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Here is a theorem found in the Falconer's book on fractal geometry:

Theorem: For any sets $E\subset \mathbb{R}^n$ and $F\subset \mathbb{R}^m$

$$ \dim_HF+\dim_HE\leq \dim_H(E\times F)\leq \dim_HE+\overline{\dim_B}F. $$

This result is also valid when $E$ and $F$ are separable metric spaces.

This kind of result is also commented in this post: Hausdorff dimension of R x X.

My question: Denote $\alpha= \dim_H(E\times F)$ and assume that the Hausdorff measures $m_{\dim_H(E)}$ and $m_{\dim_H(F)}$ are finite or $\sigma$-finite measures. Are there any conditions for the metric spaces $E$ and $F$ in order for the associated Hausdorff measure $m_{\alpha}$ to be finite or $\sigma$-finite?

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  • $\begingroup$ While I found the question itself quite interesting, as well as the result in Falconer book, and the mentioned question, I fail to see the relationship between those and your question. Finiteness of the dimensione doesn't say much about finiteness of the measure. $\endgroup$
    – Silvia Ghinassi
    Commented Mar 12, 2016 at 4:15
  • $\begingroup$ @SilviaGhinassi Maybe I have not explained well my context. But it is a problem that I came across. I have sets $E$ and $F$ where $\dim_H(E),$ $dim_H(F)$ and $\alpha=dim_H(E\times F)$ are finite. I'm looking for if $m_\alpha$ is finite or conditions over $E$ and $F$ for it. As you I can't see the relationship between those. But, this does not mean that it not exists, so i'm asking. $\endgroup$
    – user11178
    Commented Mar 12, 2016 at 4:28

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