SEQUENCES IN PROBABILISTIC NORMED SPACES
M. MURSALEEN and S.A. MOHIUDDINE
One of the generalizations of statistical convergence is I-convergence which was introduced by Kostyrko et al. [12]. In this paper, we define and study the concept of I-convergence, I∗-convergence, I-limit points and I-cluster points of double sequences in probabilistic normed space. We discuss the relationship between I2-convergence andI2∗-convergence, i.e., we show thatI2∗-convergence impliesI2- convergence in probabilistic normed space. Furthermore, we have also demon- strated through an example that, in general, I2-convergence does not imply I2∗- convergence in probabilistic normed space.
AMS 2000 Subject Classification: 40A05, 46A70.
Key words: t-norm, probabilistic normed space,I-convergence, I-limit point, I- cluster point.
1. INTRODUCTION AND PRELIMINARIES
The concept of statistical convergence for sequences of real numbers was introduced by Fast [4] and Steinhaus [22] independently in the same year 1951 and since then several generalizations and applications of this notion have been investigated by various authors, namely [3], [7], [15], [16], [17], [18], [19]. One of its interesting generalization is I-convergence which was given by Kostyrko et al. [12]. RecentlyI-convergence for sequences of functions has been studied by Balcerzak et al. [2] and by Komisarski [13].
The theory of probabilistic normed spaces [5] originated from the concept of statistical metric spaces which was introduced by Menger [14] and further studied by Schweizer and Sklar [20, 21]. It provides an important method of generalizing the deterministic results of normed linear spaces. It has also very useful applications in various fields, e.g., continuity properties [1], topologi- cal spaces [5], linear operators [8], study of boundedness [9], convergence of random variables [10] etc.
In this paper we study the concept ofI-convergence andI∗-convergence in a more general setting, i.e., in probabilistic normed spaces. We also define
MATH. REPORTS12(62),4 (2010), 359–371
I-limit points and I-cluster points in probabilistic normed space and prove some interesting results.
We recall some notations and basic definitions used in this paper.
Definition1.1 ([21]). Atriangular norm(t-norm) is a continuous mapping
∗: [0,1]×[0,1]→[0,1] such that ([0,1],∗) is an abelian monoid with unit one and c∗d≥a∗b ifc≥aand d≥bfor all a, b, c, d∈[0,1].
Definition 1.2 ([5]). A function f : R → R+0 is called a distribution function if it is non-decreasing and left-continuous with inf
t∈R
f(t) = 0 and sup
t∈R
f(t) = 1.
ByD, we denote the set of all distribution functions.
Definition 1.3 ([5]). Let X be a real linear space and ν : X → D. A probabilistic norm or ν-normis at-norm satisfying the following conditions:
(i) νx(0) = 0,
(ii) νx(t) = 1 for all t >0 iffx= 0,
(iii) ναx(t) =νx(|α|t ) for allα∈R\ {0} and for all t >0, (iv) νx+y(s+t)≥νx(s)∗νy(t) for all x, y∈X and s, t∈R+0, where νx meansν(x) and νx(t) is the value ofνx att∈R.
The (X, ν,∗) is called a probabilistic normed space (for short, PNS).
Definition 1.4 ([11]). Let (X, ν,∗) be an PNS. A sequence x = (xk) is said to be convergent to ξ ∈X with respect to the probabilistic norm ν, that is,xk−→ν ξ if for every t >0 and∈(0,1), there is a positive integerk0 such that νxk−ξ(t)>1−whenever k≥k0. In this case we write ν- limx=ξ.
Remark1.1. Let (X,k · k) be a real normed linear space, and ν(x, t) := t
t+kxk
for all x∈X and t >0. Then xn−→k·k x if and only if xn−→ν x.
Definition 1.5 ([6]). LetK be a subset of N, the set of natural numbers.
The asymptotic density of K denoted byδ(K), is defined as δ(K) = lim
n
1
n|{k≤n:k∈K}|,
where the vertical bars denote the cardinality of the enclosed set.
Definition1.6 ([4, 22]). A number sequence x= (xk) is said to be statis- tically convergent to the number `if for each >0, the set K() ={k ≤n:
|xk−`|> } has asymptotic density zero, i.e., limn
1
n|{k≤n:|xk−`|> }|= 0.
In this case we writest- limx=`.
Definition 1.7 ([11]). Let (X, ν,∗) be an PNS. A sequence x = (xk) is said to be statistically convergent to ξ ∈ X with respect to the probabilistic norm ν provided that, for everyt >0 and >0,
δ({k≤n:νxk−ξ(t)≤1−}) = 0 or, equivalently,
limn
1
n|{k≤n:νxk−ξ(t)≤1−}|= 0.
In this case we writestν- limx=ξ.
Definition1.8 ([12]). IfX is a non-empty set, a familyI ⊂2X of subsets of X is called an ideal in X if
(a) ∅ ∈I,
(b) A, B∈I impliesA∪B∈I,
(c) for each A∈I and B ⊂A we haveB ∈I.
An ideal I is called nontrivial if X6∈I.
Definition 1.9 ([12]). LetX be a non-empty set. A non-empty family of setsF ⊂P(X), the power set of X, is called a filter on X if and only if
(a) ∅ 6∈F,
(b) A, B∈F implies A∩B ∈F,
(c) for each A∈F and B ⊃A we have B ∈F.
Definition 1.10 ([12]). A non-trivial ideal I in X is called an admissible ideal if it is different fromP(N) and it contains all singletons, i.e.,{x} ∈I for each x∈X.
Let I ⊂ P(X) be a non-trivial ideal. A class F(I) = {M ⊂ X : M = X\A, for someA ∈I} is a filter on X, called the filter associated with the ideal I.
Definition1.11 ([12]). An admissible idealI ⊂P(N) is said to satisfy the condition (AP) if for every sequence (An)n∈N of pairwise disjoint sets from I there are sets Bn ⊂N,n∈N, such that the symmetric differenceAn4Bn is a finite set for every nand S
n∈N
Bn∈I.
Definition 1.12 ([12]). LetI ⊂2N be a nontrivial ideal inN. A sequence x = (xk) is said to be I-convergent toL if for every > 0, the set
k∈ N :
|xk−L| ≥ ∈I.In this case we writeI- limx=L.
2. I2-CONVERGENCE IN PNS
In this section, we study the concept of ideal convergence of double se- quences in probabilistic normed space. Throughout the paper we take I2 as a nontrivial admissible ideal in N×N.
We define
Definition 2.1. Let I be a non trivial ideal of N×N and (X, ν,∗) be a probabilistic normed space. A double sequence x= (xjk) of elements of X is said to beI2-convergent to ξ∈X with respect to the probabilistic normν (or I2ν-convergenttoξ) if for each >0 and t >0,
(1)
(j, k)∈N×N:νxjk−ξ(t)≤1− ∈I2. In this case we write I2ν- limx=ξ.
Theorem 2.1. Let (X, ν,∗) be a PNS. Then, the following statements are equivalent:
(i) I2ν-limx=ξ;
(ii) {(j, k)∈N×N:νxjk−ξ(t)≤1−} ∈I2ν for every >0 andt >0;
(iii) {(j, k)∈N×N:νxjk−ξ(t)>1−} ∈F(I2ν) for every >0 andt >0;
(iv) I2-limνxjk−ξ(t) = 1.
Theproof is standard.
Theorem2.2. Let (X, ν,∗) be a PNS. If a double sequence x= (xjk) is I2ν-convergent then I2ν-limit is unique.
Proof. Suppose that I2ν- limx=ξ1 and I2ν- limx =ξ2. Given >0 and t > 0, choose r >0 such that (1−r)∗(1−r) ≥ 1−. Then, we define the following sets as
Kν,1(r, t) =
(j, k)∈N×N:νxjk−ξ1(t)≤1−r , Kν,2(r, t) =
(j, k)∈N×N:νxjk−ξ2(t)≤1−r .
SinceI2ν- limx=ξ1, we haveKν,1(r, t)∈I2.Furthermore, usingI2ν- limx=ξ2, we get Kν,2(r, t)∈I2.Now, let Kν(r, t) =Kν,1(r, t)∪Kν,2(r, t)∈I2. Then we see thatKν(r, t)∈I2. This implies that its complementKνC(r, t) is non empty inF(I2). If (j, k)∈KνC(r, t), then we have (j, k)∈Kν,1C (r, t)∩Kν,2C (r, t), and so (2) νξ1−ξ2(t)≥νxjk−ξ1(t/2)∗νxjk−ξ2(t/2)>(1−r)∗(1−r).
Since (1−r)∗(1−r) ≥ 1−, we have νξ1−ξ2(t) > 1−. Since >0 was arbitrary, we get ν(ξ1−ξ2, t) = 1 for all t >0, which yieldsξ1=ξ2.
This completes the proof of the theorem.
Theorem 2.3. Let (X, ν,∗) be a PNS.
(i) If ν-limxjk =ξ thenI2ν-limxjk =ξ.
(ii) IfI2ν-limxjk=ξ1 andI2ν-limyjk=ξ2 thenI2ν-lim(xjk+yjk) = (ξ1+ξ2).
(iii) If I2ν-limxjk =ξ thenI2ν-limαxjk =αξ.
Proof. (i) Suppose that ν- limxjk = ξ. Then for each > 0 and t >0 there exists a positive integer N such that
νxjk−ξ(t)>1− for each j, k > N. Since the set
A(t) =
(j, k)∈N×N:νxjk−ξ(t)≤1−
is contained in {1,2,3, . . . , N −1} and the ideal I2 is admissible we have A(t)∈I2. HenceI2ν- limxjk =ξ.
(ii) Let I2ν- limxjk =ξ1 and I2ν- limyjk =ξ2. For given >0 and t >0, choose r >0 such that (1−r)∗(1−r)>1−. Define the sets
Kν,1(r, t) =
(j, k)∈N×N:νxjk−ξ1(t)≤1−r , Kν,2(r, t) =
(j, k)∈N×N:νyjk−ξ2(t)≤1−r . Since I2ν- limxjk =ξ1, we have
Kν,1(r, t)∈I2. Furthermore, using I2ν- limx=ξ2, we get
Kν,2(r, t)∈I2.
Now, let Kν(r, t) = Kν,1(r, t)∪Kν,2(r, t). Then Kν(r, t) ∈ I2 which implies that KνC(r, t) is non empty inF(I2). Now, we have to show that KνC(r, t)⊂ {(j, k) ∈ N×N: ν(xjk+yjk)−(ξ1+ξ2)(t) > 1−}. If (j, k) ∈ KνC(r, t), then we have νxjk−ξ1(2t)>1−r and νyjk−ξ2(t2)>1−r. Therefore,
ν(xjk+yjk)−(ξ1+ξ2)(t)≥νxjk−ξ1
t 2
∗νyjk−ξ2 t
2
>(1−r)∗(1−r)>1−. This shows that
KνC(r, t)⊂
(j, k)∈N×N:ν(xjk+yjk)−(ξ1+ξ2)(t)>1− . Since KνC(r, t)∈F(I2), we haveI2ν- lim(xjk+yjk) = (ξ1+ξ2).
(iii) It is trivial forα= 0. Now letα6= 0. Then for given >0 andt >0,
(3) B(t) =
(j, k)∈N×N:νxjk−ξ(t)>1− ∈F(I2).
It is sufficient to prove that for each >0 and t >0, B(t)⊂
(j, k)∈N×N:ναxjk−αξ(t)>1− .
Let (j, k)∈B(t). Then we have νxjk−ξ(t)>1−. Now, ναxjk−αξ(t) =νxjk−ξ
t
|α|
≥νxjk−ξ(t)∗ν0 t
|α|−t
=
=νxjk−ξ(t)∗1 =νxjk−ξ(t)>1−. Hence
B(t)⊂
(j, k)∈NN:ναxjk−αξ(t)>1− and from (3), we conclude that I2ν- limαxjk =αξ.
This completes the proof of the theorem.
3. I∗2-CONVERGENCE IN PNS
In this section, we introduce the concept of I2∗-convergence of double sequences in probabilistic normed space and show thatI2∗-convergence implies I2-convergence but not conversely.
Definition3.1. Let (X, ν,∗) be a probabilistic normed space. We say that a sequence x= (xjk) of elements inX is I2∗-convergent toξ ∈X with respect to the probabilistic norm ν if there exists a subset K ={(jm, km) :j1< j2<
· · · ;k1 < k2 <· · · } of N×N such that K ∈F(I2) (i.e., N×N\K ∈I2) and ν- lim
m xjmkm =ξ.
In this case we write I2∗,ν- limx = ξ and ξ is called the I2∗-limit of the double sequence x= (xjk).
Theorem 3.1. Let (X, ν,∗) be a PNS and I2 be an admissible ideal. If I2∗,ν-limx=ξ thenI2ν-limx=ξ.
Proof. Suppose that I2∗,ν- limx = ξ. Then K = {(jm, km) : j1 < j2 <
· · · ;k1 < k2 < · · · } ∈ F(I2) (i.e., N×N\K = H(say) ∈ I2) such that ν- lim
m xjmkm = ξ. But then for any > 0 and t > 0 there exists a positive integerN such thatνxjmkm−ξ(t)>1−for allm > N. Since{(jm, km)∈K : νxjmkm−ξ(t) ≤1−} is contained in {j1 < j2 <· · · < jN−1;k1 < k2 <· · ·<
kN−1} and the idealI2 is admissible, we have
(jm, km)∈K :νxjmkm−ξ(t)≤1− ∈I2. Hence
(j, k)∈N×N:νxjk−ξ(t)≤1− ⊆
⊆H∪ {j1< j2 <· · ·< jN−1;k1< k2 <· · ·< kN−1} ∈I2 for all >0 andt >0. Therefore, we conclude thatI2ν- limx=ξ.
Remark 3.1. The following example shows that the converse of Theo- rem 3.1 need not be true.
Example3.1. Let (R,| · |) denote the space of all real numbers with the usual norm, and leta∗b=abfor all a, b∈[0,1]. For allx∈Rand everyt >0, consider
νx(t) := t t+|x|. Then (R, ν,∗) is a PNS.
Let N×N = S
i,j
4ij be a decomposition of N×N such that for any (m, n) ∈N×N each 4ij contains infinitely many (i, j)0swhere i≥m, j ≥n and 4ij ∩ 4mn = ∅ for (i, j) 6= (m, n). Let I2 be the class of all subsets of N×Nwhich intersect atmost a finite number of4ij’s. ThenI2is an admissible ideal. Now, we define a double sequence xmn= ij1 if (m, n)∈ 4ij. Then
νxmn(t) = t
t+|xmn| →1 as m, n→ ∞. HenceI2ν- lim
m,nxmn= 0.
Now, suppose that I2∗,ν- lim
m,nxmn = 0. Then there exists a subset K = {(mj, nj) : m1 < m2 < · · ·;n1 < n2 < · · · } of N×N such that K ∈ F(I2) and ν- lim
j xmjnj = 0. Since K ∈ F(I2), there is a set H ∈ I2 such that K = N×N\H. Now, from the definition of I2, there exist, say p, q ∈ N, such that
H⊂ p
[
m=1
∞ [
n=1
4mn
∪ q
[
n=1
∞ [
m=1
4mn
.
But then 4p+1,q+1 ⊂K, and therefore xmjnj = 1
(p+ 1)(q+ 1) >0
for infinitely many (mj, nj)0s from K which contradicts ν- limjxmjnj = 0.
Therefore, the assumption I2∗,ν- lim
m,nxmn= 0 leads to the contradiction.
Hence the converse of the theorem need not be true.
Remark3.2. From the above result we have seen thatI2∗-convergence im- plies I2-convergence but not conversely. Now the question arises under what condition the converse may hold. The following theorem shows that the con- verse holds if the ideal I2 satisfies condition (AP).
Definition 3.2. An admissible ideal I2 ⊂P(N×N) is said to satisfy the condition (AP) if for every sequence (An)n∈N of pairwise disjoint sets fromI2
there are sets Bn ⊂N,n∈N, such that the symmetric differenceAn4Bn is a finite set for every nand S
n∈N
Bn∈I2.
Theorem 3.2. Let (X, ν,∗) be a PNS and the ideal I2 satisfy the con- dition (AP). If x = (xjk) is a double sequence in X such that I2ν-limx =ξ, then I2∗,ν-limx=ξ.
Proof. Suppose I2 satisfies condition (AP) and I2ν- limx = ξ. Then for each >0 andt >0,
(j, k)∈N×N:µxjk−ξ(t)≤1− ∈I2. We define the set Ap forp∈Nand t >0 as
Ap =
(j, k)∈N×N: 1− 1
p ≤νxjk−ξ(t)<1− 1 p+ 1
.
Obviously, {A1, A2, . . .} is countable and belongs to I2, and Ai∩Aj =∅ for i6=j. By condition (AP), there is a countable family of sets{B1, B2, . . .} ∈I2
such that the symmetric difference Ai4Bi is a finite set for each i ∈ N and B =S∞
i=1Bi ∈ I2. From the definition of associate filter F(I2) there is a set K ∈F(I2) such that K =N×N\B. To prove the theorem it is sufficient to show that the subsequence (xjk)(j,k)∈K is convergent toξ with respect to the probabilistic normν. Letη >0 andt >0. Chooseq ∈Nsuch that 1q < η. Then
(j, k)∈N×N:νxjk−ξ(t)≤1−η ⊂
⊂
(j, k)∈N×N:νxjk−ξ(t)≤1−1 q
⊂
q+1
[
i=1
Ai.
SinceAi4Bi,i= 1,2, . . . , q+1 are finite, there exists (j0, k0)∈N×Nsuch that q+1
[
i=1
Bi
∩{(j, k) :j≥j0 and k≥k0}= (4)
= q+1
[
i=1
Ai
∩{(j, k) :j ≥j0 and k≥k0}.
If j ≥j0, k ≥ k0 and (j, k) ∈K then (j, k) 6∈ Sq+1
i=1Bi. Therefore by (4), we have (j, k)6∈Sq+1
i=1Ai. Hence for every j≥j0,k≥k0 and (j, k)∈K we have νxjk−ξ(t)>1−η.
Since η >0 was arbitrary, we have I2∗,ν- limx= ξ. This completes the proof of the theorem.
Theorem 3.3. Let(X, ν,∗) be a PNS. Then the following conditions are equivalent:
(i) I2∗,ν-limx=ξ.
(ii) There exist two sequences y = (yjk) and z = (zjk) in X such that x=y+z, ν-limy =ξ and the set{(j, k) :zjk 6=θ} ∈I2, where θ denotes the zero element of X.
Proof. Suppose that the condition (i) holds. Then there exists a subset K ={(jm, km) :j1 < j2<· · ·;k1< k2 <· · · } ofN×Nsuch that
(5) K∈F(I2) and ν- lim
m xjmkm =ξ.
We define the sequences y = (yjk) andz= (zjk) as yjk =
( xjk if (j, k)∈K, ξ if (j, k)∈KC;
andzjk =xjk−yjkfor all (j, k)∈N×N. For given >0,t >0 and (j, k)∈KC, we have
νyjk−ξ(t) = 1>1−.
Using (5) we have ν- limy = ξ. Since {(j, k) : zjk 6= θ} ⊂ KC, we have {(j, k) :zjk 6=θ} ∈I2.
Let the condition (ii) hold. Then K = {(j, k) : zjk = θ} ∈ F(I2) is an infinite set. Obviously, K ∈ F(I2) is an infinite set. Let K = {(jm, km) : j1 < j2 < · · · ;k1 < k2 < · · · }. Since xjmkm = yjmkm and ν- lim
m yjmkm = ξ, ν- lim
m xjmkm = ξ. Hence I2∗,ν- lim
j,k xjk = ξ. This completes the proof of the theorem.
4. I2-LIMIT POINTS AND I2-CLUSTER POINTS IN PNS
In this section we define I2-limit points and I2-cluster points in proba- bilistic normed space analogous to the statistical limit points and statistical cluster points due to Fridy [8].
Definition 4.1. Let (X, ν,∗) be a PNS, andx = (xjk)∈X. An element ξ ∈ X is said to be a limit point of the sequence x = (xjk) with respect to the probabilistic norm ν (or a ν-limit point) if there is subsequence of the sequence x which converges toξ with respect to the probabilistic normν.
By Lν2(x), we denote the set of all limit points of the double sequence x= (xjk) with respect to the probabilistic norm ν.
Definition 4.2. Let (X, ν,∗) be a PNS, andx = (xjk)∈X. An element ξ ∈ X is said to be an I2-limit point of the sequence x with respect to the
probabilistic norm ν (or I2ν-limit point) if there is a subset K = {(jm, km) : j1 < j2 <· · ·;k1< k2 <· · · }ofN×Nsuch thatK6∈I2andν- lim
m→∞xjmkm=ξ.
We denote by ΛI,ν2 (x), the set of all I2ν-limit points of the sequence x= (xjk).
Definition 4.3. Let (X, ν,∗) be a PNS, andx = (xjk)∈X. An element ξ ∈X is said to be an I2-cluster point of x with respect to the probabilistic norm ν (orI2ν-cluster point) if for each >0 and t >0
K =
(j, k)∈N×N:νxjk−ξ(t)>1− 6∈I2.
By ΓI,ν2 (x), we denote the set of all I2ν-cluster points of the sequence x= (xjk).
Theorem 4.1. Let(X, ν,∗)be a PNS. Then for every sequencex= (xjk) in X we have ΛI,ν2 (x)⊂ΓI,ν2 (x)⊂ Lν2(x).
Proof. Let ξ ∈ ΛI,ν2 (x). Then there exists a set K = {(jm, km) : j1 <
j2 <· · ·;k1 < k2 <· · · } of N×Nsuch that K 6∈I2 and ν- lim
m→∞xjmkm =ξ.
For each >0 and t > 0 there exists N ∈N such that for j, k > N we have νxjk−ξ(t)>1−. Hence
(j, k)∈N×N:νxjk−ξ(t)>1− ⊃
jN+1, jN+2, . . .;kN+1, kN+2, . . .}
and so
(j, k)∈N×N:νxjk−ξ(t)>1− 6∈I2, which means that ξ∈ΓI,ν2 (x). Hence ΛI,ν2 (x)⊂ΓI,ν2 (x).
Letξ ∈ΓI,ν2 (x). Then for given >0 andt >0, we have (j, k)∈N×N:νxjk−ξ(t)>1− 6∈I2.
Let K ={(jm, km) : j1 < j2 < · · ·;k1 < k2 < · · · }. Then there is a subse- quence (xjk)(j,k)∈Kof (xjk) that converges toξwith respect to the probabilistic norm ν. Therefore ξ is an ordinary limit point of (xjk), that is ξ∈ Lν2(x) and hence ΓI,ν2 (x)⊂ L2ν(x). This completes the proof of the theorem.
Theorem 4.2. Let x = (xjk) be a sequence in a PN-space (X, ν,∗).
Then ΛI,ν2 (x) = ΓI,ν2 (x) ={ξ}, provided I2ν-lim
j,k xjk =ξ.
Proof. Let η ∈ ΛI,ν2 (x), where ξ 6= η. Then there exist two subsets K and K0, that is, K = {(jm, km) : j1 < j2 < · · · ;k1 < k2 < · · · } and K0 =
{(pm, qm) :p1 < p2 <· · · ;q1 < q2 <· · · } ofN×Nsuch that K6∈I2 and ν- lim
m→∞xjmkm =ξ, (6)
K0 6∈I2 and ν- lim
m→∞xpmqm =η.
(7)
By (7), given > 0 and t >0, there exists N ∈ N such that for m > N we have νpmqm−η(t)>1−. Therefore,
A=
(pm, qm)∈K0:νpmqm−η(t)≤1− ⊂
⊂ {(pm, qm) :p1 < p2 <· · ·< pN;q1 < q2<· · ·< qN}.
As I2 is an admissible ideal we have A∈I2. If we take B =
(pm, qm)∈K0 :νpmqm−η(t)>1− 6∈I2.
Otherwise, if B ∈ I2, then A∪B = K0 ∈ I2, which contradicts (7). Since I2ν- lim
j,k xjk =ξ, we have that for each >0 andt >0, C =
(j, k)∈N×N:νxjk−ξ(t)≤1− ∈I2. Therefore,
CC =
(j, k)∈N×N:νxjk−ξ(t)>1− ∈F(I2).
Since for every ξ 6= η, we have B∩CC = ∅, B ⊂ C. Since C ∈ I2 implies B ∈I2, this contradicts the fact thatB 6∈I2. Hence ΛI,ν2 (x) ={ξ}.
On the other hand, suppose thatη∈ΓI,ν2 (x), whereξ6=η. By definition, for each >0 andt >0,
A=
(j, k)∈N×N:νxjk−ξ(t)>1− 6∈I2, B =
(j, k)∈N×N:νxjk−η(t)>1− 6∈I2.
For ξ 6= η, we have A∩B = ∅ and therefore B ⊂ AC. Also, I2ν- lim
j,k xjk =ξ implies that
AC =
(j, k)∈N×N:νxjk−ξ(t)≤1− ∈I2.
Hence B ∈ I2, which is a contradiction to B 6∈ I2. Therefore, ΓI,ν2 (x) ={ξ}.
This completes the proof of the theorem.
Theorem 4.3. Let (X, ν,∗) be a PNS and for any two sequences x = (xjk), y = (yjk) in X, the set A ={(j, k) ∈N×N: xjk 6= yjk} ∈ I2. Then ΛI,ν2 (x) = ΛI,ν2 (y) and ΓI,ν2 (x) = ΓI,ν2 (y).
Proof. Let ξ∈ΛI,ν2 (x). Then there exists a subsetK={(jm, km) :j1<
j2 <· · ·;k1 < k2 <· · · } of N×Nsuch that K 6∈I2 and ν- lim
m→∞xjmkm =ξ.
Given > 0 and t > 0, there exists N ∈ N such that νxjmkm−ξ(t) > 1− for m > N. Define K1 = K ∩A and K2 = K \A. Since A ∈ I2 we have
K1 ∈ I2. As K = K1 ∪K2 and K 6∈ I2 we have K2 6∈ I2. It is clear that the subsequence (yjk)(j,k)∈K2 of the sequencey= (yjk) is convergent toξwith respect to the probabilistic normν we have. This implies thatξ∈ΛI,ν2 (y) and hence ΛI,ν2 (x)⊂ΛI,ν2 (y).
Similarly, we can show that ΛI,ν2 (y)⊂ΛI,ν2 (x). Hence ΛI,ν2 (x) = ΛI,ν2 (y).
Now, we need to show that ΓI,ν2 (x) = ΓI,ν2 (y). Let ξ ∈ΓI,ν2 (x). For each >0 andt >0 the set
B =
(j, k)∈N×N:νxjk−ξ(t)>1− 6∈I2. Given >0 and t >0, define
C=
(j, k)∈N×N:νyjk−ξ(t)>1− . Now, we have to show that C 6∈I2. LetC ∈I2. Then
CC =
(j, k)∈N×N:νyjk−ξ(t)≤1− ∈F(I2).
By hypothesis, AC = {(j, k) ∈ N×N:xjk =yjk} ∈F(I2). Since F(I2) is a filter in N×N, CC∩AC ∈F(I2). Since CC∩AC ⊂BC,BC ∈F(I2). This implies that B ∈ I2 which is a contradiction, since B 6∈ I2. So that C 6∈ I2 and therefore ΓI,ν2 (x)⊂ΓI,ν2 (y). Similarly, we can show that ΓI,ν2 (y)⊂ΓI,ν2 (x) and hence ΓI,ν2 (x) = ΓI,ν2 (y). This completes the proof of the theorem.
Acknowledgements.Research of the first author was supported by the Department of Science and Technology, New Delhi, under grant No. SR/S4/MS:505/07. The re- search of the second author was supported by the Department of Atomic Energy, Government of India under the NBHM-Post Doctoral Fellowship programme number 40/10/2008-R&D II/892.
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Received 22 August 2009 Aligarh Muslim University
Department of Mathematics Aligarh 202002, India mursaleenm@gmail.com
and
Aligarh Muslim University Department of Mathematics
Aligarh 202002, India mohiuddine@gmail.com
