The product of two Hausdorff measures

Question

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?

Answer 1

As noted in a comment, for Riemannian manifolds, the Hausdorff measures are equal to (up to a constant) the usual volumes. So this works.

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The metric case you mention can fail. There are metric spaces of Hausdorff dimension $1$ that are not "rectifiable". Every subset has either $\mathcal H^1(E) = 0$ or $\mathcal H^1(E) = \infty$. Discussion is in Chapter 3 of

Falconer, K. J., The geometry of fractal sets, Cambridge Tracts in Mathematics, 85. Cambridge etc.: Cambridge University Press. XIV, 162 p. (1985). ZBL0587.28004.

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And also note this is incorrect for general dimensions. Reference: third edition of

Falconer, Kenneth, Fractal geometry. Mathematical foundations and applications, Hoboken, NJ: John Wiley & Sons (ISBN 978-1-119-94239-9/hbk). xxx, 368 p. (2014). ZBL1285.28011.

Here are some things from Chapter 7.

Proposition 7.1
If $E \subset \mathbb R^n, F \subset \mathbb R^m$ are Borel sets with $\mathcal H^s(E), \mathcal H^t(F) < \infty$, then $$ \mathcal H^{s+t}(E \times F) \ge c \mathcal H^s(E)\;\mathcal H^t(F) \tag1$$ where $c > 0$ depends only on $s$ and $t$.

The opposite inequality $\le$ can fail.

Example 7.8
There exist sets $E, F \subset \mathbb R$ with $\dim_\mathcal H E = \dim_\mathcal H F = 0$ and $\dim_\mathcal H(F \times F) \ge 1$.

Falconer credits these results to:

Besicovitch, A. S.; Moran, P. A. P., The measure of product and cylinder sets, J. Lond. Math. Soc. 20, 110-120 (1945). ZBL0063.00354.

and

Marstrand, J. M., The dimension of Cartesian product sets, Proc. Camb. Philos. Soc. 50, 198-202 (1954). ZBL0055.05102.

plugs:

Those two papers are among those reprinted in

Edgar, Gerald A. (ed.), Classics on fractals, Reading, MA: Addison-Wesley Publishing Company. x, 366 p. (1993). ZBL0795.28007.

More on this from a student of mine:

Mullins, Edmond N., Jr, Derivation bases, interval functions, and fractal measures. Thesis (Ph.D.)–The Ohio State University. 1996. 97 pp. ISBN: 978-0591-18087-9 (ProQuest LLC)