I'm wondering where the *relative probabilistic distance* was first studied:
$$d(A,B) =\mathbb P(\overline A\cup\overline B\mid A\cup B)$$
where $\overline A$ is the complement of $A$.

A web search turned up this:

<pre>@TechReport{Yianilos91,
  author = 	 "Peter N. Yianilos",
  title = 	 "Normalized Forms for Two Common Metrics",
  institution =  "NEC Research Institute",
  year = 	 {1991,2002}
}</pre>

which contains a detailed proof, but surely this was discovered prior to 1991?