Ultrametric skeletons

We prove that for every ε ∈ (0,1) there exists C_ε ∈ (0,∞) 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 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))≤(μ(B_d(x,C_εr)) ^1-ε for all x ∈ X and r ∈ (0,∞). 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 measure theorem.