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