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We establish a convergence theorem for the vanishing discount problem for aweakly coupled system of Hamilton-Jacobi equations. The crucial step is theintroduction of Mather measures and their relatives for the system, which wecall respectively viscosity Mather and Green-Poisson measures. This is done bythe convex duality and the duality between the space of continuous functions ona compact set and the space of Borel measures on it. This is part 1 of ourstudy of the vanishing discount problem for systems, which focuses on thelinear coupling, while part 2 will be concerned with nonlinear coupling.

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