Glivenko–Cantelli classes

Use the empirical-measure convention of Empirical, centered, and Rademacher processes: for iid observations $X_1,X_2,\dots$ with common law $P$, $P_{n}f=n^{-1}{\sum }_{i=1}^{n}f(X_i)$ and $Pf=\int fdP$. Let $\mathcal{F}$ be a nonempty class of measurable real-valued functions with a nonnegative measurable integrable envelope $F$, meaning $\mathrm{\mid }f\mathrm{\mid }\le F$ $P$-almost surely for every $f\in \mathcal{F}$ and $PF<\mathrm{\infty }$. It is pointwise measurable if there is a countable subclass ${\mathcal{F}}_0$ such that every $f\in \mathcal{F}$ is the pointwise limit of a sequence from ${\mathcal{F}}_0$. For a pointwise-measurable class, dominated convergence Dominated convergence gives $P{f}_k\to Pf$ along the approximating sequence, and finite empirical averages converge pointwise, so the supremum defining the uniform law is determined by ${\mathcal{F}}_0$. The class is $P$-Glivenko–Cantelli if

${\sup }_{f\in \mathcal{F}}\mathrm{\mid }{P}_{n}f-Pf\mathrm{\mid }⟶0$

in probability. It is strongly $P$-Glivenko–Cantelli if the same convergence holds almost surely. For an uncountable class with a countable pointwise-dense subclass, the supremum is interpreted using outer probability unless the chosen version makes it measurable. For a pointwise-measurable class the supremum is determined by the countable subclass and is measurable. That subclass has a common conull envelope bound, so the supremum is finite almost surely; after choosing a real-valued version, almost-sure convergence implies convergence in probability by Almost-sure convergence implies convergence in probability.