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Hausdorff distance is a lower (or upper bound) for what probability metric?
A general note is that the answer depends heavily on the properties of $\mu$.
First a note that in general $d_H(A,B) \not \le C \cdot W_p(\mu|_A,\mu|_B)$ for $p\in[1,\infty)$ and some $C>0$. Though …