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

The authors generalize 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 Hausdorff convergence, which was introduced in [F. Hausdorff, Grundzüge der Mengenlehre. Leipzig: Veit $&$ Comp. (1914; JFM 45.0123.01)]. They give the definitons of Kuratowski, Wijsman and Hausdorff statistical convergence of sequences of sets. Then they prove a theorem that says

[i] $\left\{{A}_{k}\right\}$ is a Wijsman statistically convergent sequence,

[ii] $\left\{{A}_{k}\right\}$ is a Wijsman statistically Cauchy sequence,

[iii] $\left\{{A}_{k}\right\}$ is a sequence for which there is a Wijsman convergent sequence $\left\{{B}_{k}\right\}$ such that ${A}_{k}={B}_{k}$ almost all $k$

are equivalent.

Moreover, they give some Tauberian conditions for Wijsman and Hausdorff statistical convergence.

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

MSC:
 40J05 Summability in abstract structures 40A35 Ideal and statistical convergence