[proofplan] We approximate the [totally bounded](/page/Totally%20Bounded) class $\mathcal F$ by finitely many representatives chosen from $\mathcal F$ itself. The finite representative class is controlled by the finite-class uniform law, while the difference between an arbitrary function and its representative is controlled by stochastic equicontinuity in the $L^1(P)$ semimetric. Combining these two estimates gives an outer-probability bound whose finite part vanishes as $n\to\infty$ and whose local part is made arbitrarily small by choosing the net radius sufficiently small. [/proofplan]

[step:Choose finitely many representatives from an $L^1(P)$ net]Fix $\delta>0$. Since $(\mathcal F,d_1)$ is totally bounded, there exist an integer $N\in\mathbb N$ and functions $h_1,\dots,h_N\in\mathcal F$ such that \begin{align*} \mathcal F\subset \bigcup_{k=1}^{N}\{f\in\mathcal F:d_1(f,h_k)<\delta/2\}. \end{align*} Thus the finite set $\mathcal F_\delta:=\{h_1,\dots,h_N\}\subset\mathcal F$ has the property that for every $f\in\mathcal F$ there is $h\in\mathcal F_\delta$ with \begin{align*} d_1(f,h)<\delta/2<\delta. \end{align*} Because each $h\in\mathcal F_\delta$ belongs to $\mathcal F$, the hypothesis $P|h|<\infty$ holds for every representative.[/step]

[guided]The role of [total boundedness](/page/Total%20Boundedness) is to replace the possibly infinite class $\mathcal F$ by a finite list of nearby functions. Fix $\delta>0$. Total boundedness of $(\mathcal F,d_1)$ says that finitely many $d_1$-balls of radius $\delta/2$ cover $\mathcal F$. Therefore there are functions $h_1,\dots,h_N\in\mathcal F$, for some $N\in\mathbb N$, such that \begin{align*} \mathcal F\subset \bigcup_{k=1}^{N}\{f\in\mathcal F:d_1(f,h_k)<\delta/2\}. \end{align*} We define the finite representative class by \begin{align*} \mathcal F_\delta:=\{h_1,\dots,h_N\}. \end{align*} Then for every $f\in\mathcal F$ there is a representative $h\in\mathcal F_\delta$ satisfying \begin{align*} d_1(f,h)<\delta/2<\delta. \end{align*} It matters that the representatives are chosen from $\mathcal F$, not merely from an ambient completion, because we will apply both the finite-class law and stochastic equicontinuity to pairs of functions belonging to $\mathcal F$. Since $\mathcal F_\delta\subset\mathcal F$, each representative is measurable and $P$-integrable by the hypotheses on $\mathcal F$.[/guided]

[step:Decompose the empirical deviation into a finite part and a local increment] For each $f\in\mathcal F$, choose $h_f\in\mathcal F_\delta$ with $d_1(f,h_f)<\delta$. Then \begin{align*} (P_n-P)f=(P_n-P)h_f+(P_n-P)(f-h_f). \end{align*} Taking absolute values and using the triangle inequality gives \begin{align*} |(P_n-P)f|\le |(P_n-P)h_f|+|(P_n-P)(f-h_f)|. \end{align*} Taking the supremum over $f\in\mathcal F$ yields the pointwise bound \begin{align*} \sup_{f\in\mathcal F}|(P_n-P)f|\le \max_{h\in\mathcal F_\delta}|(P_n-P)h|+\sup_{\substack{f, g\in\mathcal F, d_1(f, g)<\delta}}|(P_n-P)(f-g)|. \end{align*} [/step]

[step:Control the finite representative class by the finite-class uniform law] The class $\mathcal F_\delta$ is finite, consists of [measurable functions](/page/Measurable%20Functions), and every $h\in\mathcal F_\delta$ satisfies $P|h|<\infty$. Therefore [citetheorem:9816] applied to $\mathcal F_\delta$ gives \begin{align*} \max_{h\in\mathcal F_\delta}|P_nh-Ph|\xrightarrow{\mathbb P}0. \end{align*} Consequently, for every $\eta>0$, \begin{align*} \lim_{n\to\infty}\mathbb P\left(\max_{h\in\mathcal F_\delta}|(P_n-P)h|>\eta/2\right)=0. \end{align*} [/step]

[step:Use stochastic equicontinuity for the local increment] Fix $\eta>0$. From the bound in the previous step and outer-probability subadditivity, for every $\delta>0$, \begin{align*} \mathbb P^*\left(\sup_{f\in\mathcal F}|(P_n-P)f|>\eta\right)\le \mathbb P\left(\max_{h\in\mathcal F_\delta}|(P_n-P)h|>\eta/2\right)+\mathbb P^*\left(\sup_{\substack{f, g\in\mathcal F, d_1(f, g)<\delta}}|(P_n-P)(f-g)|>\eta/2\right). \end{align*} Taking $\limsup_{n\to\infty}$ and using the finite-class convergence gives \begin{align*} \limsup_{n\to\infty}\mathbb P^*\left(\sup_{f\in\mathcal F}|(P_n-P)f|>\eta\right)\le \limsup_{n\to\infty}\mathbb P^*\left(\sup_{\substack{f, g\in\mathcal F, d_1(f, g)<\delta}}|(P_n-P)(f-g)|>\eta/2\right). \end{align*} Now let $\delta\downarrow0$. The stochastic equicontinuity hypothesis makes the right-hand side tend to $0$. Hence \begin{align*} \limsup_{n\to\infty}\mathbb P^*\left(\sup_{f\in\mathcal F}|P_nf-Pf|>\eta\right)=0. \end{align*} Since $\eta>0$ was arbitrary, this is exactly \begin{align*} \sup_{f\in\mathcal F}|P_nf-Pf|\xrightarrow{\mathbb P^*}0. \end{align*} Thus $\mathcal F$ is $P$-Glivenko-Cantelli in outer probability. [/step]