Is the set of all probability measures weak*-closed?
No. Take $\Omega=\mathbb N$ and $\Sigma$ the power set. As wikipedia says, then $ba(\Sigma)=ba=(\ell^\infty)^*$. However, the collection of probability measures is just the collection of $(x_n)\in\ell^1$ (as a measures are countably additive) with $x_n\geq 0$ for all $n$, and $\sum_n x_n=1$. This is not weak$^*$-closed in $(\ell^\infty)^*$. For example, any limit point of the set $\{\delta_n:n\in\mathbb N\}$, where $\delta_n\in\ell^1$ is the point mass at $n$, is a member of $ba \setminus \ell^1$.