We prove that an amart indexed by a directed set decomposes into a martingale and an amart which converges to zero in $L_1$ norm. The main theorem asserts that the underlying family of $\sigma$-algebras satisfies the Vitali condition if and only if every $L_1$ bounded amart essentially converges.
"Amarts Indexed by Directed Sets." Ann. Probab. 6 (2) 267 - 278, April, 1978. https://doi.org/10.1214/aop/1176995572