Abstract
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 ε r )) 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.
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