From https://en.wikipedia.org/wiki/Vapnik–Chervonenkis_theory

Let $X_1, ..., X_n$ be a random sample from a probability distribution P. Let $$P_nf=\frac{1}{n}\sum_{i=1}^nf(X_i), \quad Pf=\int fdP.$$ A class ${F}$ of measurable functions $f$ is called $P$-Glivenko-Cantelli if $$ \sup_{f \in {F}} |{P}_nf - Pf| \xrightarrow{as} 0. $$

The empirical process evaluated at $f$ is defined as $$ G_n f= \sqrt{n}({P}_nf - Pf). $$ This class $ {F} $ is called $ {P}$-Donsker if the sequence of processes $ \{ G_n f : f \in {F} \} $ converges in distribution to a tight limit process in the space $ l^\infty({F}) $.

A Donsker class is Glivenko–Cantelli in probability by an application of Slutsky's theorem.

How to formally prove the above claim?



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