**Context:** Given a discrete finite metric space $X$ (in my case X={0,1}$^n$ with the Hamming/L$_1$ distance), I need to define the *natural* or *canonical* metric on the set of all probability distributions over $X$ (call it $D(X)$) which extends the given metric.

I have read in several places [*] that the Earth-Mover's is the *natural*, or *induced*, or *most canonical* extension.

I would like to know whether it is proved (or can be easily proven) that it is the *only*, or indeed the *canonical* metric that can be given to the set of distributions over $X$ so that when restricted to 

D1(X) = {d in D(X)  |  d(x)=0   for all x in X, except one} 

(the distributions whose support is just one element),
you obtain the metric of $X$.

[*]
<a href="http://mathoverflow.net/questions/103115/">mathoverflow: *Distance metric between two sample distributions* [...]</a>,
or google "EMD naturally extends the notion of distance" and you'll find several papers saying a similar thing, as a "fact", without proof or reference.