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I have tried to construct a sequence of sets satisfying the following requirement. But I cannot. Could someone help me?

Let $A$ be a compact subset of $d$-dimensional $C^1$-manifold embedded in $\mathbb{R}^m$ with $\mathcal{H}_d(A)=\infty.$

Can we find a sequence of compact subsets of $A$, say $\{A_n\}_{n \in \mathbb{N}}$, such that $\mathcal{H}_d(A_n)<\infty$ for all $n\in \mathbb{N}$ and $$\lim_{n \rightarrow \infty} \mathcal{H}_d(A_n)=\infty?$$

Thank you.

Masih

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1 Answer 1

As stated, the hypotheses involve a contradiction. A compact subset of $C^1$ manifold $M$ of dimension $d$ cannot have infinite $d$-dimensional measure. Indeed, every point $p\in M$ has a neighborhood $U$ covered by $C^1$ (hence Lipschitz) chart, which implies that $\mathcal{H}_d(U)<\infty$. Every compact subset of $M$ can be covered by finitely many such neighborhoods.


Dropping the manifold assumption, it's possible to find such a sequence for quite general sets of infinite Hausdorff measure (not necessarily integer-dimensional). The proof is not easy and I refer to Geometry of sets and measures in Euclidean spaces by Mattila.

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