I've been stuck on the following question and was wondering if it'd be possible to get some help.
Let $\mu, \nu$ be two finite Borel measures on a metric space $X$ s.t. $\mu << \nu$ and let $c>0$ and suppose that $\nu$ is doubling and that $$\lim_{r \to 0} \frac{\mu(B_r(x))}{\nu(B_r(x))} = c$$ for $\nu$ almost every $x\in X$. By using the Vitali Covering theorem to show that $\mu = c\nu$. (We assume the balls are closed)
I've been able to prove this using Radon-Nikodym combined with the lebesgue differentiation theorem but wish to prove it solely using Vital's covering theorem but have been unsuccessful in even finding an appropriate Vitali cover.
Theorem(Vitali Covering Theorem)
Let X be a metric space and let $\nu$ be a doubling measure on $X$ and let $\mathcal{B}$ be a Vitali cover for $S \subset X$ then there exists a countable $\mathcal{B}' \subset \mathcal{B}$ such that all elements of $\mathcal{B}'$ are disjoint and $$\nu \bigg(S \setminus \bigcup_{B' \in \mathcal{B}'}B'\bigg)=0$$
Definition(Vitali Cover)
Let $S \subset X$ then $\mathcal{B}$ a collection of closed balls such that $\forall \epsilon >0 , \forall x \in S$ there exists $B \in \mathcal{B}$ such that $x \in B$ and $\rm{rad}(B) < \epsilon$ then we call $\mathcal{B}$ a Vitali cover for S.
Many Thanks.