  In this paper, we establish the existence and uniqueness of weak solutions to
first-order discount mean field games and a stability result to give the
existence for the ergodic problem. We show an example to illustrate the
multiplicity of weak solutions to the ergodic problem. With this motivation, we
address a selection condition, which is a necessary condition that any limit of
solutions under subsequence satisfies. As an application, we show a nontrivial
example to get the convergence of weak solutions.
