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For a metric space $M$ and the Wasserstein 1-distance $W_1$, what qualifying assumptions do we need for $M$? I have seen we only need $M$ be compact, $M$ be a Polish space, or $M$ be a Radon space. Which would be the "best" condition to put on $M$? And why one over the other? I.e. what advantages do we gain if we require $M$ be a Radon space versus a Polish space? This is not my specialty so any and all knowledge is appreciated!

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