 for all atoms \(\delta_i\) of \(T_k\alpha\) and all atoms \(\beta_j\) of \(T_l\alpha\) for all \(k \ne l\). We then refer to \(\alpha\) as the Bernoulli partition, and we call a dynamical system \([X, \mu , T]\) a Bernoulli system if \(T\) is a Bernoulli automorphism, i.e., a Bernoulli transformation mapping \(X\) onto itself.