Prove the following $L_1$ convergent proposition

Question

Suppose $X_n \geq 0$ for $n \geq 0$. Suppose further that $X_n \xrightarrow[]{\mathbb{P}} X_0$ and $E(X_n) \to E(X_0)$. The sequence {$X_n: n \geq 1$} is $L_1$-convergent to $X_0$

Answer 1

We have the identity $$\mathbb{E}(|X_n - X_0|) = \mathbb{E}(X_n)+\mathbb{E}(X_0) - 2\mathbb{E}( \min\{X_n,X_0\}).$$ Since we have convergence in measure, we can take of any sequence of natural numbers a subsequence $(n_k)_{k \in \mathbb{N}}$ such that $X_{n_k} \rightarrow X_0$ almost sure. By assumptation and Fatou's lemma we get $$\limsup_{k \rightarrow \infty} \ \mathbb{E}(|X_{n_k}-X_0|) \leq 2 \mathbb{E} X_0 - 2 \liminf_{k \rightarrow \infty} \ \mathbb{E}(\min\{X_{n_k},X_0\}) =0.$$ This implies already that $\mathbb{E}(|X_n -X_0|) \rightarrow 0$.