[proofplan] The argument is a direct application of the almost-sure-continuity form of the [Continuous Mapping Theorem](/theorems/1847) in the [metric space](/page/Metric%20Space) $\ell^\infty(F)$. First we record that $\ell^\infty(F)$, with its sup norm and Borel $\sigma$-algebra, is an admissible metric-state space and that the compositions with $\Phi$ are measurable real-valued random variables. The null-discontinuity hypothesis is exactly the hypothesis required by the continuous mapping theorem, so [weak convergence](/page/Weak%20Convergence) of the full empirical-process random elements transfers to weak convergence of the real-valued statistics. [/proofplan]

[step:Verify that the statistic is a measurable real-valued random variable] Let $\mathcal B(\ell^\infty(F))$ denote the Borel $\sigma$-algebra generated by the norm topology on $\ell^\infty(F)$, and let $\mathcal B(\mathbb R)$ denote the Borel $\sigma$-algebra on $\mathbb R$. By hypothesis, for every $n\in\mathbb N$, \begin{align*} G_n:(\Omega,\mathcal A)\to(\ell^\infty(F),\mathcal B(\ell^\infty(F))) \end{align*} is measurable, and \begin{align*} G_P:(\Omega,\mathcal A)\to(\ell^\infty(F),\mathcal B(\ell^\infty(F))) \end{align*} is measurable. Also by hypothesis, \begin{align*} \Phi:(\ell^\infty(F),\mathcal B(\ell^\infty(F)))\to(\mathbb R,\mathcal B(\mathbb R)) \end{align*} is measurable. Therefore the compositions \begin{align*} \Phi\circ G_n:(\Omega,\mathcal A)\to(\mathbb R,\mathcal B(\mathbb R)) \end{align*} and \begin{align*} \Phi\circ G_P:(\Omega,\mathcal A)\to(\mathbb R,\mathcal B(\mathbb R)) \end{align*} are measurable. Hence $\Phi(G_n)$ and $\Phi(G_P)$ are real-valued random variables. [/step]

[step:Apply the continuous mapping theorem at the limit-continuity points]The space $\ell^\infty(F)$ is a metric space under the metric \begin{align*} d_\infty(z,w):=\|z-w\|_\infty \end{align*} for $z,w\in\ell^\infty(F)$. The assumed convergence \begin{align*} G_n\xrightarrow{d}G_P \end{align*} is weak convergence of the laws of $G_n$ to the law of $G_P$ on the Borel $\sigma$-algebra of this metric space. We use the Continuous Mapping Theorem for maps continuous almost surely at the limit: if $X_n$ and $X$ are random elements in a metric space $E$, $X_n\xrightarrow{d}X$, and $\varphi:E\to\mathbb R$ is Borel measurable with discontinuity set $D_\varphi$ satisfying $\mathbb P(X\in D_\varphi)=0$, then $\varphi(X_n)\xrightarrow{d}\varphi(X)$ in $\mathbb R$. Apply this theorem with \begin{align*} E:=\ell^\infty(F), \end{align*} with $X_n:=G_n$, with $X:=G_P$, and with $\varphi:=\Phi$. The discontinuity set $D_\varphi$ is exactly $D_\Phi$, and the hypothesis gives \begin{align*} \mathbb P(G_P\in D_\Phi)=0. \end{align*} All hypotheses of the theorem are therefore satisfied. Consequently, \begin{align*} \Phi(G_n)\xrightarrow{d}\Phi(G_P) \end{align*} in $\mathbb R$.[/step]

[guided]We want to pass from convergence of the entire random element $G_n\in\ell^\infty(F)$ to convergence of the numerical statistic $\Phi(G_n)\in\mathbb R$. The tool designed for exactly this passage is the Continuous Mapping Theorem, in its version allowing the map to be discontinuous on a set that the limiting random element avoids almost surely. First, the state space is a metric space. Define \begin{align*} d_\infty:\ell^\infty(F)\times\ell^\infty(F)\to[0,\infty) \end{align*} by \begin{align*} d_\infty(z,w):=\|z-w\|_\infty \end{align*} for $z,w\in\ell^\infty(F)$. This is the metric induced by the Banach-space norm on $\ell^\infty(F)$. Thus the assumption \begin{align*} G_n\xrightarrow{d}G_P \end{align*} means weak convergence of probability laws on the Borel $\sigma$-algebra generated by this metric. Now state the exact external principle being used. The Continuous Mapping Theorem for maps continuous almost surely at the limit says: if $X_n$ and $X$ are random elements in a metric space $E$, if $X_n\xrightarrow{d}X$, and if $\varphi:E\to\mathbb R$ is Borel measurable with discontinuity set \begin{align*} D_\varphi:=\{y\in E:\varphi\text{ is not continuous at }y\} \end{align*} satisfying \begin{align*} \mathbb P(X\in D_\varphi)=0, \end{align*} then \begin{align*} \varphi(X_n)\xrightarrow{d}\varphi(X) \end{align*} as real-valued random variables. We verify the hypotheses one by one in the present setting. Take \begin{align*} E:=\ell^\infty(F). \end{align*} Take \begin{align*} X_n:=G_n \end{align*} for each $n\in\mathbb N$, and take \begin{align*} X:=G_P. \end{align*} The convergence hypothesis required by the theorem is exactly the assumed convergence \begin{align*} G_n\xrightarrow{d}G_P. \end{align*} Take \begin{align*} \varphi:=\Phi. \end{align*} The required measurability of $\varphi$ is exactly the assumed Borel measurability of $\Phi$. Finally, the discontinuity set of $\varphi$ is \begin{align*} D_\varphi=D_\Phi, \end{align*} because $\varphi$ and $\Phi$ are the same map. The required null-discontinuity condition is therefore precisely \begin{align*} \mathbb P(G_P\in D_\Phi)=0, \end{align*} which is assumed. Thus every hypothesis of the Continuous Mapping Theorem is satisfied. Its conclusion gives \begin{align*} \Phi(G_n)\xrightarrow{d}\Phi(G_P) \end{align*} in $\mathbb R$. This is exactly the desired convergence of the empirical-process statistic.[/guided]

[step:Conclude the asserted convergence of the empirical-process statistics] The first step proves that $\Phi(G_n)$ and $\Phi(G_P)$ are real-valued random variables. The second step proves that their laws converge weakly on $\mathbb R$. Hence \begin{align*} \Phi(G_n)\xrightarrow{d}\Phi(G_P), \end{align*} which is the desired conclusion. [/step]