Statistical convergence of sequences of sets.(English)Zbl 1287.40004

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] $$\{A_k\}$$ 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.

MSC:

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