By using the global deformation of almost complex structures which are
compatible with a symplectic form off a Lebesgue measure zero subset, we
construct a (measurable) Lipschitz Kahler metric such that the one-form type
Calabi-Yau equation on an open dense submanifold is reduced to the complex
Monge-Ampere equation with respect to the measurable Kahler metric. We give an
existence theorem for solutions to the one-form type Calabi-Yau equation on
closed symplectic manifolds