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⁻ᵋ 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.