# zbMATH — the first resource for mathematics

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.

##### MSC:
 40J05 Summability in abstract structures (should also be assigned at least one other classification number from Section 40-XX) 46S50 Functional analysis in probabilistic metric linear spaces 40A35 Ideal and statistical convergence