Let $\mathcal{M}(\mathbb R)$ be the space of Radon measures, equipped with topology $\tau$ generated by the following "weak convergence":

$$ \mu_n \rightarrow \mu \quad \text{iff} \quad \int f d\mu_n \rightarrow \int f d\mu \quad $$ for all continuous function $f$ with quadratic growth: $|f(x)|\leq C(1+|x|^2)$ for some $C>0$. Let $\mathcal{M}_2(\mathbb R)$ be the subspace of $\mathcal{M}(\mathbb R)$ that contains all Radon measures with finite second moment.

**I would like to know if there is a description of the topological dual of $(\mathcal{M}(\mathbb R),\tau)$ and $(\mathcal{M}_2(\mathbb R),\tau)$.**

I know $\mathcal{M}(\mathbb R)$ is the dual of $C_0(\mathbb R)$, so we have $$ (\mathcal{M}(\mathbb R),\sigma(\mathcal{M}(\mathbb R),C_0(\mathbb R))^*=C_0(\mathbb R) $$ where $\sigma(\mathcal{M}(\mathbb R),C_0(\mathbb R))$ is the weak star topology. It is also obvious that convergence $\tau$ implies convergence in the weak star topology. So I was hopping the dual of dual of $(\mathcal{M}(\mathbb R),\tau)$ or $(\mathcal{M}_2(\mathbb R),\tau)$ would just be the family of continuous functions wit quadratic growth.

I also notice that $\tau$ convergence is the same as convergence in Wasserstein 2 distance, when restricted to probability measures with finite second moment. I will also be interested to see if there is any connection.

I hope my question make sense and looking forward to any hints and ideas!