TITLE: local behavior of a finite Borel measure
QUESTION [9 upvotes]: 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., 
$\mu\left(\left\{x: \liminf_{r \to 0} r |\log \mu (B(x,r)) | > 0 \right\}\right) > 0$?

REPLY [3 votes]: If you are willing to replace $\mathbb{R}^n$ by a Banach space $B$, then it is perfectly possible. For example, the Wiener measure $\mu$ on the space of continuous function from $[0,1]$ to $\mathbb{R}$ satisfies that for $\mu$-almost every $x$, $\mu(B(x,r)) \sim \exp(-1/r^2)$ for small $r$. (The keyword for this is "small ball estimates".) If you are willing to forgo separability of $B$, then you can even construct probability measures with the property that, for every $x \in B$, $\mu(B(x,1)) = 0$! 
The canonical example for this is to take for $\mu$ the law of a sequence of i.i.d. normal random variables and for $B$ the space of sequences $x = (x_n)$ such that
$$
\|x\| := \sup_{n\ge 1} \Big(2^{-n}\sum_{k=2^{n-1}}^{2^n-1} x_k^2\Big)^{1/2} < \infty\;.
$$
The claim then follows from the strong law of large numbers.