Definition (Weak Convergence)

Let $\mathbb{X}$ be a Polish space and let $\mathcal{P}(\mathbb{X})$ denote the family of all Probability Measures on $(\mathbb{X},\mathcal{B}(\mathbb{X}))$. Let $\{ \mu_{n} \}_{n\in\mathbb{N}}\subset \mathcal{P}(\mathbb{X})$ be a sequence of Borel measures. We say $\mu_{n}\to \mu\in\mathcal{P}(\mathbb{X})$ weakly if $$\int\limits _{\mathbb{X}}c(x) \, \mu_{n}(dx)\to \int\limits _{\mathbb{X}}c(x) \, \mu(dx)$$for every Continuous and bounded $c:\mathbb{X}\to \mathbb{R}$.