Grafakos systematically proved that $A_\infty$ weights have different characterizations for cubes in Euclidean spaces in his classical text book. Very recently, Duoandikoetxea, Martín-Reyes, Ombrosi and Kosz discussed several characterizations of the $A_{\infty}$ weights in the setting of general bases. By conditional expectations, we study $A_\infty$ weights in martingale spaces. Because conditional expectations are Radon-Nikodým derivatives with respect to sub$\hbox{-}σ\hbox{-}$fields which have no geometric structures, we need new ingredients. Under a regularity assumption on weights, we obtain equivalent characterizations of the $A_{\infty}$ weights. Moreover, using weights modulo conditional expectations, we have one-way implications of different characterizations.