Integral probability metrics and their generating classes of functions

We consider probability metrics of the following type: for a class 𝔉 of functions and probability measures P, Q we define d𝔉(P,Q: = supf∈𝔉 | ∫ f dP – ∫ f dQ |. A unified study of such integral probability metrics is given. We characterize the maximal class of functions that generates such a metric. Further, we show how some interesting properties of these probability metrics arise directly from conditions on the generating class of functions. The results are illustrated by several examples, including the Kolmogorov metric, the Dudley metric and the stop-loss metric.