This exercise weakens the conditions on the optional stopping theorem. Show that if Mt is a uniformly integrable martingale that is right continuous with left limits and T is a finite stopping time,...

This exercise weakens the conditions on the optional stopping theorem. Show that if Mt is a uniformly integrable martingale that is right continuous with left limits and T is a finite stopping time, then E M_T
= E M₀.

Let W be a Brownian motion and let T be a stopping time with
  Prove that EW_T
= 0 and

 This is not an easy application of the optional stopping theorem because we do not know that
  is necessarily a uniformly integrable martingale.