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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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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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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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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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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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 ...
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. ...
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 $B$ ...