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

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 $h\ge 0$) 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}Δ\wedgeω=\int_{{\mathbf R}^n}λ\wedge dω, \end{equation*} for every $C_c^{\infty }$ 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 $E:=\{y\in {\mathbf{R}}^n|\Delta (y)=\mu (y)\}$. 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 $f$. }