# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...