Pointwise density topology

Abstract

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.

On the structure of the pointwise density sets on the real line

“…Although it is possible to find the sets from theorem 2.5, 2.6 it turns out that the family T p = {A ∈ L : A ⊂ Φ p (A)} forms topology containing T nat . The crucial properties of this topology are investigated in [2]. In the similar line of thought to the proof of Lemma 2 in [1] we can prove the below lemma.…”