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Questions tagged [hausdorff-measure]

If $(X,\rho)$ is a metric space, then for any subset $S$, we have $$ H_\delta^d(S):=\inf\ \left\{ \sum_{i=1}^\infty ({\rm diam}\ U_i)^d | S\subset \bigcup_{i=1}^\infty U_i,\ {\rm diam}\ U_i < \delta \right\}, $$ where the infimum is over all countable covers of $S$. Then we have $H^d(S):=\lim_{\delta \rightarrow 0 } H^d_\delta (S)$, which is called the $d$-dimensional Hausdorff measure.

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Is the $\alpha$-Hausdorff content of a set of diameter $\delta$ equal to $\delta^\alpha$?

Fix $\alpha>0$. The $\alpha$ dimensional Hausdorff content of a bounded set $K$ is given by $$H^{\alpha}_{\infty}(K)=\inf\left\{\sum_{i=1}^\infty \text{diam}(B_i)^\alpha:K\subset\bigcup_{i=1}^\...
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Showing particular quotient space is (not) compact, connected and Hausdorff

I came across the following question, I'm unsure about some of my answers. Let $U = \{(x, y) \in \mathbb{R}^{2} \mid y \in \{0, 1\}\}$ be a subspace of $\mathbb{R}^{2}$. We define the following ...
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Upper semicontinuity of sequence of Hausdorff measures

This is exercise 12.2 of Measure theory and integration of Michael Taylor. Let $(d_j)$ be a sequence of metrics on a compact space $X$ such that there exists $c,C>0$ with $$ cd_0(x,y) \leq d_j(x,y) ...
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Fubini for Hausdorff measures

Let $f \in L^2(S^{n-1})$, where $S^{n-1}$ denotes the $n$ dimensional sphere. Now, by using Fubini we can write $$\int_{S^{n-1}}\int_{S^{n-1}} 1_{x \cdot y = 0} |f(x)|^2 d\mathcal{H}^{n-2}(x) d\...
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Hausdorff dimension of the Sierpiński triangle calculated from definition

I already know how to calculate the Hausdorff dimension of the Sierpiński triangle the way it is presented on Hausdorff dimension of Sierpinski triangle less than log3/log2. However, my complaint is ...
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Hausdorff dimension if Hausdorff measure is infinity

I am reading the article "Cyclic Functions in Lp(R), 1≤p<∞" from Joseph M. Rosenblatt and Karen L. Shuman. In Theorem 5 they calculate the Hausdorff dimension of the set E, which is a ...
tf101097's user avatar
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Fractal Geometry: Show that the Hausdorff dimension of a set and its image under $f(x)=x^2$ are the same.

Let $f:\mathbb{R} \to \mathbb{R}$ be the function $f(x) = x^2$, and let $F$ be any subset of $\mathbb{R}$. Show that $\text{dim}_Hf(F) = \text{dim}_\mathrm{H} F$. Here $\text{dim}_\mathrm{H}$ refers ...
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Is every isometry invariant measure on Euclidean space equivalent to Hausdorff measure?

One of the properties of Hausdorff measure on $\mathbb{R}^d$ is invariance under isometry. I wonder that if every isometry-invariant measure is similar to Hausdorff measure. To be rigorous, let $\mu$ ...
Daeyoung Jeong's user avatar
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Hausdorff measure and orthogonal projection

I was reading Federer's proof of Gustin's boxing inequality and during the proof there's a step which is unclear to me. Here's the context: $A,B\subseteq \mathbb{R}^{n}$ compacts sets such that $A\cup ...
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Iterated function system with a fractional number of contractions

Iterated function systems can be used to generate fractals. One starts off with a simple geometric figure and applies the IFS infinitely many times to obtain a fractal. For example, in the case of the ...
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Convergence of Lesbegue measure of the images of a sequences converging uniformly

Good morning, my answer is related to the one introduced here Convergence of images of a sequence of converging continuous functions but with stronger hypotheses. Suppose you have a sequence of ...
Andrew Be's user avatar
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Definition of $L^2(S^{n-1})$

I'm puzzled with how we're supposed to define $L^2(S^{n-1})$ where $S^{n-1}$ is the unit sphere in $\mathbb{R}^n$. How do we even define the inner product there? the only way that comes to mind ...
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Locally bi-Lipschitz map preserves Hausdorff dimension

I was able to prove that bi-Lipschitz function $f: \mathbb R^n \rightarrow \mathbb R^m$ preserves Hausdorff dimension. In particular, the following lemma was useful: Let $E \subseteq \mathbb R^k$ be ...
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Understanding Falconer Example 4.2

Presently, I am reading Falconer's book on Geometric Measure theory. In example 4.2 he used mass distribution principle to calculate the lower bound of the Hausdorff dimension of Cantor set $C$. Now ...
Mayank's user avatar
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Understanding Equality of Hausdorff Measures in Global and Submanifold Riemannian Metrics

I am reading chapter 12 of Measure and Integration Theory of Micheal E.Taylor. I struggle with a point on the proof of the following proposition. Proposition 12.7: Let $\Omega$ be a $C^1$ manifold ...
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Why is Hausdorff Measure used on surfaces integrals?

I have a mild curiosity as to why are we using the Hausdorff measure to define surface integrals (for example the co area formula) and not use instead the Lebesgue measure, or at least on my class and ...
Ramiro genta's user avatar
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Do countable collections of points in $B^m$ have Hausdorff 1-dimensional measure zero?

Let $\mu$ denote that Hausdorff $1$-dimensional measure. Suppose $S\subset\mathbb{B}^m$, where $B^m$ is the unit ball in $\mathbb{R}^m$, $m\geq 3$. I'm not so familiar with the Hausdorff measure, but ...
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Is the Hausdorff measure of this disk equal to 0?

Consider a topological space $D^n$ homeomorphic to the closed unit ball of $\mathbb{R}^n$. Suppose that $D^n$ is equipped with a topogically compatible metric that is intrinsic. Is the $(n+1)$-...
mathplayer's user avatar
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When does the Hausdorff measure equal infinity?

I'm writing a project on fractals and I got as far with Hausdorff dimension to say that, for some surface $S \subset \mathbb{R}^n$ and $r,s\in \mathbb{R}_{\ge 0}$ such that $r>s$ that $$0 \leq H^r(...
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A question on the Hausdorff dimension of a subset of $\mathbb{R}.$ [closed]

Let $p\in [0,1].$ I am interested in showing that there exist sets $A,B\subset \mathbb{R}$ of Hausdorff dimension $p$ such that the $p$-dimensional Hausdorff measures $H_p(A)=\infty$ and $H_p(B)=0$. I'...
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Local isometry and Hausdorff dimension

I am currently reading Falconer's book on Hausdorff dimension. My question is whether Hausdorff dimension is invariant under local isometry between smooth Riemannian manifolds? I think it should be ...
User 11111's user avatar
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Embedded subshift in $[0,1]$ has dimension $0$?

I was wondering whether most embeddings of one-dimensional subshifts have zero Hausdorff dimension? Given a finite alphabet $\mathcal{A}= \{ 0,...,d-1 \}$ and $\Omega\subseteq \mathcal{A}^\mathbb{N}$, ...
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Hausdorff dimension calculation for groups

I was reading about homogeneous space i.e let $G$ be a Lie group and $\Gamma$ be a discrete subgroup of $G$ with finite covolume. Then Hausdroff dimension of $G / \Gamma$ is the Hausdorff dimension of ...
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what is the dimension of the intersection of the planar brownian range with a line

We consider two dimensional standard brownian motion $B: t\mapsto (B_{1}(t),B_{2}(t))$. Let $D$ be its range (that is the image of $[0,+\infty[$ by $B$ i.e, $B([0,+\infty[)$). Is there some known ...
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question related to hausdorff dimension and hausdorff measure [closed]

If $A$ is a subset of $\mathbb{R}^d$, we denote the diameter of $A$ by $\Delta(A)=\sup \{|x-y|: x, y \in A\} \in[0, \infty]$. For every $\alpha>0$ and every $\varepsilon>0$, we set, for every ...
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Naturalness of definition of line integral

Let $I = [a,b]$ denote some interval, and $f : I \rightarrow \mathbb{C}$ a continuous function of bounded variation, in other words, $f$ is a parameterisation of $\gamma = f(I)$, a rectifiable curve. ...
porridgemathematics's user avatar
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Do there exist continuous maps from the Sphere to the Plane which preserve the Hausdorff Measure of all sets with some Dimension between 1 and 2?

It's well known that there do exist area preserving maps between the sphere and the plane. It's well known there do NOT exist distance preserving maps between the sphere and the plane. So naturally ...
Sidharth Ghoshal's user avatar
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References on Hausdorff Dimension on Non-Euclidean Vector Spaces?

Given a fractal set in the Euclidean space $\mathbb{R}^n$, we can study it's Hausdorff dimension. Common examples include: (1) The middle-thirds Cantor set in $\mathbb{R}$. (2) The Sierpinski Gasket ...
VShaw's user avatar
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Existence of an $\alpha$-regular measure with positive measure on a binary digits do not have a limiting frequency

let $$X=\left\{ \sum_{n=1}^{\infty}a_{n}2^{-n}:a_{n}\in\left\{ 0,1\right\} ,\liminf\frac{1}{n}\sum_{i=1}^{n}a_{i}<\limsup\frac{1}{n}\sum_{i=1}^{n}a_{i}\right\} $$ I'm studying fractal geometry and ...
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Proof of the convergence part of Jarník's Theorem

Let $\psi: \mathbb{N} \to \mathbb{R}^+$ be some function and denote by $W(\psi)$ the set of numbers $x \in [0,1]$ for which $$\left\lvert x - \frac{p}{q} \right\rvert < \frac{\psi(q)}{q} \quad \...
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Hausdorff measure of $f(E)$ is greater than zero where $E$ compact and $f$ is (...)?

I'm studying example 4.1.3 of "Variational Analysis in Sobolev Spaces and BV Spaces" by Attouch, Buttazzo and Michaille. At one point it states that Given $f: R^n \to R^m \> (n\leq m)$ ...
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Convergence of the perimeter of level sets

Suppose you have a sequence of $C^1$ functions $\{\phi_n\}_{n\in \mathbb{N}}$ defined on $\mathbb{R}^n$ that converges in $C^{1}_{\mathrm{loc}}$ to a function $\phi$, as $n \to +\infty$. By $C^1_{\...
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Does there exist a function $f:\mathbb{R}\to\mathbb{R}$ where all points in the graph of f cannot be approximated by other points in $f$? [duplicate]

Does there exist an explicit function $f:\mathbb{R}\to\mathbb{R}$, such if $X$ is the graph of $f$ (i.e., $X=\left\{(x,f(x)):x\in\mathbb{R}\right\}$) then for each point $p$ on the graph of $f$, I ...
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Conditional Expectation of a mathematical function

Suppose $(X,d)$ is a metric space. If set $A\subseteq X$, let $H^{\alpha}$ be the $\alpha$-dimensional Hausdorff measure on $A$, where $\alpha\in[0,+\infty)$ and $\text{dim}_{\text{H}}(A)$ is the ...
Arbuja's user avatar
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7 votes
0 answers
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Is there a nonmeasurable subset of $\mathbb{R}^2$ that is $1-$dimensional Hausdorff measurable?

For $n\in\mathbb{N}^*$ and $s\in\mathbb{R}_{\ge 0}$, the $s-$dimensional Hausdorff measure $H^s$ is an outer measure over $\mathbb{R}^n$, and the $\sigma-$algebra of $s-$dimensional measurable subsets ...
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Prove $ \lim_{\varepsilon\to 0} \frac{dH^{n-1}(\Omega\cap\partial B(x,\varepsilon))}{dH^{n-1}(\partial B(x,\varepsilon))} = \frac{1}{2}.$

Assume $\Omega\subset \mathbb{R}^n$ is a bounded open set with Lipschitz boundary; let $d H^{n-1}$ be the Hausdorff measure on $\partial \Omega$. Since $\Omega$ has Lipschitz boundary, for $d H^{n-1}$-...
Milk's user avatar
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Is there an Approximate tangent space that is NOT a Classical tangent space?

First given $M \subset \mathbb{R}^n$, and $x \in \mathbb{R}^n$, $r>0$, we define the blow up by: $$ \Phi_{x,r}(y) = \frac{y-x}{r}.$$ We say $x \in \mathbb{R}^n$ has a $k$ dimensional approximate ...
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What is the "content" of a box?

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
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1 vote
1 answer
161 views

Hausdorff Dimension of an Intersection

Let $E$ and $F$ be two non-empty subsets of $\mathbb{R}^n$ for some $n \in \mathbb{N}$. I'm wondering if there's a simple formula or bound for the Hausdorff dimension of $E \cap F$ in terms of the ...
EndothermicIntegral's user avatar
2 votes
1 answer
100 views

Inequality in theorem proof: Hausdorff dimension and projection theorem with energy integrals (Mattila book)

I am studying Mattila's book "Fourier Analysis and Hausdorff Dimension" and I do not understand how to reach the first inequality in the proof of Theorem 4.2. It is the following: Let $2<...
Emilia's user avatar
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2 votes
0 answers
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Do smooth bijections always preserve Hausdorff dimension?

I am wondering if any smooth bijection $f \in C^\infty(\mathbb{C})$ on $\mathbb{C}$ preserves the Hausdorff dimension of any given subset $A \subset \mathbb{C}$? In particular, I am working on ...
Blue Jewel's user avatar
1 vote
0 answers
67 views

Relating an integral over a sphere to an integral of a sphere one dimension smaller "without" using polar coordinates

Since I have always been at war with polar coordinates, I wanted to find a way to compute spherical Hausdorff integrals in a different way. I know that this will get incredibly tedious if I try to ...
Sellerie's user avatar
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0 answers
221 views

Relation between Hausdorff measure and Lebesgue Measure

I have the following definition of $\alpha$-dimensional Hasudorff measure Let $E$ be subset of $\mathbb{R^n}$. Fix $\alpha \in [0, \infty)$, $\kappa \in \mathbb{R}^+$ and $\epsilon > 0$. Then $\...
Nirmal Rawat's user avatar
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53 views

Bounding the Hausdorff measure of the image of a map

Let $\phi:[0,1]^n\to [0,1]^m$ be continuous everywhere and smooth almost everywhere. The Hausdorff measure of $\phi([0,1]^n)$ is given by $$\mathcal{H}^n\left(\phi\left([0,1]^n\right)\right)=\int_{[0,...
Andrew Murdza's user avatar
3 votes
1 answer
262 views

Naïve definition of a measure on a fractal

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 the so-called Hausdorff measure $\mathcal H^...
Matheus Manzatto's user avatar
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141 views

Hausdorff dimension of the Koch curve

It is well known that the Hausdorff dimension of the Koch curve is $\log_34\approx1.26>1$. Thus by the property of Hausdorff measure, $\forall1<p<1.2$, the $p$-Hausdorff measure of the Koch ...
Liyang Shao's user avatar
1 vote
1 answer
84 views

$\| \gamma(b) - \gamma (a) \| \leq \mathcal{H}^1(\gamma([a,b]))$ if $\gamma$ is not rectifiable?

I'm dealing with Hausdorff-Measure and the length of curves. It widely known that for a rectifiable continuous curve $\gamma:[a,b] \rightarrow \mathbb{R}^n$ the inequasion $$\| \gamma(b) - \gamma (a) \...
Lukas's user avatar
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1 vote
1 answer
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Prove that if $E$ has smooth bounday $\lim_{r \to 0}{\mathcal{H}^{n-1}(\partial (E+B(0,r)))} = \mathcal{H}^{n-1}{\partial E} $

Prove that, if $E \subset \mathbb{R}^n$ is an open bounded set with $C^1$ boundary (if you want you can assume $C^\infty$ ), then the following holds $$\lim_{r \to 0}{\mathcal{H}^{n-1}(\partial (E + B(...
Paul's user avatar
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Hausdorff measure of uncountable dense subsets

As far as I know, the Hausdorff measure of a countable subset is zero (please correct me otherwise). Is it possible to say the same about the uncountable dense subsets? Is there a general statement ...
Ozzy's user avatar
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2 votes
0 answers
33 views

Show that $g \in C([a,b]) \cap BV([a,b])$ with $\mu(E) = 0 \implies \mu(g(E)) = 0$ is in $AC([a,b])$

Let $g : [0,1] \to \mathbb{R}$ be continuos and of bounded variation such that has the (N) proprety ( i.e. $\mu(E) = 0 \implies \mu(g(E)) = 0$ ), show that $g$ is absolutely continuos. My attempt : To ...
Paul's user avatar
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