arXiv:2610.28335v1 Announce Type: new Abstract: We introduce and analyze a numerical framework for a class of first-order, non-local Mean Field Game (MFG) systems with two sources of low regularity: the time-dependent data are merely measurable, and the Hamiltonian is only of class $C^{1,1}$ with respect to the momentum variable. The proposed method combines a semi-Lagrangian approximation of the Hamilton--Jacobi--Bellman equation with a Lagrange--Galerkin discretization of the continuity equation. Temporal measurability is handled by averaging the data over the time-grid intervals and using the stability theory of $L^1$-viscosity solutions. The limited regularity of the Hamiltonian is addressed through arguments from nonsmooth analysis, based in particular on Clarke's generalized differential calculus. We prove that the discrete solutions converge, up to some subsequence, to a solution of the MFG system. As a consequence, the numerical approximation also provides a constructive existence result. Under a monotonicity assumption on the coupling terms, we show that the solution to the MFG system is unique, implying the convergence of the whole sequence of discrete solutions. Finally, numerical experiments in one and two spatial dimensions illustrate the robustness of the method, including a linear--quadratic example with a running cost that is discontinuous in time and for which we compute an exact solution.