The uniform threshold class is Glivenko–Cantelli

Given: Countable Choice, the iid Uniform$[0,1]$ experiment, the threshold class $\mathcal{F}$, and the bracketing Glivenko–Cantelli theorem.

$P([0,t])=t$ for $t\in [0,1]$, and the envelope $F(x)=1$ is integrable.

Countable Choice supplies the theorem's strong-law branch and, for each fixed $t<1$, a sequence of rational approximants chosen from the intervals $(t,\min (1,t+1\mathrm{/}k))$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)).

Glivenko–Cantelli classes: uniform convergence of $P_{n}f$ to $Pf$ over the class defines the Glivenko–Cantelli property.

Covering, packing, bracketing numbers, and metric entropy: an $L^1(P)$ bracket has measurable endpoints, contains functions lying between them $P$-almost everywhere, and has width $P(u-l)$.

Empirical, centered, and Rademacher processes: for iid observations, $P_n{f}_t=n^{-1}{\sum }_{i=1}^n{f}_t(X_i)$ and $P{f}_t=\int f_{t}dP$.

Glivenko–Cantelli theorem by bracketing entropy: a pointwise-measurable class with an integrable envelope and finite $L^1(P)$ bracketing number at every positive scale is strongly $P$-Glivenko–Cantelli when Countable Choice is assumed.

$\mathbb{Q}$ is countably infinite: $\mathbb{Q}$ is countable.

Every subset of an at most countable set is at most countable: every subset of an at most countable set is at most countable.

The rationals embed densely in the reals: between any two distinct real numbers there is a rational.