Regular ultrametric skeletons

The ultrametric skeleton theorem associates with every compact metric probability space $(X,d,μ)$ a compact subset $S\subseteq X$ of ultrametric distortion $O(1/\varepsilon)$ and a probability measure $ν$ supported on $S$ such that $ν(B_d(x,r))\leqμ(B_d(x,C_\varepsilon r))^{1-\varepsilon}$. We prove a stronger version with a compatible lower estimate on $ν(B_d(x,r))$ and with $C_\varepsilon=O(1/\varepsilon)$.