Let the $s$-dimensional Hausdorff measure for a Borel set $F \subset \mathbb{R}^d$ be denoted as follows:
$$\mathcal{H}^s(F) := \lim_{\delta \to 0} \mathcal{H}^s_\delta(F) := \lim_{\delta \to 0} \inf \left\{ \sum_i |A_i|^s: \text{ $\{A_i\}_i$ is a $\delta$-cover of $F$ } \right\}, $$ where $|A_i|$ denotes the diameter of the set $A_i$ and the infimum is taken over all the $\delta$-covers of $F$.
Let us consider a bounded set $A$ with the property $0 <\mathcal{H}^s(A) < \infty$, for some $0<s<d$. I am wondering if we can obtain an upper bound of the following form: $$\mathcal{H}^s(A) \leq C |A|^s. $$
I found a density estimate in Mattila's book, Theorem 6.2 (https://i.stack.imgur.com/Uxhaj.jpg). However, I couldn't figure out how to use it. I would greatly appreciate if someone could help me on this issue.
PS: There is also a related question in this post: Growth rate of the Hausdorff measure