*Background:* I work on a SPDE problem where in order to apply Prokhorov's theorem I need that some measure space is Polish space. And additionaly it would be good if that space is Banach space. Earlier today I was reading the book [1] by Malek, Necas, Rokyta, Ruzicka, and I have a question from the Subsection 1.2.8 titled Radon measures. The definitions given bellow are taken from the same book.

On the one hand, the space of Radon measures is defined as: $$M(\mathbb{R}^d)\equiv \{ \mu : C_0 (\mathbb{R}) \to \mathbb{R}; \mu \text{ is linear}\wedge \exists c>0, |\mu (f)|\le c \|f\|_{\infty}, \forall f \in \mathcal{D}(\mathbb{R}^d)\}.$$ Here

- $C_0(\mathbb{R}^d)\equiv \{ u \in C(\mathbb{R}^d): \lim_{|x|\to \infty} u(x) = 0 \}$ and
- $C_0(\mathbb{R}^d)=\overline{\mathcal{D}(\mathbb{R}^d)}^{\|\cdot\|_{\infty}}$.

As usual $\mathcal{D}(\Omega)$ stands for the space of functions from $C^{\infty}( \overline{\Omega})$ with compact support in $\Omega$.

If we further define $\|\mu\|_{M(\mathbb{R}^d)}\equiv \sup\{|\mu(f)|: f \in \mathcal{D}(\mathbb{R}^d),\|f\|_{\infty}\leq 1 \}$, then the space $\big(M(\mathbb{R}^d), \| \cdot \|_{M(\mathbb{R}^d)}\big)$ is a Banach space.

On the other hand, let $\Omega$ be a bounded domain. We denote by $M(\Omega)$ the space of Radon measures defined as the dual space of $C(\overline{\Omega})$. Also in this case we know that $L^1(\Omega)\hookrightarrow M(\Omega)$ (and we know that $L^1(\Omega)$ is separable).

*My questions are:*

**Is the space of Radon measures separable - in the case $\Omega \subset \mathbb{R}^d$ and in the case $\mathbb{R}^d$? Or to be more precise is it a Polish space?**I have search it in a few books and in the questions here but I didn't find any concrete answer (I maybe have missed something).**Maybe some subspace of the space of Radon measure is Polish?**I've read somewhere that the space of positive Radon measures is Polish but didn't find any book to confirm that.**Are there some other spaces of measure-valued functions that are Polish**(besides the spaces mentioned above)?

I usually avoid dealing with measure-valued spaces so I don't know much about them. Help with this would be great (and I definitely need it). Thanks in advance.

**Reference**

[1] Jindřich Nečas, Josef Málek, Mirko Rokyta, Michael Růžička, *Weak and measure-valued solutions to evolutionary PDEs*, Applied Mathematics and Mathematical Computation. 13. London: Chapman & Hall, pp. vii+317 (1996), ISBN:0-412-57750-X, MR1409366, Zbl 0851.35002.

allRadon measures with the weak-* topology is not metrizable, though it is separable. $\endgroup$3more comments