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Presuming your space is separable, the measure space $(X,\mu)$ is always Lebesgue, i.e., isomorphic to a disjoint union of an interval (with the Lebesgue measure on it) and an at most countable family …

Yes, it is true for any $n$. The easiest way to see it is by using the fact that your condition means precisely that $X$ and $Y$ can be realized on the same probability space $\Omega$ in such a way th …

Yes -- the idea is to use the random delta-measure described in the first part of your question. However, in order to obtain a good approximation one has to subdivide the real line into small interval …

Assuming that your distance is convex, the question reduces just to the case when both $\mu$ and $\nu$ are delta measures, and amounts then to the inequality $$\tag {$\star$} d(\delta_x,\delta_y) \le …

Why to make it so complicated? One can see directly that in the limit these graphs have no cycles (just estimate this probability as a function of $n$). PS I was never able to understand why probabil …