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 $X$.

[*] mathoverflow: Distance metric between two sample distributions [...], 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.

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 $X$.

[*] mathoverflow: Distance metric between two sample distributions [...], 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.