Let $\xi$ be a random variable defined on $\mathbf{R}$ with $\sigma$-algebra $\mathcal{F}$. Let $\mathscr{P}$ denote the set of all probabiltiy distribution on $(\mathbf{R},\mathcal{F})$. Let $\mathcal{P}=\{P\in \mathscr{P}:\mathbf{E}_P[\xi]=0,\mathbf{E}_P[\xi^2]=1\}$. It is easy to see that $\mathcal{P}$ is not closed because one counter example is $P_n(\xi = \pm\sqrt{n})=\frac{1}{2n},P_n(\xi = 0)=1-\frac{1}{n}$. It is easy to see that $P_n\in \mathcal{P}$ and $P_n$ converges weakly to $\mathcal{P}^*$ s.t. $\mathcal{P}^*(\xi=0)=1$. However, $\mathcal{P}^*\notin \mathcal{P}$.
My question is given $\mathcal{P}=\{P\in \mathscr{P}:\mathbf{E}_P[\vec{G}(\xi)]\in K\}.$ Under what situations ($\vec{G}(\xi)\in R^n$ and $K\subset R^n$) will $\mathcal{P}$ be a closed set? For example, $G$ is an indicator function, power function ($x^n$), bounded function etc;$K$ is a closed set, convex set and bounded set etc.
Closeness can be either in weak convergence or total variation.
In particular, I am very interested in when $K$ is a closed set in $R^n$ and $\vec{G}(x)=(1[x>a],x,x^2)$