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MathJax: \dim
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Martin Sleziak
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Let $(X,d_X)$ and $(Y,d_Y)$ be two compact metric space with Hausdorff dimensions $dim_H(X)=n$$\dim_H(X)=n$ and $dim_H(Y)=m$$\dim_H(Y)=m$ and Hausdorff measures $\mathcal{H}^{n}$ and $\mathcal{H}^{m}$. Assume that $dim_H(X\times Y)=n+m$$\dim_H(X\times Y)=n+m$ for the cartesian product $(X\times Y, d)$ where $d=\sqrt{d_X^2+d_Y^2}$, then we have $(n+m)$-dimensional Hausdorff measure $\mathcal{H}^{n+m}$ on it.

Do we have $\mathcal{H}^{n+m}=\mathcal{H}^{n}\otimes\mathcal{H}^{m}$?

In particular, does the equation hold, if $Y$ is a smooth Riemannian manifold?

Let $(X,d_X)$ and $(Y,d_Y)$ be two compact metric space with Hausdorff dimensions $dim_H(X)=n$ and $dim_H(Y)=m$ and Hausdorff measures $\mathcal{H}^{n}$ and $\mathcal{H}^{m}$. Assume that $dim_H(X\times Y)=n+m$ for the cartesian product $(X\times Y, d)$ where $d=\sqrt{d_X^2+d_Y^2}$, then we have $(n+m)$-dimensional Hausdorff measure $\mathcal{H}^{n+m}$ on it.

Do we have $\mathcal{H}^{n+m}=\mathcal{H}^{n}\otimes\mathcal{H}^{m}$?

In particular, does the equation hold, if $Y$ is a smooth Riemannian manifold?

Let $(X,d_X)$ and $(Y,d_Y)$ be two compact metric space with Hausdorff dimensions $\dim_H(X)=n$ and $\dim_H(Y)=m$ and Hausdorff measures $\mathcal{H}^{n}$ and $\mathcal{H}^{m}$. Assume that $\dim_H(X\times Y)=n+m$ for the cartesian product $(X\times Y, d)$ where $d=\sqrt{d_X^2+d_Y^2}$, then we have $(n+m)$-dimensional Hausdorff measure $\mathcal{H}^{n+m}$ on it.

Do we have $\mathcal{H}^{n+m}=\mathcal{H}^{n}\otimes\mathcal{H}^{m}$?

In particular, does the equation hold, if $Y$ is a smooth Riemannian manifold?

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Jialong Deng
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The product of two Hausdorff measures

Let $(X,d_X)$ and $(Y,d_Y)$ be two compact metric space with Hausdorff dimensions $dim_H(X)=n$ and $dim_H(Y)=m$ and Hausdorff measures $\mathcal{H}^{n}$ and $\mathcal{H}^{m}$. Assume that $dim_H(X\times Y)=n+m$ for the cartesian product $(X\times Y, d)$ where $d=\sqrt{d_X^2+d_Y^2}$, then we have $(n+m)$-dimensional Hausdorff measure $\mathcal{H}^{n+m}$ on it.

Do we have $\mathcal{H}^{n+m}=\mathcal{H}^{n}\otimes\mathcal{H}^{m}$?

In particular, does the equation hold, if $Y$ is a smooth Riemannian manifold?