Let $X$ and $Y$ be random variables taking values in a metric space $(S,d)$.  The metric I have in mind is
$$d(X,Y) = \mathbb{E}[\min\{d(X,Y),1\}]$$
if $X$ and $Y$ take values in the a metric space $(S,d)$.

**Question: Does this metric have an official name?**  

[Wikipedia][1] 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][2]. 

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 $d$ (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.)

  [1]: https://en.wikipedia.org/wiki/Lp_space#L0.2C_the_space_of_measurable_functions
  [2]: http://en.wikipedia.org/wiki/L%C3%A9vy_metric