Let ((X, mathscr{A}, mu)) be a finite measure space and let (left(u_{n}, mathscr{A}_{n}ight)_{n in mathbb{N}}) be a

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Let \((X, \mathscr{A}, \mu)\) be a finite measure space and let \(\left(u_{n}, \mathscr{A}_{n}ight)_{n \in \mathbb{N}}\) be a martingale. Set \(\mathscr{A}_{0}:=\{\emptyset, X\}\) and \(u_{0}=\mu(X)^{-1} \int u_{1} d \mu\). Show that this is the only choice which makes \(\left(u_{n}, \mathscr{A}_{n}ight)_{n \in \mathbb{N}_{0}}\) a martingale.

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