Dobrushin and Steif Metrics are Equal

Abstract It is proved here that two useful and apparently different metrics on the set of Borel probabilities on countable products of Polish spaces of bounded diameters are equal. This tidies up the subject and paves the way for advances in their computation, because one is defined as a supremum and the other as an infimum. As an example of application, the distance between two stationary probabilities for Toom’s north–east–centre majority voter probabilistic cellular automaton is calculated exactly.