Counterexample for uniform integrability of an $\mathbb{L}^1$-bounded sequence

Question

I need to find an example such that $\sup\limits_{n \ge 1} \mathbb{E}[|X_n|] < \infty$ but $\{X_n\}_{n \ge 1}$ are not uniformly integrable. I can show that if $\{X_n\}_{n \ge 1}$ are uniformly integrable, we have that $\sup\limits_{n \ge 1} \mathbb{E}[|X_n|] < \infty$. However, I cannot think of a counterexample to disprove the reverse. I appreciate any insights on this.

Answer 1

Let $\Omega=[0,1]$ with Lebesgue measure, and $X_n=n1_{[0,\frac{1}{n}]}$. Then the $X_n$ are bounded in $L^1$ but not uniformly integrable.