Let $\mu$ be a fixed finite measure on $\mathbb R$. We say that $\mu$ is doubling if there exists a constant $C>0$, such that for any two adjacent intervals $I=[x−h,x]$ and $J=[x,x+h]$, $$C^{−1}\mu(I)≤\mu(J)≤C\mu(I).$$ Assuming that $\mu$ is doubling, show that there exist positive constants $B$ and $a$, such that for every interval $I$, $$\mu(I)≤B[length(I)]^a$$
By Radon-Nikodim, I solved but I use the fact that $\mu$ is absolutely continous with respect to Lebesgue measure, but I don't know to prove this last part, i.e., that $\mu$ is absolutely continuous with respect to Lebesgue.
