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

Question:

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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