  We consider the solution $\mathcal V_\delta$ of the discounted
Hamilton-Jacobi equation in the Wasserstein space arising from potential MFG
and we prove its full convergence to a corrector function $\chi_0$. We follow
the structure of the proof of the analogue result in the finite dimensional
setting provided by Davini, Fathi, Iturriaga, Zavidovique in 2017. We
characterize the limit $\chi_0$ through a particular set of smooth Mather
measures. A major point that distinguishes the techniques deployed in the
standard setting from the ones that we use here is the lack of mollification in
the Wasserstain space.
