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

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.

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)
