Regular ultrametric skeletons

The theorem of Mendel and Naor (2013) associates to every compact metric probability space a large ultrametric subset carrying a probability measure controlled from above on balls by the original measure. Building on Bartal’s Ramsey decompositions (2021) and the author’s previous proofs of the ultrametric skeleton theorem for doubling spaces (2022, 2023), we prove a regular version for arbitrary compact metric spaces with a linear dilation of the control balls.