Consider the following estimator of the distribution function $F$. Let a sequence of a random variables $X_i$ be i.i.d with distribution function $F$ and our estimator $\hat{F}_n(X) = \frac{1}{n} \sum_{i=1}^{n} I(X_i \leq x)$ and $I$ is an indicator function.
Now the strong law of large numbers states for a fixed $x$ that $\hat{F}_n(X) \rightarrow_{as} F(X)$.
The Glivenko Gantelli theorem states that $\sup_{x \in \mathbb{R}} |\hat{F}_n(x) - F(x)| \rightarrow_{as} 0$.
What I'm failing to understand is why the Glivenko-Cantelli theorem doesn't follow from the strong law of large numbers? If we show convergence for each individual $x$ wouldn't that imply convergence of the supremum? Can someone provide a counter example if this is not the case?
The law of large numbers says that for each $x$ you have a null set $N_x$ such that for all $\omega \in N^c$ $\hat F_n(x)(\omega) \to F(x)$.
Now for the convergence of the supremum you'd have to look at the union of all $N_x$ for $x\in\mathbb R$. This could be a non-null set (as the union is over an uncountable index set), thus the Glivenko-Cantelli theorem is a strict improvement over the law of large number!