Let us say that a measure $\mu$ on $\mathbb{R}^d$ is *locally doubling* if for each
$x\in\mathbb{R}^d$ there is a constant $C(x)$ such that for all $r>0$,
$\mu(B(x,2r)) \le C(x) \mu(B(x,r))$,
where $B(x,r)$ is the $r$-ball about $x$. 
Additionally, $\mu$ is *bounded* if $\mu(\mathbb{R}^d)<\infty$.



> 
> **Question.**
>  Is it true that any bounded measure on $\mathbb{R}^d$
>  is locally doubling, except on a set of small measure?
> 
> 
> 



Locally doubling measures
The answer is **no**. In fact, something far more dramatic can happen: there exist Radon measures $\mu$ on $\mathbb{R}^d$ such that for $\mu$ almost all $x$, for any arbitrary Radon measure $\nu$, the normalized restriction of $\mu$ to $B(x,r)$ "looks like" $\nu$ (it is arbitrarily close in the weak$^\*$ topology).


To be more precise, given a measure $\mu$, a point $x$ in its topological support, and $r>0$, let 
$$
\mu\_{x,r}(A) = \mu(rA+x)/\mu(B(x,r)).
$$ 
Note that $\mu\_{x,r}$ is obtained by magnifying $\mu$ around $x$ by a factor $r^{-1}$ and normalizing so that the unit ball gets unit mass. Weak$^\*$ accumulation points of $\mu\_{x,r}$ as $r\downarrow 0$ are called **tangent measures** of $\mu$ at $x$.


Now O'Neil [**O'Neil, Toby**. A measure with a large set of tangent measures. *Proc. Amer. Math. Soc.* 123 (1995), no. 7, 2217--2220] constructed a Radon measure $\mu$ which has **all** Radon measures as tangent measures at almost all points. See also [**Sahlsten, Tuomas**. Tangent measures of typical measures. *Real Anal. Exchange* 40 (2014/15), no. 1, 53--79].


In particular, let $\nu\_\delta$ be a smooth measure such that $\nu\_\delta(B(0,2/3)) \le \delta \nu\_\delta(B(0,1))$. Since $\mu$ has $\nu\_\delta$ as a tangent measure at almost all points, this means that for $\mu$-almost all $x$ there are arbitrarily small $r$ such that $\mu(B(x,r/2)) \le 2\delta \mu(B(x,r))$. As $\delta>0$ is arbitrary, $\mu$ fails to be locally doubling at almost all points, even if weaken the notion of locally doubling by requiring the doubling only for sufficiently small $r$ depending on $x$.


