Proof of Progressive Measurability of the Realized Control with Respect to the Observation Filtration

Throughout, a real-valued map on a measurable space $(\Xi,\mathcal{H})$ is called $\mathcal{H}$-measurable when it is measurable with respect to $\mathcal{H}$ and the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R})$ of the real line. We use two devices, formulated for a generic measurable space $(\Xi,\mathcal{H})$; these two letters are placeholders and are re-instantiated at each use. (D1) If $f^1,\dots,f^d:\Xi\to\mathbb{R}$ are $\mathcal{H}$-measurable and $g:\mathbb{R}^d\to\mathbb{R}$ is sequentially continuous, then $g(f^1,\dots,f^d)$ is $\mathcal{H}$-measurable, by Sequentially Continuous Functions of Measurable Euclidean Maps are Measurable; since sums, differences, products, absolute values, squares, maxima and minima are sequentially continuous, they preserve $\mathcal{H}$-measurability, the indicator of a set of $\mathcal{H}$ is $\mathcal{H}$-measurable (its preimages are $\emptyset$, the set, its complement, or $\Xi$), and $\{f^1\le f^2\}=\{f^2-f^1\ge0\}\in\mathcal{H}$ for $\mathcal{H}$-measurable $f^1,f^2$. (D2) If $(E_n)_n$ is a countable family in $\mathcal{H}$ covering $\Xi$ and $f:\Xi\to\mathbb{R}$ satisfies $E_n\cap f^{-1}(\Gamma)\in\mathcal{H}$ for every $n$ and every $\Gamma\in\mathcal{B}(\mathbb{R})$, then $f$ is $\mathcal{H}$-measurable, since $f^{-1}(\Gamma)=\bigcup_n(E_n\cap f^{-1}(\Gamma))$. Write $\lambda$ for Lebesgue measure on the real line and $\lambda_{[0,t]}$ for its restriction to $[0,t]$ as in Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval, so that $\int_{[0,t]}\cdot\,ds=\int_{[0,t]}\cdot\,d\lambda_{[0,t]}$ for $t>0$. Finally, a function $\theta:\Xi\to\mathbb{R}$ with $\{\theta\le q\}\in\mathcal{H}$ for every real $q$ is $\mathcal{H}$-measurable, by the criterion of claim 3 of Rational Intervals and Rays Generate the Borel Sigma-Algebra of the Real Line applied to the complements $\{\theta>q\}$.

Claim 1. The inclusion $\mathcal{G}_t\subseteq\mathcal{F}^{\mathrm{sys}}_t$ is clause (vii)(e) of Existence, Uniqueness, and Regularity for the Controlled N-Agent Dynamics. By the definition of the observation filtration in Solution of the Controlled N-Agent Dynamics, $\mathcal{G}_t$ is the $\sigma$-algebra generated by the random variables $\Upsilon^\upsilon_s$, $s\le t$, together with every event of $\mathcal{F}$ of probability zero (so that each such $\Upsilon^\upsilon_s$ is $\mathcal{G}_t$-measurable); so every such event lies in $\mathcal{G}_t$, and in particular $\Omega\setminus\Omega_0$, an event of $\mathcal{F}$ with $P(\Omega\setminus\Omega_0)=1-P(\Omega_0)=0$. Now let $X$, $X'$ and an event $\mathcal{Z}\in\mathcal{F}$ with $P(\mathcal{Z})=0$ be as in the claim, so that $X=X'$ off $\mathcal{Z}$, and let $\Gamma\in\mathcal{B}(\mathbb{R})$. Then

$$X^{-1}(\Gamma)=\bigl(X'^{-1}(\Gamma)\setminus\mathcal{Z}\bigr)\cup\bigl(X^{-1}(\Gamma)\cap\mathcal{Z}\bigr).$$

The first set lies in $\mathcal{G}_t$, as $X'^{-1}(\Gamma)\in\mathcal{G}_t$ and $\mathcal{Z}\in\mathcal{G}_t$. The second is an event of $\mathcal{F}$, being an intersection of two events, and is contained in $\mathcal{Z}$, so it has probability zero by monotonicity of $P$ (claim 2 of Basic Properties of a Measure); hence it lies in $\mathcal{G}_t$ as well. Thus $X^{-1}(\Gamma)\in\mathcal{G}_t$, and $X$ is $\mathcal{G}_t$-measurable.

Claim 2. Fix $j\ge1$ and $q\in[0,T]$. By claim 1 of The Realized Control of the Controlled N-Agent Dynamics as a Random Element of the Control Set, $\{\tau_j\le q\}=\{\tilde{c}_q\ge j\}$. Each observation counter $\tilde{N}^{i,\upsilon}_q$ is a random variable by clause (vii)(c) of Existence, Uniqueness, and Regularity for the Controlled N-Agent Dynamics, so the observation total $\tilde{c}_q=\sum_{i,\upsilon}\tilde{N}^{i,\upsilon}_q$ is a random variable by (D1). At every $\omega\in\Omega_0$, condition 4 of Solution of the Controlled N-Agent Dynamics gives $\Upsilon^\upsilon_q=\frac1N\sum_i\tilde{N}^{i,\upsilon}_q$ for every $\upsilon$, hence $\tilde{c}_q=N\sum_{\upsilon=1}^{\tilde{l}}\Upsilon^\upsilon_q$. The right-hand side is a $\mathcal{G}_q$-measurable random variable by (D1), each $\Upsilon^\upsilon_q$ being a generator of $\mathcal{G}_q$; it agrees with $\tilde{c}_q$ off $\Omega\setminus\Omega_0$; so $\tilde{c}_q$ is $\mathcal{G}_q$-measurable by claim 1, and $\{\tau_j\le q\}=\{\tilde{c}_q\ge j\}\in\mathcal{G}_q$, the closed ray $[j,\infty)$ being a Borel set (claim 4 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets).

The function $\theta_j=\min(\tau_j,T)$ takes values in $[0,T]$ since $\tau_j$ takes values in $[0,T]\cup\{T+1\}$. For $t\in[0,T)$ one has $\{\theta_j\le t\}=\{\tau_j\le t\}\in\mathcal{G}_t$, and $\{\theta_j\le T\}=\Omega\in\mathcal{G}_T$; so $\theta_j$ is a stopping time of $(\mathcal{G}_t)_{t\in[0,T]}$.

For the channels, fix $t\in[0,T]$, $j\ge1$ and $\upsilon$. Write $\tilde{c}^\upsilon_s=\sum_{i=1}^{N}\tilde{N}^{i,\upsilon}_s$ for the channel subtotal, a random variable by (D1), and put $\Psi^\upsilon_s=\mathbf{1}_{\Omega_0}\tilde{c}^\upsilon_s$ for $s\in[0,T]$. Since $\tilde{c}^\upsilon_s=N\Upsilon^\upsilon_s$ at every point of $\Omega_0$ by condition 4, $\Psi^\upsilon_s=\mathbf{1}_{\Omega_0}N\Upsilon^\upsilon_s$ everywhere, which is $\mathcal{G}_s$-measurable by (D1), as $\Omega_0\in\mathcal{G}_s$ by claim 1 and $\Upsilon^\upsilon_s$ is a generator of $\mathcal{G}_s$. Every path of $\Psi^\upsilon$ is right-continuous in the sense of Progressive Measurability: Sections, Right-Continuous Adapted Processes, Arithmetic, and Indefinite Time Integrals: off $\Omega_0$ it vanishes identically, and at $\omega\in\Omega_0$ it is the sum over $i$ of the paths $s\mapsto\tilde{N}^{i,\upsilon}_s(\omega)$, each of which coincides on $[0,T]$ with the restriction of a counting path by condition 3. A counting path $c$ is nondecreasing with integer values (clauses 1 and 2 of its definition), and by clause 3 the value $c(s)$ is the greatest lower bound of the set of integers $\{c(s'):s'>s\}$, which is therefore attained at some $s_0>s$ (a nonempty set of integers bounded below has a least element), so that $c$ is constant on $[s,s_0]$ by monotonicity; a finite sum of such paths is thus constant to the right of every $s<T$ on the intersection of the corresponding intervals, and every sequence in $[s,T]$ converging to $s$ eventually lies in that intersection; for $s=T$ every sequence in $[T,T]$ is constant, so the condition holds trivially. Hence $\Psi^\upsilon$ is adapted to $(\mathcal{G}_t)_{t\in[0,T]}$ with every path right-continuous, and is progressively measurable by claim 2 of the toolkit. Put $\theta_0=0$ and $\theta_i=\min(\tau_i,T)$ for $i\ge1$; these are stopping times by the preceding paragraph and claim 1 of Stopping Times on a Compact Time Interval: Elementary Operations, the Prior Sigma-Algebra, Dyadic Approximation, Sampling, and Hitting Times, so $\min(t,\theta_i)$ is a stopping time bounded by $t$ (claim 1 of that toolkit), and the sampled variable $\Psi^\upsilon_{\min(t,\theta_i)}$ is measurable with respect to $\mathcal{G}_{\min(t,\theta_i)}\subseteq\mathcal{G}_t$ by claims 4(ii) and 2 there. Consequently, by (D1),

$$H'=\bigl\{\omega\in\Omega:\ \Psi^\upsilon_{\min(t,\theta_j)}(\omega)-\Psi^\upsilon_{\min(t,\theta_{j-1})}(\omega)=1\bigr\}\in\mathcal{G}_t,\qquad H=\{\tau_j\le t\}\cap H'\cap\Omega_0\in\mathcal{G}_t .$$

We claim that $\{\tau_j\le t\}\cap\{\upsilon_j=\upsilon\}\cap\Omega_0=H$. Let $\omega\in\Omega_0$ with $\tau_j(\omega)\le t$. By claim 1 of The Realized Control of the Controlled N-Agent Dynamics as a Random Element of the Control Set, $\tilde{c}_t(\omega)\ge j$, hence $j\le\tilde{c}_T(\omega)$ by monotonicity, and $\tau_1(\omega)<\dots<\tau_{\tilde{c}_T(\omega)}(\omega)$ are the jump times in $[0,T]$ of the counting path $c$ with which $\tilde{c}(\omega)$ agrees on $[0,T]$ (condition 3), listed in condition 5 as strictly increasing; also $\tau_1(\omega)>0$, since by condition 5 these numbers are jump times of $c$ in the sense of the counting-path definition, and jump times are positive. With $\tau_0=0$ we thus have $0\le\tau_{j-1}(\omega)<\tau_j(\omega)\le t$, so $\min(t,\theta_i(\omega))=\tau_i(\omega)$ and $\Psi^\upsilon_{\tau_i}(\omega)=\tilde{c}^\upsilon_{\tau_i}(\omega)$ for $i\in\{j-1,j\}$. Next, $\tilde{c}_{\tau_j}(\omega)\ge j$ by the equivalence of claim 1, while $\tilde{c}_{\tau_j}(\omega)<j+1$: if $j+1\le\tilde{c}_T(\omega)$ this is the same equivalence together with $\tau_{j+1}(\omega)>\tau_j(\omega)$, and otherwise $\tilde{c}_{\tau_j}(\omega)\le\tilde{c}_T(\omega)=j$. As $\tilde{c}$ is integer-valued, $\tilde{c}_{\tau_j}(\omega)=j$; likewise $\tilde{c}_{\tau_{j-1}}(\omega)=j-1$, which for $j=1$ reads $\tilde{c}_0(\omega)=0$, clause 1 of the counting-path definition. Summing over channels, $\sum_{\upsilon'=1}^{\tilde{l}}\bigl(\tilde{c}^{\upsilon'}_{\tau_j}(\omega)-\tilde{c}^{\upsilon'}_{\tau_{j-1}}(\omega)\bigr)=1$, each summand being a nonnegative integer because every counter is nondecreasing and integer-valued; so exactly one channel $\upsilon^*$ has increment $1$ and every other channel has increment $0$. Finally $\upsilon^*=\upsilon_j(\omega)$: by claim 1 of the realized-control lemma, $\upsilon_j(\omega)$ is the channel of condition 5, so some counter $\tilde{N}^{i,\upsilon_j(\omega)}$ has $\tau_j(\omega)$ as a jump time, meaning that its value at $\tau_j(\omega)$ exceeds the least upper bound of its values on $[0,\tau_j(\omega))$, in particular its value at $\tau_{j-1}(\omega)$; adding the inequalities $\tilde{N}^{i',\upsilon_j(\omega)}_{\tau_j}(\omega)\ge\tilde{N}^{i',\upsilon_j(\omega)}_{\tau_{j-1}}(\omega)$ for the remaining agents $i'$ (claim 3 of Elementary Order Arithmetic in an Ordered Field) gives $\tilde{c}^{\upsilon_j(\omega)}_{\tau_j}(\omega)>\tilde{c}^{\upsilon_j(\omega)}_{\tau_{j-1}}(\omega)$, so the increment of channel $\upsilon_j(\omega)$ is positive, hence equal to $1$, and $\upsilon^*=\upsilon_j(\omega)$. Thus, at such $\omega$, $\upsilon_j(\omega)=\upsilon$ holds exactly when $\omega\in H'$, which proves the claimed identity. Hence $\{\tau_j\le t\}\cap\{\upsilon_j=\upsilon\}\cap\Omega_0=H$, and

$$\{\tau_j\le t\}\cap\{\upsilon_j=\upsilon\}=H\cup\bigl(\{\tau_j\le t\}\cap\{\upsilon_j=\upsilon\}\cap(\Omega\setminus\Omega_0)\bigr),$$

where the second set is an event of $\mathcal{F}$ ($\tau_j$ and $\upsilon_j$ being random variables) of probability zero, hence a member of $\mathcal{G}_t$ by claim 1. So the union lies in $\mathcal{G}_t$.

Claim 3. Fix $t\in[0,T]$ and $\kappa\in\{1,\dots,m\}$, put $\Xi=[0,t]\times\Omega$ and $\mathcal{H}=\mathcal{B}_{[0,t]}\otimes\mathcal{G}_t$. For $j\ge1$ define $\tau^t_j:\Omega\to[0,T+1]$ by $\tau^t_j(\omega)=\tau_j(\omega)$ if $\tau_j(\omega)\le t$ and $\tau^t_j(\omega)=T+1$ otherwise. For real $q$ the set $\{\tau^t_j\le q\}$ is $\emptyset$ if $q<0$, is $\{\tau_j\le\min(q,t)\}$ if $0\le q<T+1$, and is $\Omega$ if $q\ge T+1$; by claim 2 and the inclusion $\mathcal{G}_{\min(q,t)}\subseteq\mathcal{G}_t$ (the family being a filtration) all of these lie in $\mathcal{G}_t$, so $\tau^t_j$ is $\mathcal{G}_t$-measurable by the sublevel criterion. Moreover, for $(s,\omega)\in\Xi$ one has $\tau_j(\omega)\le s$ if and only if $\tau^t_j(\omega)\le s$, because $s\le t<T+1$.

The maps $(s,\omega)\mapsto s$ and $(s,\omega)\mapsto\tau^t_j(\omega)$ on $\Xi$ are $\mathcal{H}$-measurable: preimages of Borel sets are $(\Gamma\cap[0,t])\times\Omega$ and $[0,t]\times(\tau^t_j)^{-1}(\Gamma)$, which lie in $\mathcal{H}$ by Product Sigma-Algebra (for $t=0$ the first is $\emptyset$ or $\{0\}\times\Omega$). Hence by (D1) and the equivalence just noted, the set $\{(s,\omega)\in\Xi:\tau_j(\omega)\le s\}=\{(s,\omega)\in\Xi:\tau^t_j(\omega)\le s\}$ lies in $\mathcal{H}$. Every $\tilde{c}_s$ is integer-valued: at $\omega\in\Omega_0$ by condition 3 and clause 1 of the counting-path definition, and off $\Omega_0$ because every observation counter vanishes there (clause (vii)(c) of Existence, Uniqueness, and Regularity for the Controlled N-Agent Dynamics), so that $\tilde{c}_s(\omega)=0$; hence $\{\tilde{c}_s=k\}=\{\tilde{c}_s\ge k\}\setminus\{\tilde{c}_s\ge k+1\}$, and therefore the following sets lie in $\mathcal{H}$:

$$\Lambda_k=\{(s,\omega)\in\Xi:\tilde{c}_s(\omega)=k\}=\{(s,\omega)\in\Xi:\tau_k(\omega)\le s\}\setminus\{(s,\omega)\in\Xi:\tau_{k+1}(\omega)\le s\}\quad(k\ge1),$$

and $\Lambda_0=\Xi\setminus\{(s,\omega)\in\Xi:\tau_1(\omega)\le s\}$, the displayed identities being the equivalence of claim 1 of The Realized Control of the Controlled N-Agent Dynamics as a Random Element of the Control Set. Set $E_*=[0,t]\times(\Omega\setminus\Omega_0)$, which lies in $\mathcal{H}$ by claim 1, and for $k\ge0$ and $v\in\{1,\dots,\tilde{l}\}^k$ (the empty tuple when $k=0$) put

$$G_{k,v}=\Omega_0\cap\bigcap_{i=1}^{k}\bigl(\{\tau_i\le t\}\cap\{\upsilon_i=v_i\}\bigr)\in\mathcal{G}_t,\qquad E_{k,v}=\Lambda_k\cap\bigl([0,t]\times G_{k,v}\bigr)\in\mathcal{H},$$

using claims 1 and 2. For $(s,\omega)\in \Lambda_k$ with $\omega\in\Omega_0$ and $k\ge1$ one has $\tau_i(\omega)\le\tau_k(\omega)\le s\le t$ for $i\le k$ (the $\tau_i$ being nondecreasing in $i$), so $E_{k,v}$ is exactly the set of $(s,\omega)\in \Lambda_k$ with $\omega\in\Omega_0$ and $\upsilon_i(\omega)=v_i$ for $i\le k$. Since at every $\omega\in\Omega_0$ and $s\in[0,t]$ the count $\tilde{c}_s(\omega)$ equals exactly one $k$ and the channels $\upsilon_1(\omega),\dots,\upsilon_k(\omega)$ form some tuple $v$, the countable family consisting of $E_*$ and the sets $E_{k,v}$ covers $\Xi$. On $E_*$ the map $\hat{\alpha}^\kappa$ is the constant $a^\kappa_0$ by the definition of $\hat{\alpha}$ in claim 2 of The Realized Control of the Controlled N-Agent Dynamics as a Random Element of the Control Set, so its preimages meet $E_*$ in $\emptyset$ or $E_*$.

Fix $k\ge0$ and $v$, let $\Pi_k=[0,T]\times R_k(T)\subseteq\mathbb{R}^{1+k}$ (so $\Pi_0=[0,T]$), and let $\mathcal{Q}_{k}$ be the $\sigma$-algebra on $\Pi_k$ generated by the sets $U\cap \Pi_k$ with $U$ an open subset of $\mathbb{R}^{1+k}$; by Observation-Driven Control Policy the component $h^\kappa_k(\cdot,\cdot,v)$ is measurable with respect to $\mathcal{Q}_{k}$ and $\mathcal{B}(\mathbb{R})$. Let $Z_k(s,\omega)=(s,\tau^t_1(\omega),\dots,\tau^t_k(\omega))$ on $\Xi$. Its components are $\mathcal{H}$-measurable, so $Z_k$ is measurable with respect to $\mathcal{H}$ and the $\sigma$-algebra $\mathcal{B}_{1+k}$ of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets by claim 2 there, and $\mathcal{B}_{1+k}$ contains every open subset of $\mathbb{R}^{1+k}$ by claim 4 there. For $(s,\omega)\in E_{k,v}$ we have $\tau^t_i(\omega)=\tau_i(\omega)$ for $i\le k$, and, by the well-definedness assertion of claim 2 of The Realized Control of the Controlled N-Agent Dynamics as a Random Element of the Control Set applied at $(s,\omega)$ with $k=\tilde{c}_s(\omega)$, $Z_k(s,\omega)\in\Pi_k$ and $\hat{\alpha}^\kappa(s,\omega)=h^\kappa_k(Z_k(s,\omega),v)$ (read as $h^\kappa_0(s)$ when $k=0$). Let $\mathcal{V}$ be the family of subsets $V\subseteq \Pi_k$ with $E_{k,v}\cap Z_k^{-1}(V)\in\mathcal{H}$. Because $Z_k$ maps $E_{k,v}$ into $\Pi_k$, preimages of complements and countable unions taken within $\Pi_k$ meet $E_{k,v}$ in complements within $E_{k,v}$ and in countable unions, so $\mathcal{V}$ is a $\sigma$-algebra on $\Pi_k$; it contains every $U\cap \Pi_k$ with $U$ open, since $E_{k,v}\cap Z_k^{-1}(U\cap \Pi_k)=E_{k,v}\cap Z_k^{-1}(U)\in\mathcal{H}$. Hence $\mathcal{Q}_{k}\subseteq\mathcal{V}$. For $\Gamma\in\mathcal{B}(\mathbb{R})$ the set $V=\{p\in \Pi_k:h^\kappa_k(p,v)\in\Gamma\}$ lies in $\mathcal{Q}_{k}$, and $E_{k,v}\cap(\hat{\alpha}^\kappa)^{-1}(\Gamma)=E_{k,v}\cap Z_k^{-1}(V)\in\mathcal{H}$. By (D2), the restriction of $\hat{\alpha}^\kappa$ to $\Xi$ is $\mathcal{H}$-measurable. As $t$ was arbitrary, $\hat{\alpha}^\kappa$ is progressively measurable with respect to $(\mathcal{G}_t)_{t\in[0,T]}$.

Since $\mathcal{G}_t\subseteq\mathcal{F}^{\mathrm{sys}}_t$, every generating rectangle $S\times C$ of $\mathcal{B}_{[0,t]}\otimes\mathcal{G}_t$ is a generating rectangle of $\mathcal{B}_{[0,t]}\otimes\mathcal{F}^{\mathrm{sys}}_t$, so the former $\sigma$-algebra is contained in the latter and $\hat{\alpha}^\kappa$ is progressively measurable with respect to $(\mathcal{F}^{\mathrm{sys}}_t)_{t\in[0,T]}$ too. Finally $\hat{\alpha}^\kappa(t,\cdot)$ is $\mathcal{G}_t$-measurable by claim 1 of Progressive Measurability: Sections, Right-Continuous Adapted Processes, Arithmetic, and Indefinite Time Integrals.

Claim 4. Fix $t\in[0,T]$ and $\omega\in\Omega$. Every value of the path $s\mapsto\hat{\alpha}^{(t)}(s,\omega)$ lies in $\mathcal{A}$, so its Euclidean norm is at most $R$. Each component of the path is $\mathcal{B}_{[0,T]}$-measurable: by claim 3 of The Realized Control of the Controlled N-Agent Dynamics as a Random Element of the Control Set the components of $s\mapsto\hat{\alpha}(s,\omega)$ are, and $\hat{\alpha}^{(t),\kappa}(s,\omega)=\hat{\alpha}^\kappa(s,\omega)\mathbf{1}_{[0,t]}(s)+a^\kappa_0\mathbf{1}_{(t,T]}(s)$ for each component index $\kappa$, where $[0,t]$ and $(t,T]$ belong to $\mathcal{B}_{[0,T]}$ (they are intersections of $[0,T]$ with Borel subsets of the real line, which contains its closed and open rays by claim 4 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets), so (D1) applies. By claim 1 of Linearity and Monotonicity of the Lebesgue Integral and claim 1 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval, $\int_{[0,T]}|\hat{\alpha}^{(t)}(\cdot,\omega)|^2\,d\lambda_{[0,T]}\le R^2T<\infty$, so the path is square-integrable in the sense of claim 1 of The Lebesgue Space of Square-Integrable Vector-Valued Functions on a Compact Interval and represents an element of $\mathcal{U}_{\mathcal{A}}$ (the exceptional null set of The Set of Controls with Values in a Prescribed Subset of Euclidean Space being empty), of which it is an admissible representative in the sense of claim 2 of Stability of the Mean-Field Flow under Perturbation of the Initial State and Weak Convergence of Controls. Agreement with $\hat{\alpha}(\omega)$ on $[0,t]$ is the definition. This holds for every $\omega\in\Omega$; from here on $\omega$ is no longer fixed.

Let $\zeta\in\mathcal{U}_{\mathcal{A}}$ with an admissible representative, again written $\zeta$ (it exists by claim 2 of Stability of the Mean-Field Flow under Perturbation of the Initial State and Weak Convergence of Controls), so $|\zeta(s)|\le R$ for every $s$; fix $r\in\mathbb{N}$ and put

$$f(s,\omega)=\bigl(\hat{\alpha}^{(t)}(s,\omega)-\zeta(s)\bigr)\cdot w_r(s)\qquad((s,\omega)\in[0,T]\times\Omega).$$

We claim $f$ is measurable with respect to $\mathcal{B}_{[0,T]}\otimes\mathcal{G}_t$. Every member of $\mathcal{B}_{[0,t]}\otimes\mathcal{G}_t$ is a member of $\mathcal{B}_{[0,T]}\otimes\mathcal{G}_t$: the subsets of $[0,t]\times\Omega$ belonging to the latter form a $\sigma$-algebra on $[0,t]\times\Omega$ containing every rectangle $S\times C$ with $S\in\mathcal{B}_{[0,t]}$ and $C\in\mathcal{G}_t$, because $S=S'\cap[0,t]$ with $S'$ Borel gives $S\in\mathcal{B}_{[0,T]}$ (the trace of a Borel set on $[0,T]$ is intersected with the Borel set $[0,t]$; for $t=0$, $\mathcal{B}_{[0,0]}=\{\emptyset,\{0\}\}$ and $\{0\}\in\mathcal{B}_{[0,T]}$). Hence, for $\Gamma\in\mathcal{B}(\mathbb{R})$, the preimage of $\Gamma$ under $(s,\omega)\mapsto\hat{\alpha}^{(t),\kappa}(s,\omega)$ is the union of $\{(s,\omega)\in[0,t]\times\Omega:\hat{\alpha}^\kappa(s,\omega)\in\Gamma\}$, a member of $\mathcal{B}_{[0,t]}\otimes\mathcal{G}_t$ by claim 3, and of $(t,T]\times\Omega$ or $\emptyset$ according as $a^\kappa_0\in\Gamma$ or not; so each component of $\hat{\alpha}^{(t)}$ is $\mathcal{B}_{[0,T]}\otimes\mathcal{G}_t$-measurable. Each component of $(s,\omega)\mapsto\zeta(s)$ and of $(s,\omega)\mapsto w_r(s)$ is so measurable as well, being a $\mathcal{B}_{[0,T]}$-measurable map composed with the projection onto $[0,T]$, whose preimages are rectangles. By (D1), $f$ is $\mathcal{B}_{[0,T]}\otimes\mathcal{G}_t$-measurable. As in the realized-control lemma, $|x\cdot y|\le|x|\,|y|$ (claim 2 of Cauchy-Schwarz Inequality for a Positive Semidefinite Quadratic Form on $\mathbb{R}^n$ with the identity matrix and claim 1 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square) and $|\hat{\alpha}^{(t)}(s,\omega)-\zeta(s)|\le2R$ (claims 5 and 6 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square) give $|f(s,\omega)|\le2R|w_r(s)|$, and $\int_{[0,T]}|w_r|\,d\lambda_{[0,T]}<\infty$: the map $|w_r|$ is $\mathcal{B}_{[0,T]}$-measurable by (D1) from the components of $w_r$, $\int_{[0,T]}|w_r|^2\,d\lambda_{[0,T]}<\infty$ since $w_r$ is square-integrable (claim 1 of The Lebesgue Space of Square-Integrable Vector-Valued Functions on a Compact Interval), and claim 4 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval with the constant function $1$ gives the finiteness. (The values $w_r(s)$ and $\zeta(s)$ are those of fixed representatives; by claim 5 of The Lebesgue Space of Square-Integrable Vector-Valued Functions on a Compact Interval and claim 2 of Integrals of Functions Vanishing or Agreeing off a Null Set on a Compact Interval the integrals below, hence the value of $\rho$, do not depend on this choice.) Since $\mathcal{G}_t\subseteq\mathcal{F}$, the restriction $P|_{\mathcal{G}_t}$ of $P$ to $\mathcal{G}_t$ is a probability measure on $(\Omega,\mathcal{G}_t)$; the maps $f^{\pm}=\max(\pm f,0)$ are $\mathcal{B}_{[0,T]}\otimes\mathcal{G}_t$-measurable by (D1), hence measurable as $[0,\infty]$-valued maps, and by the statement on sections of Tonelli and Fubini Theorems each section $f^{\pm}(\cdot,\omega)$, hence also $f(\cdot,\omega)=f^{+}(\cdot,\omega)-f^{-}(\cdot,\omega)$, is $\mathcal{B}_{[0,T]}$-measurable; so the Tonelli statement of Tonelli and Fubini Theorems, applied on the product of the finite, hence $\sigma$-finite, measure spaces $([0,T],\mathcal{B}_{[0,T]},\lambda_{[0,T]})$ and $(\Omega,\mathcal{G}_t,P|_{\mathcal{G}_t})$, shows that $\omega\mapsto\int_{[0,T]}f^{\pm}(s,\omega)\,d\lambda_{[0,T]}(s)$ are $\mathcal{G}_t$-measurable as $[0,\infty]$-valued maps; their values lie in $[0,2R\int_{[0,T]}|w_r|\,d\lambda_{[0,T]}]$ by monotonicity, so they are $\mathcal{G}_t$-measurable real-valued maps. For each $\omega$ the section $f(\cdot,\omega)$ is integrable, being bounded by the integrable $2R|w_r|$, and its integral is $\int f^+-\int f^-$ by Integrable Function and the Lebesgue Integral; by claim 4 of The Lebesgue Space of Square-Integrable Vector-Valued Functions on a Compact Interval this integral is $\langle\hat{\alpha}^{(t)}(\omega)-\zeta,w_r\rangle_{L^2}$, so $\omega\mapsto\min(2^{-r},|\langle\hat{\alpha}^{(t)}(\omega)-\zeta,w_r\rangle_{L^2}|)$ is $\mathcal{G}_t$-measurable by (D1), with values in $[0,1]$. By claim 1 of Measurability of Countable Suprema, Bounded Pointwise Limits, Monotone Functions, and Continuous Functions the supremum over $r$ is $\mathcal{G}_t$-measurable, and by the defining formula of $\rho$ in The Set of Controls with Values in a Compact Convex Set is Weakly Metrizable and Compact, together with claim 1 there for the existence of the supremum, it is $\rho(\hat{\alpha}^{(t)}(\omega),\zeta)$.

By claim 3 of The Simplex, the Control Set and Their Product are Compact Separable Metric Spaces the metric space $(\mathcal{U}_{\mathcal{A}},\rho)$ is separable; let $D_{\mathcal{U}}$ be a countable dense subset, nonempty because $\mathcal{U}_{\mathcal{A}}$ contains the class of the constant path $a_0$. Applying claim 3 of Borel Sets and Measurable Maps in a Separable Metric Space on the measurable space $(\Omega,\mathcal{G}_t)$ to the separable metric datum $((\mathcal{U}_{\mathcal{A}},\rho),D_{\mathcal{U}})$, the $\mathcal{G}_t$-measurability of $\omega\mapsto\rho(\hat{\alpha}^{(t)}(\omega),q)$ for every $q\in D_{\mathcal{U}}$ shows that $\omega\mapsto\hat{\alpha}^{(t)}(\omega)$ is measurable with respect to $\mathcal{G}_t$ and the Borel $\sigma$-algebra of $(\mathcal{U}_{\mathcal{A}},\rho)$.

Claim 5. Fix $t\in[0,T]$. For each component $A^\kappa$, the map $(s,\omega)\mapsto A^\kappa_s$ on $[0,t]\times\Omega$ is $\mathcal{B}_{[0,t]}\otimes\mathcal{G}_t$-measurable: its preimage of $\Gamma\in\mathcal{B}(\mathbb{R})$ is $\{s\in[0,t]:A^\kappa_s\in\Gamma\}\times\Omega$, and $\{s\in[0,t]:A^\kappa_s\in\Gamma\}=\{s\in[0,T]:A^\kappa_s\in\Gamma\}\cap[0,t]$ belongs to $\mathcal{B}_{[0,t]}$, since $\{s\in[0,T]:A^\kappa_s\in\Gamma\}\in\mathcal{B}_{[0,T]}$ is a Borel subset of the real line by claim 1 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval. Together with claim 3 and (D1), $(s,\omega)\mapsto|\hat{\alpha}(s,\omega)-A_s|^2=\sum_{\kappa=1}^{m}(\hat{\alpha}^\kappa(s,\omega)-A^\kappa_s)^2$ is $\mathcal{B}_{[0,t]}\otimes\mathcal{G}_t$-measurable on $[0,t]\times\Omega$; as $t$ is arbitrary, the family $\chi_s=|\hat{\alpha}(s,\cdot)-A_s|^2$ is progressively measurable. Both $\hat{\alpha}(s,\omega)$ and $A_s$ lie in $\mathcal{A}$, so $|\hat{\alpha}(s,\omega)-A_s|\le2R$ by claims 5 and 6 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square, and $0\le \chi_s\le4R^2$ everywhere. Claim 4 of Progressive Measurability: Sections, Right-Continuous Adapted Processes, Arithmetic, and Indefinite Time Integrals, applied with $K=4R^2$, shows that $\mathcal{E}_t(\omega)=\int_{[0,t]}\chi_s(\omega)\,ds$ is defined for every $t$ and $\omega$, that $|\mathcal{E}_t(\omega)-\mathcal{E}_{t_0}(\omega)|\le4R^2(t-t_0)$ for $0\le t_0\le t\le T$, that every path of $\mathcal{E}$ is continuous on $[0,T]$, and that $\mathcal{E}$ is adapted and progressively measurable with respect to $(\mathcal{G}_t)_{t\in[0,T]}$; in particular $\mathcal{E}_t$ is $\mathcal{G}_t$-measurable. It remains to see that $\mathcal{E}_t(\omega)\ge\mathcal{E}_{t_0}(\omega)$ for $t_0\le t$. For $u\in\{t_0,t\}$ let $\widetilde{\chi}^{u}(\omega)$ denote the zero extension to the real line of the section $s\mapsto\chi_s(\omega)\mathbf{1}_{[0,u]}(s)$ on $[0,T]$, which is $\mathcal{B}_{[0,T]}$-measurable by claim 1 of Progressive Measurability: Sections, Right-Continuous Adapted Processes, Arithmetic, and Indefinite Time Integrals and (D1). Then $\mathcal{E}_u(\omega)=\int_{\mathbb{R}}\widetilde{\chi}^{u}(\omega)\,d\lambda$: for $u>0$ this is claim 2 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval applied twice, on $[0,u]$ and on $[0,T]$, both zero extensions being the same function on the real line; for $u=0$ both sides are $0$: $\mathcal{E}_0=0$ by the convention of the statement, while $s\mapsto\chi_s(\omega)\mathbf{1}_{[0,0]}(s)$ vanishes on $[0,T]$ off the set $\{0\}$, which is $\lambda_{[0,T]}$-null by claim 1 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval (a single point has Lebesgue measure zero, being contained in intervals of arbitrarily small length), so its integral over $[0,T]$ is $0$ by claim 1 of Integrals of Functions Vanishing or Agreeing off a Null Set on a Compact Interval, and $\int_{\mathbb{R}}\widetilde{\chi}^{0}(\omega)\,d\lambda=0$ by claim 2 of the toolkit. The two extended integrands are nonnegative and $\widetilde{\chi}^{t}(\omega)\ge\widetilde{\chi}^{t_0}(\omega)$ pointwise, so $\mathcal{E}_{t_0}(\omega)\le\mathcal{E}_t(\omega)$ by monotonicity, claim 1 of Linearity and Monotonicity of the Lebesgue Integral. Finally $0\le\mathcal{E}_t\le4R^2t\le4R^2T$ by the same monotonicity together with $\lambda_{[0,t]}([0,t])=t$ (claim 1 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval) for $t>0$, the case $t=0$ being the convention $\int_{[0,0]}\cdot\,ds=0$; so $\mathcal{E}$ takes values in $[0,4R^2T]$ and its paths are nondecreasing. $\blacksquare$