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…



Completion of spaces of measures w.r.t. weak norms
This is known as the Arens-Eells space $AE(X)$. In the nonlinear Banach space literature it's also called the Lipschitz-free space $\mathcal{F}(X)$. It is not a dual space in general, but rather the predual of the space ${\rm Lip}\_0(X)$ of Lipschitz functions which vanish at $x\_0$.


In some cases it is a dual space. For instance, if $X$ is a compact metric space and $0 < \alpha < 1$, let $X^\alpha$ be the set $X$ equipped with the "Holder" or "snowflake" metric $\rho^\alpha$. In this case $AE(X^\alpha)$ is the dual of the space of "little" Lipschitz functions which are locally flat.


The space $AE(X)$ is characterized abstractly by its universal property: there is a natural isometric embedding of $X$ in $AE(X)$ (in your formulation, take the point $p$ to the Dirac measure at $p$ minus the Dirac measure at $x\_0$), and for any Banach space $E$ and any Lipschitz function $f: X \to E$ which takes $x\_0$ to $0$ there is a unique bounded linear extension $T: AE(X) \to E$. Moreover, $\|T\| = Lip(f)$.


You can find out all you want about this space in Chapter 3 of my book *Lipschitz Algebras* (second edition).


