Let $\mathcal{M}=\mathcal{M}(\mathbb{R})$ be the space of bounded measures. Equipped with the weak convergence, the dual space of $\mathcal{M}$ is $\mathcal{C}_b(\mathbb{R})$ consisting of continuous bounded functions defined on $\mathbb{R}$.

Now we consider a subspace $\mathcal{M}_1$ of $\mathcal{M}$ such that:

$$\mathcal{M}_1=\{\mu\in\mathcal{M}: \int_{\mathbb{R}} |x|\mu(dx)<\infty\}$$

I would like to know the dual space of $\mathcal{M}_1$. In other words, which topology $\mathcal{M}_1$ should be used and for this topology which is the dual space of $\mathcal{M}_1$.

For example, we can take the topology of Wasserstein metric:

http://en.wikipedia.org/wiki/Wasserstein_metric

Which is the dual space associated to Wasserstein metric?

Does someone know the related results? Thanks a lot!

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