Solution of Monge-Amp\`ere Equation on Wiener Space for log-concave measures: General case

We show the existence of the strong solutions of the Monge-Ampere equation for the log-concave measures without any regularity assumption. In particular the measures with density of the form $\exp -f$, where $f$ is an $H$-convex function which may take the value $\mathrm{\infty }$ on a set of positive measure are included.