Let \(\left(u_{n}, \mathscr{A}_{n}ight)_{n \in \mathbb{N}}\) be a supermartingale such that \(u_{n} \geqslant 0\) and \(\lim _{n ightarrow \infty} \int u_{n} d \mu=0\). Then \(u_{n} ightarrow 0\) pointwise a.e. and in \(\mathcal{L}^{1}\).

Remark: Positive supermartingales with \(\lim _{n ightarrow \infty} \int u_{n} d \mu=0\) are called potentials.