Superdensity with respect to Radon measure on ${\mathbf R}^n$

TL;DR

This paper introduces the concepts of superdensity and density degree of sets relative to Radon measures in Euclidean space, exploring their properties and applications including set approximation and a generalization of Schwarz's theorem.

Contribution

It defines superdensity and density degree with respect to Radon measures and demonstrates their applications in set approximation and differential theorems.

Findings

Sets can be approximated by closed subsets with small density degree.

A generalization of Schwarz's theorem on cross derivatives is established.

Applications illustrate the utility of superdensity concepts in analysis.

Abstract

We introduce and investigate superdensity and the density degree of sets with respect to a Radon measure on ${\mathbf R}^n$. Some applications are provided. In particular we prove a result on the approximability of a set by closed subsets of small density degree and a generalization of Schwarz's theorem on cross derivatives.