We prove a Donsker and a Glivenko--Cantelli theorem for sequences of random discrete measures generalizing empirical measures. Those two results hold under standard conditions upon bracketing numbers of the indexing class of functions. As a byproduct, we derive a posterior consistency and a Bernstein--von Mises theorem for the Dirichlet process prior, under the topology of total variation, when the observation space is countable. We also obtain new information about the Durst--Dudley--Borisov theorem