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Ultrametric skeletons

We prove that for every $ε\in (0,1)$ there exists $C_ε\in (0,\infty)$ with the following property. If $(X,d)$ is a compact metric space and $μ$ is a Borel probability measure on $X$ then there exists a compact subset $S\subseteq X$ that embeds into an ultrametric space with distortion $O(1/ε)$, and a probability measure $ν$ supported on $S$ satisfying $ν(B_d(x,r))\le (μ(B_d(x,C_εr))^{1-ε}$ for all $x\in X$ and $r\in (0,\infty)$. 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.