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?



The product of two Hausdorff measures
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](https://zbmath.org/?q=an:0587.28004).




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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](https://zbmath.org/?q=an:1285.28011).


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**](http://dx.doi.org/10.1112/jlms/s1-20.2.110), J. Lond. Math. Soc. 20, 110-120 (1945). [ZBL0063.00354](https://zbmath.org/?q=an:0063.00354).


and


*Marstrand, J. M.*, The dimension of Cartesian product sets, Proc. Camb. Philos. Soc. 50, 198-202 (1954). [ZBL0055.05102](https://zbmath.org/?q=an:0055.05102).


**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](https://zbmath.org/?q=an:0795.28007).


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)


