Fix a compact 4-dimensional manifold with self-dual second Betti number one and with a given symplectic form. This article proves the following: The Frêchet space of tamed almost complex structures as defined by the given symplectic form has an open and dense subset whose complex structures are compatible with respect to a symplectic form that is cohomologous to the given one. The theorem is proved by constructing the new symplectic form by integrating over a space of currents that are defined by pseudo-holomorphic curves.
"Tamed to compatible: symplectic forms via moduli space integration." J. Symplectic Geom. 9 (2) 161 - 250, June 2011.