[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 $\mathbb R$.
[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]
[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]
