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: Malek, Necas, Rokyta, Ruzicka – Weak and Measure-valued Solutions to Evolutionary PDEs, 1996, 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:
$$M(\mathbb{R}^d)\equiv \{ \mu : C_0 (\mathbb{R}) \rightarrow \mathbb{R}; \mu \hspace{0.2cm} linear \hspace{0.2cm} s.t. \hspace{0.2cm} \exists c>0, |\mu (f)|\leq c ||f||_{\infty}, \forall f \in \mathcal{D}(\mathbb{R}^d)\}.$$
Here
$C_0(\mathbb{R}^d)\equiv \{ u \in C(\mathbb{R}^d): lim_{|x|\rightarrow \infty} u(x) = 0 \}$
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 $(M(\mathbb{R}^d), || \cdot ||_{M(\mathbb{R}^d)})$ 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 meaasure-valued spaces so I don't know much about them. Help with this would be great (and I definitely need it). Thanks in advance.
Best Answer
With respect to the norm topology, the space of Radon measures on a domain $\Omega$ is not seperable. Indeed, for any two distinct points $x,y$ in $\Omega$, the Dirac measures $\delta_x$ and $\delta_y$ (where $\delta_x(f)=f(x)$) satisfy $\|\delta_x-\delta_y\|=2$ since you can always find a compactly supported smooth function $f$ with $f(x)=-f(y)=1=\|f\|_\infty$. Any metric space that contains uncountably many disjoint open balls cannot be seperable. Of course there are many subspaces of Radon measures that are seperable in the norm topology, e.g., as you noted $L^1$ naturally embeds as a subspace and is seperable.
The space of Radon measures on a domain $\Omega$ is seperable in the weak$^*$ topology (This is probably the remark you allude to have read somewhere.) Indeed, consider the countable set $M_Q$ of measures of the form $\sum_{x \in S} a_x \delta_x$ where the coefficients $a_x$ are rational and $S$ runs over finite sets of points with rational coordinates. This $M_Q$ is countable and weak$^*$ dense. Also the embedding of $L^1$ as space of measures with absolutely continuous to Lebesgue measure is dense, and this gives another proof of weak$^*$ seperability.