A generalization of the Lebesgue density theorem via modulus density

Abstract: In this paper, we introduce the notion of a $γ$-density point for Lebesgue-measurable subsets of $\mathbb{R}$, where $γ$ is a modulus function, and study its basic measure-theoretic properties. We show that every $γ$-density point is a Lebesgue density point, while under Condition~(A) the two notions coincide. Consequently, for such modulus functions, the set of $γ$-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 $γ$-density topology $τγ$ and investigate its structure. In general, $τγ$ is contained in the classical Lebesgue density topology, and if $γ$ satisfies Condition~(A), then $τγ=τ_d$. We also compare $τγ$ with $ψ$-density topologies and establish several topological properties of $τγ$, including that countable sets are $τγ$-closed and that $(\mathbb{R},τ_γ)$ is nonseparable, nonregular, and nonmetrizable. Finally, we introduce $γ$-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.

• The paper introduces γ-density points by replacing linear normalization with a modulus function, recovering classical density when γ(t) = t under Condition (A).
• It develops the τγ-density topology, revealing unique non-Euclidean features such as nonseparability, nonmetrizability, and finite compact sets.
• The work extends the framework to function spaces, defining γ-approximately continuous functions and establishing a Banach space structure for bounded cases.

Generalizing Lebesgue Density via Modulus Density: Theory and Topology

Modulus Density Points and Their Measure-Theoretic Properties

The paper introduces the concept of $\gamma$-density points, extending the classical notion of Lebesgue density points by replacing the linear normalization with a modulus function $\gamma$ satisfying specific properties (zero at zero, subadditivity, right-continuity at zero, and monotonicity). For a Lebesgue-measurable set $A \subseteq \mathbb{R}$, a point $x$ is defined as a $\gamma$-density point if

$$\lim_{\alpha \to 0^+} \frac{\gamma(|(x-\alpha, x+\alpha)\cap A^c|)}{\gamma(2\alpha)} = 0,$$

which recovers the classical density point notion when $\gamma(t) = t$.

A principal result is that every $\gamma$-density point is necessarily a Lebesgue density point, but, in general, the converse fails unless $\gamma$ satisfies Condition (A), which imposes a certain vanishing ratio property near zero. Under Condition (A), $\gamma$-density points coincide with Lebesgue density points almost everywhere, and the symmetric difference between a measurable set and its set of $\gamma$0-density points is a null set. This establishes a modulus-theoretic formulation of the Lebesgue Density Theorem, with the classical theorem as a special case.

The paper further analyzes the structural properties of $\gamma$1-density point sets, verifying their measurability and showing that operators corresponding to different modulus functions are equivalent whenever the functions are comparable in a neighborhood of zero.

Modulus Density Topology: Structure and Comparisons

Building on the measure-theoretic groundwork, a topology $\gamma$2 is constructed on $\gamma$3 via $\gamma$4-density points: the open sets are those $\gamma$5 with $\gamma$6. This topology is always contained in the classical Lebesgue density topology $\gamma$7, and equality holds precisely when $\gamma$8 satisfies Condition (A).

A detailed comparison with other density-type topologies is provided. Specifically, under suitable subadditivity and growth constraints, any $\gamma$9-density topology can be realized as a $A \subseteq \mathbb{R}$0-density topology for a modulus function constructed from $A \subseteq \mathbb{R}$1. The paper proves that $A \subseteq \mathbb{R}$2 is finer than the $A \subseteq \mathbb{R}$3-density topology when $A \subseteq \mathbb{R}$4 (viewed as $A \subseteq \mathbb{R}$5) satisfies appropriate conditions.

Several non-standard topological properties are proved:

• Countable sets are closed in $A \subseteq \mathbb{R}$6; thus, $A \subseteq \mathbb{R}$7 is nonseparable, and all compact sets are finite.
• The space fails to be regular and therefore is nonmetrizable.
• The closure and interior operations are tied to density operators, yielding regular open sets as those equal to their set of $A \subseteq \mathbb{R}$8-density points.
• Nowhere dense, measure zero, category, and closed discrete subsets coincide in $A \subseteq \mathbb{R}$9 under Condition (A).

The characterization of $x$0-limit points is provided via a modulus-based limsup condition involving the Lebesgue outer measure.

Modulus Approximately Continuous Functions and Function Spaces

The framework is extended to function theory, defining $x$1-approximately continuous functions as those for which at each point $x$2, there exists a measurable set $x$3 with $x$4 a $x$5-density point of $x$6 and

$x$7

The space $x$8 of all such functions is shown to be a vector space over $x$9, with composition by continuous functions preserving $\gamma$0-approximate continuity.

The subclass of bounded $\gamma$1-approximately continuous functions, $\gamma$2, is shown to be a Banach space under the sup norm. However, the full space $\gamma$3 (on $\gamma$4) is not normed under $\gamma$5 due to the existence of unbounded approximately continuous functions constructed via discrete tall bumps on negligible intervals accumulating at a point.

Implications and Future Directions

The modulus density framework generalizes classical notions in measure theory and topology, allowing for fine-grained analysis suited to moduli that encode non-linear scaling behavior. The equivalence of $\gamma$6-density points with classical density points under Condition (A) demonstrates the flexibility of the approach, while the topological results suggest highly non-Euclidean behavior, especially the disconnect between separability and compactness.

The theoretical implications extend to operator theory, function spaces, and generalized continuity. Practically, modulus density may find application in functional analysis, probability (via moduli encoding non-standard behaviors), and real analysis where classical density notions are inadequate.

Future work includes:

• Extending modulus density to higher-dimensional and non-Euclidean settings.
• Investigating relationships with other generalized topologies and density-type operators.
• Analyzing $\gamma$7-approximately continuous function spaces in connection with Baire classes, differentiation, and integration.
• Exploring statistical and probabilistic moduli and their continuity/density notions.

Conclusion

The paper provides a comprehensive generalization of the Lebesgue density point concept via modulus functions, develops the corresponding topology and function spaces, and establishes strong structural results under suitable conditions on the modulus. The proposed framework broadens the toolkit for density analysis and paves the way for further inquiry into non-classical measure-theoretic and topological structures, as well as new classes of approximately continuous functions (2604.13626).