Abstract
A class\(\widetilde{F}\) of measurable functions on a probability space is called a Glivenko-Cantelli class if the empirical measuresPn converge to the trueP uniformly over\(\widetilde{F}\) almost surely.\(\widetilde{F}\) is a universal Glivenko-Cantelli class if it is a Glivenko-Cantelli Cantelli class for all lawsP on a measurable space, and a uniform Glivenko-Cantelli class if the convergence is also uniform inP. We give general sufficient conditions for the Glivenko-Cantelli and universal Glivenko-Cantelli properties and examples to show that some stronger conditions are not necessary. The uniform Glivenko-Cantelli property is characterized, under measurability assumptions, by an entropy condition.
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