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12 votes
2 answers
866 views

Sets that project to zero measure on all lines except one

It is a (difficult) exercise to show that there exists a measurable set $E \subset [0,1]^2$ (necessarily with zero 2-dimensional Lebesgue measure) such that the projection on every line passing ...
Castoro Moro's user avatar
1 vote
0 answers
87 views

Hausdorff distance and Hausdorff measure of symmetric difference

Let $X_n$ be a sequence of $k$-dimensional piecewise smooth submanifolds of $\mathbb{R}^m$, converging in Hausdorff distance to a $k$-dimensional piecewise smooth submanifold $Y \subset \mathbb{R}^m$, ...
Hajime S.'s user avatar
4 votes
1 answer
551 views

Is there an explicit, everywhere surjective $f:\mathbb{R}\to\mathbb{R}$ whose graph has zero Hausdorff measure in its dimension?

Suppose $f:\mathbb{R}\to\mathbb{R}$ is Borel. Let $\text{dim}_{\text{H}}(\cdot)$ be the Hausdorff dimension, and $\mathcal{H}^{\text{dim}_{\text{H}}(\cdot)}(\cdot)$ be the Hausdorff measure in its ...
Arbuja's user avatar
  • 63
3 votes
1 answer
199 views

Product of low dimensional Hausdorff measures

Let $\mathcal{H}^n$ and $\mathcal{H}^m$ be Hausdorff measures on $\mathbb{R}^n$ and $\mathbb{R}^m$. We know that the product measure $\mathcal{H}^n\otimes \mathcal{H}^m$ is the Hausdorff measure $\...
Yueqi's user avatar
  • 73
1 vote
1 answer
133 views

A question about the maximal function

Let $n>4$, $f\in C^{\infty}(\mathbb{R}^{n})$ and 0 denote the origin of $\mathbb{R}^{n}$. We define a weighted maximal function by $$Mf(x)=\sup_{0<r<1}r^{4-n}\int_{B_{r}(x)}|f|$$ which is ...
Xin Qian's user avatar
  • 155
2 votes
0 answers
146 views

The Hausdorff measure of intersection of annulus and conformal curve

Recently I came across a problem in my research. Let $g:[0,1]\to\mathbb{C}$ be a restriction of a conformal map that is defined in a simply connected domain $\Omega\subseteq\mathbb{C}$ that include $[...
mathematics is all's user avatar
2 votes
0 answers
151 views

$\mathscr{H}^{n-2}(\Sigma)< \infty$ implies $\mathscr{H}^{n-1}(\pi(\Sigma))=0$

Let $\Sigma\subset \mathbb{R}^{n+1}$ be a set with $(n-2)$-dimensional Hausdorff measure finite, i.e. $\mathscr{H}^{n-2}(\Sigma)<\infty$. Let $\pi:\mathbb{R}^{n+1}\to \mathbb{R}^n$ be the ...
No-one's user avatar
  • 1,149
2 votes
1 answer
300 views

If $\mathcal{H}^{n-1}(E)=0$ then $\mathbb{R}^n\setminus E$ is connected

Let $E\subset \mathbb{R}^n$ be a (measurable) subset with $\mathcal{H}^{n-1}(E)=0$, where $\mathcal H^{n - 1}$ is the ($n - 1$)-dimensional Hausdorff measure. I want to know if $\mathbb{R}^n\setminus ...
No-one's user avatar
  • 1,149
1 vote
2 answers
213 views

If $\mathcal{H}^{n-1}(K)=0$ then $\mathcal{H}^n(K\times \mathbb{R})=0$

I am reading a paper Simon and Wickramasekera - A Frequency Function and Singular Set Bounds for Branched Minimal Immersions where the authors seem to claim that if $K\subset\mathbb{R}^n$ is a compact ...
No-one's user avatar
  • 1,149
2 votes
2 answers
848 views

Defining a measure of uniformity for measurable subsets of $[0,1]^2$ w.r.t dimension $\alpha\in[0,2]$

Let $(X,d)$ be a metric space. If set $A\subseteq X$, let $H^{\alpha}$ be the $\alpha$-dimensional Hausdorff measure on $A$, where $\alpha\in[0,2]$ and $\text{dim}_{\text{H}}(A)$ is the Hausdorff ...
Arbuja's user avatar
  • 63
5 votes
0 answers
160 views

Naïve definition of a measure on a fractal

This question was previously posted on MSE. Let $K\subset \mathbb R^2$ be a compact fractal of Hausdorff dimension $1<d<2$. I want to define a natural measure on $K$. One option would be to use ...
Matheus Manzatto's user avatar
0 votes
1 answer
60 views

If $ \mathcal{H}^k(B_1(0)\cap S)\leq A\omega_k $ when $ \mathcal{H}^k(B_r(x)\cap S)\leq A\omega_kr^k $ for all $ 0<r<\delta $, $ x\in\mathbb{R}^n $?

Let $ S\subset\mathbb{R}^n $ is of finite $ k $-dimensional Hausdorff and $ 0<\delta<1 $ is a constant. If for any $ x\in\mathbb{R}^n $ and $ r>0 $, we hae $$ \mathcal{H}^k(S\cap B_r(x))\leq ...
Luis Yanka Annalisc's user avatar
2 votes
0 answers
64 views

A counterexample to regular boundary points for minimizers of variational integrals under subquadratic growth

Let $\Omega\subset\mathbb{R}^n$ for some $n\geq 3$ be an open bounded set with at least Lipschitz boundary. Let $p\in (1, 2), N>1$ and $f: \overline{\Omega} \times\mathbb{R}^N\times\mathbb{R}^{Nn}\...
Nirav's user avatar
  • 347
4 votes
1 answer
487 views

Finiteness of Hausdorff measure of balls

Let $(X,d)$ be an arbitrary metric space and let $\Bbb B(x,r)$ denote the closed ball with center $x \in X$ and radius $r>0$. For $p\geq 0$, let $H^p$ denote the $p$- dimensional Hausdorff measure. ...
John D's user avatar
  • 185
8 votes
1 answer
865 views

Fubini's theorem for Hausdorff measures

$B\subset \mathbb{R}^2$ is a Borel set. Define the slices $B_x:= \{y \in \mathbb{R}: (x,y) \in B \}$. If $\lambda$ denotes the Lebesgue measure on $\mathbb{R}$, presentations of Fubini's theorem often ...
Calamardo's user avatar
  • 675
5 votes
3 answers
359 views

Recovering the length metric from Hausdorff measure

The metric cannot be recovered from its Hausdorff measure in general. Now, assume that $(X,d_X)$ and $(Y, d_Y)$ are connected compact length spaces and induce $n$-dimensional Hausdorff measures $\...
Jialong Deng's user avatar
  • 1,799
4 votes
1 answer
900 views

Hausdorff dimension and surface measure

Could someone please indicate me some reference that contains the proof of the following theorem? Below $\mathcal{H}^n$ denotes the $n$-dimensional Hausdorff outer measure in $\mathbb{R}^n$. Theorem: ...
rfloc's user avatar
  • 637
3 votes
1 answer
372 views

Average of the sum of dirac measures

Let $(M^n,g)$ be a closed smooth Riemannian manifold. Consider a set $\mathcal B_{\epsilon}$ which consists of a maximal number of points in $M$ with pairwise distance no smaller than $\epsilon$. We ...
Totoro's user avatar
  • 2,535
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 ...
Redeldio's user avatar
  • 171
4 votes
0 answers
213 views

Classification of Euclidean-invariant measures?

Is there a classification of measures on $\mathbb R^n$ which are invariant under (Euclidean) isometries? Hausdorff measures of all kinds are examples -- could that be all of them? More precisely, By ...
Tim Campion's user avatar
  • 63.9k
8 votes
1 answer
213 views

How do sets with unit fractional Hausdorff measure of dimension $>1$ look like?

Triggered by the recent question How can we not know the measure of the Sierpiński triangle? I would like to ask: Let $s>1$ and $s$ not be an integer. How to construct a set $A$ with $\mathfrak{H}^...
Dirk's user avatar
  • 12.7k
1 vote
0 answers
145 views

How to show that this function is continuous (Geometric Measure Theory)

I want to prove that the function $F: \mathbb{R}_+ \to \mathbb{R}$ defined by $$F(t)=\int_{\{d=t\}} g \, d\mathcal{H}^{n-1}$$ is continuous if $g:\Omega \subset \mathbb{R}^n\to \mathbb{R}$ is ...
HighLiuk's user avatar
4 votes
0 answers
123 views

Converse on the rectifiability of products of rectifiable sets

Let $1\leq k\leq m$ and $1\leq l\leq n$ fixed integers, $\mathscr{H}^k$ the $k$ dimensional Hausdorff measure and $E\subset \mathbb{R}^m$. We say that : (1) $E$ is $k$ rectifiable if there exists $C\...
Paul-Benjamin's user avatar
5 votes
1 answer
767 views

Hausdorff approximating measures and Borel sets

Suppose $ 1 \leq m \leq n $ are integers and for each $ 0 < \delta < \infty $ let $\mathscr{H}^{m}_{\delta} $ be the size $ \delta $ approximating measure of the $ m $ dimensional Hausdorff ...
Longyearbyen's user avatar
15 votes
2 answers
3k views

Radon-Nikodym theorem for non-sigma finite measures

Let $(X,\mathcal M, \mu)$ be a measured space where $\mu$ is a positive measure. Let $\lambda$ be a complex measure on $(X,\mathcal M)$. When $\mu$ is sigma-finite, the Radon-Nikodym theorem provides ...
Bazin's user avatar
  • 16.2k
7 votes
1 answer
1k views

Generalization of area and coarea formula for fractional Hausdorff measures

Let $X,Y$ be Polish spaces, $s,t>0$ and $F:X\to Y$ locally Lipschitz continuous such that $X$ is $\sigma$-finite w.r.t. the $(s+t)$-dimensional Hausdorff measure $\mathcal{H}^{s+t}$. The Eilenberg ...
Johannes Hahn's user avatar
2 votes
1 answer
624 views

Results for Hausdorff Measure after Linear Transformation

For the Sierpinski Triangle, $S$, the $d$ dimensional Hausdorff measure is given by, $H^{d}(S)$. If a linear transformation, $W$ is applied to $S$, with $$W(x,y)=\begin{bmatrix} 1/2 & 0 \\ 0 &...
Zachary W. Robertson's user avatar
13 votes
1 answer
1k views

Are there any exact results for Hausdorff Measure?

The computation of the Hausdorff measure is extremely difficult due to the infimum appearing in its definition. This has made the calculation of the Hausdorff measure for nearly all fractals difficult ...
Zachary W. Robertson's user avatar