Borel subsets A and a probability measure P. For a class of subsets, and a class of functions define random variables where is the empirical measure, is the corresponding map, and , assuming that it exists. Definitions A class is called a Glivenko–Cantelli class (or GC class) with respect to a probability measure P if any of the following equivalent statements is true. 1. almost surely as . 2. in probability as . 3. , as (convergence in mean). The Glivenko–Cantelli classes of functions are defined similarly. A class is called a universal Glivenko–Cantelli class if it is a GC class with respect to any probability measure P on (S,A). A class is called uniformly Glivenko–Cantelli if the convergence occurs uniformly over all probability measures P on (S,A): Theorem (Vapnik and Chervonenkis, 1968) A class of sets is uniformly GC if and only if it is a Vapnik–Chervonenkis class. Examples Let and . The classical Glivenko–Cantelli theorem implies that this class is a universal GC class. Furthermore, by Kolmogorov's theorem, , that is is uniformly Glivenko–Cantelli class. Let P be a nonatomic probability measure on S and be a class of all finite subsets in S. Because , , , we have that and so is not a GC class with respect to P. See also Donsker's theorem Dvoretzky–Kiefer–Wolfowitz inequality – strengthens the Glivenko–Cantelli theorem by quantifying the rate of convergence. References Further reading Empirical process Asymptotic theory (statistics) Probability theorems Theorems in statistics