Let $\mathcal{F}$ be a class of measurable functions $X\to\Bbb{R}$. We say that $\mathcal{F}$ is a uniform Glivenko-Cantelli class if: $$ \sup_{P\in \Delta X} \sup_{f\in \mathcal{F}} | P_n f - Pf | \to 0 $$ as $n\to\infty$, where $\Delta X$ is the set of all Borel probability measures on $X$, $P_n$ are the empirical probability measures, and $Pf$ means the expectation: $$ \int_X f\, dP\;. $$
It is important that the convergence happens uniformly in $P$. (Have I understood correctly?)
Now, can someone give me examples and counterexamples of such a class? Also a reference (containing examples) would be welcome. In particular, I would like examples where $X$ is not a subset of $\Bbb{R}^n$.