[proofplan] We approximate the full index set $F$ by finite $d_P$-nets. On each finite net, the assumed finite-dimensional convergence gives [weak convergence](/page/Weak%20Convergence) of the corresponding projected empirical processes. Asymptotic equicontinuity makes the empirical process uniformly close to its finite-net projection, while the assumed uniform $d_P$-continuity of $G_P$ gives the analogous approximation for the Gaussian limit. Testing against bounded Lipschitz functions on $\ell^\infty(F)$ and then sending first $n\to\infty$ and then the net mesh to $0$ transfers the finite-dimensional convergence to weak convergence in $\ell^\infty(F)$. [/proofplan]

[step:Choose finite nets and define projection operators]Fix $\delta>0$. Since $(F,d_P)$ is [totally bounded](/page/Totally%20Bounded), there exist $m_\delta\in\mathbb N$ and functions $f_{\delta,1},\dots,f_{\delta,m_\delta}\in F$ such that \begin{align*} F\subset \bigcup_{j=1}^{m_\delta}\{f\in F:d_P(f,f_{\delta,j})<\delta\}. \end{align*} Define a map $\pi_\delta:F\to \{f_{\delta,1},\dots,f_{\delta,m_\delta}\}$ by choosing, for each $f\in F$, the least index $j$ such that $d_P(f,f_{\delta,j})<\delta$, and setting $\pi_\delta(f):=f_{\delta,j}$. Then \begin{align*} d_P(f,\pi_\delta(f))<\delta \end{align*} for every $f\in F$. Define the finite-coordinate restriction map \begin{align*} R_\delta:\ell^\infty(F)\to\mathbb R^{m_\delta}, \qquad z\mapsto (z(f_{\delta,1}),\dots,z(f_{\delta,m_\delta})). \end{align*} and define the extension map \begin{align*} E_\delta:\mathbb R^{m_\delta}\to\ell^\infty(F), \qquad a\mapsto \bigl(f\mapsto a_j\text{ where }\pi_\delta(f)=f_{\delta,j}\bigr). \end{align*} The map $E_\delta$ is continuous when $\mathbb R^{m_\delta}$ is equipped with the maximum norm and $\ell^\infty(F)$ is equipped with the supremum norm, because \begin{align*} \|E_\delta a-E_\delta b\|_{\ell^\infty(F)}\le \max_{1\le j\le m_\delta}|a_j-b_j| \end{align*} for all $a,b\in\mathbb R^{m_\delta}$. For each $n\in\mathbb N$, define the projected empirical process \begin{align*} G_{n,\delta}:=E_\delta R_\delta G_n. \end{align*} Define likewise \begin{align*} G_{P,\delta}:=E_\delta R_\delta G_P. \end{align*} Thus, for every $f\in F$, \begin{align*} G_{n,\delta}(f)=G_n(\pi_\delta(f)) \end{align*} and \begin{align*} G_{P,\delta}(f)=G_P(\pi_\delta(f)). \end{align*}[/step]

[guided]The role of [total boundedness](/page/Total%20Boundedness) is to reduce the potentially infinite index set $F$ to finitely many representatives. Fix $\delta>0$. Total boundedness of $(F,d_P)$ says precisely that finitely many $d_P$-balls of radius $\delta$ cover $F$. Hence we may choose $m_\delta\in\mathbb N$ and functions $f_{\delta,1},\dots,f_{\delta,m_\delta}\in F$ such that \begin{align*} F\subset \bigcup_{j=1}^{m_\delta}\{f\in F:d_P(f,f_{\delta,j})<\delta\}. \end{align*} For each $f\in F$, at least one of these centres lies within $d_P$-distance $\delta$ of $f$. To make a genuine function rather than a multivalued choice, define $\pi_\delta:F\to \{f_{\delta,1},\dots,f_{\delta,m_\delta}\}$ by choosing the least index $j$ such that $d_P(f,f_{\delta,j})<\delta$, and set $\pi_\delta(f):=f_{\delta,j}$. This finite tie-breaking rule gives a well-defined map and ensures \begin{align*} d_P(f,\pi_\delta(f))<\delta \end{align*} for every $f\in F$. Now we separate two operations. First, restrict a [bounded function](/page/Bounded%20Function) on $F$ to the finite net by defining \begin{align*} R_\delta:\ell^\infty(F)\to\mathbb R^{m_\delta}, \qquad z\mapsto (z(f_{\delta,1}),\dots,z(f_{\delta,m_\delta})). \end{align*} Second, extend a vector of net values back to all of $F$ by making it constant on the cells determined by $\pi_\delta$: \begin{align*} E_\delta:\mathbb R^{m_\delta}\to\ell^\infty(F), \qquad a\mapsto \bigl(f\mapsto a_j\text{ where }\pi_\delta(f)=f_{\delta,j}\bigr). \end{align*} This extension is continuous. Indeed, if $a,b\in\mathbb R^{m_\delta}$, then for each $f\in F$ the values $(E_\delta a)(f)$ and $(E_\delta b)(f)$ are two coordinates with the same index $j$, so \begin{align*} |(E_\delta a)(f)-(E_\delta b)(f)|\le \max_{1\le j\le m_\delta}|a_j-b_j|. \end{align*} Taking the supremum over $f\in F$ gives \begin{align*} \|E_\delta a-E_\delta b\|_{\ell^\infty(F)}\le \max_{1\le j\le m_\delta}|a_j-b_j|. \end{align*} We then define \begin{align*} G_{n,\delta}:=E_\delta R_\delta G_n \end{align*} and \begin{align*} G_{P,\delta}:=E_\delta R_\delta G_P. \end{align*} These are the finite-net approximations to $G_n$ and $G_P$. Pointwise on $F$, they satisfy \begin{align*} G_{n,\delta}(f)=G_n(\pi_\delta(f)) \end{align*} and \begin{align*} G_{P,\delta}(f)=G_P(\pi_\delta(f)). \end{align*}[/guided]

[step:Obtain weak convergence on each finite net] By the finite-dimensional convergence hypothesis applied to the finite set $\{f_{\delta,1},\dots,f_{\delta,m_\delta}\}$, \begin{align*} R_\delta G_n\xrightarrow{d}R_\delta G_P \end{align*} in $\mathbb R^{m_\delta}$. The vector $R_\delta G_n$ is an ordinary finite-dimensional random vector in the measurable case; in the outer-probability case this finite-dimensional convergence is part of the hypothesis and is interpreted in the usual Borel sense on $\mathbb R^{m_\delta}$. Since $E_\delta:\mathbb R^{m_\delta}\to\ell^\infty(F)$ is continuous, the [continuous mapping theorem](/theorems/1847), or equivalently the bounded-Lipschitz form of its outer-probability version, gives \begin{align*} G_{n,\delta}=E_\delta R_\delta G_n\xrightarrow{d}E_\delta R_\delta G_P=G_{P,\delta} \end{align*} in $\ell^\infty(F)$ for each fixed $\delta>0$. [/step]

[step:Control the empirical projection error by asymptotic equicontinuity] For each $n\in\mathbb N$ and $\delta>0$, \begin{align*} \|G_n-G_{n,\delta}\|_{\ell^\infty(F)} =\sup_{f\in F}|G_n(f)-G_n(\pi_\delta(f))|. \end{align*} Since $d_P(f,\pi_\delta(f))<\delta$ for every $f\in F$, \begin{align*} \|G_n-G_{n,\delta}\|_{\ell^\infty(F)} \le \sup_{\substack{f, g\in F, d_P(f, g)<\delta}}|G_n(f)-G_n(g)|. \end{align*} Therefore, for every $\eta>0$, \begin{align*} \lim_{\delta\downarrow0}\limsup_{n\to\infty}\mathbb P^*\left(\|G_n-G_{n,\delta}\|_{\ell^\infty(F)}>\eta\right)=0 \end{align*} by asymptotic equicontinuity. [/step]

[step:Control the Gaussian projection error by uniform continuity] Because $G_P$ has bounded uniformly $d_P$-continuous sample paths on the [probability space](/page/Probability%20Space) $(\Omega_P,\mathcal A_P,\mathbb P_P)$ from the theorem statement, for $\mathbb P_P$-almost every $\omega\in\Omega_P$ the map $G_P(\omega):F\to\mathbb R$, defined by $f\mapsto G_P(f)(\omega)$, is bounded and uniformly continuous with respect to $d_P$. Hence, for such $\omega$, \begin{align*} \|G_P(\omega)-G_{P,\delta}(\omega)\|_{\ell^\infty(F)} = \sup_{f\in F}|G_P(f)(\omega)-G_P(\pi_\delta(f))(\omega)| \end{align*} is bounded above by \begin{align*} \sup_{\substack{f, g\in F, d_P(f, g)<\delta}}|G_P(f)(\omega)-G_P(g)(\omega)|. \end{align*} The right-hand side tends to $0$ as $\delta\downarrow0$ by uniform $d_P$-continuity of the sample path. Thus \begin{align*} \|G_P-G_{P,\delta}\|_{\ell^\infty(F)}\xrightarrow{a.s.}0 \end{align*} as $\delta\downarrow0$. [/step]

[step:Transfer convergence from finite nets to $\ell^\infty(F)$] Let $\varphi:\ell^\infty(F)\to\mathbb R$ be a bounded [Lipschitz function](/page/Lipschitz%20Function). Define \begin{align*} \|\varphi\|_\infty:=\sup_{z\in\ell^\infty(F)}|\varphi(z)| \end{align*} and let $L_\varphi\ge0$ be a Lipschitz constant for $\varphi$ with respect to the supremum norm on $\ell^\infty(F)$. Fix $\varepsilon>0$. By the Gaussian projection estimate, choose $\delta>0$ so small that \begin{align*} \mathbb E\left[\min\{1,\|G_P-G_{P,\delta}\|_{\ell^\infty(F)}\}\right]<\varepsilon. \end{align*} By the empirical projection estimate, after possibly decreasing $\delta$, we also have \begin{align*} \limsup_{n\to\infty}\mathbb P^*\left(\|G_n-G_{n,\delta}\|_{\ell^\infty(F)}>\varepsilon\right)<\varepsilon. \end{align*} Using the Lipschitz property and boundedness of $\varphi$, we obtain \begin{align*} |\varphi(G_n)-\varphi(G_{n,\delta})| \le L_\varphi\|G_n-G_{n,\delta}\|_{\ell^\infty(F)} \end{align*} on the event $\{\|G_n-G_{n,\delta}\|_{\ell^\infty(F)}\le\varepsilon\}$, while the absolute difference is at most $2\|\varphi\|_\infty$ everywhere. Hence \begin{align*} \limsup_{n\to\infty}\left|\mathbb E[\varphi(G_n)]-\mathbb E[\varphi(G_{n,\delta})]\right| \le L_\varphi\varepsilon+2\|\varphi\|_\infty\varepsilon. \end{align*} Similarly, \begin{align*} \left|\mathbb E[\varphi(G_P)]-\mathbb E[\varphi(G_{P,\delta})]\right| \le L_\varphi\mathbb E\left[\min\{\|G_P-G_{P,\delta}\|_{\ell^\infty(F)},1\}\right]+2\|\varphi\|_\infty\mathbb P(\|G_P-G_{P,\delta}\|_{\ell^\infty(F)}>1), \end{align*} and the right-hand side tends to $0$ as $\delta\downarrow0$. For the fixed $\delta$ chosen above, the finite-net convergence gives \begin{align*} \lim_{n\to\infty}\mathbb E[\varphi(G_{n,\delta})] = \mathbb E[\varphi(G_{P,\delta})]. \end{align*} Combining the three estimates gives \begin{align*} \limsup_{n\to\infty}\left|\mathbb E[\varphi(G_n)]-\mathbb E[\varphi(G_P)]\right| \le L_\varphi\varepsilon+2\|\varphi\|_\infty\varepsilon+o_\delta(1), \end{align*} where $o_\delta(1)$ denotes a quantity that tends to $0$ as $\delta\downarrow0$. Since $\varepsilon>0$ was arbitrary, this proves \begin{align*} \lim_{n\to\infty}\mathbb E[\varphi(G_n)] = \mathbb E[\varphi(G_P)] \end{align*} for every bounded Lipschitz $\varphi:\ell^\infty(F)\to\mathbb R$. In the Borel measurable case, the bounded-Lipschitz characterization of weak convergence on metric spaces applies to the Borel laws of $G_n$ and $G_P$. Therefore \begin{align*} G_n\xrightarrow{d}G_P \end{align*} in $\ell^\infty(F)$. In the outer-probability case, replace every expectation involving $G_n$ in the preceding bounded-Lipschitz estimate by outer expectation $\mathbb E^*$ and every probability involving $G_n$ by outer probability $\mathbb P^*$. The projection estimate for $G_n$ was already proved with $\mathbb P^*$, and the finite-net convergence term is ordinary because it concerns the finite-dimensional vector $R_\delta G_n$ and the continuous finite-net image $E_\delta R_\delta G_n$. Asymptotic measurability ensures that the bounded Lipschitz functions $\varphi(G_n)$ have outer and inner expectations asymptotically equal, so the preceding estimate gives \begin{align*} \mathbb E^*\varphi(G_n)\to\mathbb E\varphi(G_P) \end{align*} for every bounded Lipschitz $\varphi:\ell^\infty(F)\to\mathbb R$. By the bounded-Lipschitz outer-expectation criterion stated in the theorem, this is precisely \begin{align*} G_n\xrightarrow{d}G_P \end{align*} in $\ell^\infty(F)$ under the standard empirical-process convention. This completes the proof. [/step]