Let be a uniformly integrable martingale, . Let and ( if for all ). One can check is a stopping time of .

How do you prove that is also a uniformly integrable martingale?

The martingale part of the problem comes immediately from: , when , using the Optional Sampling Theorem.

But how about the uniform integrability? For and all , one can write , but how do you get a bound for the integral remainder which is independent of ?

Thanks for your help.