Let X1,...,Xn be i.i.d. P on S. Suppose S is countable and let E be the collection of all subsets of S. Let Pˆn be the empirical measure; that is, for any subset E of E, Pˆn(E) is the proportion of observations Xi that fall in E. Prove, with probability one, sup E∈E

|Pˆn(E) − P(E)| → 0 .