On the countable generator theorem

Let T be a finite entropy, aperiodic automorphism of a nonatomic probability space. We give an elementary proof of the existence of a finite entropy, countable generating partition for T . In this short article we give a simple proof of Rokhlin’s countable generator theorem [Ro], originating from considerations in [Se] which use standard techniques in ergodic theory. We hope that these considerations will be useful for elementary expositions in the future. For other proofs see [Pa]. Let (X,A, μ) be a nonatomic probability space whose σ-algebra A is generated modulo μ by a countable collection {A1, A2, . . .} of elements of A. Let T be an aperiodic automorphism of (X,A, μ) with finite entropy. For the definitions and properties of entropy and generators used in the sequel, we refer the reader to Billingsley [Bill] and Walters [Wa]. Theorem. (X,A, μ, T ) has a countable generating partition of finite entropy. Our proof is based on the following lemma. Lemma. Let P be a finite partition of (X,A, μ, T ), A an element of A, and ε > 0. Set P̃ := P ∨ {A,Ac} and g := h̃− h, where h := h(T,P) and h̃ := h(T, P̃) denote the respective mean entropies of the partitions P and P̃. Then there exists a finite partition Q of (X,A, μ, T ) such that (1) P Q, 1991 Mathematics Subject Classification: 28D05, 28D20.