**Edit:** Changed from "Hausdorff" to "metric" spaces.


Let $\mathcal{M}(\Omega)$ denote the space of signed regular Borel measures on a compact metric space $\Omega$. By Riesz-Markov, this is the dual space of $C(\Omega)$, the space of all continuous real valued functions on $\Omega$. Denote by $$\mathcal{P}(\Omega) = \{\mu\in\mathcal{M}(\Omega)\ :\ \mu\geq 0, \mu(\Omega)=1\}$$
i.e. the set of all probability measures in $\mathcal{M}$.


The weak convergence (also called weak\* convergence) in $\mathcal{M}(\Omega)$ is defined by duality and it is known that weak convergence in $\mathcal{P}(\Omega)$ can be metrizised by, e.g. the [Prokhorov](https://en.wikipedia.org/wiki/Prokhorov_metric) metric $d\_P$ or the [Wasserstein metrics](https://en.wikipedia.org/wiki/Wasserstein_metric) $d\_W$.


Obviously, both metrics do not metrizise weak convergence on $\mathcal{M}(\Omega)$: For the Wasserstein metric we have $d\_W(\mu,\nu)=\infty$ if $\mu(\Omega)\neq\nu(\Omega)$ and for the Prokhorov metric we do not even have $d\_P(\mu,\mu)=0$, as far as I see.


Googling and searching MSC did not produce any results on my question:



> 
> Are there any metrics available which metrizise weak convergence of signed regular Borel measures?
> 
> 
> 


I would be surprised if there weren't (or are there any fundamental obstructions?).


I would also be happy with metrics for weak convergence of non-negative measures (but not normed ones) or for uniformly bounded measures and would also like to know the answer to the same question for vector valued Radon measures on a metric space.


**2dn edit:** Thanks for the great answers. I had forgotten the general procedure to define a metric for weak(\*) convergence on bounded set, but in fact I had a more "geometric" metric in mind, something in the direction of R Ws and Dans answers.



Metrization of weak convergence of signed measures
Of course, there are many ways of metrizing the weak topology on $\mathcal M(\Omega)$ by using various tools of functional analysis. However, as it has already been pointed out by Dan, the most natural way is to use the transportation metric on the space of measures. [It is much more natural than the Prokhorov metric. I don't want to go into historical details here - they can be easily found elsewhere, but I insist that the transportation metric should really be related with the names of Kantorovich (in the first place) and his collaborator Rubinshtein]. Dan gives its dual definition in terms of Lipschitz functions, however its "transport definition" is actually more appropriate here. Let me remind it. 


Given two probability measures $\mu\_1,\mu\_2$ on $\Omega$
$$
\overline d(\mu\_1,\mu\_2) = inf\_M \int d(x\_1,x\_2) dM(x\_1,x\_2) \;,
$$
where $d$ is the original metric on $\Omega$, and the infimum (which is in fact attained) is taken over all probability measures $M$ on $\Omega\times\Omega$ whose marginals ($\equiv$ coordinate projections) are $\mu\_1$ and $\mu\_2$. One should think about such measures as "transportation plans" between distributions $\mu\_1$ and $\mu\_2$, while the integral in the RHS of the definition is the "cost" of the plan $M$. 


It is obvious that the above definition makes sense not just for probability measures, but for any two positive measures $\mu\_1,\mu\_2$ with the same mass. Moreover, $\overline d(\mu\_1,\mu\_2)$ actually depends on the difference $\mu\_1-\mu\_2$ only, so that one can think about it as a "weak norm" 
$$
|||\mu\_1-\mu\_2||| = \overline d(\mu\_1,\mu\_2) 
$$
of the signed measure $\mu\_1-\mu\_2$ (clearly, it is homogeneous with respect to multiplication by scalars).


Let now $\mu=\mu\_1-\mu\_2$ be an arbitrary signed measure, where $\mu\_1,\mu\_2$ are the components of its Hahn decomposition. The only reason why the definition of the weak norm does not work in this situation is that the measure $\mu$ need not to be "balanced" in the sense that the total masses $\|\mu\_1\|$ and $\|\mu\_2\|$ need not be the same any more. However, this can be easily repaired in the following way: extend the original space $\Omega$ 
to a new metric space $\Omega'$ by adding to it an "ideal point" $o$ and putting $d(\omega,o)=1$ for any $\omega\in\Omega$. Then the measure 
$$
\mu'=\mu - (\|\mu\_1\|-\|\mu\_2\|)\delta\_o \;,
$$
where $\delta\_o$ is the unit mass at the point $o$, is now balanced, so that $|||\mu'|||$ is well defined. Therefore, one can extend the definition of the weak norm $|||\cdot|||$ to arbitrary signed measures $\mu$ by putting
$$
|||\mu|||=|||\mu'||| \;.
$$ 


It is now easy to see that the distance $|||\mu\_1-\mu\_2|||$, where $\mu\_1,\mu\_2$ are two arbitrary signed measures, metrizes the weak topology on $\mathcal M(\Omega)$. 


