[Lemma 4 of JPS07]. Let be a random variable, such that for all . Let . Let be a natural filtration generated by . Then,
1) is a stopping time w.r.t. .
2) is a non-uniform-integrable martingale with .
Proof. 1) It can be seen from .
2) For all bounded stopping time , one can show a) ; b) . Thus, by Proposition II.3.5 of [RY99], is a martingale. Since , it is not a uniformly integrable martingale by Theorem II.3.1 of [RY99].