In the theory of probability, the Glivenko–Cantelli theorem (sometimes referred to as the Fundamental Theorem of Statistics), named after Valery Ivanovich Glivenko and Francesco Paolo Cantelli, determines the asymptotic behaviour of the empirical distribution function as the number of independent and identically distributed observations grows. The uniform convergence of more general empirical measures becomes an important property of the Glivenko–Cantelli classes of functions or sets. The Glivenko–Cantelli classes arise in Vapnik–Chervonenkis theory, with applications to machine learning. Applications can be found in econometrics making use of M-estimators. Statement Assume that are independent and identically distributed random variables in with common cumulative distribution function . The empirical distribution function for is defined by where is the indicator function of the set . For every (fixed) , is a sequence of random variables which converge to almost surely by the strong law of large numbers. Glivenko and Cantelli strengthened this result by proving uniform convergence of to . Theorem almost surely. This theorem originates with Valery Glivenko and Francesco Cantelli, in 1933. Remarks If is a stationary ergodic process, then converges almost surely to . The Glivenko–Cantelli theorem gives a stronger mode of convergence than this in the iid case. An even stronger uniform convergence result for the empirical distribution function is available in the form of an extended type of law of the iterated logarithm. See asymptotic properties of the empirical distribution function for this and related results. Proof For simplicity, consider a case of continuous random variable . Fix such that for . Now for all there exists such that . Note that Therefore, Since by strong law of large numbers, we can guarantee that for any positive and any integer such that , we can find such that for all , we have . Combined with the above result, this further implies that , which is the definition of almost sure convergence. Empirical measures One can generalize the empirical distribution function by replacing the set by an arbitrary set C from a class of sets to obtain an empirical measure indexed by sets Where is the indicator function of each set . Further generalization is the map induced by on measurable real-valued functions f, which is given by Then it becomes an important property of these classes whether the strong law of large numbers holds uniformly on or . Glivenko–Cantelli class Consider a set with a sigma algebra of