Completion (7). Let \(\left(u_{n}, \mathscr{A}_{n}ight)_{n \in \mathbb{N}}\) be a submartingale and denote by \(\overline{\mathscr{A}}_{n}\) the completion of \(\mathscr{A}_{n}\). Then \(\left(u_{n}, \overline{\mathscr{A}}_{n}ight)_{n \in \mathbb{N}}\) is still a submartingale.