# Completion of spaces of measures w.r.t. weak norms

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 $$\newcommand{\norm}{\|#1\|} \newcommand{\Lip}{\mathrm{Lip}} \norm{\sigma}_0^* = \sup\{\int_X f\,d\sigma\ :\ \Lip f\leq 1,\ f(x_0)=0\}$$ 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…

• This is a good question. But the norm $\lVert\sigma\rVert^\ast_0$ as defined here is not the Kantorovich-Rubinstein norm as defined in Bogachev's book. And the completeness of $M_0(X)$ with $\lVert\sigma\rVert^\ast_0$ does not have much to do with the completeness of $X$. For example, $M_0(X)$ is not complete when $X$ is the unit interval $[0,1]$ with the usual metric. – PassingThru Mar 2 '16 at 14:55
• @PassingThru As far as I see, the norm $\|\sigma\|_0^*$ is exactly as Bogachev defines it in §8.10 for the difference of two Borel probability measures with finite first moments. In fact he has another Kantorovich-Rubinshtein norm in §8.3 but I thought that I could go with this setting as well. – Dirk Mar 2 '16 at 15:44
• Yes, Bogachev defines the "modified K-R metric" in 8.10 on the space $M_1(X)$ of those signed measures that integrate every Lipschitz function (not on the space $M_0(X)$). And yes, that modified K-R metric agrees with your definition. In 8.3 he defines the K-R norm on the space of signed Borel measures, and that norm can be restricted to the space $M_0(X)$; but that norm is in general different from the "modified" one. – PassingThru Mar 2 '16 at 18:43
• @PassingThru Ok, thanks for clarifying. As much as I try to love these two volumes of Bogachev, I always struggle with the notation and diversity of concepts in there. – Dirk Mar 3 '16 at 8:09

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 2 of my book Lipschitz Algebras.

• I edited out the comment that the universal property was independently discovered by Kadets and me. Actually Kadets did not state it, but it could be easily deduced from his work. So maybe the correct attribution is a little murky. – Nik Weaver Mar 1 '16 at 17:50
• Nik, what references do Godefroy and Kalton give? – Yemon Choi Mar 1 '16 at 18:01
• And this was a space that had been discovered 50 years earlier by Arens and Eells, vastly predating everyone else, so I also feel some sense of historical injustice ... – Nik Weaver Mar 1 '16 at 18:18
• Let me point out that people working in the theory of Markov processes in Polish spaces call this the Fortet--Mourier norm on the space of measures. – Tomasz Kania Mar 1 '16 at 20:16
• @Dirk, I am not an expert but it is mainly used to metrise the weak convergence of probability measures. I am not sure if they care about the completion. – Tomasz Kania Mar 2 '16 at 9:16