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Statistical convergence of sequences of sets. (English) Zbl 1287.40004
The authors generalize statistical convergence using Kuratowski convergence, which was introduced in [{\it 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 [{\it 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 [{\it 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] $\{A_k\}$ 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 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.

40J05Summability in abstract structures
40A35Ideal and statistical convergence