On ideal convergence of double sequences in probabilistic normed spaces. (English) Zbl 1240.40032

An ideal in a nonempty set \(X\) is a family \(I\) of subsets of \(X\) such that (a) \(X\in I\); (b) \(A,B\in I\Rightarrow A\cup B\in I\), and (c) \(A\in I\) and \(B\subset I\Rightarrow B\in I\). If \(I\) is an ideal in \(X\), then \(F(I)=\{X\setminus A : A\in I\}\) is a filter on \(X\), called the filter associated to the ideal \(I\). A sequence \((x_k)\) in \(\mathbb R\) is called \(I\)-convergent to \(\xi\in\mathbb R\) if \(|[k\in\mathbb N:| x_k-L|\geq\epsilon\}\in I\) for every \(\epsilon > 0\). \(I\)-convergence was considered first by P. Kostyrko, T. Šalát and W. Wilczyński [Real Anal. Exch. 26, No. 2, 669–685 (2001; Zbl 1021.40001)], and independently by F. Nuray and W. H. Ruckle [J. Math. Anal. Appl. 245, No. 2, 513–527 (2000; Zbl 0955.40001)], called by them generalized statistical convergence.
The authors consider this type of convergence for double sequences in a probabilistic normed space (PNS) \((X,\nu,*)\), where \(*\) is a triangle function on \([0,1]\) and \(\nu\) a probabilistic norm on the real vector space \(X\) (see [B. Schweizer and A. Sklar, Probabilistic metric spaces. New York-Amsterdam-Oxford: North-Holland (1983; Zbl 0546.60010)]). For an ideal \(I_2\) in \(\mathbb N\times\mathbb N\), a double sequence \(x=(x_{k,j})\) in the PNS \((X,\nu,*)\) is said to be \(I^\nu_2\)-convergent to \(\xi\in X\) if \(\; \{(j,k)\in\mathbb N\times\mathbb N:\nu_{x_{j,k}-\xi}(t)\leq 1-\epsilon\}\in I_2\) for all \(\epsilon,t>0.\) If the double sequence \(x=(x_{k,j})\) is \(\nu\)-convergent to \(\xi\in X\), then it is \(I_2^\nu\)-convergent to \(\xi\).
The authors study the basic properties of this type of convergence-characterization in terms of the associated filter, uniqueness, algebraic operations. They consider also a weaker type of convergence, called \(I^{*\nu}_2\)-convergence, and in the last part of the paper study \(I_2\) limit and cluster points for double sequences in PNSs.


40J05 Summability in abstract structures
46S50 Functional analysis in probabilistic metric linear spaces
40A35 Ideal and statistical convergence