By definition, if $\{ s_n, \mathcal{F}_n,n\ge 1\}$ is a submartingale then $\mathbb{E} (|s_n|)<\infty$ for all $n\in \mathbb{N}^*$. But according to a arcticle, in a theorem, author add addition: $\mathbb{E} (s_1^-)<\infty$. I don't understand why they did it. This is content of that theorem:
