  Let (M, \omega) be a compact symplectic 4-manifold with a compatible almost
complex structure J. The problem of finding a J-compatible symplectic form with
prescribed volume form is an almost-K\"ahler analogue of Yau's theorem and is
connected to a programme in symplectic topology proposed by Donaldson. We call
the corresponding equation for the symplectic form the Calabi-Yau equation.
Solutions are unique in their cohomology class. It is shown in this paper that
a solution to this equation exists if the Nijenhuis tensor is small in a
certain sense. Without this assumption, it is shown that the problem of
existence can be reduced to obtaining a C^0 bound on a scalar potential
function.
