This question came up (unexpectedly) in a problem I was working on a few years ago. It may not be too difficult but I never got around to figuring out the answer, because all I needed at that time was a constant multiple of Haar measure.

Let $k\ge 2$ and suppose that $p_1,\ldots , p_k$ are *distinct* primes. Let $\mu$ be the Haar probability measure on $\mathbb{Z}_{p_1}\times\cdots\times\mathbb{Z}_{p_k}$, and let $\mathrm{H}^k$ denote $k-$dimensional Hausdorff measure on the same space with respect to the metric $d$ defined by
$$d(\mathbf{x},\mathbf{y})=\max_{1\le i\le k}\{|x_i-y_i|_{p_i}\}.$$
It is easy to verify that $\mathrm{H}^k$ is finite, regular, and translation invariant and therefore $\mathrm{H}^k=c\mu$ for some real constant $c$. Furthermore if $k=2$ then it is not too difficult to show that $c=1$. Is it always true that $c=1$?