# Decomposition of supermartingale into UI martingale and supermartingale

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.

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.