For the sake of concreteness denote by $M_0(X)$ the linear space of all signed Borel measures $\sigma$ with $\sigma(X)=0$ on some metric space $(X,d)$ and fix some base point $x_0\in X$. On this space define the norm $$ \newcommand{\norm}[1]{\|#1\|} \newcommand{\Lip}{\mathrm{Lip}} \norm{\sigma}_0^* = \sup\{\int_X f\,d\sigma\ :\ \Lip f\leq 1,\ f(x_0)=0\} $$ where the supremum is taken over all Lipschitz functions on $X$ and $\Lip f$ denotes the Lipschitz constant of $f$. In Bogachev's "Measure theory" (§8.10) this norm is called Kantorovich-Rubinshtein norm and it is shown that convergence in the Kantorovich-Rubinshtein metric implies weak convergence. Moreover, it is stated that $M_0(X)$ is not complete with this norm provided that $X$ is not complete. This can be seen by as follows: Assume that there exist sequences $x_k,y_k\in X$ which do not converge, but $d(x_k,y_k)\to 0$. Then define the sequence $\sigma_k = \delta_{x_k}-\delta_{y_k}$ and observe that $$ \norm{\sigma_k}_0^* \leq d(x_k,y_k)\to 0 $$ in other words $\sigma_k\to 0$ w.r.t $\norm{\cdot}_0^*$. However, $\sigma_k$ does not convergence weakly to zero.

My question is:

### What is the completion of $M_0(X)$ w.r.t. $\norm{\cdot}_0^*$?

I would like some description as a dual space or derived from some space of measures.

It feels like it should be something like a dual space of differentiable functions, or derivatives of some measure…