History of the Jaccard distance $d(A,B) = \mathbb P(\overline A\cup\overline B\mid A\cup B)$ I'm wondering where the relative probabilistic distance or Jaccard 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:
@TechReport{Yianilos91,
  author =   "Peter N. Yianilos",
  title =    "Normalized Forms for Two Common Metrics",
  institution =  "NEC Research Institute",
  year =     {1991,2002}
}
which contains a detailed proof that  $d $ obeys the triangle inequality, but surely that was discovered prior to 1991?
On a seemingly related note, Cathy O'Neil mentions at 5:00 in Deciphering recommendation engines, http://youtu.be/lzavwJy1SgQ that 
$$\mathbb P(A\mid A\cup B)$$
is a mathematically interesting notion of closeness.
 A: If I remember correctly, it was by Menger, in this series of papers.
Menger, 1942(12), statistical metrics, PNAS 28(12):535-537.
Menger, 1951(3), probabilistic theory of relations, PNAS 37(3):178-180.
Menger, 1951(4), probabilistic geometry, PNAS 37(4):226-229.
Menger et al, 1959(au), probabilistic metrics and numerical metrics with probability, czechoslovakian mathematical journal 9(3):459-466.
If each pair of coordinates points is supposed to nowhere exist strictly a distance $x$ apart, for all possible $x$, and they only have a probability $z$ of being separated by such a distance (or by a lesser distance), then special cases exist, one per distinguishable $x$:---probabilities for one and only one $x$ are defined in each case and undefined for other distances. There the $z$ corresponding to each pair of points is itself FAPP a distance between any two such points. From each such metric, at most one further constraint is required for a metric of the sort in the question title to result.
[The formal part of this answer, I agree it needs rewriting, temporarily gone. Also, I'm checking another possible source, slightly more recent, where the desired metric is explicitly stated unless my memory is wrong.]
