3_rahmat.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 2, No. 2, 2009, (195-212)

ISSN 1307-5543 – www.ejpam.com

On I -Convergence in the Topology Induced by Prob-

abilistic Norms

M. R. S. Rahmat∗ and Harikrishnan K. K.

School of Applied Mathematics, The University of Nottingham Malaysia Campus

Jalan Broga, 43500 Semenyih, Selangor Darul Ehsan, Malaysia

Abstract. The concepts of I -convergence is a natural generalization of statistical conver-

gence and it is dependent on the notion of the ideal of subsets of N of positive integer set.

In this paper we study the I -convergence of sequences, I -convergence of sequences of

functions and I -Cauchy sequences in probabilistic normed spaces and prove some important

results.

AMS subject classifications: 47H10, 54E17, 54E50, 54E70

Key words: probabilistic norms, ideal Convergence, statistical convergence, ideal Cauchy

sequences, F-topology

∗Corresponding author.

Email addresses: Mohd.Rafi�nottingham.edu.my (M. Rahmat),harikrishnan.kk�nottingham.edu.my (Harikrishnan K.)

http://www.ejpam.com 195 c© 2009 EJPAM All rights reserved.



M. Rahmat and Harikrishnan K. / Eur. J. Pure Appl. Math, 2 (2009), (195-212) 196

1. Introduction

The concepts of statistical convergence was introduced (independently) by Fast

[7] and Steinhause [25]. In their studies, the concept of ordinary convergence of

sequence of real numbers was extended to statistical convergence in the following

way: a sequence {xn} ⊂ R is said to be statistically convergent to the real number

x0 ∈ R provided that each ε neighborhood Nε(x0) of x0, the set consisting of all

elements not contained by Nε(x0) has natural density zero for any ε > 0. The no-

tion of natural density here can be described as a function δ : 2N → [0, 1] and given

by δ(K) := limn→∞ n−1|{k ∈ K : k ≤ n}| where K ⊂ N, and |A| denotes the car-

dinality of the set A. The concept of statistical convergence was further discussed

and developed by many authors including [1,4,8–11,19,21]. Statistical convergence

has also been discussed in more general abstract spaces such as the fuzzy number

spaces [22], locally convex spaces [18], Banach spaces [15] and characterization

of Banach spaces [5]. Recently, Karakus [13] has extended the concept of statisti-

cal convergence for sequences in probabilistic normed spaces (PN space) and proved

several interesting results. In another paper, Karakus and Demirci [14] studied the

concept of statistical convergence of double sequences on PN spaces. The idea of I

-convergence for sequences, was inspired by the concept of statistical convergence

introduced in [7], see Kostyrko et al. [16] for a comprehensive bibliography. It is a

natural generalization of the concept of statistical convergence. The I -convergence

is based on the notion of the ideal I of subsets of N, the set of positive integers. Here,

a sequence {xn} ⊂ R is said to be I -convergent to the real number x0 ∈ R provided

that each ε neighborhood Nε(x0) of x0, the set consisting of all elements not con-

tained by Nε(x0) belongs to I for any ε > 0. Further works on ideal convergence

can be found in [2,3,6,12,17,20] The work of Karakus [13] inspired us to study the

I -convergence and other related properties in PN spaces. In this context, we obtain



M. Rahmat and Harikrishnan K. / Eur. J. Pure Appl. Math, 2 (2009), (195-212) 197

some results that parallel to the one given in [3,12,20,24].

Now we recall some notation and definitions used in this paper (see [26]).

Definition 1.1. A function f : R → R+0 is called a distribution function if it is non-

decreasing and left-continuous with inft∈R f (t) = 0 and supt∈R f (t) = 1. We denote the

set of all distribution function by ∆+.

Definition 1.2. A t-norm T is a continuous mapping T : [0, 1]× [0, 1] → [0, 1] such

that for all a, b, c, d ∈ [0, 1]

(i) T (a, b) = T (b, a);

(ii)T(a,T(b,c))=T(T(a,b),c);

(iv) T (a, b) ≤ T (c, d) whenever a ≤ c and b ≤ d;

(v) T (a, 1) = a.

Example 1.1. The operation T (a, b) = ab, T (a, b) = max(a+ b−1, 0) and T (a, b) =

min(a, b) on [0, 1] are t-norms.

The following definition is due to A. N. Šerstnev [23].

Definition 1.3. A probabilistic normed space (briefly, a PN space) is a triplet (X , F, T ),

where X is a real linear space, T is a continuous t-norm, and F (called probabilistic

norm) is a mapping from X into ∆+ (writing F(x) as Fx), the following conditions hold

for every x , y ∈ X and every s, t > 0:

(N1) Fx(t) = 1 if and only if x = θ(the null vector of X);

(N2) Fαx(t) = Fx(
t

|α|
) for α 6= 0;

(N3) Fx+y(s+ t)≥ T (Fx(s), Fy(t));

Example 1.2. Let (X ,‖ · ‖) is a normed space and T (a, b) = ab (or T(a,b)=min(a,b)).

Define

Fx(t) =
t

t + ‖x‖



M. Rahmat and Harikrishnan K. / Eur. J. Pure Appl. Math, 2 (2009), (195-212) 198

where x ∈ X and t > 0. Then (X , F, T ) is a PN space.

Let (X , F, T ) be a PN space. Since T is a continuous t-norm, the system of (ε,λ)-

neighborhoods of θ (the null vector in X )

{Nθ (ε,λ): ε > 0,λ ∈ (0, 1)}, (1.1)

where

Nθ(ε,λ) = {x ∈ X : Fx(ε)> 1−λ} (1.2)

determines a first countable Hausdorff topology on X , called the F-topology. Thus,

the F -topology can be completely specified by means of F -convergence of sequences.

It is clear that x − y ∈ Nθ means y ∈ Nx and vice-versa.

A sequence (xn) is said to be F -convergent to ξ ∈ X if for every ε > 0, and for

every λ ∈ (0, 1) there exists a number N ∈ N such that

xn− ξ ∈ Nθ (ε,λ) for all n ≥ N .

or equivalently,

xn ∈ Nξ(ε,λ) for all n ≥ N .

In this case we write F − lim xn = ξ.

Lemma 1.1. Let (X ,‖ ·‖) be a real normed space and (X , F, T ) be a PN space induced by

the probabilistic norm Fx(t) =
t

t+‖x‖
, where x ∈ X and t > 0. Then for every {xn} in X

lim xn = ξ⇒ℑ− lim xn = ξ.

Proof. Let suppose that lim xn = ξ. Then for every t > 0 there exists a positive

integer N = N(t) such that

‖xn− ξ‖ < t for all n ≥ N .

We observe that for any given ε > 0,



M. Rahmat and Harikrishnan K. / Eur. J. Pure Appl. Math, 2 (2009), (195-212) 199

ε+ ‖xn− ξ‖

ε
<
ε+ t

ε

which is equivalent to

ε

ε+ ‖xn− ξ‖
>
ε

ε+ t
= 1−

t

ε+ t
.

Therefore, by letting λ = t

ε+t
∈ (0, 1) we have

Fxn−ξ(ε)> 1−λ for all n ≥ N .

This implies that xn ∈Nξ(ε,λ) for all n ≥ N as required.

We recall the definition and notations of ideal.

Definition 1.4. A non-empty subset I of 2N is called an ideal on N if

(i) B ∈ I whenever B ⊆ A for some A∈ I (closed under subsets),

(ii)A∪ B ∈ I whenever A, B ∈ I (closed under unions).

An ideal called proper ifN /∈ I . An ideal called admissible if its proper and contains

all finite subsets.

Filter F is a dual notion to ideal I -it is closed under supersets and intersections.

It holds that {N\A: A∈ I } is a filter if and only if I is ideal. The filter F (I ) is called

the filter associated with the ideal I . Thus, one can write

A∈ I ⇔ Ac ∈ F (I ).

where Ac denotes the complement of A.

Ideal can be viewed as a way to describe which sets will be considered "small",

i.e., finite. Filter is collection of all "large" sets.



M. Rahmat and Harikrishnan K. / Eur. J. Pure Appl. Math, 2 (2009), (195-212) 200

2. I -Convergence for Sequences in PN Spaces

In this section we define the ideal convergence of a sequence in (X , F, T ) and prove

some important results.

Definition 2.1. Let I ⊆ 2N be a proper ideal in N and (X , F, T ) be a PN space. The

sequence (xn) in X is said to be I F − conver gent to x ∈ X (I − conver gent to x ∈ X

with respect to F-topology) if for each ε > 0, and λ ∈ (0, 1)

{n ∈ N : xn /∈ Nx(ε,λ)} ∈ I .

The vector x is called the I F− l imit of the sequence {xn} and we write I F− lim xn = x.

Definition 2.2. Let (X , F, T ) be a PN space and I be an admissible ideal in N. The

sequence {xn} in X is said to be I F∗ -convergent to ξ ∈ X (i.e., I F∗− lim xn = ξ) if and

only if there exists a set M = {m1 < m2 < · · · } ∈ F (I ) such that ℑ− lim xmk
= ξ.

Lemma 2.1. Let (X , F, T ) be a PN space. I F − l imit of any sequence if exists is unique.

Proof. Let {xn} be any sequence and suppose that I F−lim xn = ξ, I F−lim xn = η

where ξ 6= η. Since ξ 6= η, select ε > 0 and λ ∈ (0, 1) such thatNξ(ε,λ) andNη(ε,λ)

are disjoint neighborhoods of ξ and η. Since ξ and η both are I F − l imit of the

sequence {xn}, we have A= {n ∈ N : xn /∈ Nξ(ε,λ)} and B = {n ∈ N : xn /∈ Nη(ε,λ)}

are both belongs to I . This implies that the sets Ac = {n ∈ N : xn ∈ Nξ(ε,λ)} and

Bc = {n ∈ N : xn ∈ Nη(ε,λ)} belongs to F (I ). Since F (I ) is a filter in N, we have

Ac ∩ Bc is a nonempty in F (I ). In this way we obtain a contradiction to the fact that

the neighborhoods Nξ(ε,λ) and Nη(ε,λ) of ξ and η are disjoints. Hence we have

ξ = η. This completes the proof.



M. Rahmat and Harikrishnan K. / Eur. J. Pure Appl. Math, 2 (2009), (195-212) 201

Lemma 2.2. Let (X , F, T ) be a PN space and Ifin be Fréchet ideal (finite subsets on N).

Then F − conver gence implies I F
fin
− conver gence.

Proof. Let ε > 0 and λ ∈ (0, 1). Suppose that {xn} is F − conver gent to ξ. Then,

there exists a number N ∈ N such that xn ∈ Nξ(ε,λ) for every n ≥ N . This implies

that the set A= {n ∈ N : xn /∈ Nξ(ε,λ)} ⊆ {1, 2, · · · , N − 1}. Since the right hand side

belongs to I f , we have A∈ Ifin. This shows that {xn} is I F
fin− conver gent to ξ.

The following example shows that the converse of above theorem is not valid.

Example 2.1. By letting X = R in Example 1, we have (R, F, T ) is a PN space induced

by the probabilistic norm Fx(ε) =
ε

ε+‖x‖
. Let us suppose that A∈ Ifin. Define a sequence

{xn} in R via

xn =







1, if n ∈ A

0, otherwise.

Then, for every ε > 0 and λ ∈ (0, 1), let K = {n ∈ N : xn /∈ Nθ (ε,λ)}. We observe

that

xn /∈Nθ (ε,λ) ⇒ Fxn
(ε)≤ 1−λ

⇒
ε

ε+ ‖xn‖
≤ 1−λ

⇒ ‖xn‖ ≥
ελ

1−λ
> 0.

Hence, we have

K = {n ∈ N : ‖xn‖> 0}

= {n ∈ N : xn = 1}

= A∈ I f .

Therefore I F
fin
− lim xn = θ . But the sequence {xn} is not convergent to θ in (R,‖ · ‖). By

Lemma 1, this implies that F − lim xn 6= θ .



M. Rahmat and Harikrishnan K. / Eur. J. Pure Appl. Math, 2 (2009), (195-212) 202

Lemma 2.3. Let (X , F, T ) be a PN space and I is an admissible ideal on X . Then I F
fin

-convergence implies I F -convergence.

Proof. For I be an admissible ideal, we have
⋃

I = N. This implies that Ifin ⊂ I .

So, I F
fin -convergence implies I F -convergence.

The following lemma is an immediate consequence of definition of statistical con-

vergence sequence.

Lemma 2.4. Let (X , F, T ) be a PN space. If Iδ = {A⊆ N : δ(A) = 0} where δ(A) be the

density of A, then I F
δ

-convergence coincide with statistical convergence.

Lemma 2.5. If {xn} and {yn} are two sequences in (X , F, T ) with T (a, a) > a for every

a ∈ (0, 1), then

(i) If I F − lim xn = ξ and I F − lim yn = η, then I F − lim(xn+ yn) = ξ+η.

(ii)If I F − lim xn = ξ and α ∈ R, then I F − limαxn = αξ.

(iii) If I F − lim xn = ξ and I F − lim yn = η, then I F − lim(xn− yn) = ξ−η.

Proof. (i) Let ε > 0 and λ ∈ (0, 1). Since I F − lim xn = ξ and I F − lim yn = η, the

sets A = {n ∈ N : xn /∈ Nξ(
ε

2
,λ)} and B = {n ∈ N : xn /∈ Nη(

ε

2
,λ)} are belongs to I .

Let C = {n ∈ N : xn+ yn /∈ Nξ+η(ε,λ)}. Since I is an ideal it is sufficient to show that

C ⊂ A∪ B. This is equivalent to show that C c ⊃ Ac ∩ Bc where Ac and Bc are belongs

to F (I ). Let n ∈ Ac ∩ Bc, i.e., n ∈ Ac and n ∈ Bc then by (N4) we have

F(xn+yn)−(ξ+η)(ε) ≥ τT (Fxn−ξ, Fyn−η)(ε)

≥ T

�

Fxn−ξ(
ε

2
), Fyn−η(

ε

2
)

�

> T (1−λ, 1−λ)

> 1−λ.

Hence, n ∈ C c ⊃ Ac ∩ Bc ∈ F (I ) which implies C ⊂ A∪ B ∈ I and the result follows.



M. Rahmat and Harikrishnan K. / Eur. J. Pure Appl. Math, 2 (2009), (195-212) 203

(ii)Let ε > 0 and λ ∈ (0, 1). Since I F − lim xn = ξ, we have A = {n ∈ N : xn /∈

Nξ(ε,λ)} ∈ I . This implies that Ac = {n ∈ N : xn ∈ Nξ(ε,λ)} ∈ F (I ). Let n ∈ Ac.

For the case α = 0, We have

F0xn−0ξ(ε) = F0ε= 1> 1−λ

and for the case α 6= 0, we have

Fαxn−αξ(ε) = Fxn−ξ(
ε

|α|
)

≥ T

�

Fxn−ξ(ε), F0(
ε

|α|
− ε)
�

> T (1−λ, , 1)

= 1−λ.

This shows that {n ∈ N : αxn /∈ Nαξ(ε,λ)} ∈ I and consequently we have I ℑ −

limαxn = αξ.

(iii) The proof is obvious from (i) and (ii).

Definition 2.3. Let (X , F, T ) be a PN space. A subset A = {xn} of X is said to be I F -

bounded on PN spaces if for every λ ∈ (0, 1), there exists ε > 0 such that

{n ∈ N : xn /∈ Nθ(ε,λ)} ∈ I .

Let (X , F, T ). We denote I F
b
(X ) the set of all I F -bounded I F − conver gent se-

quences on X and lF
∞(X ) the set of all I F -bounded sequences on X .

Theorem 2.1. Let (X , F, T ) be a PN space such that T (a, a) > a for every a ∈ (0, 1). Let

I ⊂ 2N be an admissible ideal in N. Then I F
b
(X ) is a closed linear subspace of the set

lF
∞(X ).

Proof. In view of Lemma (), it is clear that the set I F
b
(X ) is a linear subspace of

the set lF
∞(X ). So to prove the result it is sufficient to prove that I F

b
(X ) = I F

b
(X ). It



M. Rahmat and Harikrishnan K. / Eur. J. Pure Appl. Math, 2 (2009), (195-212) 204

is clear that I F
b
(X ) ⊂ I F

b
(X ). Now we show that I Fm

b
(X ) ⊂ I F

b
(X ). Let y ∈ I F

b
(X ).

We notice that since Ny(ε,λ)∩I
F
b
(X ) 6= ; for every ε > 0 and λ ∈ (0, 1), there exists

an x ∈ Ny(ε,λ)∩ I
F
b
(X ) such that the set K = {n ∈ N : x /∈ Ny(

ε

2
,λ)} belongs to I .

This implies that K c = {n ∈ N : x ∈ Ny(
ε

2
,λ)} ∈ F (I ). Now let n ∈ K c, then by (N4),

we have

Fyn
(ε) = Fyn−xn+xn

(ε)

≥ T

�

Fyn−xn
(
ε

2
), Fxn

(
ε

2
)

�

> T (1−λ, 1−λ)

> 1−λ.

Thus, we have {n ∈ K c : yn ∈ Nθ(ε)> 1−λ} ∈ F (I ) which implies that {n ∈ N : yn /∈

Nθ(ε)> 1−λ} ∈ I . Thus y ∈ I F
b
(X ) and this completes the proof.

Lemma 2.6. If a sequence in a PN space (X , F, T ) is I F∗ -convergent, then it is I F
f

-convergent to the same limit.

Proof. Let I F∗ − lim xn = ξ, then by definition, there exists M = {m1 < m2 <

· · · } ∈ F (I ) such that F − lim xmk
= ξ. Let ε > 0 and λ ∈ (0, 1) be given. Since

F − lim xmk
= ξ, there exists N ∈ N such that xmk

∈ Nξ(ε,λ) for every k ≥ N . Let

A = {k ∈ N : xmk
/∈ Nξ(ε,λ)}. Then it is clear that A ⊂ {1, 2, · · · , N − 1} ∈ I f .

Therefore, the sequence {xn} is I F − lim xn = ξ.

3. I -Convergence for Continuous Functions in PN Spaces

In this short section, we extend the study of ideal convergence to a sequence of

function fn in (X , F, T ) and prove a theorem about ideal convergence. We begin with

the following definition.



M. Rahmat and Harikrishnan K. / Eur. J. Pure Appl. Math, 2 (2009), (195-212) 205

Definition 3.1. Let (X , F, T ) be a PN spaces and I be an arbitrary admissible ideal

in N. We say that a sequence of functions fn : X → X is I F -convergent to a function

f : X → X ) denoted I F − lim fn = f , if for every x ∈ X , ε > 0 and λ ∈ (0, 1) the set

{n ∈ N : fn(x)− f (x) /∈ Nθ (ε,λ)} belongs to I .

Theorem 3.1. Let (X , F, T ) be a PN spaces such that supa<1 T (a, a) = 1 and let I be an

arbitrary admissible ideal in N. Let I F − lim fn = f (on X) where fn : X → X , n ∈ N,

are equi-continuous (on X) and f : X → X . Then f is F -continuous (on X).

Proof. Let x0 ∈ X and x − x0 ∈ Nθ(ε,λ) be fixed. By equi-continuity of fn’s, for

every ε > 0, there exists a γ ∈ (0, 1) with γ < λ such that

fn(x)− fn(x0) ∈ Nθ(
ε

3
,γ)

for every n ∈ N. Since I F − lim fn = f , the set

K = {n ∈ N : fn(x0)− f (x0) /∈ Nθ(
ε

3
,γ)}
⋃

{n ∈ N : fn(x)− f (x)) /∈ Nθ (
ε

3
,γ)}

is in I and different from N. Hence, there exists n ∈ F (K) such that

fn(x0)− f (x0) ∈Nθ (
ε

3
,γ)) and fn(x)− f (x) ∈Nθ (

ε

3
,γ).

It follows that

F f (x0)− f (x)(ε) ≥ T

�

F f (x0)− fn(x0)
(
ε

3
), T (F fn(x0)− fn(x)

(
ε

3
), F fn(x)− f (x)(

ε

3
)

�

> T (1− γ, T (1− γ, 1− γ))

> T (1− γ, 1− γ)

> 1− γ

> 1−λ.

This implies that f is F -continuous (on X ).



M. Rahmat and Harikrishnan K. / Eur. J. Pure Appl. Math, 2 (2009), (195-212) 206

4. I -Continuity of a Function in PN Spaces

We begin with the definition of continuity an important type of sequential conti-

nuity in PN space.

Definition 4.1. Let I be an ideal and (X , F, T ) be a PN space. A map f : X → X is

called F − cont inuous at a point ξ ∈ X , if

F − lim xn = ξ =⇒ F − lim f (xn) = f (ξ).

This means for every ε > 0 and λ ∈ (0, 1), there exists a number N ∈ N such that for

n ≥ N, we have xn− ξ ∈ Nθ(ε,λ) implies f (xn)− f (ξ) ∈Nθ (ε,λ).

Definition 4.2. Let I be an ideal and (X , F, T ) be a PN space. A map f : X → X is

called I F − cont inuous at a point ξ ∈ X , if

I F − lim xn = ξ =⇒ I F − lim f (xn) = f (ξ).

Theorem 4.1. Let (X , F, T ) be a PN space and I be an arbitrary ideal in N. If f : X → X

is F -continuous then it is I F -continuous.

Proof. Let {xn} ∈ X and I F − lim xn = ξ. Then by F -continuity of f at ξ ∈ X we

means for every ε > 0 and λ ∈ (0, 1), we have xn−ξ ∈ Nθ (ε,λ) implies f (xn)− f (ξ) ∈

Nθ(ε,λ). Thus {n ∈ N : f (xn) − f (ξ) /∈ Nθ (ε,λ)} ⊂ {n ∈ N : xn − ξ /∈ Nθ (ε,λ)}.

Since I F − lim xn = ξ, we have {n ∈ N : xn − ξ /∈ Nθ(ε,λ)} ∈ I . This implies that

{n ∈ N : f (xn)− f (ξ) /∈ Nθ(ε,λ)} ∈ I which means I F − lim f (xn) = f (ξ). Hence,

f is an I F -continuous.

Theorem 4.2. Let (X , F, T ) be a PN space and I be an arbitrary admissible ideal in N.

If f : X → X is I F -continuous then f is I F
fin

-continuous.



M. Rahmat and Harikrishnan K. / Eur. J. Pure Appl. Math, 2 (2009), (195-212) 207

Proof. Let f is I F -continuous at ξ ∈ X . Suppose that f is not I F
f

-continuous,

then the set A= {n ∈ N : f (xn)− f (ξ) /∈ Nθ(ε,λ)} 6∈ I f , i.e., A is infinite set whenever

{n ∈ N : xn − ξ /∈ Nθ(ε,λ)} ∈ I f . Let {yn} be the subsequence of {xn} given by the

subset A of N. Then {n ∈ N : f (yn)− f (ξ) /∈ Nθ (ε,λ)} = N. Also, the subsequence

{yn} holds I F
f
− lim yn = ξ. By Lemma 4, this implies I F − lim yn = ξ. Thus, by

I F continuity of f , we have I F − lim f (yn) = f (ξ). Hence {n ∈ N : f (yn)− f (ξ) /∈

Nθ(ε,λ)} = N ∈ I , a contradiction. Therefore f is I F
f

-continuous.

From theorem 4.3 and 4.4, we can easily prove the following lemma.

Lemma 4.1. Let (X , F, T ) be a PN space and I be an arbitrary admissible ideal in N. If

f : X → X is a map, then the following implication hold:

F − cont inuous⇒I F − cont inuous⇒Ifin− cont inuous

5. I -Cauchy Sequences in PN Spaces

Definition 5.1. Let (X , F, T ) be a PN space. A sequence {xn} in X is said to be F -Cauchy,

if for every ε > 0 and λ ∈ (0, 1), there exists a number N = N(ε,λ) ∈ N such that

xn− xm ∈ Nθ (ε,λ) for every n, m ≥ N .

Definition 5.2. Let (X , F, T ) be a PN space and I be an admissible ideal. Then a

sequence (xn) in X is called I F−Cauchy sequence in X if for every ε > 0 and λ ∈ (0, 1),

there exists M = M(ε,λ) ∈ N such that

{n ∈ N : xn− xM /∈ Nθ(ε,λ)} ∈ I .

Definition 5.3. Let (X , F, T ) be a PN space and I be an admissible ideal. Then a

sequence (xn) in X is called I F∗−Cauchy sequence in X if for every ε > 0 and λ ∈ (0, 1),



M. Rahmat and Harikrishnan K. / Eur. J. Pure Appl. Math, 2 (2009), (195-212) 208

there exists a set M = {m1 < m2 < · · · < mk, · · · } ∈ F (I ) such that the subsequence

xM = (xmk
) is F − Cauchy in X , i.e. there exists a number k0 ∈ N such that

xmk
− xmp

∈ Nθ(ε,λ) for every k, p ≥ k0.

Theorem 5.1. Let (X , F, T ) be a PN space and I in N is an admissible ideal. If {xn} in

X is I F∗ − Cauchy then it is I F − Cauchy.

Proof. Let {xn} be a I F∗ − Cauchy sequence. Then for every ε > 0 and λ ∈ (0, 1)

there exists a set M = {m1 < m2 < · · · < mk, · · · } ∈ F (I ) and a number k0 ∈ N such

that xmk
− xmp

∈ Nθ(ε,λ) for every k, p ≥ k0. Now, fix N = mk0+1. Then for every

ε > 0 and λ ∈ (0, 1), we have xmk
− xN ∈ Nθ (ε,λ) for every k ≥ k0. Let H = N\M .

It is obvious that H ∈ I and A(ε,λ) = {n ∈ N : xn − xN /∈ Nθ (ε,λ)} ⊂ H ∪ {m1 <

m2 < · · · < mk0
}. Clearly, the right hand side of the last argument is belongs to I .

Therefore, for every ε > 0 and λ ∈ (0, 1) we can find N = N(ε,λ) ∈ N such that

A(ε,λ) ∈ I , i.e., {xn} is I F − Cauchy sequence in X .

Theorem 5.2. Let (X , F, T ) be a PN space such that T (a, a) > a for every a ∈ (0, 1) and

I be an admissible ideal. A sequence {xn} in X is I F -convergent if and only if it is I F

-Cauchy.

Proof: Necessity: Suppose that {xn} is I F -convergent to ξ ∈ X . Let ε > 0 and λ ∈ (0, 1)

be given. Since I F − lim xn = ξ, we have A= {n ∈ N : xn /∈ Nξ(
ε

2
,λ)} ∈ I . This implies

that Ac = {n ∈ N : xn ∈ Nξ(
ε

2
,λ)} ∈ F (I ). Now, by (N4), for every n, m ∈ Ac,

νxn−xm
(ε) ≥ T

�

νxn−ξ(
ε

2
),νxm−ξ(

ε

2
)

�

> T (1−λ, 1−λ)

> 1−λ.

Hence, {n ∈ N : xn− xm ∈ Nθ (ε,λ)} ∈ F (I ). This implies that {n ∈ N : xn− xm /∈

Nθ(ε,λ)} ∈ I , i.e., {xn} is a I F -Cauchy sequence.



M. Rahmat and Harikrishnan K. / Eur. J. Pure Appl. Math, 2 (2009), (195-212) 209

Proof. Sufficiency: Assume that {xn} is a I F -Cauchy sequence. We shall prove that

{xn} is I F -convergent sequence. For this, let {εp} be a strictly decreasing sequence

of positive real numbers such that εp → 0 as p → ∞. Since {xn} is a I F -Cauchy

sequence, there exists a strictly increasing sequence {mp} of positive integers such

that

Ap = {n ∈ N : xn− xmp
/∈Nθ (εp,λ)} ∈ I p = 1, 2, 3, · · · .

This implies that

; 6= {n ∈ N : xn− xmp
∈Nθ (εp,λ)} ∈ F (I ) p = 1, 2, 3, · · · . (5.1)

Let p and q be two positive integers such that p 6= q. Then by (3), both the sets

{n ∈ N : xn − xmp
∈ Nθ(εp,λ)} and {n ∈ N : xn − xmq

∈ Nθ (εq,λ)} are nonempty

elements of F (I ). Since F (I ) is a filter on N, therefore

; 6= {n ∈ N : xn− xmp
∈Nθ (εp,λ)} ∩ {n ∈ N : xn− xmq

∈ Nθ (εq,λ)} ∈ F (I ).

Thus, for each p and q with p 6= q, we can select np, nq ∈ N such that xnp
− xmp

∈

Nθ(εp,λ) and xnq
− xmq

∈ Nθ (εq,λ). Let ε = εp + εq. Then by (N4), we have

νxmp
−xmq
(ε) ≥ T (νxnp

−xmp
(εp),νxnp

−xmq
(εq))

> T (1−λ, 1−λ)

> 1−λ.

This implies that {xmp
} is a F−Cauchy sequence and satisfies the Cauchy criterion.

Say lim xmp
= ξ. Also we have ε → 0 as p → ∞, so for each ε > 0 we can choose

p0 ∈ N such that εp0
< ε

2
and

xmp
∈ Nξ(

ε

2
,λ) for p ≥ p0.

Next we prove that A = {n ∈ N : xn /∈ Nξ(ε,λ)} ⊂ Ap0
= {n ∈ N : xn − xmp0

/∈

Nθ(εp0
,λ)}. Since A and Ap0

are both in I , it is sufficient to show that Ac ⊃ Ac
p0

.



REFERENCES 210

Let n ∈ Ac
p0

, then we have

νxn−ξ(ε) ≥ T

�

νxn−xmp0
(
ε

2
),νxmp0

−ξ(
ε

2
)

�

> T (1−λ, 1−λ)

> 1−λ.

This implies that n ∈ Ac. Therefore A⊂ Ap0
. Since Ap0

⊂ I , we conclude that A⊂ I .

This proves that the sequence (xn) is I F -convergent to ξ.

ACKNOWLEDGEMENTS. The authors would like to thank the referee for giving

useful comments and suggestions for the improvement of this paper.

References

[1] S. Aytar and S. Pehlivan, Statistically monotonic and statistically bounded sequences of

fuzzy numbers. Information Sciences, 176: 734-744 (2006).

[2] V. Balaz, J. Cervennansky, T. Kostyrko, T. Salat, I -convergence and I -continuity of real

functions. Acta Mathematica, Faculty of Natural Sciences, Constantine the Philosopher

University Nitra 5: 43-50 (2002).

[3] M. Balcerzak, K. Dems, A. Komisarski, Statistical convergence and ideal convergence for

sequences of functions. J. Math. Anal. Appl. 328: 715-729 (2007).

[4] J. Connor, The statistical and strong p-Cesaro convergence of sequences. Analysis 8: 47-

63 (1988).

[5] J. Connor, M. Ganichev, V. Kadets, A characterization of Banach spaces with separable

duals via weak statistical convergence. J. Math. Anal. Appl. 244: 251-261 (2000).

[6] K. Dems, On I -Cauchy sequences. Real Anal. Exch. 30: 123-128 (2004/2005).

[7] H. Fast, Sur la convergence statistique. Colloq. Math. 2: 241-244 (1951).

[8] A. R. Freedman and J. J. Sember, Densities and Summability. Pasific J. Math. 95: 293-305

(1981).



REFERENCES 211

[9] J. A. Fridy, On statistical convergence. Analysis 5: 301-313 (1985).

[10] J. A. Fridy, Statistical limit points. Proc. Amer. Math. Soc. 118: 1187-1192 (1993).

[11] J. A. Fridy and C. Orhan, Statistical limit superior and limit inferior. Proc. Amer. Math.

Soc. 125: 3625-3631 (1997).

[12] J. Jasinski, I. Reclaw, Ideal convergence of continuous functions. Topology and its Ap-

plications, 153: 3511-3518 (2006).

[13] S. Karakus, Statistical convergence on probabilistic normed spaces. Math. Comm. 12:

11-23 (2007).

[14] S. Karakus, K. Demirci, Statisitcval convergence of double sequences on probabilistic

normed spaces. Int. J. Math. Math. Sci. 2007: 11 pages (2007)

[15] E. Kolk, The statistical convergence in Banach spaces. Acta Comment. Univ. Tartu. 928:

41-52 (1991).

[16] P. Kostyrko, T. Salat, W. Wilczynski, I -convergence. Real Anal. Exch. 26, 2: 669-686

(2000/2001).

[17] P. Kostyrko, M. Macaj, T. Salat, M. Sleziak. I -convergence and extremal I -limit point.

Math. Slovaca 55: 443-464 (2005).

[18] I. J. Maddox, Statistical convergence in a locally convex space. Math. Proc. Cambridge

Phil. Soc. 104: 141-145 (1988).

[19] Mursaleen and O. H. H. Edely, Statistical convergence of double sequences. J. Math.

Anal. Appl. 288: 223-231 (2003).

[20] A. Nabiev, S. Pehlivan, M. Gurdal, On I -Cauchy sequence. Taiwanese J. Math. 11, 2:

569-576 (2007).

[21] T. Šalát, On statistical convergent sequences of real numbers. math. Slovaca 30: 139-

150 (1980).

[22] E. Savaş, On statistically convergent sequences of fuzzy numbers. Information Sciences,

137: 277-282 (2001).

[23] A. N. Šerstnev, On the notion of a random normed spaces. Dokl. Akad. nauk SSSR 149:

280-283 (1963).

[24] M. Sleziak, I -continuity in topological spaces. Preprint.



REFERENCES 212

[25] H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique. Collog.

Math. 2: 73-74 (1951).

[26] B. Schweizer, A. Sklar, Probabilistic metric spaces, Elsevier/North-Holand. New York,

1983.