Some remarks on points of Lebesgue density and density-degree functions (2407.12343v1)
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 $\rnn$ 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 $C1$ 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 $f$. }