Non uniformly integrable sequence

Question

I am looking for a sequence of random variable $X_1, X_2, \cdots$ on $([0,1], \mathbb B, \lambda)$ such that $$X_n\to 0 \quad \text{a.s.}$$ $$EX_n\to 0$$ but such that $(X_n)_n$ is not uniformly integrable.

I already showed that the sequence can't have $X_n\geq 0$ for all $n$. Can we maybe achieve it with linear combinations of indicator functions $\mathbb 1_{[\frac{i-1}{n}, \frac{i}{n}]}$ ?

Answer 1

Consider $$X_n := n 1_{(0,1/n)}-n 1_{(1/n,2/n)}$$