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Statistical convergence of sequences of sets. (English) Zbl 06174708

In this paper, the authors generalized statistical convergence using Kuratowski convergence, which was introduced in [K. Kuratowski, Topology. Vol. I. New edition, revised and augmented. New York-London: Academic Press; Warszawa: PWN-Polish Scientific Publishers. (1966; Zbl 0158.40802)], Wijsman convergence, which was introduced in [R. A. Wijsman, Bull. Am. Math. Soc. 70, 186–188 (1964; Zbl 0121.39001), Trans. Am. Math. Soc. 123, 32–45 (1966; Zbl 0146.18204)] and Hausdorf convergence, which was introduced in [F. Hausdorff, Grundzüge der Mengenlehre. Mit 53 Figuren im Text. Leipzig: Veit & Comp. (1914; Zbl 45.0123.01)]. They give the definitons of Kuratowski, Wijsman and Hausdorff statistical convergence of sequences of sets. Then they proved the theorem that says, description

[i] {A k } is a Wijsman statistically convergent sequence,

[ii] {A k } is a Wijsman statistically Cauchy sequence,

[iii] {A k } is a sequence for which there is a Wijsman convergent sequence {B k } such that A k =B k almost all k. are equivalent.

Moreover, they give some Tauberian conditions for Wijsman and Hausdorf statistical convergences.

Finally, they introduced Cesàro summable, strongly Cesàro summable, strongly p-Cesàro summable, almost converget, strongly almost convergent and strongly p-almost convergent for above definitions, and then they give some basic theorems of those new notions.

40A05Convergence and divergence of series and sequences
46E25Rings and algebras of continuous, differentiable or analytic functions