Statistical convergence and ideal convergence for sequences of functions

Statistical Convergence and Ideal Convergence for Sequences of Functions

Let I ⊂ P(N) stand for an ideal containing finite sets. We discuss various kinds of statistical convergence and I-convergence for sequences of functions with values in R or in a metric space. For real valued measurable functions defined on a measure space (X,M, μ), we obtain a statistical version of the Egorov theorem (when μ(X) < ∞). We show that, in its assertion, equi-statistical convergence...

Statistical Convergence and Ideal Convergence of Sequences of Functions in 2-Normed Spaces

Wijsman Statistical Convergence of Double Sequences of Sets

In this paper, we study the concepts of Wijsman statistical convergence, Hausdorff statistical convergence and  Wijsman statistical Cauchy double sequences of sets and investigate the relationship between them.

On Ideal Version of Lacunary Statistical Convergence of Double Sequences

For any double lacunary sequence θrs = {(kr, ls)} and an admissible ideal I2 ⊆ P(N×N), the aim of present work is to define the concepts of Nθrs(I2)− and Sθrs(I2)−convergence for double sequence of numbers. We also present some inclusion relations between these notions and prove that Sθrs(I2)∩`∞ and S2(I2)∩ `∞ are closed subsets of `∞, the space of all bounded double sequences of numbers.

Ideal convergence of bounded sequences

We generalize the Bolzano-Weierstrass theorem (that every bounded sequence of reals admits a convergent subsequence) on ideal convergence. We show examples of ideals with and without the Bolzano-Weierstrass property, and give characterizations of BW property in terms of submeasures and extendability to a maximal P-ideal. We show applications to Rudin-Keisler and Rudin-Blass orderings of ideals ...