Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space, $(\mathcal{X},\mathcal{F}_{\mathcal{X}})$ be a measurable space and $X,X_1,...,X_n,...$ be a sequence of $\mathbb{P}$-i.i.d. random variables from $(\Omega,\mathcal{F})$ into $(\mathcal{X},\mathcal{F}_{\mathcal{X}})$.
If $A\in\mathcal{F}_{\mathcal{X}}$ then we know from the law of large numbers that $$\exists F\in\mathcal{F}, \left(\mathbb{P}(F)=0\right)\land\left(\forall\omega\in F^c, \mathbb{P}(X\in A)=\lim_{n\to+\infty}\frac{1}{n}\sum_{k=1}^n\chi_A(X_k(\omega))\right).$$
Then, by Egoroff theorem, we also know that $$\forall\varepsilon>0, \exists F\in\mathcal{F}, \left(\mathbb{P}(F)<\varepsilon\right)\land\left(\lim_{n\to+\infty}\left\|\mathbb{P}(X\in A)-\frac{1}{n}\sum_{k=1}^n\chi_A(X_k)\right\|_{L^\infty(F^c)}=0\right).$$ So $$\forall A\in\mathcal{F}_{\mathcal{X}}, \forall\varepsilon>0, \exists F\in\mathcal{F},\\ \left(\mathbb{P}(F)<\varepsilon\right)\land\left(\lim_{n\to+\infty}\left\|\mathbb{P}(X\in A)-\frac{1}{n}\sum_{k=1}^n\chi_A(X_k)\right\|_{L^\infty(F^c)}=0\right).$$
I'm wondering if this result holds uniformly in $A$, i.e.
is it true that $$\forall\varepsilon>0, \exists F\in\mathcal{F},\\ \left(\mathbb{P}(F)<\varepsilon\right)\land\left(\forall A\in\mathcal{F}_{\mathcal{X}},\lim_{n\to+\infty}\left\|\mathbb{P}(X\in A)-\frac{1}{n}\sum_{k=1}^n\chi_A(X_k)\right\|_{L^\infty(F^c)}=0\right)?$$
This question fits the framework of empirical processes. So let $\mathcal{G}\subset\mathcal{F}_{\mathcal{X}}$ and let $\mathbb{P}_n(\,\cdot\,):=n^{-1}\sum_{i=1}^n 1\{X_i\in \cdot\,\}$ denote the empirical measure. Then $$ \left(\sup_{G\in\mathcal{G}}|\mathbb{P}_n(G)-\mathbb{P}(G)|\right)^*\to 0 \quad\text{a.s.} $$ provided that $\mathcal{G}$ is a VC class of sets (VC stands for Vapnik–Chervonenkis), where $(A)^{*}$ denotes the minimal measurable majorant of $A$ (note that $(A)^*=A$ if $A$ is measurable). Now you can apply Egorov's theorem to conclude that there exists $F\in\mathcal{F}$ with $\mathbb{P}(F)<\epsilon$ s.t. the convergence of $|\mathbb{P}_n(G)-\mathbb{P}(G)|$ is uniform on $F^c$ for any $G\in\mathcal{G}$.
Examples of VC classes include semi-infinite intervals and closed intervals in $\mathbb{R}$, i.e. $\{(-\infty,t]:t\in \mathbb{R}\}$ and $\{[s,t]:s,t\in \mathbb{R},s<t\}$. In the first case one gets the Glivenko-Cantelli theorem.