Let $X$ and $Y$ be random variables taking values in a separable metric space $(S,d)$. The metric I have in mind is $$d(X,Y) = \mathbb{E}[\min\{d(X,Y),1\}]$$$$\rho(X,Y) = \mathbb{E}[\min\{d(X,Y),1\}]$$ if $X$ and $Y$ take values in the a metric space $(S,d)$.
Question: Does this metric have an official name?
Wikipedia calls it the "Lévy metric for $L^0$". Also, someone once told me they thought it was called the Lévy metric. I can't find it anywhere else. (If you Google™ it, don't use anything written by me as evidence that it is called the Lévy metric!) Moveover, Lévy metric is the name for another metric.
I also know this metric metricizes convergence in probability (measure), and is equivalent to the Ky Fan metric (which I have also seen called the probability distance): $$K(X,Y) = \inf\{\varepsilon \geq 0: P(d(X,Y) > \varepsilon) \leq \varepsilon\}.$$
I have also seen this very similar metric to mine $$K^*(X,Y) = \mathbb{E}\left[\frac{d(X,Y)}{1 + d(X,Y)} \right]$$ called one of the "Ky Fan metrics".
I know I could just switch to the Ky Fan metric, but I wrote a long paper using this metric $d$$\rho$ (calling it the Lévy metric), and I don't want to have to go through the whole thing and switch to the Ky Fan metric (and change my calculations) just because I know don't know what to properly call it. (Also, I like this metric since it is reminiscent of the $L^1$-norm.)
