Is integration against an indicator Wasserstein-Continuous

Question

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?

Answer 1

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.