Let $A\subseteq [0,1]^d$ be a Borel measurable set. Let $d_H(A)$ be the Hausdorff dimension of $A$. Let $\mathcal{H}^{d_H(A)}$ be the Hausdorff measure w.r.t to the dimension of $A$.
My question is, in what generality is it true that $\mathcal{H}^{d_H(A)}|_A$ (i.e. restricted to $A$) is a finite and positive measure? I am sorry if this question is trivial, but I have searched in several geometric measure theory books, and online, but haven’t found an answer. All the results I found deal with Hausdorff measure of integer dimension, and restricted to smooth submanifold, and not fractals for some reason.
Thanks in advance