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Questions tagged [hausdorff-measure]

If $(X,\rho)$ is a metric space, then for any subset $S$, we have $$ H_\delta^d(S):=\inf\ \{ \sum_{i=1}^\infty ({\rm diam}\ U_i)^d | S\subset \bigcup_{i=1}^\infty U_i,\ {\rm diam}\ U_i < \delta \} $$ 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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Measurability of Hausdorff measure of a given level set for stochastic process

I have this question and I can't find a reasonable answer. Let $(\Omega, F, P)$ be a probability space, and $X$ be a continuous stochastic process from $\mathbb{R}^m$ to $\mathbb{R^n}$. That is, for $...
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Anisotropic perimeter as derivative of measures.

Fix $K$ open bounded convex subset of $\mathbb R^n$ containing the origin and $\lvert x\rvert$ the euclidean norm of $x$. We define for every $x\in \mathbb R^n$ these quantities $$ \lVert x\rVert=\inf\...
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1answer
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Does the domain where a harmonic map is conformal has Hausdorff dimension smaller than one?

Let $\mathbb{D}^2$ be the closed unit disk, and let $f:\mathbb{D}^2 \to \mathbb{R}^2$ be a harmonic map, satisfying $\det df \neq 0$ almost everywhere on $\mathbb{D}^2$. Suppose also that $$U= \{ p \...
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1answer
51 views

Why are Hausdorff dimension and Minkowski (box) dimension not equivalent?

I am putting the finishing touches on my master's essay for graduation this semester and I want to end my paper with a proof of why Hausdorff dimension and Minkowski (box) dimension are different. ...
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51 views

Is it possible to have a Hausdorff dimension less than the topological dimension?

"Normal" geometric shapes have Hausdorff dimensions equal to their topological dimensions. Mandelbrot defined fractals as shapes that have a Hausdorff dimension greater than their topological ...
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44 views

Every set is contained in a $G_\delta$ set of the same Hausdorff dimension

I have been trying to prove that if $(X,d)$ is a metric space, then every subset of $X$ is contained in a $G_\delta$ set of the same Hausdorff dimension. I know that this is obvious if a set is open ...
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50 views

Self-similarity dimension for IFS or attractor of IFS

We can have same attractor from different iterated function systems. So i wonder about self-similarity dimension concept is for IFS or its attractor. We know that when IFS satisfies the open set ...
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Show that $\dim_{H}(A) = \dim_{H}(\Phi(A))$ for diffeomorphism $\Phi$

I have a question about Hausdorff dimensions and hope some of you can help me. I'm quite new with this topic so I hope this is not a too stupid question. For a given $\Phi: \Omega \rightarrow \mathbb{...
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Can't prove these are equivalent.

This is (part of) Exercise 1.15.20 in An Epsilon of Room I, Terence Tao. Exercise 1.15.20. Let $0<d\leq n$, and let $E\subset \mathbb{R}^n$ be a compact set. Show that $\mathrm{dim}_H(E)\geq d$ ...
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Convolution square of the Cantor set

For $0\leq d\leq 1$, let $\eta_d$ be the $d$-dimensional Hausdorff measure on $\mathbb{R}$ (for some normalization); recall that it is translation-invariant. Motivation for what follows: Up to ...
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Smooth function $f$ with large Hausdorff dimension of $\{x: \ x=0,\ \nabla f(x)\ne0\}$

Is it possible to construct a function $f:[0,1]^2 \to \mathbb R$ twice continuously differentiable (or in the Sobolev space $H^2((0,1)^2)$ such that the set $$ \{ x: \ f(x)=0 , \ \nabla f(x)\ne 0\} $$ ...
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Question regarding measurable set, Hausdorff-Space and almost everywhere properties of measurable functions f,g

I've been given the following task Let $(X,\mathscr{M}_X,\mu)$ a measure space. Two measurable mappings $f,g:X \to Y$ into a measurable space $(Y,\mathscr{M}_Y)$ are called equal almost ...
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1answer
52 views

Understanding a remark for the Hausdorff measure in Wolff's lecture notes

In the chapter of Hausdorff measures in Wolff's notes on harmonic analysis, I'm trying to understand a piece of remark. Fix $\alpha>0$, and let $E\subset\mathbb{R}^n$. For $\epsilon>0$, one ...
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Hausdorff $2$-dimensional measure of $\mathbf{R}$ [closed]

I know that $\mathcal{H}^2(\mathbf{R}) = 0$, but what is any easy way to see it? I tried coming up with reasonable covers, but they all seem to give upper bounds that are too large.
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Is there a technique to exactly calculate the Hausdorff dimension of the border of this fractal?

I came up with a self-similar fractal that fits into itself like a jigsaw puzzle. While the surface area is clearly 2-dimensional, it is unclear to me how to compute the Hausdorff dimension of the ...
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I can't find this theorem about Hausdorff dimension and finite many contractions.

I am preparing for an exam on measure theory and I can't find a theorem, that we used to calculate the Hausdorff dimension of certain sets in $\mathbb{R}^n$. It used contractions and their ...
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108 views

Exact value of Hausdorff measure of two dimensional Cantor set

Let $\mathcal{C}$ denote the classical Cantor set, then it is well-known that $\mathcal{C}$ has Hausdorff dimension $\alpha = \ln 2 /\ln 3$, and its $\alpha$-dimensional Hausdorff measure is $\mathcal{...
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Hölder criterion and Hausdorff dimension

I've recently learned that the Cantor function is Hölder continuous with parameter $\alpha$ where $\alpha$ is the Hausdorff dimension of the Cantor set. My intuition is that this fact stems from the ...
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31 views

Calculate Lebesgue and Hausdorf measure of a hexagon

Given this hexagon $P$, I've got to calculate the Lebesgue measure $\lambda_{2}(P)$ and the Hausdorff measure $\mathscr{H}^1(\partial P)$. My thoughts are: You can leave out the 6 line segments of ...
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26 views

Estimate/inequality releated to the Hausdorff measure

Define the diameter of a subset $Y \subseteq \mathbb R^n$ of the metric space $(\mathbb R^n, d)$ with the standard metric $d$ to be $$\operatorname{diam}(Y) := \sup_{\mathbf x,\mathbf y \in Y} d(\...
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Hausdorff Dimension Infinity

What are some examples of non-trivial metric spaces that have Hausdorff Dimension of infinity? I could only think of $\mathbb{R}$ with the discreet metric.
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0-dimensional Hausdorff dimension of a set

There is an easy proof to show that for a finite object of any size $H^0(M) = |M|$, its 0-Hausdorff dimension is equal to its size. This is done by covering the set by sufficiently small balls and ...
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Do compacta in $\mathbb R^n$ have finite Hausdorff measure? [duplicate]

For $A\subset \mathbb R^n$ let $\mathcal H^s(A)$ be the $s$-dimensional Hausdorff measure with respect to the Euclidean metric. The Hausdorff dimension of $A$ is given by $$\dim_H(A) = \inf\{s>0\...
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35 views

Hausdorff measure (property)

If $\mathcal{H^s}$ is the Hausdorff measure, how to show: $\bigcup_{i=1}^{\infty} E_{i}$ with $\mathcal{H^s}(E_i)<\infty$? I tried to use the properties of the Hausdorff measure to show this. I ...
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1answer
33 views

Positive Hausdorff measure and $L^{2}$ convergence. [closed]

i would like to know the relation between the positive Hausdorff measure and $L^{2}$ convergence in the ((1.3) existente theorem)
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1answer
41 views

How to define “eventually a subset”?

Let $(C_n)_n$ be a sequence of nonempty subsets of $[0,1]$ (we can assume that the $A_n$ are open/closed if it helps). Let $P$ be a nonempty subset if $[0,1]$. What is a natural way to formalize ...
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1answer
26 views

Are there simple criteria for determining if there is a finite, non-zero Hausdorff measure of a set?

It’s kind of a wide question, so I’d like to motivate it: When trying to determine the Hausdorff dimension of a set, I found that it is often reasonably easy to find it heuristically, but very ...
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68 views

Question about Hausdorff dimension and Lipschitz homeomorphism.

Is it true that the Hausdorff dimension of a subset of a segment does not change under a segment homeomorphism satisfying the Lipschitz condition? Please help, i proved fact, that for $C$-lipschitz ...
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1answer
51 views

$\delta$-approximative-p-dimensional Hausdorff measure a measure? [closed]

Might someone explain why it is necessary to define the Hausdorff-measure as the limit of the outer measure depictured below?
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1answer
74 views

Hausdorff measure on non separable spaces

In his book Geometry of Sets and Measures in Euclidean Spaces, Pertti Mattila defines the Hausdorff measures via the Carathéodory's construction (chap.4). My doubt ...
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35 views

Upper bound on Hausdorff dimension of set

I'm trying to solve the following question: Let $2\leq b\in \mathbb{N}$, $A\subseteq [0,1)$ and $\mu$ be a finite Borel measure such that $\mu(A)>0$. Denote $I_{n,b}^j:=\Big[ \frac{j-1}{b^n}, \...
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Capacity and Hausdorff measure

My question arise from Thm 4.16 page 179 of the book by Evans & Gariepy, Measure Theory and Fine Properties of Functions (revised edition). I want to prove the following: If $\mathcal{H}^{n-p}(A) ...
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39 views

Convergence of $H_\delta^s(A_n) \rightarrow H_\delta^s(A)$ for $A_n \to A$

Let X be a metric space, define $H_\delta^s(E):= \inf \sum ( \operatorname{diam}(E_i))^s$ where the infimum is taken over all countable coverings $(E_i)_{i\in \mathbb{N}}$ of E with $ \operatorname{...
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How to prove that the Hausdorff Measure is $\infty$ for all $t<s$?

I have recently started to study measure theory and in particular, right now, the Hausdorff measure. This question concerns a pdf-file I am reading: https://www.math.cuhk.edu.hk/course_builder/1415/...
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1answer
75 views

Hausdorff measure a radon measure

Can someone help me to prove that the Hausdorff measure is a Radon measure (see here for Radon measure)? I do not know how to start, so I would be thankful for some advice.
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Existence of dimension function (i.e., exact gauge function)

Motivation: Some sets have Hausdorff dimension $\alpha$ but have zero $\alpha$-dimensional Hausdorff measure. These sets may have another dimension function; i.e., a function $h:[0,\infty)\to[0,\infty)...
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Hausdorff measure approximation by $G_{\delta}$ sets.

Let $H^s$ denote the s-dimensional Hausdorff measure of an arbitrary set $A\subset\mathbb{R}^n$ given by $$ H^s(A)=\lim_{\epsilon\rightarrow0}H^s_{\epsilon}(A) $$ Where $H_\epsilon^s$ is an auxiliary ...
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1answer
67 views

Hausdorff measure of a smooth curve

I want to find a refrence to the following question: If $f:(a,b)\rightarrow \mathbb{R}^d$ is a parameterization of a smooth curve, then: $\int_a^b\vert f'(t)\vert dt=\mathcal{H}_1\Big( f\big[ (a,b) \...
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Letting $\delta \rightarrow \infty$ in the $\delta$-Hausdorff premeasure

This may come off as a naive question but it's something I've always wondered. We define the Hausdorff measure as $\mathcal H^s(A) = \lim_{\delta\rightarrow0}\mathcal H^s_\delta(A)$ for some $A\...
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1answer
127 views

Set of Hausdorff dimension $\alpha$

I would like to construct a set $\Sigma\subset\mathbb{R}^n$ such that $$ mr^\alpha\leq\mathscr{H}^\alpha(B(0,r)\cap\Sigma)\leq M r^\alpha, $$ for all $0\leq r\leq1$ and some $0<m\leq M<\infty$. ...
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1answer
57 views

Calculating the length of a curve $(x,x^2)$ for $x\in [0,2]$ using the Hausdorff measure of the set $A=\{(x,x^2)\in\mathbb{R}^2\mid x\in[0,2] \}$?

Let $A$ be the set of points given by the graph of $f(x) = x^2$ on the interval $[0,1]$: $$ A = \{(x,x^2) \in \mathbb{R}^2\bigg| \ x \in[0,2] \}. $$ I want to calculate the length of this one ...
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The $m$-dimensional Hausdorff measure agrees with the Lebesgue measure on $\mathbb{R}^n$ when $m=n$?

Let $\alpha_m$ be the Lebesgue measure of the closed unt ball in $\mathbb{R}^m$. For $A \subset \mathbb{R}^n$, the $m$-dimesnsional Hausdorff measure of $A$ is defined as: $$ H^m(A) = \lim_{\delta \to ...
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2answers
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Does this fractal have a Hausdorff dimension of 1?

So I'm familiar with the simple methods of calculating the Hausdorff dimension of a fractal, but when I try to apply them for this case I get into trouble. What I mean with the simple method is using ...
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1answer
32 views

Sufficient condition for the growth of Hausdorff measure which implies sigma-finiteness

The Hausdorff meausure of $E\subset\mathbb R^n$ is $H^s(E)=\lim\limits_{\delta\to0}H^s_\delta(E)$ where $H_{\delta }^{s}(E)=\inf {\Bigl \{}\sum _{{i=1}}^{\infty }(\operatorname {diam}\;U_{i})^{s}:\...
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Propeties of the Hausdorff Measure

Using the definition of the Hausdorff measure as $$H^\alpha(C) = \lim_{\delta\rightarrow\infty} H_\delta^\alpha(C) = \lim_{\delta\rightarrow\infty} \left( \inf\sum_{i=1}^\infty |\operatorname{diam}(...
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1answer
60 views

Hausdorff measure of the Cartesian square of a Cantor like set

Define $K_{0}=[0,1] $ $K_{1}=[0,1/8]\cup[1-1/8,1] $ $K_{2}=[0,1/8^{2}]\cup[1/8-1/8^{2},1/8]\cup[7/8,7/8+1/8^{2}]\cup[1-1/8^{2},1] $ and so on... then put $K=\bigcap_{i=0}^{\infty}K_i$ [Claim: ...
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The Hausdorff, packing, and Minkowski dimensions of general Cantor-type sets

I want to find a reference for the proof of the following statements: If $C$ is a Cantor middle $\{\alpha_n\}$-set, $0\leq \alpha_n\leq 1$, $a_n:=\prod_{j=1}^n(1-\alpha_j)/2$, then: $\dim_P(C)={\...
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104 views

One-dimensional Hausdorff measure of a line segment [closed]

Let $\mathcal{H}^1$ be the one-dimensional Hausdorff measure in $\mathbb{R}^n$ and let $[uv] = \{ u + t(v-u) : t \in [0,1] \}$ be the segment joining the vectors $u,v \in \mathbb{R}^n$. How do we show ...
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2answers
86 views

Let $\gamma\in C(I,\Bbb R^n)$. Show that $\dim_H(\Gamma)=1$

Let $I:=[a,b]$ a perfect interval and $\gamma\in C(I,\Bbb R^n)$ an injective path such that $\Gamma:=\gamma(I)$ is rectifiable. Show that $\dim_H(\Gamma)=1$. Here $\dim_H$ is the Hausdorff dimension. ...
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64 views

For $A\subset\Bbb R^n$ and $B\subset \Bbb R^m$ show that $\dim_H(A\times B)=\dim_H(A)+\dim_H(B)$

Im stuck with this exercise For $A\subset\Bbb R^n$ and $B\subset \Bbb R^m$ show that $\dim_H(A\times B)=\dim_H(A)+\dim_H(B)$ where $\dim_H$ is the Hausdorff dimension. I know that when $A$ and $B$ ...