In Stochastic Integration Theory by Peter Medvegyev, the author assumes that only the "negative part" $M^-:=\max(-M,0)$ of a supermartingale $M$ is integrable. Analogously, he assumes that only the "positive part" $N^+:=\max(N,0)$ of a submartingale $N$ is integrable.
Clearly, that's enough to ensure that all occuring integrals are well-defined. But what's the point of this assumption? Why he doesn't assume that $M^+$ and $N^-$ are integrable?
Submartingales where originally defined as such and that is a weaker condition that leads to the convergence theorem (see Doob's Stochastic processes). Nowadays most authors just adopt the stricter condition of absolute integrability, which is what you propose.