Some remarks on points of Lebesgue density and density-degree functions

TL;DR

This paper investigates properties of Lebesgue density points and density-degree functions, providing new results on differential forms and the approximation of measurable functions by density-degree functions.

Contribution

It introduces novel theorems relating differential forms and density points, and shows how measurable functions can be approximated by density-degree functions.

Findings

Density points relate to differential form properties.

Measurable functions with specific values can be approximated by density-degree functions.

New conditions for density points and degree functions are established.

Abstract

Some properties of $m$-density points and density-degree functions are studied. Moreover the following main results are provided: \vskip2mm \begin{itemize} \item {\it Let $\lambda$ be a continuous differential form of degree $h$ in ${\mathbf R}^n$ (with $h\geq 0$) having the following property: There exists a continuous differential form $\Delta$ of degree $h+1$ in $\rn^n$ such that \begin{equation*} \int_{{\mathbf R}^n}\Delta\wedge\omega =\int_{{\mathbf R}^n}\lambda\wedge d\omega, \end{equation*} for every $C^\infty_c$ differential form $\omega$ of degree $n-h-1$ in ${\mathbf R}^n$. Moreover let $\mu$ be a $C^1$ differential form of degree $h+1$ in ${\mathbf R}^n$ and set $E:=\{y\in {\mathbf R}^n\,\vert\, \Delta (y)=\mu(y)\}$. Then $d\mu (x) = 0$ whenever $x$ is a $(n+1)$-density point of $E$.} \vskip2mm \item {\it Let $f:{\mathbf R}^n\to\overline {\mathbf R}$ be a measurable function such that $f(x)\in \{0\}\cup [n,+\infty]$ for a.e. $x\in {\mathbf R}^n$. Then there exists a countable family $\{F_k\}_{k=1}^\infty$ of closed subsets of ${\mathbf R}^n$ such that the corresponding sequence of density-degree functions $\{d_{F_k}\}_{k=1}^\infty$ converges almost everywhere to $f$. }