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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28 views

Can the Hausdorff dimension of a subset of the 2D plane be greater than 2?

Although I've gotten the gist, I'm relatively unfamiliar with the bounds of a Hausdorff measure. From my understanding, very loosely speaking, it's a way of taking a geometric object, usually a ...
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29 views

Hausdorff dimension of Borel set of positive measure in $\mathbb{R}^n$

Let $H_{\alpha}$ be the $\alpha$-Hausdorff measure, and $m_n$ be the $n$-dimensional Lebesgue measure. How do I prove that if $𝐴\subset \mathbb{R}^n$ is a Borel set of positive measure $m_n(A)>0,$...
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Sets of infinite Hausdorff dimension in a second countable metric space

I am wondering if there exists an example of a second countable metric space $X$ containing a set $A$ with infinite Hausdorff dimension.
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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 ...
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Hausdorff dimension of product omega-languages

In this question I will refer to Hausdorff dimension and measure of regular $\omega$-languages as defined here. Hereby I give a quick recap of the definitions before diving into the question. Recap An ...
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155 views

Lower bound for upper $\pi/2$ angular density

This is exercise 2.3 in Falconer's book 'The Geometry of Fractal Sets'. Let $E\subset \mathbb{R}^n$ be an $\textit{s}$-set. That is, it is measurable for the s-dimensional Hausdorff measure $H^s$ and ...
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Multifractal scalling exponent function \tau(q) for a Cauchy r.v.

The following function is called Multifractal scalling exponent function $\tau(q)$ $$ \tau(q)=\lim_{n\rightarrow\infty}\log_{n}\sum_{k=1}^{n}\left|\mathscr{H}\left(\left[\tfrac{k-1}{n},\tfrac{k}{n}\...
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Hausdorff measure of Cantor set

I am trying to find a set $A\subset\mathbb{R}$ with Hausdorff dimension $\log2/\log3=:s$ but has $H^s(A)=\infty$. I suspect this is the Cantor set, but im struggling to show that it has Hausdorff ...
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The Hausdorff measure of the unit interval

I am trying to calculate the Hausdorff measure of the unit interval. Here's my attempt: Fix $\epsilon>0$. consider the open balls $B(x,\epsilon)$ with $x\in[0,1]$. How many can cover the unit ...
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Inequality of Hausdorff measures for convex sets $\mathfrak{H}^{n-1}(\partial E)\le \mathfrak{H}^{n-1}(\partial F)$

I'm preparing for an exam on the calculus of variations and I need help in solving this exercise from an old exam text (actually it's only a part of a bigger exercise but its parts are quite ...
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Integral equality implies Hausdorff measure equality?

Suppose that the following equality holds true: $$\int_{\mathbb{R}}\mathcal{H}^{n-1}\left(B \cap A_r \right)dr=\int_{\mathbb{R}}\mathcal{H}^{n-1}\left(B \cap C_r \right)dr, \quad \forall \, B \in \...
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Contraction Proof - Prove that $d_H(S(A), S(B)) ≤ cd_H(A, B)$

I need a little help with this simple problem whilst revising; Let S be a contraction on D with constant 0 < c < 1 and A, B ∈ S. Prove that $d_H(S(A), S(B)) ≤ cd_H(A, B)$ where $d_H$ is the ...
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Hausdorff Measure of Linear orthogonal map

Question: Suppose $T:\mathbb{R}^n\rightarrow\mathbb{R}^m$ is a linear orthogonal map $(n<m)$. Prove that $$\mathcal{H}^s(T(A))=\mathcal{H}^s(A)$$ for all $0\le s$. $\\$ ${\color{red}{\text{What ...
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Show for a $\gamma-$holder map $\mathcal{H}^{s/\gamma}(f(A))\le MH^s(A)$

Suppose $f: \mathbb{R}^n\rightarrow \mathbb{R}^m$ is a $\gamma-$Holder continuous map $(0<\gamma\le1)$, that is:} $$|f(x)-f(y)|\le C |x-y|^\gamma, \ \ \ \ \text{for some }C>0, \text{ and all }x,...
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Hausdorff Measure - δ-covers by closed sets

I'm struggling with this exercise; Consider the following definition of the s-dimensional Hausdorff measure of F; $$\mathcal{H}^s(F) := \lim_{\delta \to 0} \mathcal{H}^s_\delta(F) := \lim_{\delta \...
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Hausdorff Measure in $R^n and R^m$

I'm struggling with this exercise at the moment; Let F ⊂ Rn ⊂ Rm for some integers 0 ≤ n < m. Let s ≥ 0. Show that the s-dimensional Hausdorff measure of F is the same whether F is regarded either ...
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$(n-1)$-Hausdorff measure of orthogonal projection onto an hyperplane of compact sets of $\Bbb R^n$

Let be $F$ a compact set of $\Bbb R^n$ and $\mu$ the $(n - 1)$-dimensional Hausdorff measure. If I denote $P_V : \Bbb R^n \to \Bbb R^n$ the orthogonal projection on $V$ for $V \subset \Bbb R^n$ a ...
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Upper bound on the exact Hausdorff measure

Let the $s$-dimensional Hausdorff measure for a Borel set $F \subset \mathbb{R}^d$ be denoted as follows: $$\mathcal{H}^s(F) := \lim_{\delta \to 0} \mathcal{H}^s_\delta(F) := \lim_{\delta \to 0} \inf ...
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Growth rate of the Hausdorff measure

Let the $s$-dimensional Hausdorff measure for a Borel set $F \subset \mathbb{R}^d$ be denoted as follows: $$\mathcal{H}^s(F) := \lim_{\delta \to 0} \mathcal{H}^s_\delta(F) := \lim_{\delta \to 0} \inf ...
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“continuity” of a regular measure on a locally compact Hausdorff space

Let $X$ be a locally compact Hausdorff space and $\mu$ a regular atom-free measure (i.e. $\mu(\{x\})=0$) for every $x\in X$. Show that if $B\subset X$ is a $\sigma$-finite Borel set and $a$ a real ...
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Hausdorff dimension from above

Let $A_n$ be a set of Hausdorff dimension $1-\frac{1}{n}$ then, the set $$A=\cup_n A_n$$ Has Hausdorff dimension $1$ (nevertheless having $H_1(A)=0$). My question is: can we do the same thing from ...
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Hausdorff dimension for bounded set

Let $F$ be the set of of numbers $x\in [0,1]$ with base 3 expansions $0.a_1a_2...$ for which there exists an integer k such that $a_i\neq 1$ for all $i\geq k$. Find the Hausdorff dimension of $F$. ...
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Hausdorff dimension and mass distribution

I'm struggling to understand part of a proof in Falconer's book on Fractal Geometry. It's Theorem 4.13 (a): Let $F$ be a subset of $\mathbb{R}^n$. If there is a mass distribution $\mu$ on $F$ with $...
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Does Hausdorff-measurability depend on the choice of Riemannian metric?

Let $M$ be a smooth (second-countable) manifold and let $g, g'$ be smooth Riemannian metrics on $M$ which induce metrics (as in "metric space") $d$ and $d'$ on $M$. Fix $j ≥ 0$ and let $H$ and $H'$ ...
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Hausdorff-dimension of non measurable sets?

The hausdorff-outer-measure is defined for all subsets of a metric space. The hausdorff measure is defined as the restriction to caratheodory measurable sets. I actually don't know how the set of ...
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Infinite Hausdorff dimension in discrete metric spaces

I was searching for a metric space that has infinite Hausdorff dimenion . I stumbled upon the example of $\mathbb{R}$ with discrete metric. $\mathbb{R}$ should then have infinite dimension but I ...
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Compact set of the plane with boundary of finite 1-Hausdrof meausre as intersection of open set with piewise smooth boundary.

I am considering a compact set $K$ of the plane with $\mathcal H^1(\partial K)=1$. Does there exists open sets with smooth boundary $O_n$, $n\in \mathbb N$ such that $K=\bigcap_n O_n$ and $\mathcal H^...
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Hausdorff dimension of the relative complement of a set contained in a $G_\delta$

We have the following result: Every set is contained in a $G_\delta$ set of the same Hausdorff dimension I was wondering how tight can this inclusion be made, complement-wise. Is true that: Let ...
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18 views

Are there interesting general properties held by sets with small Hausdorff dimension?

Let us focus on the plane. I am wondering whether the sets with small (but non zero) Hausdorff dimension share some interesting properties. Let $\epsilon\in(0,1)$ and assume that a set $F\subset \...
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46 views

Hausdorff measure finite and positive

Let $A\subseteq [0,1]^d$ be a Borel measurable set. Let $d_H(A)$ be the Hausdorff dimension of $A$. Let $\mathcal{H}^{d_H(A)}$ be the Hausdorff measure w.r.t to the dimension of $A$. My question is, ...
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52 views

Complement of strictly less than (n-1)-dimensional set is connected?

Let $n\in \mathbb{N}$ and $A\subseteq \mathbb{R}^n$ be a subset such that its Hausdorff dimension is strictly less than $n-1$. Is it then true that $\mathbb{R}^n\setminus A$ is (path-)connected? ...
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Calculating Haursdorff measure of Cantor Dust and Sierpinski Carpet

I am a masters student and I have taken a course on measure theory ( it is my 1 st measure theory course). In one of initial classes professor gave defination of Haursdorff measure similar to as ...
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Last step in proof of Hausdorff outer measure $\mathcal{H}^{s,*} (cA) = c^s \mathcal{H}^{s,*}(A)$

The problem is just showing that the Hausdorff outer measure scales as follows: $$\mathcal{H}^{s,*} (cA) = c^s \mathcal{H}^{s,*}(A)$$ For some covering $\{E_k\}$ of $A$, we get that $$cA \subset \...
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Showing that function satisfying Hölder condition with $\alpha=2$ is constant using Hausdorff measure?

Suppose we are given the following: Let $f:X\rightarrow Y$ be such that $|f(x)-f(y)| \leq c|x-y|^\alpha$ for all $x,y \in X$. Let $U \subset X$. Then, for all $s$, $H^{s/\alpha}(f(U)) \leq c^{s/\...
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Frostman's Lemma

Say $h_\alpha$ is Hausdorff measure. Recall Frostman's Lemma. Suppose $K\subset\Bbb R^d$ is compact. Then $h_\alpha(K)>0$ if and only if there exists a (regular Borel) probability measure $\...
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Are the Hausdorff measures inner regular?

I've been looking all over the usual textbooks on geometric measure theory, and I can't seem to find a discussion of the fact whether the Hausdorff measures $\{ H^s : 0 < s < d \}$ are inner ...
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Scaling property proof of Hausdorff measure

I need to prove that $\mathcal H^s(\lambda F) = \lambda^s\mathcal H^s(F)$. Now my argument is as follows: Let $\{U_i\}$ be a $\delta$-cover of $F$, then $\{\lambda U_i\}$ is a $\lambda\delta$-cover ...
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50 views

Looking for Clarification Regarding Hausdorff Dimension (Lay)

I am reading an introductory book on Fractals where the following intuition is given for Hausdorff dimension of a smooth curve: if we define "length" as $L(r)=N(r)\cdot r$, with $N(r)$ straight line ...
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Hausdorff dimension of compact set

Let $(X, d)$ be a metric space and $S \subset X$. Let $\DeclareMathOperator{\diam}{diam}\diam(S)$ denote diameter of $S$, that is $\diam(S) = \sup \{ d(x, y) \colon \: x, y \in S \}$. Let $\delta > ...
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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.

Let $K$ be an open bounded convex subset of $\mathbb R^n$ containing the origin and $\lvert x\rvert$ be the euclidean norm of $x$. We define the quantities $$ \begin{split} &\lVert x\rVert=\inf\...
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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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161 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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180 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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99 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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122 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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29 views

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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84 views

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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67 views

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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59 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 ...