Measure Concentration and the Weak Pinsker Property

Let pX,μq be a standard probability space. An automorphism T of this space has the weak Pinsker property if for every ε ą 0 it is isomorphic to a direct product of a Bernoulli shift and an automorphism of entropy less than ε. This property was introduced by Thouvenot, who conjectured that it holds for all measure-preserving systems. This paper proves Thouvenot’s conjecture. The proof applies with little change to give a relative version of the conjecture, according to which any given factor map from pX,μ, T q to another measure-preserving system can be enlarged by arbitrarily little entropy to become relatively Bernoulli. With this relative version in hand, known results about relative orbit equivalence quickly give the analogous result for all free measure-preserving actions of a countable amenable group. The key to this proof is a new result in the study of measure concentration. Consider a probability measure μ on a product space A with its Hamming metric. Our new result here gives an efficient decomposition of μ into summands which mostly exhibit a strong kind of measure concentration, and where the number of summands is bounded in terms of the difference between the Shannon entropy of μ and the combined Shannon entropies of its marginals. This result provides a new approach to measure concentration on product spaces.