The vanishing discount problem for monotone systems of Hamilton-Jacobi equations. Part 1: linear coupling

We establish a convergence theorem for the vanishing discount problem weakly coupled system of Hamilton-Jacobi equations. The crucial step is introduction Mather measures and their relatives system, which we call respectively viscosity Green-Poisson measures. This done by convex duality between space continuous functions on compact set Borel it. part 1 our study systems, focuses linear coupling, while 2 will be concerned with nonlinear coupling.