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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 metric $d_P$ or the Wasserstein metrics $d_W$.

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.

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 metric $d_P$ or the Wasserstein metrics $d_W$.

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.

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 metric $d_P$ or the Wasserstein metrics $d_W$.

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.

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 metric $d_P$ or the Wasserstein metrics $d_W$.

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.

Let $\mathcal{M}(\Omega)$ denote the space of signed regular Borel measures on a compact Hausdorff 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 metric $d_P$ or the Wasserstein metrics $d_W$.

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 Hausdorff space.

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 metric $d_P$ or the Wasserstein metrics $d_W$.

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.