Let $X_n$ denote a sequence of random variables. Here is Polya's theorem:
Suppose that $X_n \rightsquigarrow X$ (convergence in law) for a random vector $X$ with a continuous distribution function. Then $\sup_x | P(X_n \leq x) - P(X \leq x) | \rightarrow 0$.
The empirical CDF $F_n(x) = \frac{1}{n}\sum_{i=1}^n 1\{X_i \leq x\}$ is random. Is it true that Polya's theorem implies:
$$\sup_x | F_n(x) - P(X \leq x) | \stackrel{a.s.}{\rightarrow} 0$$
so long as $X$ has a continuous distribution function?
Actually if $F_n(x) \to F(x)$ on a dense set and $F$ is continuous then $F_n(x) \to F(x)$ uniformly. (This is an easy generalization of Polya's Theorem) . There is a null set $E$ such that $F_n (x,\omega) \to P(X\leq x)$ whenever $x$ is rational for $\omega \notin E$. Hence the conclusion is true.