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 I⊆P(ℕ) (with Fin⊆I), Lebesgue measurable set A⊆ℝ we introduce a definition of a density point of A with respect to 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).