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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?

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?

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: Denoting $\alpha= \dim_H(E\times F)$, 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?

In view of the comments of, Silvia Ghinassi, I decide to added the more general question that It may be more appropriate:

More General question Are there any conditions over the the metric space $X$ with finite $\alpha$ Hausdorff dimension in order the associated Hausdorff measure $m_{\alpha}$ to be finite or $\sigma$-finite?

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?

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: Denoting $\alpha= \dim_H(E\times F)$, 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?

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: Denoting $\alpha= \dim_H(E\times F)$, 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?

In view of the comments of, Silvia Ghinassi, I decide to added the more general question that It may be more appropriate:

More General question Are there any conditions over the the metric space $X$ with finite $\alpha$ Hausdorff dimension in order the associated Hausdorff measure $m_{\alpha}$ to be finite or $\sigma$-finite?