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 \(\sup _{n \in \mathbb{N}} \xi_{n}=\infty\).