  In the presence of a common noise, we study the convergence problems in mean
field game (MFG) and mean field control (MFC) problem where the cost function
and the state dynamics depend upon the joint conditional distribution of the
controlled state and the control process. In the first part, we consider the
MFG setting. We start by recalling the notions of $measure$--$valued$ MFG
equilibria and of approximate $closed$--$loop$ Nash equilibria associated to
the corresponding $N$--player game. Then, we show that all convergent sequences
of approximate $closed$--$loop$ Nash equilibria, when $N \to \infty,$ converge
to $measure$--$valued$ MFG equilibria. And conversely, any $measure$--$valued$
MFG equilibrium is the limit of a sequence of approximate $closed$--$loop$ Nash
equilibria. In other words, $measure$--$valued$ MFG equilibria are the
accumulation points of the approximate $closed$--$loop$ Nash equilibria.
Previous work has shown that $measure$--$valued$ MFG equilibria are the
accumulation points of the approximate $open$--$loop$ Nash equilibria.
Therefore, we obtain that the limits of approximate $closed$--$loop$ Nash
equilibria and approximate $open$--$loop$ Nash equilibria are the same. In the
second part, we deal with the MFC setting. After recalling the $closed$--$loop$
and $open$--$loop$ formulations of the MFC problem, we prove that they are
equivalent. We also provide some convergence results related to approximate
$closed$--$loop$ Pareto equilibria.
