 where the indices \(j_i\) range over \(1, \ldots ,r\). Since we are free to chose the sets A\(_0\), A\(_1,\ldots\), A\(_r\) as we please, we can always chose them so that they are the past history of the system: the system was in \(A_{j_0}\) \(k\) time steps back, in \(A_{j_1}\) \( k+1\) time steps back, etc. Call this the (coarse-grained) remote past of the system—‘remote’ because we only consider states that are more than \(k\) time steps back. The K-mixing condition then says that the system’s entire remote past history becomes irrelevant to predicting what happens in the future as time tends towards infinity. Typically Bernoulli systems are compared with K-systems by focussing on the events in the Bernoulli partition. With respect to that partition K is weaker than Bernoulli.  The difference is both in the limit and the remote history. In a Bernoulli system the future is independent of the entire past (not only the remote past), and this is true without taking a limit (in the case of K-mixing independence only obtains in the limit).  However, this only holds for the Bernoulli partition; it may or may not hold for other partitions—the definition of a Bernoulli system says nothing about that case.[22] 