Almost sure convergence of L1 bounded sequence implies L1 convergence

Question

I wanted to ask whether the following statement is true:

Assume random variables $X_n \to X$ almost surely, and that $\left\lvert X_n\right\rvert \leq Y$ with $Y\in L^1$, then $X_n \to X$ in $L^1$

Answer 1

You have $|X_n|\to|X|$ almost surely, and $|X_n|\le Y$ with $Y$ integrable. So by dominated convergence, conclude $E|X_n|\to E|X|$. Now you can apply Scheffé's lemma.