Bracketing entropy implies empirical-process asymptotic equicontinuity

Choose a countable pointwise-dense subclass ${\mathcal{F}}_0\subseteq \mathcal{F}$. For $\delta >0$, set ${\mathcal{H}}_{\delta }=\{(f-g)-P(f-g):f,g\in {\mathcal{F}}_0,{\rho }_P(f,g)\le \delta \}$. Every $h\in {\mathcal{H}}_{\delta }$ has $\mathrm{\parallel }h{\mathrm{\parallel }}_{P,2}\le \delta$ and is bounded in absolute value by $H:=2F+2PF$, with $\mathrm{\parallel }H{\mathrm{\parallel }}_{P,2}\le 4\mathrm{\parallel }F{\mathrm{\parallel }}_{P,2}$.

If $f$ and $g$ lie in $L^2(P)$ brackets $[l_i,u_i]$ and $[l_j,u_j]$ of width at most $r\mathrm{\parallel }F{\mathrm{\parallel }}_{P,2}$, respectively, then $(f-g)-P(f-g)$ lies in the bracket $[l_i-u_j-P(u_i-l_j),u_i-l_j-P(l_i-u_j)]\mathrm{.}$ Its width is $w_i+w_j+P(w_i+w_j)$, where $w_i=u_i-l_i$ and $w_j=u_j-l_j$; Cauchy–Schwarz gives its $L^2(P)$ norm at most $2(\mathrm{\parallel }{w}_i{\mathrm{\parallel }}_{P,2}+\mathrm{\parallel }{w}_j{\mathrm{\parallel }}_{P,2})\le 4r\mathrm{\parallel }F{\mathrm{\parallel }}_{P,2}$. Thus $N_{[]}(4r\mathrm{\parallel }F{\mathrm{\parallel }}_{P,2},{\mathcal{H}}_{\delta },L^2(P))\le N_{[]}(r\mathrm{\parallel }F{\mathrm{\parallel }}_{P,2},\mathcal{F},L^2(P){)}^2$.

Fix $0<\delta <\mathrm{\parallel }H{\mathrm{\parallel }}_{P,2}\mathrm{/}2$ and choose $q_0\ge 1$ with $\delta \mathrm{/}2<2^{-q_0}\mathrm{\parallel }H{\mathrm{\parallel }}_{P,2}\le \delta$. For each $q\ge q_0$, take a finite bracketing cover of ${\mathcal{H}}_{\delta }$ at width $2^{-q}\mathrm{\parallel }H{\mathrm{\parallel }}_{P,2}$ and partition the class by assigning each function to the first bracket containing it. Refine these finite partitions cumulatively through level $q$; by [A1], the countable family of finite covers and their representatives can be selected. If $b_q$ is the unrefined level-$q$ count, step 1.2 gives $b_q\le N_{[]}(r_q\mathrm{\parallel }F{\mathrm{\parallel }}_{P,2},\mathcal{F},L^2(P){)}^2$ for $r_q=2^{-q}\mathrm{\parallel }H{\mathrm{\parallel }}_{P,2}\mathrm{/}(4\mathrm{\parallel }F{\mathrm{\parallel }}_{P,2})$, and the refined count satisfies $m_q\le {\prod }_{p=q_0}^q{b}_p$. Consequently, using $\sqrt{1+\sum x_p}\le 1+\sum \sqrt{x_p}$ for $x_p\ge 0$ and summing the resulting geometric weights, ${\sum }_{q\ge q_0}{2}^{-q}\sqrt{1+\log (2m_q)}\le C_1{\int }_0^{C_2\delta \mathrm{/}\mathrm{\parallel }F{\mathrm{\parallel }}_{P,2}}\sqrt{1+\log N_{[]}(ϵ\mathrm{\parallel }F{\mathrm{\parallel }}_{P,2},\mathcal{F},L^2(P))}dϵ$ for universal constants $C_1,C_2$: the unrefined entropy terms at scale $r_q$ are bounded by the integrand on the adjacent dyadic interval $[r_q\mathrm{/}2,r_q]$ by monotonicity of bracketing numbers, and cumulative refinement contributes only a geometric sum of earlier terms. The added $1$ contributes a geometric sum of order $2^{-q_0}$, which is also bounded by the displayed integral because its integrand is at least $1$. On each refined cell $C$, define ${\mathrm{\Delta }}_{q,C}={\sup }_{h,h^{\mathrm{\prime }}\in C}\mathrm{\mid }h-h^{\mathrm{\prime }}\mathrm{\mid }$ and choose ${\pi }_q(C)\in C$. Countability makes this pointwise supremum measurable; containment in a level-$q$ bracket gives ${\mathrm{\Delta }}_{q,C}\le u-l$ $P$-almost everywhere, so it has the required $L^2(P)$ bound. Intersecting partitions makes these oscillations nonincreasing on child cells. Thus these partitions and widths satisfy the hypotheses of [F2] when the local maximal inequality is applied.

Apply [F2] to ${\mathcal{H}}_{\delta }$ with envelope $H$ and this nested partition. Its coarse term is at most $C\delta \sqrt{1+\log (2m_{q_0})}$, its dyadic tail is bounded by the right-hand side of step 2.1, and its final envelope-tail term tends to zero as $n\to \mathrm{\infty }$ for this fixed $\delta$: for each fixed $a>0$, $\sqrt{n}P(H{\mathbf{1}}_{\{H>a\sqrt{n}\}})\le a^{-1}P(H^2{\mathbf{1}}_{\{H>a\sqrt{n}\}})\to 0$ by [F4].

The entropy integral in the statement makes the integral in step 2.1 tend to zero with $\delta$. It also gives $\delta \sqrt{\log N_{[]}(c\delta \mathrm{\parallel }F{\mathrm{\parallel }}_{P,2},\mathcal{F},L^2(P))}\to 0$ for each fixed $c>0$: if $h(ϵ)=\sqrt{\log N_{[]}(ϵ\mathrm{\parallel }F{\mathrm{\parallel }}_{P,2},\mathcal{F},L^2(P))}$, monotonicity yields $(ϵ\mathrm{/}2)h(ϵ)\le {\int }_{ϵ\mathrm{/}2}^{ϵ}h(u)du\to 0$. Therefore step 3.1 implies ${\lim }_{\delta \downarrow 0}{\mathrm{lim\hspace{0.17em}sup}}_{n}E{\sup }_{h\in {\mathcal{H}}_{\delta }}\mathrm{\mid }{\mathbb{G}}_n(h)\mathrm{\mid }=0$.

By [F5], Markov's inequality converts the expected-supremum limit in step 4.1 into asymptotic equicontinuity for pairs in ${\mathcal{F}}_0$. For arbitrary $f,g\in \mathcal{F}$, choose pointwise approximants $f_k,g_k\in {\mathcal{F}}_0$. The common envelope gives $L^1(P)$ convergence by [F4]; applying [F4] to $\mathrm{\mid }{f}_k-f{\mathrm{\mid }}^2\le 4F^2$ and $\mathrm{\mid }{g}_k-g{\mathrm{\mid }}^2\le 4F^2$ also gives $L^2(P)$ convergence. Pointwise convergence gives $P_n{f}_k\to P_{n}f$ and $P_n{g}_k\to P_{n}g$ for each fixed finite sample, so ${\rho }_P(f_k,g_k)\to {\rho }_P(f,g)$ and the empirical-process increment converges pointwise to the desired increment. Hence a pair with ${\rho }_P(f,g)<\delta \mathrm{/}2$ is a limit of pairs from ${\mathcal{F}}_0$ with distance below $\delta$. The outer-probability modulus over $\mathcal{F}$ is therefore bounded by the countable-class modulus over ${\mathcal{F}}_0$ at radius $\delta$, proving the asserted asymptotic equicontinuity.