  We obtain the effective action of four dimensional quantum gravity, induced
by N massless matter fields, by integrating the RG flow of the relative
effective average action. By considering the leading approximation in the large
N limit, where one neglects the gravitational contributions with respect to the
matter contributions, we show how different aspects of quantum gravity, as
asymptotic safety, quantum corrections to the Newtonian potential and the
conformal anomaly induced effective action, are all represented by different
terms of the effective action when this is expanded in powers of the curvature.
