  We study a mean-field version of Bank-El Karoui's representation theorem of
stochastic processes. Under different technical conditions, we establish some
existence and uniqueness results. As motivation and first applications, our
mean-field representation results provide a unified approach to study different
Mean-Field Games (MFGs) in the setting with common noise and multiple
populations, including the MFG of timing, the MFG with singular control, etc.
As a crucial technical step, we provide a stability result on the classical
Bank-El Karoui's representation theorem, which has its own interests and other
applications, such as in deriving stability results of the optimizers (in the
strong sense) for a class of optimal stopping problems and singular control
problems.
