Consider the probability space ((mathbb{N}, mathscr{P}(mathbb{N}), mathbb{P})) with [mathbb{P}{n}:=frac{1}{n}-frac{1}{n+1}] Set [mathscr{A}_{n}:=sigma({1},{2}, ldots,{n},[n+1, infty) cap mathbb{N})] and show

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Consider the probability space \((\mathbb{N}, \mathscr{P}(\mathbb{N}), \mathbb{P})\) with

\[\mathbb{P}\{n\}:=\frac{1}{n}-\frac{1}{n+1}\]

Set

\[\mathscr{A}_{n}:=\sigma(\{1\},\{2\}, \ldots,\{n\},[n+1, \infty) \cap \mathbb{N})\]

and show that \(\xi_{n}:=(n+1) \mathbb{1}_{[n+1, \infty) \cap \mathbb{N}}, n \in \mathbb{N}\), is a positive martingale such that \(\int \xi_{n} d \mathbb{P}=1, \lim _{n ightarrow \infty} \xi_{n}=0\) but \(\sup _{n \in \mathbb{N}} \xi_{n}=\infty\).

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