Given that $(X_n)_{n \in \mathbb(N)}$ converges in distribution to $X$, I am trying to show that $$E(|X|) \le \lim_{n \to \infty} \inf (E|X_n|).$$
I though of using Portmanteau theorem since part e) in my notes claims the following
- for each $U \subset S $ open $\lim_{n \to \infty} \inf \mu_n(U) \ge \mu(U)$.
But how do I come to expectations from probability measures ?