# Approximate in $W_1$ sense, an empirical distribution with restriction of true distribution on a set

Let $\mu$ be a probability distribution on a metric space $X=(X,d)$ (to avoid unnecessary complications, assume the full filtration $2^X$) and let $x_1,x_2,\ldots,x_N$ be a sample of size $N$ drawn i.i.d from $\mu$. Consider the empirical measure $\hat{\mu}_N := (1/N)\sum_{i=1}^N\delta_{x_i}$ thus obtained.

# Question

Given $\epsilon > 0$, is it possible to construct a subset $B$ of $X$ such that $$W_1(\hat{\mu}_N,\mu|_B) \le \epsilon?$$

N.B.: Here, $\mu|_B: A \mapsto \dfrac{\mu(A \cap B)}{\mu(B)}$ defines the restriction of $\mu$ on $B$, and $W_1$ is the Wassertstein distance.

# Observation

On second though, it seems the $\epsilon$-blowup of $S_N:=\{x_1,\ldots,x_N\}$ defined by $S_N^\epsilon := \{x \in X | \min_{i=1,\ldots,N} d(x,x_i) \le \epsilon\} = \cup_{i=1}^N \operatorname{Ball}_X(x_i;\epsilon)$ might do the trick.

# Question 2

Can Every distribution $\nu$ on $X$ with $W_1(\hat{\mu}_N,\nu) \le \epsilon$ be realized in such a way (i.e by constructing some appropriate $B \subseteq X$ and setting $\nu = \mu|_B$) ?