All Questions
Tagged with lebesgue-measure geometric-measure-theory
8 questions with no upvoted or accepted answers
6
votes
0
answers
271
views
Existence of a limit of alpha-difference quotient for Hölder functions
Let $f:\mathbb{R}\to \mathbb{R}^d,d\geq 1,$ be an Hölder function with exponent $\alpha\in (0,1)$, meaning that
\begin{equation}
\sup_{x, y \in \mathbb R, \,x\neq y}\frac{|f(x)-f(y)|}{|x-y|^\alpha}<...
- 61
5
votes
0
answers
140
views
Measure of the boundary of an BV-extension domain: do we have $|\nabla Eu|(\partial \Omega)=0?$
Let $\Omega\subset \Bbb R^d$ be open. The space $BV(\Omega)$ consists in functions $u\in L^1(\Omega)$ with bounded variation, i.e. $|u|_{BV(\Omega) }<\infty$ where
\begin{align}\label{eq:bounded-...
- 1,101
4
votes
0
answers
306
views
Continuity of the Lebesgue measure w.r.t the Hausdorff metric
I have a question linked to Interplay of Hausdorff metric and Lebesgue measure. Let us denote as $\mathcal K(\mathbb R^n)$ the space of compact subsets of $\mathbb R^n$ endowed with the Hausdorff ...
- 171
2
votes
0
answers
263
views
Is a maximal set of rectangles known for which Lebesgue’s Differentiation Theorem holds true?
Lebesgue's differentiation theorem states that if $x$ is a point in $\mathbb{R}^n$ and $f:\mathbb{R}^n\rightarrow\mathbb{R}$ is a Lebesgue integrable function, then the limit of $\frac{\int_B f d\...
- 4,597
0
votes
0
answers
234
views
Lebesgue measure of a neighbourhood of a curve
Let $\Omega\subseteq\mathbb{R}^N$ be an open, bounded and with smooth boundary (e.g. Lipschitz boundary or more if necessary).
For any function $\phi:\Omega\to\mathbb{R},\ \phi\in C^1(\overline{\Omega}...
- 1,759
0
votes
0
answers
88
views
Sequence of open sets converge in characteristic function to an open set?
Let $\Omega\subseteq\mathbb{R}^N$ be an open and bounded set with Lipschitz boundary. Consider a sequence of open sets $\omega_n\subseteq\Omega,\ n\in\mathbb{N}^*$ such that there is a Lebesgue ...
- 1,759
0
votes
0
answers
216
views
Signed distance function
Let $\Omega\subseteq\mathbb{R}^N$ be an open and bounded set with uniform Lipschitz boundary. Consider the signed distance function:
$d:\mathbb{R}^N\to\mathbb{R},\ d(x)=\begin{cases} \mathrm{dist}(x,\...
- 1,759
0
votes
0
answers
552
views
Egorov's and Lusin's Theorem in the space with infinite measure
Both the fundamental Egorov's and Lusin's Theorem in measure theory are given on any measurable space $X$ whose measure is finite.
On the measurable space whose measure is infinite, does there ...
- 4,058