Martingale not uniformly integrable
Solution 1:
Let $\xi_j$ be i.i.d. with $(\forall j \in \mathbb{N}) \mbox{} \mathbb{P}(\xi_j=0)=\mathbb{P}(\xi_j=2)=1/2$, and define $M_n=\prod_{j=1}^n \xi_j$ for $n\geq 0$. Then $(M_n)$ is a non-negative martingale with $\mathbb{E}(M_n)=1$ for all $n$. But $M_n\to 0$ almost surely, so $(M_n)$ cannot be uniformly integrable.