  Let $(X,\mu)$ be a standard probability space. An automorphism $T$ of
$(X,\mu)$ has the weak Pinsker property if for every $\varepsilon > 0$ it has a
splitting into a direct product of a Bernoulli shift and an automorphism of
entropy less than $\varepsilon$. This property was introduced by Thouvenot, who
asked whether it holds for all ergodic automorphisms.
  This paper proves that it does. The proof actually gives a more general
result. Firstly, it gives a relative version: any factor map from one ergodic
automorphism to another can be enlarged by arbitrarily little entropy to become
relatively Bernoulli. Secondly, using some facts about relative orbit
equivalence, the analogous result holds for all free and ergodic
measure-preserving actions of a countable amenable group.
  The key to this work is a new result about measure concentration. Suppose now
that $\mu$ is a probability measure on a finite product space $A^n$, and endow
this space with its Hamming metric. We prove that $\mu$ may be represented as a
mixture of other measures in which (i) most of the weight in the mixture is on
measures that exhibit a strong kind of concentration, and (ii) the number of
summands is bounded in terms of the difference between the Shannon entropy of
$\mu$ and the combined Shannon entropies of its marginals.
