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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!

Update:

According to the example given in the comments, there can be $\mu\neq \nu$ and $\nu'$ such that $\mu*\nu' = \nu*\nu'$. I have the following additional question:

(1') Suppose $\mu\neq \nu$ satisfy that for any $\nu'$, $\mu*\nu'\neq \mu*\nu'$. Then does there exist a sequence $\mu_i$ such that $\lim_{i\to\infty} W_2(\mu*\mu_i,\nu*\mu_i)=0$?

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!

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!

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!

Update:

According to the example given in the comments, there can be $\mu\neq \nu$ and $\nu'$ such that $\mu*\nu' = \nu*\nu'$. I have the following additional question:

(1') Suppose $\mu\neq \nu$ satisfy that for any $\nu'$, $\mu*\nu'\neq \mu*\nu'$. Then does there exist a sequence $\mu_i$ such that $\lim_{i\to\infty} W_2(\mu*\mu_i,\nu*\mu_i)=0$?

I have two related questions. Let $\mu$ and $\nu$ be two distinct probability measures on $\mathbb{R}^n$, 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!

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!