Let $\mathcal{P}\_p(X)$ denote the Wasserstein space over a compact metric space $X$, and $1\leq p<\infty$. Fix a non-empty closed subset $C\subseteq X$. Then is the map:
$$
\mathbb{P} \mapsto \int\_{x \in X} I\_C(x) d\mathbb{P}(x),
$$
continuous on $P\_p(X)$ with respect to the Wasserstein distance?



Is integration against an indicator Wasserstein-Continuous
Assuming $C$ isn't also open, find a sequence $(x\_n)$ in $X\setminus C$ which converges to a point $x$ in $C$. Then the point measures $\delta\_{x\_n}$ converge to $\delta\_x$ but their integrals against $1\_C$ do not.


