Lebesgue density and statistical convergence
type: Item
instance of: scholarly article
title: Lebesgue density and statistical convergence
zbMATH Open document ID: 1484.40002
author: Marek Bienias
author: Szymon Glab
published in: Real Analysis Exchange
publication date: 2021-12-03T00:00:00Z
full work available at URL: realanalexch.46.2.0495.full
review text: The notion of density points of a Lebesgue measurable subset of real line is well known, as well as the famous Lebesgue Density Theorem. Many authors considered several generalizations of the concept in different directions (see works of \textit{B.~Aniszczyk} and \textit{R.~Frankiewicz} [Bull. Pol. Acad. Sci., Math. 34, 211--213 (1986; Zbl 0591.54002)] and \textit{S.~J. Taylor} [Fundam. Math. 46, 305--315 (1959; Zbl 0086.04601)]). The paper under review is devoted to the generalization by applying the idea of ideal convergence which had been deeply investigated by \textit{P.~Kostyrko} et al. [Real Anal. Exch. 26, No.~2, 669--685 (2001; Zbl 1021.40001)]. For an ideal $I$ of subsets of $\mathbb N$, the set of positive integers, and for a measurable subset $A$ of the set of real numbers, the authors introduce the notion of a density point. The notion of density points w.r.t. $I_{\{d\}}$ coincides with the classical Lebesgue density, where $I_{\{d\}}$ is the ideal of density zero sets (Theorem~4).
Mathematics Subject Classification ID: 40A35
Mathematics Subject Classification ID: 40A05
zbMATH DE Number: 7438313
zbMATH Keywords: Lebesgue density
zbMATH Keywords: statistical convergence
reviewed by: Hüseyin Çakallı
MaRDI profile type: Publication
OpenAlex ID: W3212188864
cites work: Q3719458
cites work: Sur la convergence statistique
cites work: \(\mathcal I\)-convergence
cites work: On strengthening the Lebesgue Density Theorem
DOI: 10.14321/REALANALEXCH.46.2.0495
description: scientific article; zbMATH DE number 7438313