Let $\mu$ be a finite Borel measure on $\mathbb{R}^n$. I am interested in how does $\mu(B(x,r))$ behave, where $B(x,r)$ is the open ball of radius $r$ centered at $x$. For instance, as far as I recall, for each $\alpha \in [0,n]$, there exists finite $\mu$ so that $\mu(B(x,r)) \sim r^{\alpha}$ for $\mu$-a.e. $x$, which are called dimensionally-regular measures (with constant local dimension $\alpha$).

Here is my question: does there exist finite Borel measure$\mu$ such that that $\mu(B(x,r))$ vanishes superpolynomially fast, say, $\sim e^{-\frac{1}{r}}$, i.e.,

$\mu\left(\left\{x: \liminf_{r \to 0} r |\log \mu (B(x,r)) | > 0 \right\}\right) > 0$?