Selçuk J. Appl. Math. Selçuk Journal of Vol. 14. No. 1. pp. 57-70, 2013 Applied Mathematics
Statistical Convergence of Double Sequences in Probabilistic Metric Spaces
Kaustubh Dutta1, Prasanta Malik2, Manojit Maity3
1Department of Mathematics, Jadavpur University, Jadavpur, Kolkata-700032, India.
e-mail: kaustubh.dutta@ gm ail.com
2Department of Mathematics, Ramakrishna Mission Vidyamandira, Belur Math,
Howrah-711202, India.
e-mail:pm jupm @ yaho o.co.in
3Assistant Teacher in Mathematics, Boral High School, Kolkata-700154, India.
e-mail:m epsilon@ gm ail.com
Received Date: February 25, 2012 Accepted Date: March 20, 2012
Abstract. In this paper we study the concepts of strongly statistically conver-gent and strongly statistically Cauchy double sequences in a probabilistic metric space endowed with the strong topology. We also introduce the notions of strong statistical limit points and cluster points of double sequences and study some of its basic properties.
Key words: Probabilistic metric space; Strong topology; Double sequences; Strong statistical convergence; Strong statistically Cauchy sequences; Strong statistical limit and cluster points.
AMS Classi…cation: 54E70. 1. Introduction
Menger [10] had started the theory of probabilistic metric spaces under the name of “Statistical metric spaces”. Here the distance between two points p; q is de…ned as a distribution function Fpq instead of a nonnegative real number. For a positive number t, Fpq(t) is interpreted as the probability that the distance between the points p and q is less than t.
The theory of probabilistic metric spaces was brought to prominence by path breaking works of Schweizer and Sklar [20-23], Tardi¤ [29] among others. A detailed history and discussions of these spaces can be seen from the beautifully written book by Schweizer and Sklar [24]. Though several topologies can be de…ned on a PM space, the most common is the strong topology as a lot of investigations have been done based on this topology.
On the other hand recall that the usual notions of convergence was extended to statistical convergence by Fast [6] and Schoenberg [19] independently. For the last four decades a lot of work has been done on this convergence. The notion of statistical convergence has been very recently studied in PM spaces by ¸Sençimen and Pehlivan [26] in 2008. (For more works see [27,28].)
The notion of statistical convergence was introduced for double sequences by Mursaleen and Edely [12] and independently by Móricz [11] using the idea of double natural density. For more recent works on double sequences one can see [2,9] where many more references can be found.
Following the line of ¸Sençimen and Pehlivan [26], in this paper we investigate the notion of strong statistical convergence of double sequences in PM spaces. We also introduce the notions of strong statistical limit and cluster points of double sequences and study some of its basic properties.
2. Preliminaries
First we recall some basic concepts related to the probabilistic metric spaces (or PM spaces) which can be studied in detail from the fundamental book [24] by Schweizer and Sklar.
De…nition 2.1. A nondecreasing function F : R ! [0; 1] de…ned on R with F ( 1) = 0 and F (1) = 1; where R = [ 1; 1], is called a distribution function.
The set of all left continuous distribution function over ( 1; 1) is denoted by .
We consider the relation ‘ ’on de…ned by F G if and only if F (x) G(x) for all x 2 R. It can be easily veri…ed that the relation ‘ ’is a partial order on
.
De…nition 2.2. For any a 2 [ 1; 1] the unit step at a is denoted by "a and is de…ned to be a function in given by
"a(x) = 0; 1 x a = 1; a < x 1:
De…nition 2.3. A sequence fFngn2Nof distribution functions converges weakly to a distribution function F and we write Fn
w
! F if and only if the sequence fFn(x)gn2Nconverges to F (x) at each continuity point x of F .
De…nition 2.4. The distance between F and G in is denoted by dL(F; G) and is de…ned as the in…mum of all numbers h 2 (0; 1] such that the inequalities
F (x h) h G(x) F (x + h) + h and G(x h) h F (x) G(x + h) + h
hold for every x 2 ( h1; 1 h).
It is known that dL is a metric on and for any sequence fFngn2N in and F 2 , we have
Fn w
! F if and only if dL(Fn; F ) ! 0:
Here we will be interested in the subset of consisting of those elements F that satisfy F (0) = 0.
De…nition 2.5. A distance distribution function is a nondecreasing function F de…ned on R+ = [0; 1] that satis…es F (0) = 0 and F (1) = 1 and is left continuous on (0; 1).
The set of all distance distribution functions is denoted by +.
The function dLis clearly a metric on +. The metric space ( +; dL) is compact and hence complete.
Theorem 2.1. Let F 2 + be given. Then for any t > 0, F (t) > 1 t if and only if dL(F; "0) < t.
Note 2.1. Geometrically, dL(F; "0) is the abscissa of the point of intersection of the line y = 1 x and the graph of F (completed, if necessary, by the addition of vertical segments at discontinuities).
De…nition 2.6. A triangle function is a binary operation on +, : + + ! + that is commutative, associative, nondecreasing in each place, and has "0 as an identity.
De…nition 2.7. A probabilistic metric space, brie‡y PM space, is a triplet (S;F; ) where S is a nonempty set whose elements are the points of the space, F is a function from S S into +, is a triangle function and the following conditions are satis…ed for all x; y; z 2 S:
(1)F(x; x) = "0
(2)F(x; y) 6= "0 if x 6= y (3)F(x; y) = F(y; x)
(4)F(x; z) (F(x; y); F(y; z)).
From now on we will denoteF(x; y) by Fxy and its value at p by Fxy(p). De…nition 2.8. Let (S;F; ) be a PM space. For x 2 S and t > 0, the strong t-neighbourhood of x is de…ned as the set
Nx(t) = fy 2 S : Fxy(t) > 1 tg:
The collectionNx= fNx(t) : t > 0g is called the strong neighbourhood system at x and the union N = S
x2S
Nx is called the strong neighbourhood system for S.
By Theorem 2.1 we can write
Nx(t) = fy 2 S : dL(Fxy; "0) < tg:
If is continuous, then the strong neighbourhood systemN determines a Haus-dor¤ topology for S. This topology is called the strong topology for S.
De…nition 2.9. Let (S;F; ) be a PM space. Then for any t > 0, the subset U(t) of S S given by
U(t) = f(x; y) : Fxy(t) > 1 tg is called the strong t-vicinity.
Theorem 2.2. Let (S;F; ) be a PM space and be continuous. Then for any t > 0, there is an > 0 such that
U( ) U( ) U(t)
whereU( ) U( ) = f(x; z) : for some y, (x; y) and (y; z) 2 U(t)g.
Note 2.2. Under the hypothesis of Theorem 2.2 we can say that for any t > 0, there is an > 0 such that Fxz(t) > 1 t whenever Fxy( ) > 1 and Fyz( ) > 1 . Equivalently it can be written as: for any t > 0, there is an > 0 such that dL(Fxz; "0) < t whenever dL(Fxy; "0) < and dL(Fyz; "0) < . In a PM space (S;F; ), if is continuous then the strong neighbourhood system N determines a Kuratowski closure operation which is called the strong closure and for any subset A of S the strong closure of A is denoted by (A) and for any nonempty subset A of S
(A) = fx 2 S : for any t > 0, there is a y 2 A such that Fxy(t) > 1 tg:
Remark 2.1. Throughout the rest of the article in a PM space (S;F; ) we always assume that is continuous and S is endowed with the strong topology. De…nition 2.10. A subset B of S is said to be strongly closed if its complement is a strongly open set.
De…nition 2.11. A family C of strongly open sets of S is said to be a strong open cover of P S ifC covers P .
A subset B of S is said to be strongly compact if every strongly open cover of B has a …nite subcover.
Theorem 2.3. In a PM space (S;F; ), a strongly closed subset of a strongly compact set is strongly compact.
Proof. Let A be a strongly closed subset of a strongly compact set B. Let C = fO : 2 g be any strong open cover of A. Then A S
2 O . Let us consider the collectionC0 =C [fAcg where Acis the complement of the set A in S. Since A is strongly closed, Ac is a strongly open subset of S. ThenC0 forms a strong open cover of B. Since B is strongly compact so there exists a …nite sub collectionC00 ofC0 such thatC00 covers B and so covers A. Since no point of A is covered by Ac so C00n fAc
g also covers A. Hence A is strongly compact. De…nition 2.12. [26] Let (S;F; ) be a PM space. A sequence fxngn2Nin S is said to be strongly convergent to a point 2 S if for any t > 0, there exists a natural number N such that xn 2 N (t) where n N and we write xn ! or lim
n!1xn= .
Similarly a sequence fxngn2Nin S is called a strong Cauchy sequence if for any t > 0, there exists a natural number N such that (xm; xn) 2 U(t) whenever m; n N .
By the convergence of a double sequence we mean convergence in Pringsheim’s sense [17]. A real double sequence x = fxjkgj;k2N is said to be convergent to a real number if for every " > 0, there exists a N 2 N such that jxjk j < " whenever j; k N .
A real double sequence x = fxjkgj;k2N is said to be a Cauchy sequence if for every " > 0, there exist N; M 2 N such that for all j; p N and k; q M , jxjk xpqj < ".
De…nition 2.13. Let (S;F; ) be a PM space. A double sequence x = fxjkgj;k2Nin S is said to be strongly convergent to a point 2 S if for any t > 0, there exists a natural number M such that xjk2 N (t) whenever j; k M . In this case we write xjk! or lim
j!1 k!1
xjk = .
Similarly a double sequence x = fxjkgj;k2Nin S is called a strong Cauchy double sequence if for any t > 0, there exist natural numbers N; M such that for all j; p N ; k; q M , (xjk; xpq) 2 U(t).
We now recall some basic concepts related to statistical convergence of double sequences (see [4; 5; 9; 12; 18] for more details).
Let K N N. Let K(n; m) be the number of (j; k) 2 K such that j n; k m. The number d2(K) = lim sup
m!1 n!1
K(n;m)
nm is called the upper double natural density of K. If the sequence fK(n;m)nm gn;m2Nhas a limit in Pringsheim’s sense then we say that K has the double natural density and it is denoted by
d2(K) = lim m!1
n!1
K(n; m) nm :
De…nition 2.14. [12] A double sequence x = fxjkgj;k2N of real numbers is said to be statistically convergent to 2 R if for every " > 0, we have d2(A(")) = 0 where A(") = f(j; k) 2 N N : jxjk j "g. In this case we write st lim
j!1 k!1
xjk= .
A statistically convergent double sequence of elements in a metric space (X; ) is de…ned essentially in the same way using (xjk; ) " instead of jxjk j ". De…nition 2.15. [12] Let (X; ) be a metric space. A double sequence x = fxjkgj;k2Nin X is said to be statistically Cauchy if for every " 0, there exist natural numbers N = N (") and M = M (") such that for all j; p N and k; q M ,
d2(f(j; k) 2 N N : (xjk; xpq) "g) = 0
Theorem 2.4. [12] A double sequence x = fxjkgj;k2Nin a metric space (X; ) is statistically convergent to 2 X if and only if there exists a subset K = f(j; k) 2 N Ng of N N such that d2(K) = 1 and lim
j!1 k!1 (j;k)2K
xjk= .
Theorem 2.5. [12] For a sequence x = fxjkgj;k2Nin a metric space (X; ), the following statements are equivalent:
(1)x is statistically convergent to l 2 X. (2)x is statistically Cauchy.
(3)There exists a subset M N N such that d2(M ) = 1 and fxjkg(j;k)2M converges to l.
3. Strong Statistical Convergence of Double Sequences
In this section we introduce the concepts of strong statistically convergent dou-ble sequences and strong statistically Cauchy doudou-ble sequences in a PM space (S;F; ) and present some of its basic properties.
De…nition 3.1. Let (S;F; ) be a PM space. A double sequence x = fxjkgj;k2N in S is said to be strongly statistically convergent to a point 2 S if for any t > 0,
d2(f(j; k) 2 N N : Fxjk (t) 1 tg) = 0:
In this case we write xjk stat
! or st lim
j!1 k!1
xjk = . We call the strong statistical limit of x = fxjkgj;k2N.
The above de…nition may be restated as follows: st lim j!1 k!1
xjk= if and only if for each t > 0, d2(f(j; k) 2 N N : xjk2 N (t)g) = 0.=
Theorem 3.1. For a double sequence x = fxjkgj;k2Nin a PM space (S;F; ), the following statements are equivalent:
(1)st lim j!1 k!1 xjk= . (2)For each t > 0, d2(f(j; k) 2 N N : dL(Fxjk ; "0) tg) = 0. (3)st lim j!1 k!1 dL(Fxjk ; "0) = 0.
Proof. The proof of the theorem directly follows from Theorem 2.1 and De…-nition 2.14.
Theorem 3.2. Let (S;F; ) be a PM space. If x = fxjkgj;k2N and y = fyjkgj;k2N are double sequences in S such that st jlim
!1 k!1 xjk = and st lim j!1 k!1
yjk= then we have st lim j!1 k!1
dL(Fxjkyjk; F ) = 0.
Proof. It is known that F is uniformly continuous mapping from S S into + if is continuous and S is endowed with the strong topology (see [24]). So for any t > 0, there exists a = (t) > 0 such that dL(F ; Fpq) < t whenever p 2 N ( ) and q 2 N ( ). We assume that st lim
j!1 k!1 xjk= and st lim j!1 k!1 yjk= . Then for any t > 0, we have f(j; k) 2 N N : dL(Fxjkyjk; F ) tg
f(j; k) 2 N N : xjk 2 N ( )g [ f(j; k) 2 N= N : yjk 2 N ( )g. Therefore= d2(f(j; k) 2 N N : dL(Fxjkyjk; F ) tg) d2(f(j; k) 2 N N : xjk 2= N ( )g) + d2(f(j; k) 2 N N : yjk2 N ( )g). Since x= jk stat ! and yjk stat ! , each set on the right hand side of the above inequality has double natural density zero. So we have d2(f(j; k) 2 N N : dL(Fxjkyjk; F ) tg) = 0 for each t > 0.
Hence st lim j!1 k!1
dL(Fxjkyjk; F ) = 0.
Theorem 3.3. Let (S;F; ) be a PM space and x = fxjkgj;k2N be a double sequence in S. Then x is strongly statistically convergent to if and only if there is another double sequence y = fyjkgj;k2N such that xjk = yjk for all (j; k) 2 N N except for a set of double natural density zero and that is strongly convergent to .
Proof. Let st lim j!1 k!1 xjk= . Then st lim j!1 k!1 dL(Fxjk ; "0) = 0. Consequently
lim j!1 k!1 (j;k)2D
dL(Fxjk ; "0) = 0 i.e. fxjkg(j;k)2D is strongly convergent to . Therefore
for any t > 0, there exists a N (t) 2 N such that (j; k) 2 D and j N (t), k N (t) imply xjk2 N (t). Now we de…ne a double sequence y = fyjkgj;k2N by
yjk = xjk; if (j; k) 2 D = ; if (j; k) =2 D:
Then y is strongly convergent to and xjk= yjk for all (j; k) 2 N N except for a set D of double natural density zero.
Conversely let x = fxjkgj;k2Nand y = fyjkgj;k2Nbe two double sequences in S such that xjk = yjk for all (j; k) 2 N N except for a set A of double natural density zero and let y be strongly convergent to 2 S. Then for t > 0 we have f(j; k) 2 N N : xjk 2 N (t)g= f(j; k) 2 N N : xjk 6= yjkg [ f(j; k) 2 N N : yjk 2 N (t)g = A [ f(j; k) 2 N= N : yjk 2 N (t)g. Since y strongly= converges to the later set is contained in the union of a …nite number of rows and columns of N N and so d2(f(j; k) 2 N N : yjk 2 N (t)g) = 0. Also= because d2(A) = 0 we get d2(f(j; k) 2 N N : xjk2 N (t)g) = 0 for each t > 0= which implies that x is strongly statistically convergent to .
De…nition 3.2. Let (S;F; ) be a PM space. A double sequence x = fxjkgj;k2N in S is said to be strongly statistically Cauchy if for every t > 0, there exist N = N (t) 2 N and M = M(t) 2 N such that for all j; p N and k; q M ,
d2(f(j; k) 2 N N : Fxjkxpq(t) 1 tg) = 0:
As a consequence of De…nition 3.2 and Theorem 2.1, the above de…nition may be restated as follows:
A double sequence x = fxjkgj;k2Nin S is said to be strongly statistically Cauchy if and only if for every t > 0, there exist N = N (t) 2 N and M = M(t) 2 N such that for all j; p N and k; q M ,
d2(f(j; k) 2 N N : dL(Fxjkxpq; "0) tg) = 0:
Theorem 3.4. In a PM space (S;F; ) every strongly statistically convergent double sequence is also a strongly statistically Cauchy sequence.
Proof. Let x = fxjkgj;k2Nbe a double sequence in S that is strongly statisti-cally convergent to 2 S. Now by Theorem 2.2 we can say that for any t > 0, there is an > 0 such that dL(FxjkxN M; "0) < t whenever dL(Fxjk ; "0) <
and dL(F xN M; "0) < . Since st lim
j!1 k!1
d2(f(j; k) 2 N N : dL(Fxjk ; "0) g) = 0. Now choose two natural
num-bers N = N ( ) and M = M ( ) such that dL(F xM N; "0) . Therefore for
any t > 0, there exists (t) > 0 and N (t); M (t) 2 N (choosing N( ) = N(t), M ( ) = M (t)) such that
f(j; k) 2 N N : j N; k M and dL(FxjkxN M; "0) tg
f(j; k) 2 N N : dL(Fxjk ; "0) g [ f(N; M)g:
Then d2(f(j; k) 2 N N : j N; k M and dL(FxjkxN M; "0) tg) = 0. This
shows that x = fxjkgj;k2Nis strongly statistically Cauchy.
Theorem 3.5. Let x = fxjkgj;k2Nbe a double sequence in a PM space (S;F; ). If x is strongly statistically Cauchy then for every t > 0, there exists a set At N N with d2(At) = 0 such that Fxjkxrs(t) > 1 t whenever (j; k); (r; s) =2 At.
Proof. Using Theorem 2.2 we can say that for any t > 0, there exists a (t) > 0 such that
(1) Fpr(t) > 1 t whenever Fpq( ) > 1 and Fqr( ) > 1
Now let t > 0 be given. Choose = (t) > 0 such that (1) holds. Since x is strongly statistically Cauchy there exist N = N ( ), M = M ( ) 2 N such that
d2(f(j; k) 2 N N : FxjkxN M( ) 1 g) = 0:
Now put A = f(j; k) 2 N N : FxjkxN M( ) 1 g. Then d2(A ) = 0,
FxjkxN M( ) > 1 and FxrsxN M( ) > 1 whenever (j; k); (r; s) =2 A . Thus
for every t > 0, there exists a set A = At N N with d2(At) = 0 and Fxjkxrs(t) > 1 t whenever (j; k); (r; s) =2 At.
Corollary 3.1. If x = fxjkgj;k2Nis a strongly statistically Cauchy sequence in a PM space (S;F; ) then for every t > 0, there exists a set Bt N N with d2(Bt) = 1 such that Fxmnxrs(t) > 1 t for any (m; n); (r; s) 2 Bt.
Proof. Proof directly follows from the above Theorem.
Theorem 3.6. Let (S;F; ) be a PM space. If x = fxjkgj;k2N and y = fyjkgj;k2Nare strongly statistically Cauchy sequences in S then fFxjkyjkgj;k2N
is a statistically Cauchy sequence in ( +; d L).
Proof. The proof is straightforward and so it has been omitted.
4. Strong Statistical Limit Points and Strong Statistical Cluster Points for Double Sequences
In this section we introduce the ideas of strong statistical limit points, strong statistical cluster points of double sequences in a PM space and establish some of their properties.
Throughout the section we denote the PM space (S;F; ) by S only.
De…nition 4.1. Let K be a subset of N N such that for each (i; j) 2 N N, there exists a (m; n) 2 K such that m > i; n > j. If x = fxjkgj;k2Nis a double sequence in S then we de…ne fxgK = fxjkg(j;k)2K as a subsequence of x. If d2(K) = 0 then we say that fxgKis a thin subsequence of x. In case d2(K) > 0 or d2(K) does not exist then, fxgK is called a non-thin subsequence of x. De…nition 4.2. An element l 2 S is said to be a strong limit point of a double sequence x = fxjkgj;k2N in S if there exists a subsequence x which is strongly convergent to l.
De…nition 4.3. Let x = fxjkgj;k2Nbe a double sequence in a PM space S. An element 2 S is said to be a strong statistical limit point of x if there exists a non-thin subsequence of x that strongly converges to .
We denote by S2(Lx) and S2( x) the set of all strong limit points and the set of all strong statistical limit points of x = fxjkgj;k2Nrespectively.
De…nition 4.4. An element 2 S is said to be a strong statistical cluster point of a double sequence x = fxjkgj;k2Nin S if and only if for each t > 0,
d2(f(j; k) 2 N N : Fxjk (t) > 1 tg) > 0:
We denote the set of all strong statistical cluster points of x = fxjkgj;k2N by S2( x). We shall now study the relationship among S2(Lx), S2( x) and S2( x). Theorem 4.1. For any double sequence x = fxjkgj;k2Nin S we have S2( x) S2( x) S2(Lx).
Proof. Let 2 S2( x). Then there exists a set M = f(mj; mk) : j; k 2 Ng such that d2(M ) > 0 and
lim mj!1
mk!1
xmjmk = :
Let t > 0 be given. Then by (2) there exists a K0 2 N such that for mj K0; mk K0 we have xmjmk 2 N (t) i.e. Fxmj mk (t) > 1 t. So we have
f(j; k) 2 N N : Fxmj mk (t) > 1 tg M n f(mj; mk) : either mj (K0 1) or mk (K0 1)g. Since d2(f(mj; mk); either mj (K0 1) or mk (K0 1)g) = 0, d2(f(j; k) 2 N N : Fxjk (t) > 1 tg) d2(M ) 0 > 0. This
implies that 2 S2( x). Therefore S2( x) S2( x).
Now let 2 S( x). Then for every t > 0, d2(f(j; k) 2 N N : Fxjk (t) >
1 tg) > 0. This shows that f(j; k) 2 N N : Fxjk (t) > 1 tg = M (say) is
not included in the union of a …nite number of rows and columns of N N. So there exists a set M = f(mj; mk) : j; k 2 Ng N N such that fxgM strongly converges to i.e. x 2 S2(Lx). This completes the proof.
Theorem 4.2. Let x = fxjkgj;k2Nbe a double sequence in S. If x is strongly statistically convergent to 2 S then S2( x) = S2( x) = f g.
Proof. Let st lim j!1 k!1
xjk= . So for every t > 0, d2(f(j; k) 2 N N : Fxjk (t) >
1 tg) = 1. Therefore 2 S2( x). Now assume that there exists at least one 2 S2( x) such that 6= . Then there are t; t0 > 0, such that f(j; k) 2 N N : Fxjk (t) 1 tg f(j; k) 2 N N : Fxjk (t0) > 1 t0g holds. Hence we get
d2(f(j; k) 2 N N : Fxjk (t) 1 tg) d2(f(j; k) 2 N N : Fxjk (t0) > 1 t0g).
Since st lim j!1 k!1
xjk= we have d2(f(j; k) 2 N N : Fxjk (t) 1 tg) = 0 and
therefore d2(f(j; k) 2 N N : Fxjk (t0) > 1 t0g) = 0 which contradicts that
2 S2( x). Hence we conclude that S2( x) = f g. Since st lim j!1 k!1
xjk= , from Theorem 3.3 we have 2 S2( x). Now by Theorem 4.1 S2( x) = S2( x) = f g. Theorem 4.3. For any double sequence x = fxjkgj;k2 N in S, the set S2( x) is strongly closed.
Proof. Let 2 (S2( x)) where denotes the strong closure. If t > 0, S2( x) contains some point r 2 N (t). Consider t0 > 0 such that Nr(t0) N (t). Since p 2 S2( x) we have d2(f(j; k) 2 N N : xjk 2 Nr(t0)g) > 0 and so d2(f(j; k) 2 N N : xjk 2 N (t)g) > 0. This implies 2 S2( x) i.e.
(S2( x)) S2( x). Hence S2( x) is strongly closed in S.
Theorem 4.4. If x = fxjkgj;k2Nand y = fyjkgj;k2Nare two double sequences in S such that xjk= yjk for all (j; k) 2 N N except for a set of double natural density zero then S2( x) = S2( y) and S2( x) = S2( y).
Proof. Let x = fxjkgj;k2N, y = fyjkgj;k2Nbe two double sequences in S such that d2(f(j; k) 2 N N : xjk6= yjkg) = 0. Let 2 S2( x). Then there exists a set M = f(mj; mk); j; k 2 Ng such that d2(M ) > 0 and lim
mj!1
mk!1
xmjmk = . Since
d2(f(j; k) 2 N N : (j; k) 2 M and xjk6= yjkg) = 0 and d2(M ) > 0 we have d2(f(j; k) 2 N N : (j; k) 2 M and xjk= yjkg) > 0. Let M0= f(j; k) 2 N N : (j; k) 2 M and xjk = yjkg. Then fygM0 is a non-thin subsequence of fygM
which strongly converges to . Therefore 2 S2( y) and S2( x) S2( y). Now by symmetry S2( y) S2( x). Hence S2( x) = S2( y). Similarly we can show that S2( x) = S2( y). This completes the proof of the theorem.
Lemma 4.5. Let x = fxjkgj;k2Nbe a double sequence in a PM space (S;F; ). Let A S be a strongly compact set and A \ S2( x) = ;. Then the set f(j; k) 2 N N : xjk2 Ag has double natural density zero.
Proof. Since A \ S2( x) = ;, for every 2 A there is a positive number t = t(") > 0 such that d2(f(j; k) 2 N N : F xjk(t) > 1 tg) = 0. Let B (t) =
fy 2 S : Fy(t) > 1 tg. Then the set of strongly open sets fB (t) : 2 Ag forms a strong open cover of A. Since A is strongly compact so there is a …nite subcover of fB (t) : 2 Ag, say fB1(t1); B2(t2); :::; B p(tp)g. Then A
p S i=1
B i(ti) and d2(f(j; k) 2 N N : F ixjk(ti) > 1 tig) = 0 for every i = 1; 2; 3; :::; p. Now
we can write jf(j; k) 2 N N : j n; k m and xjk 2 Agj p P i=1jf(j; k) 2 N N : j n; k m and F ixjk(ti) > 1 tigj. So lim n!1 m!1 1 nmjf(j; k) 2 N N : j n; k m and xjk 2 Agj p P i=1 lim n!1 m!1 1 nmjf(j; k) 2 N N : j n; k m and Fixjk(ti) > 1 tigj = 0 which gives d2(f(j; k) 2 N N : xjk2 Ag) = 0 . This
completes the proof.
De…nition 4.6. A double sequence x = fxjkgj;k2N in a PM space (S;F; ) is called strongly bounded if there exists a strongly compact set B such that xjk2 B for all (j; k) 2 N N.
Theorem 4.6. If a double sequence x = fxjkgj;k2Nin a PM space (S;F; ) has a strongly bounded non-thin subsequence then the set S2( x) is a nonempty strongly closed set.
Proof. Let fxjkg(j;k)2K (K N N) be a strongly bounded non-thin sub-sequence of x i.e. d2(K) 6= 0 and there is a strongly compact set A such that xjk 2 A for each (j; k) 2 K. Now if S2( x) = ; then A \ S2( x) = ; and so by Lemma 4.5 we have d2(f(j; k) 2 N N : xjk 2 Ag) = 0. But jf(j; k) 2 N N : j n; k m and (j; k) 2 Kgj jf(j; k) 2 N N : j n; k m and xjk 2 Agj which gives d2(K) = 0. This contradicts our assumption and therefore S2( x) 6= ;. Now S2( x) is strongly closed by Theorem 4.3 which completes the proof.
De…nition 4.7. A double sequence fxjkg(j;k)2Nin a PM space (S;F; ) is said to be strongly statistically bounded if there exists a strongly compact set B such that d2(f(j; k) 2 N N : xjk2 Bg) = 0=
Obviously in this case d2(f(j; k) 2 N N : xjk 2 Bg) = 1. Clearly, every strongly bounded sequence is strongly statistically bounded.
Corollary 4.7. If a double sequence fxjkg(j;k)2N in a PM space (S;F; ) is strongly statistically bounded then the set S2( x) is nonempty and strongly compact.
Proof. Let B be a strongly compact set such that d2(f(j; k) 2 N N : xjk 2= Bg) = 0. Then d2(f(j; k) 2 N N : xjk 2 Bg) = 1 6= 0 which implies that B contains a non-thin subsequence of x. Hence by Theorem 4.6 S2( x) is nonempty and strongly closed. Now to prove that S2( x) is strongly compact we need to show that S2( x) B. If possible let us assume that 2 S2( x) but 2 B.=
Since B is strongly compact there exists a t > 0 such that N (t)\B = ;. In this case we have f(j; k) 2 N N : F xjk(t) > 1 tg f(j; k) 2 N N : xjk 2 Bg.=
Therefore d2(f(j; k) 2 N N : F xjk(t) > 1 tg) = 0 which contradicts the
fact that 2 S2( x). Therefore S2( x) B. This completes the proof of the theorem.
5. Acknowledgement:
The authors are grateful to Prof. Pratulananda Das for his advice during the preparation of this paper. The work of the …rst author was supported by Coun-cil of Scienti…c and Industrial Research, HRDG, India, under the project No. 25(0186)/10/EMR-II. The authors are also thankful to the referee for his valu-able suggestions.
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