Let $E \subset \mathbb{R}^n$ be a closed set, not reduced to a point, and let $\mu$ be a (positive Borel) measure supported on $E$. If there is a constant $C \ge 1$ such that $C^{-1} r^d ≤ μ(E \cap B(x, r)) ≤ C r^d$ for $x \in E$ and $0 < r < \textrm{diameter}(E)$, then $E$ is Ahlfors-regular of dimension $d$, and there is a constant $C$ such that $C^{-1}μ(A) ≤ H^d (E ∩ A) ≤ C μ(A)$ for all Borel sets $A \subset R^n$ .
Can anyone give me a reference for this lemma ?
This is Exercise 8.11 in Juha Heinonen's "Lectures on Analysis on Metric Spaces" (and the paragraphs immediately preceding it give some ideas for the proof), and 1.4.3 in John Mackay and Jeremy Tyson's "Conformal Dimension, Theory and Applications".