I have two related questions. Let $\mu$ and $\nu$ be two **distinct** probability measures on $\mathbb{R}^n$ with finite second moments, and $W\_2(\cdot,\cdot)$ be the $2$-Wasserstein metric. The question is:
(1) As $\mu$ and $\nu$ are distinct, hence $W\_2(\mu,\nu)\neq 0$. Is it possible that there is a sequence of distributions $\mu\_i, i\geq 1$ on $\mathbb{R}^n$ such that $\lim\_{i\to\infty} W\_2(\mu\*\mu\_i,\nu\*\mu\_i)=0$? Here $\*$ is the convolution.


While I was trying to figure out the answer using Prokhorov's theorem, I realized that I need to answer the following more elementary looking question (though I do not have an answer yet):


(2) If $\mu$ and $\nu$ are two distinct probability measures on $\mathbb{R}^n$, then is it possible that there is another probability measure $\mu'$ such that $\mu\*\mu' = \nu\*\mu'$?


Some special cases can be easily dealt with (to give a negative answer), using for example Fourier transform. However, I am unsure about the general case.


Any comments and references to these two problems are highly appreciated!



2-Wasserstein metric on convolution of probability distributions
The answer is yes to the second question, and hence yes to the first question as well.


Indeed, it is easy to check that the functions $f$ and $g$ given by
$$f(t):=\max(0,1-|t|)$$
and
$$g(t):=\sum\_{k=-\infty}^\infty f(t-2k)$$
for real $t$ are characteristic functions:
$$f(t)=\int\_{-\infty}^\infty e^{itx}\mu(dx)$$
and
$$g(t)=\int\_{-\infty}^\infty e^{itx}\nu(dx),$$
where
$$\mu(B):=\int\_B dx\,\frac{1-\cos x}{\pi x^2}$$
and
$$\nu(B):=\frac{1(0\in B)}2
+\frac2{\pi^2}\,\sum\_{k=-\infty}^\infty 
\frac{1((2k-1)\pi\in B)}{(2k-1)^2}
$$
for all Borel sets $B\subseteq\mathbb R$.


We also have $$gf=f^2=ff.$$


So, if $\mu':=\mu$, then we have $\mu\ne\nu$ but $\mu\*\mu'=\nu\*\mu'$, as claimed.


