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I am looking for examples of non-empty metric spaces which are compact with Hausdorff dimension $\alpha$, and have either its $\alpha$-dimensional Hausdorff measure equal to $0$ or its $\alpha$-dimensional Hausdorff measure equal to $\infty$.

Without compactness, examples with the second propery are easy to find, as the real line has dimension $1$, but $1$-dimensional measure equal to $\infty$. That's why I added compactness.

(I assume that this question has already been asked on this site, but I could not find it ...)

Cosine
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1 Answers1

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It might be possible with $\{0,1\}^{\mathbb{N}}$, if you try different metrics on it. Given $a\in (0,\infty)^{\mathbb{N}}$ and $\gamma>0$ put $d_{a,\gamma(x,y)}=a_{\min\{n\in\mathbb{N}\mid x(n)\neq y(n)\}}2^{-\gamma\min\{n\in\mathbb{N}\mid x(n)\neq y(n)\}}$. If $a_n2^{-\gamma n}> a_m2^{-\gamma m}$ for $n<m$, then this defines a metric generating the product topology on $\{0,1\}^{\mathbb{N}}$. If $a_n$ is bounded by a polynomial in $n$ then the Hausdorff dimension of $d_{a,\gamma(x,y)}$ is $\frac{1}{\gamma}$. If you choose $a_n=n$ for sufficiently large $n$ then the measure will be infinite and if you choose $a_n=\frac{1}{n}$ for sufficiently large $n$ then the measure will be zero. Of course some details have to be filled in, but I hope this will work.

dialegou
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