 Solution

Let $({\tau }_n)$ be a localizing sequence for $M$. For $s\le t$, the optional sampling theorem givesBoth sides converge almost surely to $M_t$ and $M_s$, and $|M_{u\wedge {\tau }_n}|\le Z$. Conditional dominated convergence therefore yields $\mathbb{E}[M_t\mid {\mathcal{F}}_s]=M_s$, so $M$ is a martingale. Moreover, the family $(M_t{)}_{t\ge 0}$ is dominated by the integrable random variable $Z$, hence is uniformly integrable. Thus $M$ is a uniformly integrable martingale.