Question

Deduce from Theorem 4.10 that if $F$ is a Borel subset of $\mathbb{R}^n$ with $\mathcal{H}^s(F)=\infty$ and $c$ is a positive number, then there is a Borel subset $E$ of $F$ with $\mathcal{H}^s(E)=c$.

    Deduce from Theorem 4.10 that if $F$ is a Borel subset of $\mathbb{R}^n$ with $\mathcal{H}^s(F)=\infty$ and $c$ is a positive number, then there is a Borel subset $E$ of $F$ with $\mathcal{H}^s(E)=c$.

Fractal Geometry: Mathematical Foundations and Applications

Fractal Geometry: Mathematical Foundations and Applications

Kenneth Falconer 2nd Edition

Chapter 4, Problem 10

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We are given that $F$ is a Borel subset of $\mathbb{R}^n$ and that the $s$-dimensional Hausdorff measure of $F$, denoted $\mathcal{H}^s(F)$, is infinite. We are also given a positive number $c$. The goal is to find a Borel subset $E$ of $F$ such that Show more…

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Deduce from Theorem 4.10 that if $F$ is a Borel subset of $\mathbb{R}^n$ with $\mathcal{H}^s(F)=\infty$ and $c$ is a positive number, then there is a Borel subset $E$ of $F$ with $\mathcal{H}^s(E)=c$.

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