All Questions

Let $\mathcal{H}^n$ and $\mathcal{H}^m$ be Hausdorff measures on $\mathbb{R}^n$ and $\mathbb{R}^m$. We know that the product measure $\mathcal{H}^n\otimes \mathcal{H}^m$ is the Hausdorff measure $\...

Let $\Sigma\subset \mathbb{R}^{n+1}$ be a set with $(n-2)$-dimensional Hausdorff measure finite, i.e. $\mathscr{H}^{n-2}(\Sigma)<\infty$. Let $\pi:\mathbb{R}^{n+1}\to \mathbb{R}^n$ be the ...

Let $E\subset \mathbb{R}^n$ be a (measurable) subset with $\mathcal{H}^{n-1}(E)=0$, where $\mathcal H^{n - 1}$ is the ($n - 1$)-dimensional Hausdorff measure. I want to know if $\mathbb{R}^n\setminus ...

The metric cannot be recovered from its Hausdorff measure in general. Now, assume that $(X,d_X)$ and $(Y, d_Y)$ are connected compact length spaces and induce $n$-dimensional Hausdorff measures $\...

Let $X,Y$ be Polish spaces, $s,t>0$ and $F:X\to Y$ locally Lipschitz continuous such that $X$ is $\sigma$-finite w.r.t. the $(s+t)$-dimensional Hausdorff measure $\mathcal{H}^{s+t}$. The Eilenberg ...