  Here, we address a uniqueness structure of viscosity solutions for ergodic
problems of weakly coupled Hamilton-Jacobi systems. In particular, we study
comparison principle with respect to generalized Mather measures as a
generalization of the result proved by Mitake and Tran, which addressed the
case of a single equation. To get the main result, it is important to construct
Mather measures effectively. We overcome this difficulty by nonlinear adjoint
methods.
