The Monge-Amp\`ere operator and geodesics in the space of K\"ahler potentials

It is shown that geodesics in the space of K"ahler potentials can be uniformly approximated by geodesics in the spaces of Bergman metrics. Two important tools in the proof are the Tian-Yau-Zelditch approximation theorem for K"ahler potentials and the pluripotential theory of Bedford-Taylor, suitably adapted to K"ahler manifolds.