It is well known that $C^\*_0(\mathbb{R})$ (the continuous dual space of $C_0(\mathbb{R})$, which is all continuous functions on $\mathbb{R}$ that vanish at $\pm \infty$) can be identified with the space of all regular signed measures. Equip this space with the weak-* topology, i.e. where $\mu_n$ converges weakly to $\mu$ if $\int f d\mu_n \rightarrow \int f d\mu$ for all $f \in C_0(\mathbb{R})$. I'm looking at the set of all probability measures in this space (positive measures for which $\mu(\mathbb{R}) = 1$). This set is not closed in the weak-* topology, since sequences such as $\delta_n \to 0$ (zero measure) as $n \to \infty$.

Consider then a convex subset of probability measures $A$ that is closed relative to the set of all probability measures. I'm wondering if $\forall \mu \in$ closure($A$), does there exists a positive measure $\nu$ such that $\mu + \nu \in A$?

By Prokhorov's theorem, I know that mass "escapes to infinity" only when $A$ is not tight. I know that by the Banach–Alaoglu theorem, the set {$\mu : 0 \leq \mu(\mathbb{R}) \leq 1 $} is closed in the weak-*, so closure($A$) can't be bigger than that set. Intuitively, it seems that this should be true, but I can't seem to see the right path. Any suggestions or opinions on the correct direction to proceed are much appreciated.