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For a metric space $M$, I'll write $Prob(M)$ for the Borel probability measures on $M$.
I am interested in metrics on $Prob(M)$, such as the Kantorovich distance (or other metrics).

If $f: M \rightarrow M'$ is continuous, is the induced map $Prob\ f : Prob(M) \rightarrow Prob(M')$ also continuous? Is it true when "continuous" is replaced by "uniformly continuous"?

Is this known for the Kantorovich metric? Or for any other metric?

I'm also interested case of discrete metrics. (This is the main application that I have in mind.)

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    $\begingroup$ It works for the topology of weak convergence of measures, using the change of variable theorem for image measures: If $g\in C_b(M)$, then $\int g~\mathrm d\mu\circ f^{-1}=\int g\circ f~\mathrm d\mu$. $\endgroup$
    – Michael Greinecker
    Commented Jan 11, 2018 at 21:19
  • $\begingroup$ I wrote two papers on this topic in 1995, see arxiv.org/abs/1206.1727 and arxiv.org/abs/1112.6161 $\endgroup$
    – Taras Banakh
    Commented Jan 13, 2018 at 16:57

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There is the Fisher-Rao metric. Here is a paper on its uniqueness. It plays a big role in information geometry. There are discrete versions of it also, see the references.

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