  We present a new notion of solution for mean field games master equations.
This notion allows us to work with solutions which are merely continuous. We
prove first results of uniqueness and stability for such solutions. It turns
out that this notion is helpful to characterize the value function of mean
field games of optimal stopping or impulse control and this is the topic of the
second half of this paper. The notion of solution we introduce is only useful
in the monotone case. We focus in this paper in the finite state space case.
