  Grafakos systematically proved that $A_\infty$ weights have different
characterizations for cubes in Euclidean spaces in his classical text book.
Very recently, Duoandikoetxea, Mart\'{\i}n-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-Nikod\'{y}m derivatives with
respect to sub$\hbox{-}\sigma\hbox{-}$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.
