  We give two applications of the Aleksandrov-Bakelman-Pucci estimate to the
Calabi-Yau equation on symplectic four-manifolds. The first is solvability of
the equation on the Kodaira-Thurston manifold for certain almost-Kahler
structures assuming $S^1$-invariance, extending a result of
Buzano-Fino-Vezzoni. The second is to reduce the general case of Donaldson's
conjecture to a bound on the measure of a superlevel set of a scalar function.
