Ultrametric skeletons

We prove that for every $ϵ\in (0,1)$ there exists $C_\epsilon\in (0,\infty)$withthefollowingproperty\mathrm{.}If$(X,d)$ is a compact metric space and $\mu$ is a Borel probability measure on $X$ then there exists a compact subset $S\subseteq X$ that embeds into an ultrametric space with distortion $O(1\mathrm{/}ϵ)$, and a probability measure $\nu$ supported on $S$ satisfying $\nu (B_d(x,r))\le (\mu (B_d(x,C_{ϵ}r){)}^{1-ϵ}$ for all $x\in X$ and $r\in (0,\mathrm{\infty })$. The dependence of the distortion on $ϵ$ is sharp. We discuss an extension of this statement to multiple measures, as well as how it implies Talagrand’s majorizing measures theorem.