While skipping through the class notes I noticed one exercice that I couldn't solve:
Suppose we have $\mu$ - probability distribution in $\mathbb{R}$.
Recalling that $\mu_k \rightarrow \mu$ iff $\int_{\mathbb{R}} f(x)\mu_k(dx) \rightarrow \int_{\mathbb{R}} f(x) \mu(dx)$, $\forall f \in C(\mathbb{R})$, $f$ bounded $k\rightarrow \infty$. We have to show:
- $\exists \mu_k$ - series of continuous distributions that $\mu_k \rightarrow \mu$.
- $\exists \mu_k$ - series of discrete distributions that $\mu_k \rightarrow \mu$.