 With these notions in hand we can now define a Bernoulli transformation: a transformation \(T\) is a Bernoulli transformation iff there exists a partition \(\alpha\) of \(X\) so that the images of \(\alpha\) under \(T\) at different instants of time are independent; that is, the partitions \(\ldots ,T_{-1}\alpha , T_0 \alpha , T_1 \alpha ,\ldots\) are all  independent.[10]  In other words, \(T\) is a Bernoulli transformation iff