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

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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

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