On some geometrization of compact metric spaces: A solution to the Banach-Ulam conjecture

Abstract

We propose a geometrization of compact metric spaces that is based on ideas of S. Banach and J. Mycielski. Then we prove the following conjecture of S. Banach and S. Ulam from 1935: in every compact metric space there exists a finitely additive probability measure, invariant under congruences. Moreover, our techniques allow us to solve a problem of M. Talagrand related to the Marczewski problem and the Banach-Tarski paradox with pieces having the property of Baire. We give also a very simple proof of the conjecture of Ulam about the product Lebesgue measure in the Hilbert cube and explain the existing results about congruence-invariant Borel measures in the language of our geometrization.