Density by moduli and Wijsman statistical convergence (Q1676406)

The paper under review is about Wijsman statistical convergence of sequences of sets in a metric space. In recent years, a number of generalizations of the concept of convergence have been made and applied to different problems. The most popular of these generalizations is statistical convergence. Statistical convergence depends on the natural density of subsets of the set \( \mathbb{N}=\left\{ 1,2,3,\dots\right\}\). The natural density \(d(K)\) of the set \( K\subseteq\mathbb{N}\) is defined by \[ d(K)=\underset{n\rightarrow \infty }{\lim }\frac{1}{n}\left| \left\{ k\leq n:k\in K\right\} \right| , \] where \(\left| \left\{ k\leq n:k\in K\right\} \right| \) denotes the cardinality of this the set. Statistical convergence can then be defined as follows [\textit{H. Fast}, Colloq. Math. 2, 241--244 (1951; Zbl 0044.33605)]: a~sequence \(x=(x_{k})\subset\mathbb{R}\) is said to be statistical convergent to \(L\in \mathbb{R}\) if, for each \(\varepsilon >0\), the set \(\left\{ k\in\mathbb{N} :\left| x_{k}-L\right| \geq \varepsilon \right\} \) has the natural density zero. Obviously, since the natural density of finite sets is zero, the usual convergence of a sequence implies its statistical convergence. The converse is not true. A new concept of density by moduli was introduced by \textit{A. Aizpuru} et al. [Quaest. Math. 37, No. 4, 525--530 (2014; Zbl 1426.40002)] that provides a non-matrix method of convergence, namely, \(f\)-statistical convergence, which is a generalization of statistical convergence. Definition. A~function \(f:\left[ 0,\infty \right) \rightarrow \left[ 0,\infty \right) \) is called a modulus function if (i) \(f(x)=0\) if and only if \(x=0,\) (ii) \(f(x+y)\leq f(x)+f(y)\) for all \(x,y\in \left[ 0,\infty \right)\), (iii) \(f\) is increasing, (iv) \(f\) is continuous. Definition. Let \(f:\left[ 0,\infty \right) \rightarrow \left[ 0,\infty \right) \) be an unbounded modulus function. The \(f\)-density \(d^{f}(K)\) of a set \( K\subseteq\mathbb{N}\) is defined as \[ d^{f}(K)=\underset{n\rightarrow \infty }{\lim }\frac{1}{f(n)}f\left\{ \left| \left\{ k\leq n:k\in K\right\} \right| \right\} \] if this limit exists. Depending on this definition, a sequence \(x=(x_{k})\subset\mathbb{R}\) is said to be \(f\)-statistical convergent to \(L\in \mathbb{R} \) if, for each \(\varepsilon >0,\) the set \(\left\{ k\in\mathbb{N}:\left| x_{k}-L\right| \geq \varepsilon \right\} \) has \(f\)-density zero. Let \(\left( X,\rho \right) \) be a metric space. For any \(x\in X\) and non-empty set \(A\subset X\), the distance \(x\) to \(A\) is defined by \[ d(x,A)=\underset{y\in A}\inf \rho (x,y). \] The set of all non-empty closed subsets of the metric space \(\left( X,\rho \right)\) is denoted by \(CL(X)\). Definition. Let \(\left( X,\rho \right) \) be a metric space, \((A_{k})\subset CL(X)\) and \( A\in CL(X)\). Then \((A_{k})\) said to be: (a) Wijsman convergent to \(A\in CL(X)\) if the numerical sequence \(\left( d(x,A_{k})\right) \) is convergent to \(d(x,A)\) for each \(x\in X\); (b) Wijsman statistically convergent to \(A\in CL(X)\) if the numerical sequence \(\left( d(x,A_{k})\right) \) is statistically convergent to \(d(x,A) \) for each \(x\in X\); (c) Wijsman bounded if \[ \sup_k d(x,A_{k})<\infty \] holds for each \(x\in X\); (d) Wijsman Cesàro summable to \(A\in CL(X)\) if the numerical sequence \( \left( d(x,A_{k})\right) \) is Cesàro summable to \((d(x,A)\) for each \(x\in X\); (e) Wijsman strongly Cesàro summable to \(A\in CL(X)\) if the numerical sequence \(\left( d(x,A_{k})\right) \) is strongly Cesàro summable to \( d(x,A)\) for each \(x\in X\). By combining Wijsman convergence and \(f\)-density, \(f\)-Wijsman statistical convergence is defined as follows: Definition. (\(f\)-Wijsman statistical convergence) Let \(\left( X,\rho \right) \) be a metric space, \((A_{k})\subset CL(X)\) and \(f:\left[ 0,\infty \right) \rightarrow \left[ 0,\infty \right) \) be an unbounded modulus function. The sequence \((A_{k})\) is said to be \(f\)-Wijsman statistical convergent to \(A\in CL(X)\) if the sequence \((\left( d(x,A_{k})\right) \) is \(f\)-statistical convergent to \(d(x,A)\) for each \(x\in X\). This is denoted by \([WS^{f}]-\lim A_{k}=A\). In this study, the authors prove that the Wijsman convergent sequences are precisely those sequences which are \(f\)-Wijsman statistically convergent for any unbounded modulus function. Also, by defining Wijsman strong Cesàro summability with respect to a modulus, it is shown that if a sequence is Wijsman strongly Cesàro summable, then it is Wijsman strongly Cesàro summable with respect to all moduli \(f\). Finally, the relation between Wijsman strong Cesàro summability with respect to a modulus \(f\) and \(f\)-Wijsman statistical convergence is studied.