Key Theorems
Gilvenko-Cantelli
The claim is that \(\underset{x}{\textrm{sup}}\ g\) (defined below!) converges "almost surely" to \(0\).
\[\begin{align*}
&g :: \mathcal{X} \to \Omega \to \mathcal{R} \\
&g \ x \ \omega = F_n(x, \omega) - F(x) \\ \\
&\underset{x}{\textrm{sup}}\ g :: \Omega \to \mathcal{R}
\end{align*}\]