Show that \(\left(u_{n}ight)_{n \in \mathbb{N}}\) is a submartingale if, and only if, \(u_{n} \in \mathcal{L}^{1}\left(\mathscr{A}_{n}ight)\) for all \(n \in \mathbb{N}\) and

\[\int_{A} u_{n} d \mu \leqslant \int_{A} u_{k} d \mu \quad \forall n

Find similar statements for martingales and supermartingales.