Decomposition of supermartingale into UI martingale and supermartingale

Question

I am trying to prove the following (source, Question 1(b)):

Let $X_n$ be a $L^1$ bounded supermartingale. Then show that there are $M,Y$ s.t.

• $X_n=M_n+Y_n$
• $M$ is a uniformly integrable martingale
• $Y$ is a supermartingale converging to 0 almost surely

The question it comes from has previous parts outlining the decomposition of a submartingale into a martingale and an $L^1$ bounded supermartingale. I am not sure if that this previous result is useful here, since we now want the integrability condition on the martingale, not the supermartingale. Any help would be appreciated.

Answer 1

Here's a start: The $L^1$ bounded supermartingale $X_n$ has an a.s. limit $X_\infty$, and the r.v. $X_\infty$ is integrable, by Fatou. Define $M_n:=\Bbb E[X_\infty|\mathcal F_n]$. Show that if $Y_n:=X_n-M_n$ then $Y_n$ is a supermartingale.