Suppose $W_r(\mu_n,\mu)\to0$, where $\mu_n$ and $\mu$ are discrete probability measures on some metric space $\Omega$, and that all measures have the same number of atoms $d$ (but not the *same* atoms):

$$\mu_n = \sum_{i=1}^d p_{n,i}\delta_{\theta_{n,i}}, \quad \mu = \sum_{i=1}^d p_{i}\delta_{\theta_{i}}.$$

$W_r$ here is the usual $L_r$-Wasserstein metric (a.k.a. Kantorovich–Rubinstein, optimal transport distance). *Assume further that the atoms are labeled such that $\theta_{n,i}\to\theta_i$ for each $i$.*

Now let $S\subset\{1,\ldots,d\}$ be any subset and consider the restriction of $\mu_n$ and $\mu$ to $S$. Let $\widetilde{\mu}_n(S)$ and $\widetilde{\mu}(S)$ denote these measures after normalization (i.e. so that each has total mass 1), i.e.

$$\widetilde{\mu}_n(S) \propto \sum_{i\in S} p_{n,i}\delta_{\theta_{n,i}}, \quad \widetilde{\mu}(S) \propto \sum_{i\in S} p_{i}\delta_{\theta_{i}}.$$

**Is it true that $W_r(\widetilde{\mu}_n(S), \widetilde{\mu}(S))\to 0$ for any $S$?**

*Note.* These can be interpreted as "conditional probabilities" where we are conditioning on the atoms in $S$.