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Analysis demonstrates that the Glivenko-Cantelli theorem fails in infinite dimensions, indicating complexities in probability theory.
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.
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В. В. Сазонов (1963) studied this question.
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