 In the case where "X" is a Polish space, the total variation metric coincides with the Radon metric.If μ and ν are both probability measures, then the total variation distance is also given by

The equivalence between these two definitions can be seen as a particular case of the Monge-Kantorovich duality.From the two definitions above, it is clear that the total variation distance between probability measures is always between 0 and 2.