LEBESGUE DENSITY AND STATISTICAL CONVERGENCE

Abstract

The paper presents a generalization of the density point’s notion to the ideal-convergence framework. For an ideal $\mathcal{I}\subseteq \mathcal{P}\left(\mathrm{\mathbb{N}}\right)$ (with $\mathrm{Fin}\subseteq \mathcal{I}$), Lebesgue measurable set $A\subseteq \mathrm{ℝ}$ we introduce a definition of a density point of A with respect to $\mathcal{I}$; we prove that the classical approach fits into this generalization (Theorem 4); we construct a family of Cantorlike sets showing that Lebesgue Density Theorem cannot be maximally improved in this direction (Theorem 8).