  This paper studies the convergence problem for mean field games with common
noise. We define a suitable notion of weak mean field equilibria, which we
prove captures all subsequential limit points, as $n\to\infty$, of closed-loop
approximate equilibria from the corresponding $n$-player games. This extends to
the common noise setting a recent result of the first author, while also
simplifying a key step in the proof and allowing unbounded coefficients and
non-i.i.d. initial conditions. Conversely, we show that every weak mean field
equilibrium arises as the limit of some sequence of approximate equilibria for
the $n$-player games, as long as the latter are formulated over a broader class
of closed-loop strategies which may depend on an additional common signal.
