Lebesgue density and statistical convergence
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).
scientific article; zbMATH DE number 7438313