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This is a follow up to this post, where I wish to verify whether one of the statements (in the post) is true but first let's recap the definitions: Let $(X,d)$ be a metric space. If set $A\subseteq X$,...

Let $(X,d)$ be a metric space. If set $A\subseteq X$, let $H^{\alpha}$ be the $\alpha$-dimensional Hausdorff measure on $A$, where $\alpha\in[0,+\infty)$ and $\text{dim}_{\text{H}}(A)$ is the ...