 There are two further notions that are crucial to the theory of Bernoulli systems, the property of being weak Bernoulli and very weak Bernoulli. These properties play a crucial role in showing that certain transformations are in fact Bernoulli. The baker’s transformation is one of the few examples that have a geometrically simple Bernoulli partition, and so one often cannot prove directly that a system is a Bernoulli system. One then shows that a certain geometrically simple partition is weak Bernoulli and uses a theorem due to Ornstein to the effect that if a system is weak Bernoulli then there exists a Bernoulli partition for that system. The mathematics of these notions and the associated proofs of equivalence are intricate and a presentation of them is beyond the scope of this entry. The interested reader is referred to Ornstein (1974) or Shields (1973).