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Statistical convergence in probabilistic generalized metric spaces w.r.t. strong topology
Journal of Inequalities and Applications volume 2021, Article number: 134 (2021)
Abstract
In this paper, the concept of probabilistic gmetric space with degree l, which is a generalization of probabilistic Gmetric space, is introduced. Then, by endowing strong topology, the definition of ldimensional asymptotic density of a subset A of \(\mathbb{N}^{l}\) is used to introduce a statistically convergent and Cauchy sequence and to study some basic facts.
Introduction
The theory of probabilistic metric space (PMspace) as a generalization of ordinary metric space was introduced by Menger in [12]. In this space, distribution functions are considered as the distance of a pair of points in statistics rather than deterministic.
The concept of the generalized metric space (briefly Gmetric space) was introduced by Mustafa and Sims in 2006 [16]. Then, in 2014, Zhou et al. [26] generalized the notion of PMspace to the Gmetric spaces and defined the probabilistic generalized metric space which is denoted by PGMspace.
In [3], Choi et al. proposed a generalization of Gmetric space named gmetric space with degree l, in which the distance function with degrees \(l=1,2\) is equivalent to ordinary and Gmetric, respectively.
The idea of statistical convergence was first introduced by Steinhaus [25] for real sequences and developed by Fast [7], then reintroduced by Shoenberg [22]. Many authors, such as [4, 6, 8, 9, 17, 21], have discussed and developed this concept. The theory of statistical convergence has many applications in various fields such as approximation theory [5], finitely additive set functions [4], trigonometric series [27], and locally convex spaces [11].
In 2008, Sencimen and Pehlivan [24] introduced the concepts of statistically convergent sequence and statistically Cauchy sequence in the probabilistic metric space endowed with strong topology.
The purpose of this paper is to develop a concept to generalize the probabilistic Gmetric space to the probabilistic gmetric space with degree l. Here, the notation of the generalized space is still referred as PGMspace. The ldimensional asymptotic density of a subset A of \(\mathbb{N}^{l}\) defined previously by the author in [1] is used to introduce the statistically convergent and Cauchy sequences with respect to strong topology, and some basic facts are studied. Note that in this definition \(l=1\) and \(l=2\) values coincide exactly with the statistical convergence in PMspace and PGMspace (related to Gmetric), respectively. Thus, the definitions and the obtained results show that this study is more comprehensive.
Preliminaries
In this section, some basic definitions and results related to PMspace, PGMspace, and statistical convergence are presented and discussed. First, recall the definition of triangular norm (tnorm) as follows.
Definition 2.1
([23])
A mapping \(T:[0,1]\times [0,1] \to [0,1]\) is called a continuous tnorm if T satisfies the following conditions:

(i)
T is commutative and associative, i.e., \(T(a,b)=T(b,a)\) and \(T(a,T(b,c))=T(T(a,b),c)\) for all \(a,b,c\in [0,1]\);

(ii)
T is continuous;

(iii)
\(T(a,1)=a\) for all \(a\in [0,1]\);

(iv)
\(T(a,b)\leq T(c,d)\) whenever \(a\leq c\) and \(b\leq d\) for all \(a,b,c,d\in [0,1]\).
A distribution function F is a map from extended reals \(\mathbb{R}_{\infty }:=\mathbb{R}\cup \{\infty , \infty \}\) into \([0, 1]\) such that it is nondecreasing, leftcontinuous at every real number, and \(F(\infty )=0\) and \(F(\infty )=1\). The set of all distribution functions is denoted by Δ and \(\Delta ^{+}:=\{F\in \Delta : F(0)=0\}\).
Definition 2.2
([23])
A Menger probabilistic metric space (PMspace) is a triple \((X,F,T)\), where X is a nonempty set, T is a continuous tnorm. and F is a mapping from \(X\times X\to \Delta ^{+}\) satisfying the following conditions:
(\(F_{(x,y)}\) denotes the value of F at the pair \((x,y)\))

(i)
\(F_{(x,y)}(t)=1\) for all \(x,y\in X\) and \(t>0\) if and only if \(x=y\);

(ii)
\(F_{(x,y)}(t)=F_{(y,x)}(t)\);

(iii)
\(F_{(x,y)}(t+s)\geq T (F_{(x,z)}(t), F_{(z,y)}(s) )\) for all \(x,y,z\in X\) and \(t,s\geq 0\).
Definition 2.3
([16])
Let X be a nonempty set and \(G:X\times X \times X \to \mathbb{R}^{+}\), be a function satisfying:

1)
\(G(x,y,z)=0\) if \(x=y=z\);

2)
\(0< G(x,x,y)\) for all \(x,y \in X\) with \(x\neq y\);

3)
\(G(x,x,y)\leq G(x,y,z)\) for all \(x,y,z\in X\) with \(z\neq y\);

4)
\(G(x,y,z)=G(x,z,y)=G(y,z,x)=\cdots\) (symmetry in all three variables);

5)
\(G(x,y,z)\leq G(x,a,a)+G(a,y,z)\) for all \(x,y,z,a \in X\).
Then the pair \((X, G)\) is called Gmetric space.
The following definition is a developing of PMspace on Gmetric.
Definition 2.4
([26])
A Menger probabilistic Gmetric space (PGMspace) is a triple \((X, G^{*}, T)\), where X is a nonempty set, T is a continuous tnorm, and \(G^{*}\) is a mapping from \(X\times X \times X\) into \(\Delta ^{+}\), satisfying the following conditions:

(i)
\(G^{*}_{(x,y,z)}(t)=1\) for all \(x, y,z \in X\) and \(t>0\) if and only if \(x=y=z\);

(ii)
\(G^{*}_{(x,x,y)}(t)\geq G^{*}_{(x,y,z)}(t)\) for all \(x,y\in X\) with \(z\neq y\) and \(t>0\);

(iii)
\(G^{*}_{(x,y,z)}(t)=G^{*}_{(x,z,y)}(t)=G^{*}_{(y,x,z)}(t)=\cdots\) (symmetry in all three variables);

(iv)
\(G^{*}_{(x,y,z)}(t+s)\geq T(G^{*}_{(x,a,a)}(t), G^{*}_{(a,y,z)}(s))\) for all \(x,y,z,a\in X\) and \(s,t\geq 0\).
Definition 2.5
([26])
Let \((X, G^{*}, T)\) be a PGMspace and \(x_{0}\in X\). For \(\epsilon >0\) and \(0<\delta <1\), the \((\epsilon , \delta )\)neighborhood of \(x_{0}\) is defined as follows:
Definition 2.6
([26])

(i)
A sequence \(\{x_{n}\}\) in a PGMspace \((X, G^{*}, T)\) is said to be convergent to a point \(x\in X\) if, for every \(\epsilon >0\) and \(0<\delta <1\), there exists a positive integer \(M_{\epsilon ,\delta }\) such that \(x_{n}\in N_{x}(\epsilon , \delta )\) whenever \(n>M_{\epsilon ,\delta }\).

(ii)
A sequence \(\{x_{n}\}\) in a PGMspace \((X, G^{*}, T)\) is called a Cauchy sequence if, for every \(\epsilon >0\) and \(0<\delta <1\), there exists a positive integer \(M_{\epsilon ,\delta }\) such that \(G^{*}_{(x_{m}, x_{n},x_{l})}(\epsilon )>1\delta \) whenever \(m, n, l>M_{\epsilon ,\delta }\).

(iii)
A PGMspace \((X, G^{*}, T)\) is said to be complete if every Cauchy sequence in X converges to a point in X.
In the following, some basic concepts of statistical convergence are discussed.
Definition 2.7
([7])
Let \(A\subset \mathbb{N}\) and \(A(n)=\{k\in A ; k\leq n\}\). Then the asymptotic density of A is defined as follows:
For a subset A of \(\mathbb{N}\), if \(\delta (A)=1\), then it is said to be statistically dense. It is clear that \(\delta (\mathbb{N}A)=1\delta (A)\).
Definition 2.8
([7])
A sequence \(\{x_{n}\}\) in \(\mathbb{R}\) is said to be statistically convergent to a point x in \(\mathbb{R}\) if, for each \(\epsilon >0\),
For more information about statistical convergence, the references [2, 4, 7–10, 13–15, 18–20] can be addressed.
Main results
In this section the main definitions and results are introduced and discussed. First of all, consider the following definition which is a generalization of a Gmetric space to an ldimensional case, where \(l\in \mathbb{N}\).
Definition 3.1
([3])
Let X be a nonempty set. A function \(g:X^{l+1}\longrightarrow \mathbb{R}_{+}\) is called a gmetric with degree l on X if it satisfies the following conditions:

g1)
\(g(x_{0}, x_{1},\ldots, x_{l})=0 \) if and only if \(x_{0}=x_{1}=\cdots=x_{l}\),

g2)
\(g(x_{0}, x_{1},\ldots, x_{l})=g(x_{\sigma (0)}, x_{\sigma (1)},\ldots,\ldots,x_{ \sigma (l)})\) for permutation σ on \(\{0, 1,\ldots, l\}\),

g3)
\(g(x_{0}, x_{1},\ldots, x_{l})\leq g(y_{0}, y_{1},\ldots, y_{l})\) for all \((x_{0}, x_{1},\ldots, x_{l}), (y_{0}, y_{1},\ldots, y_{l})\in X^{l+1}\) with \(\{x_{i}: i=0, 1,\ldots, l\}\subseteq \{y_{i}: i=0, 1,\ldots, l\}\),

g4)
For all \(x_{0}, x_{1},\ldots, x_{s}, y_{0}, y_{1},\ldots, y_{t}, w\in X\) with \(s+t+1=l\),
$$ g(x_{0}, x_{1},\ldots, x_{s}, y_{0}, y_{1},\ldots, y_{t})\leq g(x_{0}, x_{1},\ldots, x_{s}, w, w,\ldots, w)+g(y_{0}, y_{1},\ldots, y_{t}, w, w,\ldots, w). $$
The pair \((X,g)\) is called a gmetric space. It is noteworthy that, if \(l=1\) (resp. \(l=2\)), then it is equivalent to an ordinary metric space (resp. Gmetric space).
Definition 3.2
([3])
Let \((X,g)\) be a gmetric space, \(x\in X\) be a point, and \(\{x_{k}\}\subseteq X\) be a sequence.

1)
\(\{x_{k}\}\) gconverges to x if for all \(\epsilon >0\) there exists \(N\in \mathbb{N}\) such that
$$ i_{1},\ldots, i_{l}\geq N\quad \Longrightarrow \quad g(x, x_{1},\ldots, x_{l})< \epsilon . $$ 
2)
\(\{x_{k}\}\) is said to be gCauchy if for all \(\epsilon >0\) there exists \(N\in \mathbb{N}\) such that
$$ i_{0}, i_{1},\ldots, i_{l}\geq N\quad \Longrightarrow\quad g(x_{i_{0}}, x_{i_{1}},\ldots, x_{i_{l}})< \epsilon . $$ 
3)
\((X, g)\) is complete if every gCauchy sequence in \((X, g)\) is gconvergent in \((X, g)\).
Now, by equipping Definition 2.4 with gmetric, we introduce the following definition that is a generalization.
Definition 3.3
A Menger probabilistic gmetric space (still is denoted as PGMspace) is a triple \((X, F, T)\), where X is a nonempty set, T is a continuous tnorm, and F is a mapping from \(X^{l+1}\) into \(\Delta ^{+}\), satisfying the following conditions:

(i)
\(F_{(x_{0}, x_{1},\ldots, x_{l})}(t)=1\) for all \(x_{0}, x_{1},\ldots, x_{l} \in X\) and \(t>0\) if and only if \(x_{0}= x_{1}= \cdots= x_{l}\);

(ii)
\(F_{(x_{0}, x_{1},\ldots, x_{l})}(t)\geq F_{(y_{0}, y_{1},\ldots, y_{l})}(t)\) for all \((x_{0}, x_{1},\ldots, x_{l}), (y_{0}, y_{1},\ldots, y_{l})\in X^{l+1}\) with \(\{x_{i}: i=0, 1,\ldots, l\}\subseteq \{y_{i}: i=0, 1,\ldots, l\}\);

(iii)
\(F_{(x_{0}, x_{1},\ldots, x_{l})}(t)=F_{(x_{\sigma (0)}, x_{\sigma (1)},\ldots,\ldots,x_{\sigma (l)})}(t)\) for permutation σ on \(\{0, 1,\ldots, l\}\);

(iv)
For all \(x_{0}, x_{1},\ldots, x_{m}, y_{0}, y_{1},\ldots, y_{n}, w\in X\) with \(m+n+1=l\),
$$ F_{(x_{0}, x_{1},\ldots, x_{m}, y_{0}, y_{1},\ldots, y_{n})}(t+s)\geq T\bigl(F_{(x_{0}, x_{1},\ldots, x_{m}, w, w,\ldots, w)}(t), F_{(y_{0}, y_{1},\ldots, y_{n}, w, w,\ldots, w)}(s)\bigr). $$
In the following, according to the generalization of asymptotic density given in [1], statistically convergent and Cauchy sequences in a PGMspace are introduced.
Definition 3.4
Let \((X, F,T)\) be a PGMspace. For any \(\epsilon >0\), \(0<\delta <1\) and \(x\in X\), the strong \((\epsilon , \delta )\)vicinity of x is defined by the subset \(M_{x}(\epsilon , \delta )\) of \(X^{l}\) as follows:
Next, we generalize the concept of asymptotic density of a set in an ldimensional case.
Definition 3.5
Let \(K\subset \mathbb{N}^{l}\), the ldimensional asymptotic density of K is defined by
Definition 3.6
Let \((X, F,T)\) be a PGMspace.

(i)
A sequence \(\{x_{n}\}\) in X is statistically convergent to a point x in X w.r.t. strong topology if, for any \(\epsilon >0\) and \(0<\delta <1\),
$$ \delta _{l}\bigl(\bigl\{ (i_{1}, i_{2},\ldots, i_{l})\in \mathbb{N}^{l}: F_{(x_{i_{1}}, x_{i_{2}},\ldots, x_{i_{l}}, x)}(\epsilon )\leq 1 \delta \bigr\} \bigr)=0, $$and is denoted by \(x_{n} \overset{st}{\longrightarrow } x\) or \(st\lim_{n\to \infty }x_{n}=x\).

(ii)
\(\{x_{n}\}\) is said to be statistically Cauchy w.r.t. strong topology if, for all \(\epsilon >0\) and \(0<\delta <1\), there exists \(i_{\epsilon }\in \mathbb{N}\) such that
$$ \delta _{l}\bigl(\bigl\{ (i_{1}, i_{2},\ldots, i_{l})\in \mathbb{N}^{l}: F_{(x_{i_{1}}, x_{i_{2}},\ldots, x_{i_{l}}, x_{i_{\epsilon }})}(\epsilon )\leq 1 \delta \bigr\} \bigr)=0. $$
We can restate part \((i)\) of the above definition as follows:
 (\(i^{\prime }\)):

\(x_{n} \overset{st}{\longrightarrow } x\) if and only if, for any \(\epsilon >0\) and \(0<\delta <1\),
$$ \delta _{l}\bigl(\bigl\{ (i_{1}, i_{2},\ldots, i_{l})\in \mathbb{N}^{l}: (x_{i_{1}}, x_{i_{2}},\ldots, x_{i_{l}})\notin M_{x}(\epsilon , \delta )\bigr\} \bigr)=0. $$
Theorem 3.7
Let \(\{x_{n}\}\) be a sequence in a PGMspace \((X, F, T)\) such that \(x_{n} \overset{st}{\longrightarrow } x\) and \(x_{n} \overset{st}{\longrightarrow } y\), then \(x=y\).
Proof
Let \(\epsilon >0\) and \(0<\delta <1\), by the continuity of T, there exists \(0<\delta _{0}<1\) such that
Set
and
Since \(x_{n} \overset{st}{\longrightarrow } x\) and \(x_{n} \overset{st}{\longrightarrow } y\), so \(\delta _{l}(A(\epsilon ,\delta ))=\delta _{l}(B(\epsilon ,\delta ))=0\) and hence \(\delta _{l}(C(\epsilon ,\delta ))=0\), therefore \(\delta _{l}(C^{c}(\epsilon ,\delta ))=1\). Suppose \((i_{1}, i_{2},\ldots, i_{l})\in C^{c}(\epsilon ,\delta )\), then by parts (ii) of Definition 3.3 and (iv) of Definition 2.1 we have
Since \(\delta >0\) is arbitrary, we conclude that \(F_{(x, y, y,\ldots, y)}(\epsilon )=1\), and therefore \(x=y\). □
Theorem 3.8
Every convergent sequence in a PGMspace is statistically convergent.
Proof
Let \(\{x_{n}\}\) be a sequence in the PGMspace \((X,F, T)\) that converges to a point \(x\in X\). For \(\epsilon >0\) and \(0<\delta <1\), there exists \(n_{0}\in \mathbb{N}\) such that, for all \(i_{1}, i_{2},\ldots, i_{l}\geq n_{0}\),
Set
then
and
so
□
Example 3.9 shows that the converse of the above theorem is not valid.
Example 3.9
Let \(X=\mathbb{R}\) and \(G:\mathbb{R}\times \mathbb{R}\times \mathbb{R}\longrightarrow \mathbb{R}^{+}\) be a Gmetric on \(\mathbb{R}\) defined by
\((T= \min )\) Define a function \(F:\mathbb{R}\times \mathbb{R}\times \mathbb{R}\longrightarrow \mathbb{R}^{+}\) as follows:
where \(H(t)\) and \(\mathcal{D}(t)\) are distribution functions as follows:
Now, consider the following sequence in \(\mathbb{R}\):
It is clear that \(\{x_{n}\}\) statistically converges to 1 but it is not convergent normally.
Definition 3.10
A set \(A=\{n_{k}: k\in \mathbb{N}\}\) is said to be statistically dense in \(\mathbb{N}\) if the set
has asymptotic density 1, i.e.,
Theorem 3.11
Let \(\{x_{n}\}\) be a sequence in the PGMspace \((X, F, T)\). Then the following are equivalent:

(i)
\(\{x_{n}\}\) statistically converges to a point \(x \in X\).

(ii)
There is a sequence \(\{y_{n}\}\) in X such that \(x_{n}=y_{n}\) for almost all n, and \(\{y_{n}\}\) converges to x.

(iii)
There is a statistically dense subsequence \(\{x_{n_{k}}\}\) of \(\{x_{n}\}\) such that \(\{x_{n_{k}}\}\) is convergent.

(iv)
There is a statistically dense subsequence \(\{x_{n_{k}}\}\) of \(\{x_{n}\}\) such that \(\{x_{n_{k}}\}\) is statistically convergent.
Proof
\((i\Longrightarrow \mathit{ii})\) Let \(\{x_{n}\}\) be a sequence that converges to x, so
For each \(k\in \mathbb{N}\), we can choose an increasing sequence \(\{n_{k}\}\) such that, for every \(n>n_{k}\),
Define the sequence \(\{y_{n}\}\) as follows:
Choose \(k\in \mathbb{N}\) such that \(\frac{1}{2^{k}}<\delta \). It is clear that \(\{y_{m}\}\) converges to x. Fix \(n\in \mathbb{N}\), for \(n_{k}< n\leq n_{k+1}\), we have
Hence,
so
\((\mathit{ii} \Longrightarrow \mathit{iii})\) Let \(\{y_{n}\}\) be a convergent sequence in X and \(A=\{n\in \mathbb{N}: y_{n}\neq x_{n}\}\). We have \(\delta _{l}(A)=1\), so the sequence \(\{y_{n}\}\) is a statistical dense subsequence of \(\{x_{n}\}\) that is convergent.
\((\mathit{iii} \Longrightarrow \mathit{iv})\) It is obvious from Theorem 3.8.
\((\mathit{iv} \Longrightarrow i)\) Let \(\{x_{n_{k}}\}\) be a statistically dense subsequence of \(\{x_{n}\}\) that is statistically convergent to a point \(x\in X\). Set \(A=\{n_{k}: k\in \mathbb{N}\}\), so \(\delta _{l}(A)=1\). For ϵ> and \(0<\delta <1\),
Hence,
So,
Therefore \(\{x_{n}\}\) statistically converges to x. □
The following corollary is a direct consequence of the above theorem.
Corollary 3.12
Every statistically convergent sequence in a PGMspace has a convergent subsequence.
Theorem 3.13
Every statistically convergent sequence in a PGMspace is statistically Cauchy.
Proof
Suppose that \(\{x_{n}\}\) is a sequence that statistically converges to a point x. Let \(\epsilon >0\) and \(0<\delta <1\). Since T is continuous, there are \(0<\delta _{1}<1\) and \(0<\delta _{2}<1\) such that \(T(1\delta _{1}, 1\delta _{2})>1\delta \). On the other hand, there exists \(i_{\epsilon }\) such that
Since
so
Hence
Since \(\{x_{n}\}\) is statistically convergent, so the righthand side of the previous inequality is zero. Therefore it shows that the sequence \(\{x_{n}\}\) is statistically Cauchy. □
Definition 3.14
Let \((X, F,T)\) be a PGMspace. If every statistically Cauchy sequence is statistically convergent, then \((X, F,T)\) is said to be statistically complete.
Corollary 3.15
Every statistically complete PGMspace is complete.
Proof
Let \((X,F,T)\) be a statistically complete PGMspace. Suppose that \(\{x_{n}\}\) is a Cauchy sequence in \((X,F,T)\), so it is a statistically Cauchy sequence. Since X is statistically complete, so \(\{x_{n}\}\) is statistically convergent. By Corollary 3.12, there is a subsequence \(\{x_{n_{k}}\}\) of \(\{x_{n}\}\) that converges to a point \(x\in X\). By the continuity of T, for \(0<\delta <1\), there exist \(0<\delta _{1}, \delta _{2}, \delta _{3}, \delta _{4}<1\) such that
Let \(\delta _{5}:=\max \{\delta _{2}, \delta _{3}\}\), then we have
For \(\epsilon >0\), since \(\{x_{n}\}\) is Cauchy, then there exist \(N_{1}\in \mathbb{N}\) and \(x_{i_{\epsilon }}\in \{x_{n}\}\) such that, for all \(i_{1}, i_{2},\ldots, i_{l}\geq N_{1}\),
and since \(x_{n_{k}}\longrightarrow x\), there exists \(N_{2}\geq N_{1}\) such that, for \(i_{n_{1}}, i_{n_{2}},\ldots, i_{n_{l}}\geq N_{2}\),
For \(i_{1}, i_{2},\ldots, i_{l}, i_{n_{1}}, i_{n_{2}},\ldots, i_{n_{l}}\geq N_{2}\), we have
The third inequality arises from part \((\mathit{ii})\) of Definition 3.3 and the nondecreasing property of F. So, \(\{x_{n}\}\) is convergent and therefore \((X,F,T)\) is complete. □
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Abazari, R. Statistical convergence in probabilistic generalized metric spaces w.r.t. strong topology. J Inequal Appl 2021, 134 (2021). https://doi.org/10.1186/s1366002102669w
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MSC
 40A35
 54E70
 54E35
Keywords
 Probabilistic metric space
 Generalized metric space
 Statistical convergence
 Statistical Cauchy sequence
 Strong topology