Density measures and applications

Abstract: The paper, that continuous some previous work of Schönherr & Schuricht, treats density measures on ${\mathbb R}^n$ that concentrate in any neighborhood of a Lebesgue null set. Such measures are typical for purely finitely additive measures. We study their basic properties and investigate related integrals. Measures taking only the values 0 and 1 are considered as special case. The results are first applied to weak convergence in $\mathcal{L}^\infty(Ω)$. Then we derive integral representations by means of such measures for several notions of differentiability for integrable functions and we show a kind of mean value theorem for some class of Sobolev functions. Finally we provide a new approach to the generalized Jacobians in the sense of Clarke.

• The paper introduces a novel framework for density measures using purely finitely additive measures that concentrate mass near Lebesgue null sets.
• The paper establishes key analytical properties, including weak* convergence, integral representations, and the role of 0-1 extreme measures in nonsmooth analysis.
• The paper extends classical differentiability and Clarke’s Jacobian by offering a measure-theoretic approach that refines traditional calculus in geometric measure theory.

Density Measures and Their Analytical Applications

Introduction and Motivation

This paper develops the systematic study of density measures on $\mathbb{R}^n$—purely finitely additive measures that concentrate mass in arbitrarily small neighborhoods of Lebesgue null sets. This framework generalizes the familiar Dirac measures and classical densities from geometric measure theory, such as the pointwise density $\dens_x A = \lim_{\delta \downarrow 0} \lambda(A \cap B_\delta(x))/\lambda(B_\delta(x))$ when the limit exists. Crucially, these density measures are not $\sigma$-additive, exhibit support in the vicinity (but not necessarily at) null sets, and, thus, serve as canonical examples of pure finitely additive measures in the dual of $L^\infty$.

The paper provides a comprehensive theory of such measures, constructing their analytical properties and exhibiting their utility in weak convergence, integral representations of differentiability, and trace theorems. Special attention is devoted to 0-1 valued measures (analogous to extreme points in $\text{ba}(\Omega)$) and their role in the description of Clarke Jacobians for nonsmooth analysis.

Preliminaries on Purely Finitely Additive Measures and Integration

The integration theory underlying this work is based on the duality between $L^\infty(\Omega)$ and $\operatorname{ba}(\Omega, \mathcal{B}(\Omega))$, where not all bounded additive measures are $\sigma$-additive. Pure measures are defined by the property that any $\sigma$-additive part dominated by their variation must vanish. The paper develops machinery to integrate with respect to such measures, relying on convergence in measure rather than almost everywhere, and establishes corresponding properties (e.g., dominated convergence, identification of equivalence classes).

The notion of the “core” of a measure is emphasized; for pure measures, the core need not coincide with the measure’s “support” in the classical sense (in fact, pure density measures typically vanish on their core). Measures can be nontrivial but have empty support and localize mass only in arbitrarily small neighborhoods of prescribed null sets. This subtle localization is pivotal in their applications to traces and local properties of integrable functions.

Density Measures: Properties and Existence

The central objects of study, density measures $\mu \in \operatorname{ba}_\lambda(\Omega)$, are nonnegative, normalized ($\dens_x A = \lim_{\delta \downarrow 0} \lambda(A \cap B_\delta(x))/\lambda(B_\delta(x))$0), and satisfy $\dens_x A = \lim_{\delta \downarrow 0} \lambda(A \cap B_\delta(x))/\lambda(B_\delta(x))$1 for all $\dens_x A = \lim_{\delta \downarrow 0} \lambda(A \cap B_\delta(x))/\lambda(B_\delta(x))$2, where $\dens_x A = \lim_{\delta \downarrow 0} \lambda(A \cap B_\delta(x))/\lambda(B_\delta(x))$3 is a closed Lebesgue null set and $\dens_x A = \lim_{\delta \downarrow 0} \lambda(A \cap B_\delta(x))/\lambda(B_\delta(x))$4 is its $\dens_x A = \lim_{\delta \downarrow 0} \lambda(A \cap B_\delta(x))/\lambda(B_\delta(x))$5-neighborhood. These measures are supported (in the sense of their core) on arbitrary small neighborhoods of the null set and are constructed via limit procedures (Hahn-Banach extension from densities on balls, neighborhoods, or lower-dimensional structures).

A key property is that the set of all such density measures $\dens_x A = \lim_{\delta \downarrow 0} \lambda(A \cap B_\delta(x))/\lambda(B_\delta(x))$6 is convex, weak*-compact in the dual of $\dens_x A = \lim_{\delta \downarrow 0} \lambda(A \cap B_\delta(x))/\lambda(B_\delta(x))$7, and all members are purely finitely additive. Existence is characterized via geometric density: $\dens_x A = \lim_{\delta \downarrow 0} \lambda(A \cap B_\delta(x))/\lambda(B_\delta(x))$8 if and only if every neighborhood of $\dens_x A = \lim_{\delta \downarrow 0} \lambda(A \cap B_\delta(x))/\lambda(B_\delta(x))$9 meets $\sigma$0 in positive Lebesgue measure. These measures also admit generalization to those supported near infinity (via one-point compactification).

Integration with respect to density measures is tied to local “averaging” near $\sigma$1, discriminating them from the Dirac measure: balls $\sigma$2 concentrate mass around but not at $\sigma$3. The paper proves sharp extremal inequalities for such integrals using essential supremum/infimum over vanishing neighborhoods.

Structure of Extreme Points: 0-1 Measures and Ultrafilters

A detailed analysis is given of $\sigma$4-$\sigma$5 valued (zo) density measures, which take only values in $\sigma$6 on Borel sets, and are shown to be precisely the extreme points of the convex set $\sigma$7. The Krein-Milman theorem then yields that any density measure is a barycenter (convex combination, possibly infinite) of zo measures.

The structure of zo measures is linked to ultrafilters on the algebra of Borel sets of positive measure, with the measure of a set $\sigma$8 given by $\sigma$9 if and only if $L^\infty$0 belongs to the ultrafilter. This characterization reveals an enormous multiplicity of such measures, even for fixed singularities and directions, reflecting the continuum of choices for “aura” sets and localizations.

Furthermore, each zo measure is uniquely associated with a localization at a point $L^\infty$1 and, even more finely, a direction—particularly in higher dimensions, via localization in cones or cusps. This permits the construction of zo measures with prescribed concentration behavior along sequences or geometrically defined regions.

Applications

Weak$L^\infty$2 Convergence in $L^\infty$3 and Necessity of Local Convergence

A major application is to the characterization of weak convergence in $L^\infty$4. The paper proves that weak convergence of $L^\infty$5 to $L^\infty$6 is equivalent to pointwise convergence of the integrals of $L^\infty$7 against all density measures, and in fact, it suffices to test convergence against all zo density measures. Stronger, pointwise necessary conditions are derived: for all density measures $L^\infty$8 at all $L^\infty$9 (not just almost every $\text{ba}(\Omega)$0), $\text{ba}(\Omega)$1. This subsumes and strengthens classical characterizations that only require convergence almost everywhere or on average. Explicit counterexamples show the difference between pointwise and norm convergence, and cases where even essential supremum convergence fails to capture weak convergence.

Integral Characterization of Differentiability and Sobolev Regularity

For $\text{ba}(\Omega)$2 (and Sobolev) functions, density measures yield pointwise integral representations of classical differentiability notions (Lebesgue points, approximate and essential differentiability). Specifically, the precise representative of a function (the limit of average values over balls) can be characterized as the integral with respect to a density measure at almost every $\text{ba}(\Omega)$3.

The paper gives analytic characterizations of approximate and essential derivatives: e.g., approximate differentiability at $\text{ba}(\Omega)$4 is equivalent to the vanishing of the density integral of a suitable difference quotient (see equation (66)). The existence, uniqueness, and calculus rules for such derivatives—sum, product, and chain rules—are all furnished using integration with respect to density measures, rather than direct pointwise approaches.

Further, using extreme zo density measures, the paper obtains refined integral mean value theorems for Sobolev functions: the difference $\text{ba}(\Omega)$5 along a segment is the integral of derivatives (in the density measure sense) within the segment, generalizing classical results to locally non-smooth and non-representable objects.

New Representation of Clarke’s Generalized Jacobian

The paper provides a new approach to Clarke’s generalized Jacobian: for a locally Lipschitz $\text{ba}(\Omega)$6, the generalized Jacobian at $\text{ba}(\Omega)$7 is precisely the convex hull of the density integrals of the classical derivative $\text{ba}(\Omega)$8 with respect to all density measures at $\text{ba}(\Omega)$9. That is,

$L^\infty(\Omega)$0

This introduces a measure-theoretic, analytic representation of what is classically introduced as a set-theoretic or limiting object. The support function of this set coincides with the essential limit superior of directional derivatives, establishing new, concise proofs of calculus rules for the generalized Jacobian (sum, chain, and mean value theorems) in finite dimensions.

Conclusion

This paper extends the classical integration and differentiability theory to the setting of purely finitely additive, density measures concentrated near Lebesgue null sets. By harnessing the analytic and topological structure of these measures (especially the 0-1 extreme points), it provides foundational tools for local analysis in $L^\infty(\Omega)$1, BV, and Sobolev spaces, as well as new approaches to generalized derivatives and nonsmooth optimization. The work yields necessary and sufficient conditions for weak convergence in $L^\infty(\Omega)$2, integral representations for traces and differentiability, and a refined, measure-based description of Clarke Jacobians. These methods open new directions for analyzing fine properties of integrable and Sobolev functions, with potential future applications in non-smooth optimization, geometric measure theory, and partial differential equations.