  We provide an approximation scheme for first-order stationary mean field
games with a separable Hamiltonian. First, we discretize Hamilton-Jacobi
equations by discretizing in time, and then prove the existence of minimizing
holonomic measures for mean field games. At last, we obtain two sequences of
solutions $\{u_i\}$ of discrete Hamilton-Jacobi equations and minimizing
holonomic measures $\{m_i\}$ for mean field games and show that $(u_i,m_i)$
converges to a solution of the stationary mean field games.
