Past exam of the probability, measure and martingales course of the University of Oxford / 2023 / 2 / b / iii / Solution

A family $({\xi }_i{)}_{i\in I}$ of integrable random variables is uniformly integrable whenThe Lipschitz condition gives, on $\{X<1\}$,Our chosen zero value on $\{X=1\}$ satisfies the same bound. Hence the random variables $(Z_n)$ are integrable, and for $K\ge L$ every tail expectation in the definition vanishes. Therefore the family is uniformly integrable.