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Density Measures

Moritz Schönherr, Friedemann Schuricht

Abstract

The paper treats density measures as typical examples of finitely additive measures in $\mathbb{R}^n$. We study their structure and derive basic properties. In addition, estimates for related integrals are provided. The results are applied to the precise representative of general integrable functions and then they are specialized to functions of bounded variation. Moreover, a new representation of the generalized gradients in the sense of Clarke is given for the finite dimensional case.

Density Measures

Abstract

The paper treats density measures as typical examples of finitely additive measures in . We study their structure and derive basic properties. In addition, estimates for related integrals are provided. The results are applied to the precise representative of general integrable functions and then they are specialized to functions of bounded variation. Moreover, a new representation of the generalized gradients in the sense of Clarke is given for the finite dimensional case.
Paper Structure (7 sections, 22 theorems, 183 equations, 2 figures)

This paper contains 7 sections, 22 theorems, 183 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries about measures and integration
  3. Density Measures
  4. Extreme points
  5. Precise representative
  6. More explicit examples to some special problems
  7. Generalized gradients

Key Result

Theorem 3.1

Let $\Omega \in \mathcal{B}(\mathbb{R}^n)$, let $C\subset\overline{\Omega}$ be a density set, and let $\mu\in\operatorname{Dens}_C$. Then and $\mu$ is pure. Moreover for all $f\in\mathcal{L}^\infty(\Omega)$.

Figures (2)

  • Figure 1: Existence of a density measure at a cusp.
  • Figure 2: Cones $K_j=K(x,v,\alpha_j)$ for a large angle $\alpha_1$ and a smaller angle $\alpha_2$.

Theorems & Definitions (44)

  • Example 2.1
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Proposition 3.3
  • proof
  • Corollary 3.4
  • proof
  • Proposition 3.5
  • ...and 34 more