Doob decomposition. Let \(\left(X, \mathscr{A}, \mathscr{A}_{n}, \muight)\) be a \(\sigma\)-finite filtered measure space and let \(\left(u_{n}, \mathscr{A}_{n}ight)_{n \in \mathbb{N}}\) be a submartingale. Define \(u_{0}:=0\) and \(\mathscr{A}_{0}:=\{\emptyset, X\}\). Show that there exists an a.e. unique martingale \(\left(m_{n}, \mathscr{A}_{n}ight)_{n \in \mathbb{N}}\) and an increasing sequence of functions \(\left(a_{n}ight)_{n \in \mathbb{N}}\) such that \(a_{n} \in L^{1}\left(\mathscr{A}_{n-1}ight)\) for all \(n \geqslant 2\) and

\[u_{n}=m_{n}+a_{n}, \quad n \in \mathbb{N}\]

[ set \(m_{0}:=u_{0}, m_{n+1}-m_{n}:=u_{n+1}-\mathbb{E}^{\mathscr{A}_{n}} u_{n+1}\) and \(a_{0}:=0, a_{n+1}-a_{n}:=\mathbb{E}^{\mathscr{A}_{n}} u_{n+1}-u_{n}\). For uniqueness assume \(\tilde{m}_{n}+\tilde{a}_{n}\) is a further Doob decomposition and study the measurability properties of the martingale \(M_{n}:=m_{n}-\tilde{m}_{n}=\tilde{a}_{n}-a_{n}\).]