We incorporate into the empirical measure the auxiliary information given by a finite collection of expectation in an optimal information geometry way. This allows to unify several methods exploiting a side information and to uniquely define an informed empirical measure. These methods are shown to share the same asymptotic properties. Then we study the informed empirical process subject to a true information. We establish the Glivenko-Cantelli and Donsker theorems for the informed empirical measure under minimal assumptions and we quantify the asymptotic uniform variance reduction. Moreover, we prove that the informed empirical process is more concentrated than the classical empirical process for all large $n$. Finally, as an illustration of the variance reduction, we apply some of these results to the informed empirical quantiles.