Dynamical Triangulations provide us with a lattice regularization of four-dimensional Euclidean quantum gravity within the realm of ordinary quantum field theory. We add a local measure term, which can also serve as a generalized higher curvature term, and explore an extended coupling constant space. We determine the phase diagram of this model using non-degenerated triangulations. A first order phase transition line is observed, but no second order transition point is located. In consequence we cannot attribute any continuum physics interpretation to the so-calledcrinkled phase of 4D dynamical triangulations.