can be obtain on the similar lines as that of the above theorem and therefore is omitted here.Theorem 15.
For any lacunary refinementθ′ of a lacunary sequence θ=(kr), ΓSθ(Δm(X))⊆ΓSθ′(Δm(X)) and ΛSθ(Δm(X))⊆ΛSθ′(Δm(X)).Proof.
Suppose eachIr of θ contains the points {kr,i′}i=1v(r) of θ′ so that kr-1<kr,1′<kr,2′<⋯<kr,v(r)′=kr, where Ir′=(kr,i-1′,kr,i′]. Note that for all r, v(r)≥1. Let (Ij*)j=1∞ be the sequence of abutting intervals Ir,i′ ordered by increasing right end points. Let Y0∈ΓSθ(Δm(X)), then for each ϵ>0,
(36)lim⁡sup⁡r→∞⁡1hr|{k∈Ir:d¯(ΔmXk,Y0)<ϵ}|>0.
As before, write hr,i′=kr,i′-kr,i-1′ and hr,1′=kr,1′-kr-1′. Now for each ϵ>0, we can write
(37)1hr|{k∈Ir:d¯(ΔmXk,Y0)<ϵ}|=1hr∑Ij*⊆Ir‍hj*1hj*|{k∈Ij*:d¯(ΔmXk,Y0)<ϵ}|=1hr∑Ij*⊆Ir‍hj*(Cθ′χK)j,
where χK is the characteristics function of the set K={k∈Ij*:d¯(ΔmXk,Y0)<ϵ} and (Cθ′χK)j=|K∩Ij*|/hj*. Suppose limj→∞(Cθ′χK)j=0. Then the right side of above expression is a regular weighted mean transform of (Cθ′χK)j and therefore tends to zero as j→∞ which contradicts (36). Thus limj→∞(Cθ′χK)j≠0, which shows that Y0∈ΓSθ′(Δm(X)). Hence ΓSθ(Δm(X))⊆ΓSθ′(Δm(X)).
Similarly, we can proveΛSθ(Δm(X))⊆ΛSθ′(Δm(X)).



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