[proofplan] We prove conditional [weak convergence](/page/Weak%20Convergence) by testing against an arbitrary bounded Lipschitz functional on $\ell^\infty(\mathcal F)$. [Total boundedness](/page/Total%20Boundedness) lets us replace the whole index class by a finite $d_P$-net. Conditional asymptotic equicontinuity controls the error of this finite-net replacement for $Z_n^*$, while the almost sure uniform $d_P$-continuity of $Z$ controls the corresponding error for the limit process. The finite-net functional depends on finitely many coordinates, so the assumed conditional finite-dimensional convergence applies; then the approximation errors are sent to zero. [/proofplan]

[step:Fix a bounded Lipschitz test functional and reduce the goal to it] Let $\Phi:\ell^\infty(\mathcal F)\to\mathbb R$ be bounded and Lipschitz. We use $\mathbb E^*[\cdot\mid\mathcal G_n]$ for the conditional outer expectation given $\mathcal G_n$, as declared in the theorem statement. Since multiplying by a positive constant does not affect convergence to $0$, it is enough to prove the assertion under the normalization \begin{align*} |\Phi(z)|\le 1 \end{align*} for every $z\in\ell^\infty(\mathcal F)$ and \begin{align*} |\Phi(z)-\Phi(w)|\le \|z-w\|_{\mathcal F} \end{align*} for all $z,w\in\ell^\infty(\mathcal F)$, where \begin{align*} \|z-w\|_{\mathcal F}:=\sup_{f\in\mathcal F}|z(f)-w(f)|. \end{align*} We must prove that \begin{align*} \mathbb E^*[\Phi(Z_n^*)\mid\mathcal G_n]-\mathbb E[\Phi(Z)]\xrightarrow{\mathbb P}0. \end{align*} [/step]

[step:Approximate the index class by a finite $d_P$-net]Fix $\delta>0$. Since $(\mathcal F,d_P)$ is [totally bounded](/page/Totally%20Bounded), there exist $N_\delta\in\mathbb N$ and functions $f_{\delta,1},\dots,f_{\delta,N_\delta}\in\mathcal F$ such that \begin{align*} \mathcal F\subset \bigcup_{j=1}^{N_\delta}\{f\in\mathcal F:d_P(f,f_{\delta,j})<\delta\}. \end{align*} Choose one such finite net. Define a map \begin{align*} \pi_\delta:\mathcal F\to\{f_{\delta,1},\dots,f_{\delta,N_\delta}\} \end{align*} by choosing, for each $f\in\mathcal F$, one point $\pi_\delta(f)$ in the net satisfying \begin{align*} d_P(f,\pi_\delta(f))<\delta. \end{align*} For $z\in\ell^\infty(\mathcal F)$, define the finite-net projection \begin{align*} R_\delta z:\mathcal F\to\mathbb R \end{align*} by \begin{align*} (R_\delta z)(f):=z(\pi_\delta(f)). \end{align*} Then $R_\delta z\in\ell^\infty(\mathcal F)$ and \begin{align*} \|z-R_\delta z\|_{\mathcal F}\le \sup_{\substack{f, g\in\mathcal F, d_P(f, g)<\delta}}|z(f)-z(g)|. \end{align*}[/step]

[guided]The reason for introducing $R_\delta$ is that weak convergence in $\ell^\infty(\mathcal F)$ is infinite-dimensional, while the hypothesis only gives finite-dimensional convergence. Total boundedness supplies the bridge: every $f\in\mathcal F$ is within $d_P$-distance $\delta$ of one of finitely many points. Formally, because $(\mathcal F,d_P)$ is totally bounded, for the fixed number $\delta>0$ there are functions $f_{\delta,1},\dots,f_{\delta,N_\delta}\in\mathcal F$ such that every $f\in\mathcal F$ satisfies \begin{align*} d_P(f,f_{\delta,j})<\delta \end{align*} for at least one index $j\in\{1,\dots,N_\delta\}$. We choose one such index for each $f$ and denote the chosen net point by $\pi_\delta(f)$. This defines a map \begin{align*} \pi_\delta:\mathcal F\to\{f_{\delta,1},\dots,f_{\delta,N_\delta}\}. \end{align*} Given a [bounded function](/page/Bounded%20Function) $z:\mathcal F\to\mathbb R$, define \begin{align*} R_\delta z:\mathcal F\to\mathbb R \end{align*} by \begin{align*} (R_\delta z)(f):=z(\pi_\delta(f)). \end{align*} This replacement is constant on the cells of the finite net, so it depends only on the finitely many coordinates $z(f_{\delta,1}),\dots,z(f_{\delta,N_\delta})$. The approximation error is controlled by the local $d_P$-oscillation of $z$. Indeed, for every $f\in\mathcal F$ the chosen net point satisfies $d_P(f,\pi_\delta(f))<\delta$, hence \begin{align*} |z(f)-(R_\delta z)(f)|=|z(f)-z(\pi_\delta(f))|\le \sup_{\substack{u, v\in\mathcal F, d_P(u, v)<\delta}}|z(u)-z(v)|. \end{align*} Taking the supremum over $f\in\mathcal F$ gives \begin{align*} \|z-R_\delta z\|_{\mathcal F}\le \sup_{\substack{u, v\in\mathcal F, d_P(u, v)<\delta}}|z(u)-z(v)|. \end{align*} This is the exact point where asymptotic equicontinuity will enter.[/guided]

[step:Control the conditional finite-net approximation error for $Z_n^*$] For $\varepsilon>0$, define the [random variable](/page/Random%20Variable) \begin{align*} \omega_{n,\delta}^*:\Omega\to[0,\infty] \end{align*} by \begin{align*} \omega_{n,\delta}^*:=\sup_{\substack{f, g\in\mathcal F, d_P(f, g)<\delta}}|Z_n^*(f)-Z_n^*(g)|. \end{align*} By the Lipschitz property of $\Phi$ and the estimate from the preceding step, \begin{align*} |\Phi(Z_n^*)-\Phi(R_\delta Z_n^*)|\le \omega_{n,\delta}^*. \end{align*} Since $|\Phi|\le 1$, the left-hand side is bounded by $2$. For every $\varepsilon>0$, \begin{align*} \mathbb E^*[|\Phi(Z_n^*)-\Phi(R_\delta Z_n^*)|\mid\mathcal G_n]\le \varepsilon+2\,\mathbb P^*(\omega_{n,\delta}^*>\varepsilon\mid\mathcal G_n). \end{align*} The conditional asymptotic equicontinuity hypothesis therefore implies \begin{align*} \lim_{\delta\downarrow 0}\limsup_{n\to\infty}\mathbb P\left(\mathbb E^*[|\Phi(Z_n^*)-\Phi(R_\delta Z_n^*)|\mid\mathcal G_n]>2\varepsilon\right)=0. \end{align*} Consequently, the conditional finite-net replacement error vanishes in probability as $n\to\infty$ and then $\delta\downarrow 0$. [/step]

[step:Control the finite-net approximation error for the limiting process] Define \begin{align*} \omega_\delta:\Omega\to[0,\infty] \end{align*} by \begin{align*} \omega_\delta:=\sup_{\substack{f, g\in\mathcal F, d_P(f, g)<\delta}}|Z(f)-Z(g)|. \end{align*} Since the sample paths of $Z$ are uniformly continuous with respect to $d_P$ almost surely, \begin{align*} \omega_\delta\to 0 \end{align*} almost surely as $\delta\downarrow 0$. The Lipschitz property of $\Phi$ gives \begin{align*} |\Phi(Z)-\Phi(R_\delta Z)|\le \omega_\delta. \end{align*} Also $|\Phi(Z)-\Phi(R_\delta Z)|\le 2$. By bounded convergence, \begin{align*} \mathbb E[|\Phi(Z)-\Phi(R_\delta Z)|]\to 0 \end{align*} as $\delta\downarrow 0$. [/step]

[step:Apply conditional finite-dimensional convergence on the finite net] For the fixed finite net, define \begin{align*} T_\delta:\mathbb R^{N_\delta}\to\ell^\infty(\mathcal F) \end{align*} by \begin{align*} (T_\delta x)(f):=x_j \end{align*} whenever $\pi_\delta(f)=f_{\delta,j}$, where $x=(x_1,\dots,x_{N_\delta})\in\mathbb R^{N_\delta}$. Define \begin{align*} \varphi_\delta:\mathbb R^{N_\delta}\to\mathbb R \end{align*} by \begin{align*} \varphi_\delta(x):=\Phi(T_\delta x). \end{align*} The function $\varphi_\delta$ is bounded by $1$ and Lipschitz for the sup norm on $\mathbb R^{N_\delta}$ with Lipschitz constant at most $1$. Moreover, \begin{align*} \Phi(R_\delta Z_n^*)=\varphi_\delta(Z_n^*(f_{\delta,1}),\dots,Z_n^*(f_{\delta,N_\delta})) \end{align*} and \begin{align*} \Phi(R_\delta Z)=\varphi_\delta(Z(f_{\delta,1}),\dots,Z(f_{\delta,N_\delta})). \end{align*} Applying the assumed conditional finite-dimensional convergence to the finite list $f_{\delta,1},\dots,f_{\delta,N_\delta}$ and to the bounded [Lipschitz function](/page/Lipschitz%20Function) $\varphi_\delta$ yields \begin{align*} \mathbb E^*[\Phi(R_\delta Z_n^*)\mid\mathcal G_n]-\mathbb E[\Phi(R_\delta Z)]\xrightarrow{\mathbb P}0 \end{align*} for every fixed $\delta>0$. [/step]

[step:Combine the three errors and pass from finite nets to the full process] For every $n\in\mathbb N$ and every $\delta>0$, the triangle inequality gives \begin{align*} \left|\mathbb E^*[\Phi(Z_n^*)\mid\mathcal G_n]-\mathbb E[\Phi(Z)]\right| \end{align*} bounded above by the sum of \begin{align*} \mathbb E^*[|\Phi(Z_n^*)-\Phi(R_\delta Z_n^*)|\mid\mathcal G_n], \end{align*} \begin{align*} \left|\mathbb E^*[\Phi(R_\delta Z_n^*)\mid\mathcal G_n]-\mathbb E[\Phi(R_\delta Z)]\right|, \end{align*} and \begin{align*} \mathbb E[|\Phi(R_\delta Z)-\Phi(Z)|]. \end{align*} Let $\gamma>0$ and $\alpha>0$ be fixed. Choose $\varepsilon>0$ so that $3\varepsilon<\gamma$. By the preceding two approximation steps, choose $\delta>0$ so that \begin{align*} \mathbb E[|\Phi(R_\delta Z)-\Phi(Z)|]<\varepsilon \end{align*} and \begin{align*} \limsup_{n\to\infty}\mathbb P\left(\mathbb E^*[|\Phi(Z_n^*)-\Phi(R_\delta Z_n^*)|\mid\mathcal G_n]>\varepsilon\right)<\alpha/2. \end{align*} For this fixed $\delta$, finite-dimensional convergence gives \begin{align*} \mathbb P\left(\left|\mathbb E^*[\Phi(R_\delta Z_n^*)\mid\mathcal G_n]-\mathbb E[\Phi(R_\delta Z)]\right|>\varepsilon\right)\to 0. \end{align*} Hence, for all sufficiently large $n$, the probability that the middle term exceeds $\varepsilon$ is less than $\alpha/2$. On the event where both random error terms are at most $\varepsilon$, the displayed triangle bound is less than $3\varepsilon<\gamma$. Therefore \begin{align*} \limsup_{n\to\infty}\mathbb P\left(\left|\mathbb E^*[\Phi(Z_n^*)\mid\mathcal G_n]-\mathbb E[\Phi(Z)]\right|>\gamma\right)\le \alpha. \end{align*} Since $\alpha>0$ was arbitrary, this proves \begin{align*} \mathbb E^*[\Phi(Z_n^*)\mid\mathcal G_n]-\mathbb E[\Phi(Z)]\xrightarrow{\mathbb P}0. \end{align*} Since $\Phi:\ell^\infty(\mathcal F)\to\mathbb R$ was an arbitrary bounded Lipschitz functional, this is precisely conditional weak convergence in probability in $\ell^\infty(\mathcal F)$. Hence \begin{align*} Z_n^*\rightsquigarrow Z \end{align*} conditionally in probability in $\ell^\infty(\mathcal F)$. [/step]