Some remarks on points of Lebesgue density and density-degree functions
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910531008331776 |
|---|---|
| author | Delladio, Silvano |
| author_facet | Delladio, Silvano |
| contents | 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 $λ$ 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 $Δ$ 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 $C^\infty_c$ differential form $ω$ of degree $n-h-1$ in ${\mathbf R}^n$. Moreover let $μ$ be a $C^1$ differential form of degree $h+1$ in ${\mathbf R}^n$ and set $E:=\{y\in {\mathbf R}^n\,\vert\, Δ(y)=μ(y)\}$. Then $dμ(x) = 0$ 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 $f$. } |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_12343 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Some remarks on points of Lebesgue density and density-degree functions Delladio, Silvano Functional Analysis 28A75, 28A05, 31C40 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 $λ$ 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 $Δ$ 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 $C^\infty_c$ differential form $ω$ of degree $n-h-1$ in ${\mathbf R}^n$. Moreover let $μ$ be a $C^1$ differential form of degree $h+1$ in ${\mathbf R}^n$ and set $E:=\{y\in {\mathbf R}^n\,\vert\, Δ(y)=μ(y)\}$. Then $dμ(x) = 0$ 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 $f$. } |
| title | Some remarks on points of Lebesgue density and density-degree functions |
| topic | Functional Analysis 28A75, 28A05, 31C40 |
| url | https://arxiv.org/abs/2407.12343 |