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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 $\Bbb R$ is called $I$-convergent to $\xi\in\Bbb R$ if $\vert[k\in\Bbb N:\vert x_k-L\vert\ge\epsilon\}\in I$ for every $\epsilon > 0$. $I$-convergence was considered first by {\it P. Kostyrko}, {\it T. Šalát} and {\it W. Wilczyński} [Real Anal. Exch. 26, No. 2, 669--685 (2001; Zbl 1021.40001)], and independently by {\it F. Nuray} and {\it 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 [{\it B. Schweizer} and {\it A. Sklar}, Probabilistic metric spaces. New York-Amsterdam-Oxford: North-Holland (1983; Zbl 0546.60010)]). For an ideal $I_2$ in $\Bbb N\times\Bbb N$, 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 $\; \{(j,k)\in\Bbb N\times\Bbb N:\nu_{x_{j,k}-\xi}(t)\le 1-\epsilon\}\in I_2$ 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.

40J05Summability in abstract structures
46S50Functional analysis in probabilistic metric linear spaces
40A35Ideal and statistical convergence