The Bernoulli property is the property of an invertible measure-preserving transformation on a probability space of being measurably isomorphic to a two-sided Bernoulli scheme. Here the isomorphism is a measure-preserving bijection between invariant subsets of full measure. Both the map and its inverse are measurable functions. It carries the original transformation to the shift on sequences. Thus the dynamics admits a complete coding by independent coordinates with a common statistical distribution, even when the original evolution rule is deterministic (Walters 1982, Liverani).
A transformation with the Bernoulli property is a mixing transformation and hence an ergodic transformation. The Bernoulli property therefore gives a stronger statistical conclusion than ergodicity alone.
