Let $ba:=ba(\Omega,\mathcal{F})$ the space of all finitely additive set functions $\mu: \mathcal{F} \to \mathbb{R}$ with finite total variation $\|\cdot\|_{var}$ where $$\|\mu\|_{var}:=\sup\left\{\sum_{i=1}^n |\mu(A_i)| : A_1,\dots,A_n\in\mathcal{F}\,\,\text{ disjoint sets, } n\in\mathbb{N}\right\}.$$ The space $ba$ can be identified with the topological dual of the Banach space $\mathcal{X}$ that is the space of all real, bounded and measurable function on $(\Omega,\mathcal{F})$ equipped with the supremum norm $\|X\|:=\sup_{\omega\in\Omega}|X(\omega)|$.
Now, I have to prove that the subspace $$\mathcal{M}_{1,f}:=\left\{\mu\in ba : \mu(\Omega)=1 \text{ e } \mu(A)\geq0 \text{ for all } A\in\mathcal{F}\right\}\subset ba$$ is weakly$^*$ closed.
I don't really know how to do this. Maybe someone can suggest me some theorems that might come in handy?