A generalized conditional expectation and the corresponding martingale is defined in terms of the Kolmogorov $A$-integral. It is proved that the uniform $A$-integrability of a martingale sequence is a sufficient condition for the sequence to be closed on the right by the $A$-integrable last element.
"A Martingale Closure Theorem for A-Integrable Martingale Sequences." Real Anal. Exchange 24 (2) 815 - 820, 1998/1999.