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\begin{document}$${\mathcal{F}}$$\end{document} be a separable uniformly bounded family of measurable functions on a standard measurable space \documentclass[12pt]{minimal}
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\begin{document}$${(X, \mathcal{X})}$$\end{document}, and let \documentclass[12pt]{minimal}
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\begin{document}$${N_{[]}(\mathcal{F}, \varepsilon, \mu)}$$\end{document} be the smallest number of \documentclass[12pt]{minimal}
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\begin{document}$${\varepsilon}$$\end{document} -brackets in L1(μ) needed to cover \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{F}}$$\end{document}. The following are equivalent: \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{F}}$$\end{document} is a universal Glivenko–Cantelli class.\documentclass[12pt]{minimal}
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\begin{document}$${N_{[]}(\mathcal{F},\varepsilon,\mu) < \infty}$$\end{document} for every \documentclass[12pt]{minimal}
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\begin{document}$${\varepsilon > 0}$$\end{document} and every probability measure μ.\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{F}}$$\end{document} is totally bounded in L1(μ) for every probability measure μ.\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{F}}$$\end{document} does not contain a Boolean σ-independent sequence. It follows that universal Glivenko–Cantelli classes are uniformity classes for general sequences of almost surely convergent random measures.
