The purpose of this note is to show that the set of $L^1$-bounded "martingales in the limit", unlike the set of $L^1$-bounded "amarts", is not a vector lattice.
A. Bellow. A. Dvoretzky. "On Martingales in the Limit." Ann. Probab. 8 (3) 602 - 606, June, 1980. https://doi.org/10.1214/aop/1176994730