Key Theorems

Gilvenko-Cantelli

The claim is that \(\underset{x}{\textrm{sup}}\ g\) (defined below!) converges "almost surely" to \(0\).

\[\begin{align*} &g :: \mathcal{X} \to \Omega \to \mathcal{R} \\ &g \ x \ \omega = F_n(x, \omega) - F(x) \\ \\ &\underset{x}{\textrm{sup}}\ g :: \Omega \to \mathcal{R} \end{align*}\]