On the Lebesgue density theorem (Q5920643)

This paper presents a variation of the definition of density point and announces several related results whose proofs appear elsewhere [e.g., \textit{V. Aversa} and the author, Rend. Circ. Mat. Palermo, II. Ser. 53, No. 3, 344--352 (2004; Zbl 1194.26002)]. As defined by the author, \(x\) is a simple density point of a Lebesgue measurable set \(A\) of real numbers if \(\chi_{n\cdot (A-x)\cap [-1,1]}\) converges to \(\chi_{[-1,1]}\) almost everywhere, whereas for density points it converges in measure. Let \(\Phi (A)\) be the set of density points and \(\Phi_s (A)\) be the set of simple density points of a measurable set \(A\). It is stated that \(\{A:A \subseteq \Phi_s(A)\}\) is a topology that is strictly stronger than the standard topology but weaker than the density topology \(\{A: A \subseteq \Phi (A)\}\), and that there are sets of Lebesgue positive measure for which \(\Phi_s(A) = \emptyset\). The notion of simple density point is generalized using an unbounded increasing sequence \(\{t_n\}\) of positive terms and the condition that \(\chi_{t_n \cdot (A-x)\cap [-1,1]}\) converge to \(\chi_{[-1,1]}\) almost everywhere, with similar results.