Let (left(u_{n}, mathscr{A}_{n}ight)_{n in mathbb{N}}) be a supermartingale and (u_{n} geqslant 0) a.e. Prove that (u_{k}=0) a.e.
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Let \(\left(u_{n}, \mathscr{A}_{n}ight)_{n \in \mathbb{N}}\) be a supermartingale and \(u_{n} \geqslant 0\) a.e. Prove that \(u_{k}=0\) a.e. implies that \(u_{k+n}=0\) a.e. for all \(n \in \mathbb{N}\).
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