Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 210 / 1 / b

The Glivenko-Cantelli theorem states that for the empirical distribution function $F_n$ of iid real observations with distribution function $F$,almost surely.

Apply part (a) to $\mathcal{G}=\{1_{(-\infty ,t]}:t\in \mathbb{R}\}$. For each $\epsilon >0$, choose finitely many quantile cutpoints so that the $P$-mass between consecutive cutpoints is at most $\epsilon$, treating atoms as cutpoints themselves. Indicators at adjacent cutpoints give finite $L^1(P)$ brackets of width at most $\epsilon$. Using the strong law for the finite bracket endpoints and then intersecting the probability-one events for $\epsilon =1/m$ gives the almost-sure conclusion.