Let $\mu$ be a probability measure on $\mathbb{R}^2$ and $x \in \text{supp}(\mu)$. Let $(C_n)$ be a sequence of rectangles of length / width $a_n \leq b_n$ such that $x \in C_n$ for all n, $\ \text{diam}(C_n) \rightarrow 0$ and $\frac{\log(a_n)}{\log(b_n)} \rightarrow 1$.
Suppose that $\frac{\log(\mu(C_n))}{\log(b_n)} \rightarrow \ell$ when $n \rightarrow +\infty$. I would like to show that $\frac{\log(\mu(B(x,r))}{\log(r)} \rightarrow \ell$ when $r \rightarrow 0^+$. Any suggestions? Thanks in advance.