Bernoulli Schemes and Their Isomorphisms

In these lectures I shall present some of the ideas and methods used in the proof of the celebrated theorem stating that two Bernoulli schemes are isomorphic if and only if they have the same entropy. I shall concentrate, because of lack of time, on a well known partial result, Sinai's theorem, which states that two Bernoulli schemes are weakly isomorphic if and only if they have the same entropy. In these lectures I shall present some of the ideas and methods used in the proof of the celebrated theorem stating that two Bernoulli schemes are isomorphic if and only if they have the same entropy. I shall concentrate, because of lack of time, on a well known partial result, Sinai's theorem, which states that two Bernoulli schemes are weakly isomorphic if and only if they have the same entropy. I shall outline Ornstein's proof of Sinai's theorem which gives an insight into the coding problems that arise in connection with the isomorphism problems. Let me start with some basic notions. A Bernoulli scheme is a mathematical model for a " head and tail " game: we have k symbols (1, 2,. .. , k) and a machine that extracts out of a box one of these symbols, randomly, with respective probability p 1 , p 2 ,. .. , p k and so that successive extractions are independent.