A generalization of the Lebesgue density theorem via modulus density

TL;DR

This paper generalizes the Lebesgue density theorem by introducing $\gamma$-density points and explores their measure-theoretic and topological properties, including the structure of the associated $\gamma$-density topology.

Contribution

It defines $\gamma$-density points and topology, compares them with classical notions, and studies their measure-theoretic and topological properties, including the Banach space of $\gamma$-approximately continuous functions.

Findings

Every $\gamma$-density point is a Lebesgue density point.

The $\gamma$-density topology is contained in the Lebesgue density topology.

The space of bounded $\gamma$-approximately continuous functions is a Banach space.

Abstract

In this paper, we introduce the notion of a $\gamma$-density point for Lebesgue-measurable subsets of $\mathbb{R}$, where $\gamma$ is a modulus function, and study its basic measure-theoretic properties. We show that every $\gamma$-density point is a Lebesgue density point, while under Condition~(A) the two notions coincide. Consequently, for such modulus functions, the set of $\gamma$-density points of a measurable set differs from the set itself only by a null set, yielding a modulus version of the Lebesgue Density Theorem. We then define the associated $\gamma$-density topology $\tau_\gamma$ and investigate its structure. In general, $\tau_\gamma$ is contained in the classical Lebesgue density topology, and if $\gamma$ satisfies Condition~(A), then $\tau_\gamma=\tau_d$. We also compare $\tau_\gamma$ with $\psi$-density topologies and establish several topological properties of $\tau_\gamma$, including that countable sets are $\tau_\gamma$-closed and that $(\mathbb{R},\tau_\gamma)$ is nonseparable, nonregular, and nonmetrizable. Finally, we introduce $\gamma$-approximately continuous functions, prove that they form a vector space, and show that the bounded class of such functions is a Banach space under the supremum norm.