Let $\Omega$ be a compact metric space and $\mathscr{P}(\Omega,\mathcal{F})$ the space of all probability measures definid on $\sigma$-field Borel of $\mathbb{X}$. $UC(\Omega,\mathbb{R})$ stands for the space of all bounded real valued continuous functions on $\Omega$.
A metric to $\mathscr{P}(\Omega,\mathcal{F})$ is
$ d(\mu,\nu)=\sum_{\varphi_n\in\mathcal{A}}\frac{1}{2^n}\left|\int_\Omega \varphi_n \;d\mu -\int_\Omega \varphi_n \;d\nu \right| $
where $\mathcal{A}$ is countable and dense set of functions in $UC(\Omega,\mathbb{R})$. The symmetry and the triangle inequality can be easily verified. To check $d(\mu,\nu)=0 \Leftrightarrow \mu=\nu$ we use the Riesz-Markov theorem.
Question: How can we prove that this metric generates the weak* topology in $\mathscr{P}(\Omega,\mathcal{F})$?