Bernoulli shift is an invertible measure preserving transformation on the unit interval with Lebesgue measure that admits an independent generator, i.e., there exists a partition of the unit interval into a finite or countable number of disjoint sets of positive measure such that distinct iterates under the transformation of sets in the partition are independent and the sigma algebra generated by all the iterates of sets in the partition is the sigma algebra of all measurable sets (see Section 2). D. S. Ornstein has shown that Bernoulli shifts with the same entropy are isomorphic [9, IO, 121.
In [4] it was shown that each ergodic measure preserving transformation induces mixing transformations on a dense class of subsets (see Section 2). In general an ergodic measure preserving transformation cannot induce Bernoulli shifts since they have positive entropy [ 1, 21. Our purpose is to prove that each Bernoulli shift, with finite or infinite entropy, induces Bernoulli shifts on a dense class of subsets. It is an open question whether each transformation with positive entropy induces Bernoulli shifts. By Ornstein's isomorphism theorem, it suffices to prove the above result for one Bernoulli shift with entropy h for each h, 0 < h < co. The construction in [5-71 is generalized to obtain a mixing Markov shift with the desired property. Since mixing Markov shifts are isomorphic to Bernoulli shifts [3], the result follows.
In [9] P. Shields showed that the transformations constructed in [7] were Bernoulli shifts by applying the result that an increasing family of Bernoulli shifts is a Bernoulli shift [lo]. We shall also apply this result to the generalized construction. The main point of the construction is that it yields a Bernoulli shift with specified entropy for which it is * Partially supported by National Foundation Grants GP-12043 and SD GU-3171.