  This paper considers discounted infinite horizon mean field games by
extending the probabilistic weak formulation of the game as introduced by
Carmona and Lacker (2015). Under similar assumptions as in the finite horizon
game, we prove existence and uniqueness of solutions for the extended infinite
horizon game. The key idea is to construct local versions of the previously
considered stable topologies. Further, we analyze how sequences of finite
horizon games approximate the infinite horizon one. Under a weakened
Lasry-Lions monotonicity condition, we can quantify the convergence rate of
solutions for the finite horizon games to the one for the infinite horizon game
using a novel stability result for mean field games. Lastly, applying our
results allows to solve the invariant mean field game as well.
