Let $X$ be a second-countable locally compact Hausdorff space. Denote by $C_c(X) \subseteq C_0(X) \subseteq C_b(X)$ the spaces of continuous functions with compact support, vanishing at infinity and those that are bounded respectively. Denote by $M(X) = C_0(X)' = C_c(X)'$ the space of signed Radon measures on $X$, where $C_0(X)$ or $C_c(X)$ carry the supremum norm ($C_c$ is dense in $C_0$). On $M(X)$ we can consider the weak$^*$ topologies $w^*_c \subseteq w^*_0 \subseteq w^*_b$ relative to $C_c$, $C_0$ and $C_b$. On the set of positive Radon measures $M^+(X)$, the topologies $w^*_c$ and $w^*_b$ are Polish (i.e. completely metrizable and separable).
- A compatible complete metric for $w^*_b$ on $M^+$ is given by the Kantorovich-Rubinstein norm [Bogachev, "Measure Theory II", Theorem 8.3.2].
- A compatible complete metric for $w^*_c$ on $M^+$ is typically constructed from a countable dense subset $\{ f_n \mid n \in \mathbb{N} \} \subseteq C_c(X)$ ($C_0$ and $C_c$ are separable), e.g. $d(\mu, \nu) = \sum_{n=1}^\infty \frac{1}{2^n} min(1, |\int f_n d\mu - \int f_n d\nu|)$.
In the literature on measure and probablity theory I mostly find discussions around these two topologies ($w^*_c$ is called the vague topology and $w^*_b$ the narrow topology). Of course, $w^*_0 = w^*_c$ on norm-bounded sets of $M(X)$, but the topology $w^*_0$ seems to be more natural on $M(X)$, simply because $(M(X), w^*_0)$ is the weak$^*$-dual of a Banach space ($C_0$); $C_c$ is only a normed space. On $M^+(X)$ these three topologies are generally different. For instance, if $X = \mathbb{N}$, then (i) $n \delta_n \to 0$ for $w^*_c$ but not for $w^*_0$ ($n \delta_n$ is not norm bounded) and (ii) $\delta_n \to 0$ for $w^*_0$ but not for $w^*_b$ ($\delta_n$ is not tight).
Questions: Is the $w^*_0$ topology also Polish on $M^+(X)$? Can anyone provide a reference for properties of this topology (on all of $M(X)$ or $M^+(X)$)?
[Note that the one-point compactification $X \to X_\infty$ induces an embedding of $M^+(X)$ as a closed subset of the Polish space $(M^+(X_\infty), w^*)$ with $w^* = \sigma(M(X_\infty), C(X_\infty))$, but the subspace topology induced on $M^+(X)$ is finer than $w^*_0$ (it can be shown that $w^* = w^*_b$ on $M^+(X)$).]