# description of dual space of space of Radon measure equipped with topology of weak convergence

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!

• Why should the intergal $\int fd\mu$ exist for $\mu\in\mathcal M$ and $f$ of quadratic growth? It does exist for $\mu\in \mathcal M_2$ and on that space $\tau$ is a weak topology $\tau=\sigma(\mathcal M_2, L)$ for the space $L$ of continuous functions of quadratic growth so that $(\mathcal M,\tau)^*=L$. – Jochen Wengenroth May 24 '18 at 6:52
• Thanks a lot. I overlook the obvious fact that for any dual pair $(X,Y)$, it always holds that $(X,\sigma(X,Y))^*=Y$. – Ryan May 24 '18 at 15:33