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It is found that the Glivenko-Cantelli theorem does not admit a generalisation to the infinite dimensional case. In fact under general conditions, with probability one the empirical distributions do not converge to the theoretical distribution uniformly on all half-spaces. Sufficient conditions for the validity of uniform convergence over a family of half-spaces with probability one are given.
В. В. Сазонов (Tue,) studied this question.
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