In this paper we shall study a density-type topology generated by the convergence everywhere except for a finite set similarly as the classical density topology is generated by the convergence in measure. Among others it is shown that the set of finite density points of a measurable set need not be measurable.

Abstract:The paper presents a new type of density topology on the real line generated by the pointwise convergence, similarly to the classical density topology which is generated by the convergence in measure. Among other things, this paper demonstrates that the set of pointwise density points of a Lebesgue measurable set does not need to be measurable and the set of pointwise density points of a set having the Baire property does not need to have the Baire property. However, the set of pointwise density points of any Borel set is Lebesgue measurable.

ABSTRACT. The paper presents a pointwise density topology with respect to admissible σ-algebras on the real line. The properties of such topologies, including the separation axioms, are studied. PreliminariesLet R be the set of reals, N the set of positive integer numbers and Q the set of rational numbers. By λ we shall denote the Lebesgue measure over R. The capitals L and L denote the σ-algebra of Lebesgue measurable sets on R and the σ-ideal of Lebesgue null sets. Let T nat be the natural topology on R. If T is a topology on R, then we fix the notation: B(T ) − σ-algebra of all Borel sets with respect to T, B a (T ) − σ-algebra of all sets having the Baire property with respect to T, K(T ) − σ-ideal of all meager sets with respect to T.If T = T nat , then we use the following short form symbols: B, B a , K. Let A stand for the complement of A in R. By χ A we shall denote the characteristic function of a set A ⊂ R. IntroductionRecall that x 0 ∈ R is a density point of the set A ∈ L if and only if

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The paper concerns some local properties of the sets with pointwise density points in terms of measure and category on the real line. We also construct nonmeasurable and not having the Baire property sets with pointwise density point. 1. Preliminaries Let R be the set of real numbers, N be the set of positive integers, Q denote the set of rational numbers and λ be the Lebesgue measure on R. By λ * and λ * we shall denote the inner and outer Lebesgue measure on R, respectively. Let L be the σ-algebra of Lebesgue measurable sets and B a the σ-algebra of sets having the Baire property on R. We say that a set has the Baire property if it is a symmetric difference between an open set and a set of the first category. Let L denote the σ-ideal of Lebesgue null sets, K denote the σ-ideal of the first category sets on the real line. Let T nat be the natural topology on R. If A ⊂ R and α, x ∈ R, then αA = {αa : a ∈ A}, A − x = {a − x : a ∈ A} and A denote the complement of A in R. We shall denote the characteristic function of a set A ⊂ R by χ A .