In this question I will refer to Hausdorff dimension and measure of regular $\omega$-languages as defined here. Hereby I give a quick recap of the definitions before diving into the question.
Recap An $\omega$-language $F \subseteq \Sigma^\omega$ is any set of infinite sequences of symbols from the finite alphabet $\Sigma$ of size $r = |\Sigma|$. We call a cover of F a set $W \subseteq \Sigma^*$ of finite sequences of symbols in the same alphabet such that $W \cdot \Sigma^\omega \supseteq F$ (where $W \cdot \Sigma^\omega$ denotes the set of all the infinite sequences obtained by concatenating a sequence in $W$ with an infinite sequence in $F$). We denote by $\mathcal{L}_l$ the set of all $W$s whose sequences have length at least $l$.
The $\alpha$-dimensional Hausdorff measure of $F$ is defined as $$m_\alpha(F) = \lim_{n \rightarrow \infty} \inf_{W \in \mathcal{L}_l} \sum_{w \in W} r^{-\alpha |w|}\;.$$ The Hausdorff dimension of $F$ is the unique number $$\dim(F) = d = \sup \{\alpha|m_\alpha(F)=\infty\} = \inf \{\alpha|m_\alpha(F)=0\} \;.$$
Question I am interested in computing the Hausdorff dimension of the product of two $\omega$-languages $F_1$ and $F_2$ over disjoint alphabets $\Sigma_1$ and $\Sigma_2$.
Let $\Sigma = \Sigma_1 \times \Sigma_2$ of sizes $r_1$ and $r_2$ respectively, and two languages $F_1$ and $F_2$ such that $\dim(F_1) = d_1$ and $\dim(F_2)=d_2$ (where the measures are computed using $r_1$ and $r_2$ respectively). Let $$F=\{f=(f_{11},f_{12})(f_{21},f_{22})(f_{31}f_{32})\dots|f_{11}f_{21}f_{31}\dots\in F_1 \land f_{12}f_{22}f_{32}\dots\in F_2\} \;.$$ What is the Hausdorff dimension of $F$ as an $\omega$-language with alphabet $\Sigma$?
My reasoning so far I think I should reason by considering the Hausdorff measure of $F$ as the product measure of the Hausdorff measures of $F_1$ and $F_2$. But is it true that $$m_d(F) = m_{d_1}(F_1)m_{d_2}(F_2)$$ (considering also $r_1$ and $r_2$ as hidden parameters of $m_{d_1}$ and $m_{d_2}$) for some $d$? If so, for which $d$?
I regret I do not have an education in measure theory and I may be missing something, but indeed it vaguely looks to me that the measure of F can be seen as a joint probability of independent events (picking $f_{11}f_{21}\dots$ from $F_1$ and $f_{12}f_{22}\dots$ from $F_2$), and I know $m_d$ is a measure function. So it may be the case that it factorizes over the measures of $F_1$ and $F_2$. But I would not know how to simplify the product of the two $\lim \inf$. Just by means of changing the bases $r_1$ and $r_2$ into $r$ and by the fact that $d$ is in the exponent of the definition, it seems to me the answer may be $$\dim(F) = d_1 \log_r r_1 + d_2 \log_r r_2 \;.$$
If anyone had some pointers on how to address this question, I would be very grateful.