$X_{n}$ and $Z$ be random variables if $X_{n} \ge Z$ then $ E[\liminf_{n\to \infty} X_{n}] \le \liminf_{n\to \infty} E[X_{n}] $

Question

Let $X_{n}$ and $Z$ be random variables on probability space $(\Omega ,\mathcal F,P)$ and $Z$ be integrable. Show that $$X_{n} \ge Z \qquad \Longrightarrow \qquad E[\liminf_{n\to \infty} X_{n}] \le \liminf_{n\to \infty} E[X_{n}] $$

Answer 1

Fatou's lemma applied to the non-negative sequence $X_n-Z$.