> [!theorem] > > Let $\seq{\mu_n}$ be such that $\mu_n \to \mu$ [[Weak Convergence of Measures|weakly]], then the [[Characteristic Function|characteristic functions]] $\hat \mu_n \to \hat \mu$ converges [[Pointwise Convergence|pointwise]]. Therefore weak convergence admits a unique limit point. > [!theorem] > > Let $\seq{\mu_n}$ be such that $\mu_n \to \mu$ weakly, and $\seq{\varphi_n} \subset BC(\real^d)$ such that $\sup \norm{\varphi_n}_u < \infty$ and $\varphi_n \to \varphi$ [[Uniform Convergence on Compact Sets|uniformly on compact sets]]. Then > $ > \limv{n}\int \varphi_n d\mu_n = \int \varphi d\mu > $ > *Proof*. By weak convergence, $\mu_n(\varphi) \to \mu(\varphi)$, and $\seq{\mu_n}$ is [[Tight|tight]]. Let $\varepsilon > 0$, then there exists $K \subset \real$ [[Compactness|compact]] such that $\sup_{n \in \nat}\mu_n(K^c) < \varepsilon$, which bounds > $ > \begin{align*} > \abs{\mu_n(\varphi_n) - \mu(\varphi)} &\le \abs{\mu_n(\varphi_n) - \mu_n(\varphi)} + \underbrace{\abs{\mu_n(\varphi) - \mu(\varphi)}}_{\to 0} > \end{align*} > $ > and > $ > \begin{align*} > \abs{\mu_n(\varphi) - \mu_n(\varphi)} &\le \int_{K}\abs{\varphi_n - \varphi}d\mu_n + \int_{K^c}\abs{\varphi_n - \varphi}d\mu_n \\ > &\le \underbrace{\int_{K}\abs{\varphi_n - \varphi}d\mu_n}_{\to 0} + 2\varepsilon \sup \norm{\varphi_n}_u > \end{align*} > $ > since $\varepsilon$ was arbitrary, $\abs{\mu_n(\varphi_n) - \mu(\varphi)} \to 0$.