Proof of The Observation Filtration of a Solution of the Controlled N-Agent Dynamics is Generated, up to Null Sets, by the Observation Record up to that Time

Throughout, measurability of a real-valued map is with respect to the named $\sigma$-algebra and the Borel $\sigma$-algebra, and we use freely claims 1--5 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions (constants, indicators, sums, products, absolute values, pointwise limits).

Step 1: claim 1. Suppose first $s<T$. By claims 1 and 2 of Restriction of a Solution of the Controlled N-Agent Dynamics to a Shorter Horizon: the Truncated Policy, the Restricted Solution, Its Filtrations, and Its Record as the Prefix of the Record the truncated policy $h^{(s)}$ is an $\mathcal{A}$-valued observation-driven control policy with horizon $s$, the restricted families form a solution of the controlled $N$-agent dynamics on $[0,s]$ for $h^{(s)}$ on the same driving system, its observation filtration satisfies $\mathcal{G}^{(s)}_u=\mathcal{G}_u$ for $u\in[0,s]$, and its observation record is $W^{(s)}=\pi_s\circ W$. Apply clause (f) of Measurable Reconstruction of the Controlled N-Agent Dynamics from Observation Records to this restricted solution, with the reconstruction data fixed in the statement: the record $W^{(s)}$ is measurable with respect to $\mathcal{F}$ and $\mathcal{R}(s,\tilde{l})$, and $(W^{(s)})^{-1}(A)\in\mathcal{G}^{(s)}_s=\mathcal{G}_s$ for every $A\in\mathcal{R}(s,\tilde{l})$. Hence $\sigma(W^{(s)})\subseteq\mathcal{G}_s$. When $s=T$ the map $\pi_s$ is the identity, $W^{(s)}=W$, and the same clause applied to the given solution yields the assertion directly.

Step 2: the event times count the observation total. Fix $\omega\in\Omega_0$ and $u\in[0,T]$. By claim 1 of The Realized Control of the Controlled N-Agent Dynamics as a Random Element of the Control Set the numbers $\tau_j(\omega)$ are nondecreasing in $j$ and satisfy

Write $J_u=\{j\ge1:\tau_j(\omega)\le u\}$. By the displayed equivalence, $j\in J_u$ if and only if $j\le\tilde{c}_u(\omega)$; since $\tilde{c}_u(\omega)$ is a nonnegative integer (condition 3 of Solution of the Controlled N-Agent Dynamics: the observation total coincides on $[0,T]$ with the restriction of a counting path, whose values are nonnegative integers), $J_u=\{1,\dots,\tilde{c}_u(\omega)\}$ is finite with $\#J_u=\tilde{c}_u(\omega)$. This is the first identity of claim 2.

Step 3: each channel subtotal counts its own events. Keep $\omega\in\Omega_0$ fixed and write $K=K_T(\omega)=\tilde{c}_T(\omega)$, $t_j=\tau_j(\omega)$ and $w_j=\upsilon_j(\omega)$ for $1\le j\le K$; by condition 5 of Solution of the Controlled N-Agent Dynamics and claim 1 of The Realized Control of the Controlled N-Agent Dynamics as a Random Element of the Control Set these are the jump times of the observation total in $[0,T]$, listed in increasing order $t_1<\dots<t_K$, and $w_j$ is the unique channel such that some observation counter with channel $w_j$ jumps at $t_j$.

Each map $u\mapsto\tilde{N}^{i,\upsilon}_u(\omega)$ coincides on $[0,T]$ with the restriction of a counting path (condition 3), hence is nondecreasing with values in the nonnegative integers and vanishes at $u=0$; therefore so does each channel subtotal $u\mapsto\tilde{c}^{\upsilon}_u(\omega)$, being a finite sum of such maps, and $\tilde{c}_u(\omega)=\sum_{\upsilon}\tilde{c}^{\upsilon}_u(\omega)$.

Say that a nondecreasing map $f$ on $[0,T]$ increases at $\theta\in(0,T]$ if $f(\theta)>f(\theta')$ for every $\theta'\in[0,\theta)$. We record two facts.

(a) A counter with channel $\upsilon$ increases at $\theta$ if and only if $\tilde{c}^{\upsilon}$ increases at $\theta$. If some $\tilde{N}^{i,\upsilon}$ increases at $\theta$, then adding to the strict inequality $\tilde{N}^{i,\upsilon}_\theta(\omega)>\tilde{N}^{i,\upsilon}_{\theta'}(\omega)$ the inequalities $\tilde{N}^{i',\upsilon}_\theta(\omega)\ge\tilde{N}^{i',\upsilon}_{\theta'}(\omega)$ for the remaining indices gives $\tilde{c}^{\upsilon}_\theta(\omega)>\tilde{c}^{\upsilon}_{\theta'}(\omega)$ for every $\theta'<\theta$. Conversely, if no counter with channel $\upsilon$ increases at $\theta$, then for each $i$ there is $\theta_i\in[0,\theta)$ with $\tilde{N}^{i,\upsilon}_\theta(\omega)=\tilde{N}^{i,\upsilon}_{\theta_i}(\omega)$; taking $\theta'$ to be the largest of these finitely many numbers, which lies in $[0,\theta)$, monotonicity gives $\tilde{N}^{i,\upsilon}_\theta(\omega)=\tilde{N}^{i,\upsilon}_{\theta'}(\omega)$ for every $i$, hence $\tilde{c}^{\upsilon}_\theta(\omega)=\tilde{c}^{\upsilon}_{\theta'}(\omega)$.

(b) The increase of $\tilde{c}^{\upsilon}$ across $t_j$ is $1$ if $w_j=\upsilon$ and $0$ otherwise; and $\tilde{c}^{\upsilon}$ is constant on each interval between consecutive event times. By Step 2, $\tilde{c}_u(\omega)=\#\{j:t_j\le u\}$, so $\tilde{c}(\omega)$ is constant on $[0,t_1)$, on each $[t_j,t_{j+1})$ and on $[t_K,T]$, and $\tilde{c}_{t_j}(\omega)-\tilde{c}_{\theta'}(\omega)=1$ for $\theta'\in[t_{j-1},t_j)$ (with $t_0=0$). Since the $\tilde{c}^{\upsilon}(\omega)$ are nondecreasing with sum $\tilde{c}(\omega)$, each is constant on every interval on which $\tilde{c}(\omega)$ is constant, and

a sum of nonnegative integers equal to $1$; hence exactly one channel contributes, and it contributes $1$. For that channel the strict inequality $\tilde{c}^{\upsilon}_{t_j}(\omega)>\tilde{c}^{\upsilon}_{\theta'}(\omega)$ holds for $\theta'\in[t_{j-1},t_j)$, and it extends to every $\theta'\in[0,t_j)$: for $i<j$ the map $\tilde{c}^{\upsilon}(\omega)$ is constant on $[t_{i-1},t_i)$ by the previous sentence, so $\tilde{c}^{\upsilon}_{\theta'}(\omega)\le\tilde{c}^{\upsilon}_{\theta''}(\omega)$ for $\theta'\in[t_{i-1},t_i)$ and any $\theta''\in[t_{j-1},t_j)$ by monotonicity. So $\tilde{c}^{\upsilon}$ increases at $t_j$ for that channel and no other, and by (a) it is the unique channel at which some observation counter jumps at $t_j$, namely $w_j$ by condition 5.

Combining, for $u\in[0,T]$ the value $\tilde{c}^{\upsilon}_u(\omega)$ is obtained from $\tilde{c}^{\upsilon}_0(\omega)=0$ by adding $1$ for each $j$ with $t_j\le u$ and $w_j=\upsilon$, and nothing else; that is, $\tilde{c}^{\upsilon}_u(\omega)=\#\{j:t_j\le u,\ w_j=\upsilon\}$, which is the second identity of claim 2, the set being a subset of the finite set $\{1,\dots,K\}$.

Step 4: claim 3. Define $\chi^{\upsilon}_u$ on $\mathbf{R}(s,\tilde{l})$ by the displayed formula, with $\chi^{\upsilon}_u(r_\emptyset)=0$ for the empty record. It is measurable with respect to $\mathcal{R}(s,\tilde{l})$: by The Observation Record Space the record $\sigma$-algebra is that of the countable disjoint union of the cells, so by claim 4(c) of that lemma it suffices that the restriction of $\chi^{\upsilon}_u$ to each cell be measurable for the cell's own $\sigma$-algebra. (That claim is stated for $[0,\infty]$-valued maps and measurability in the sense of Lebesgue Integral of a Nonnegative Measurable Function; since $\chi^{\upsilon}_u$ takes values in $[0,\infty)$, that notion agrees with measurability with respect to the Borel $\sigma$-algebra of the real line, as recorded in Lebesgue Integral of a Nonnegative Measurable Function.) On $C_\emptyset$ the map is constant. On the cell $C_{k,v}$, which carries the transport along $\mathbf{t}\mapsto(k,\mathbf{t},v)$ of the restriction of the Borel $\sigma$-algebra $\mathcal{B}_k$ to the ordered time simplex $D_k(s)$, the mark vector $v$ is fixed, so

a finite sum of constants times indicators of the sets $\{\mathbf{t}\in D_k(s):t_j\le u\}$, which belong to $\mathcal{B}_k$ restricted to $D_k(s)$ because the $j$-th coordinate projection is measurable (claim 1 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets) and $D_k(s)\in\mathcal{B}_k$ (The Ordered Time Simplex: Borel Measurability and Volume); transporting along the bijection preserves measurability by claim 2 of Assembly of Measure Spaces: Restriction, Transport, One-Point Spaces, and Countable Disjoint Unions.

Now fix $\omega\in\Omega_0$, $\upsilon$ and $u\in[0,s]$. If $K_s(\omega)=\tilde{c}_s(\omega)=0$ then by Step 2 no $t_j$ lies in $[0,s]$, so $\tilde{c}^{\upsilon}_u(\omega)=0$ by Step 3; also $W^{(s)}(\omega)$ is the empty record, since by claim 2 of Restriction of a Solution of the Controlled N-Agent Dynamics to a Shorter Horizon: the Truncated Policy, the Restricted Solution, Its Filtrations, and Its Record as the Prefix of the Record the restricted solution has observation-event count $K_s$ and event times $\tau_1<\dots<\tau_{K_s}$, and The Observation Record of a Solution of the Controlled N-Agent Dynamics assigns the empty record when that count is $0$; hence $\chi^{\upsilon}_u(W^{(s)}(\omega))=0=\Upsilon^{\upsilon}_u(\omega)$. If $K_s(\omega)=k\ge1$, then again by claim 2 of Restriction of a Solution of the Controlled N-Agent Dynamics to a Shorter Horizon: the Truncated Policy, the Restricted Solution, Its Filtrations, and Its Record as the Prefix of the Record and The Observation Record of a Solution of the Controlled N-Agent Dynamics,

so by the formula for $\chi^{\upsilon}_u$ and by Step 3,

the second equality because every $j$ with $\tau_j(\omega)\le u\le s$ satisfies $j\le\tilde{c}_s(\omega)=k$ by Step 2, and the third by condition 4 of Solution of the Controlled N-Agent Dynamics.

Step 5: claim 4. By Solution of the Controlled N-Agent Dynamics the $\sigma$-algebra $\mathcal{G}_s$ is generated by the random variables $\Upsilon^{\upsilon}_u$ with $u\in[0,s]$ and $\upsilon\in\{1,\dots,\tilde{l}\}$ together with every event of $\mathcal{F}$ of probability zero, that is, by the family $\mathcal{C}\cup\mathcal{N}$ with $\mathcal{C}=\{(\Upsilon^{\upsilon}_u)^{-1}(B):u\in[0,s],\ \upsilon,\ B\in\mathcal{B}(\mathbb{R})\}$.

Write $\mathcal{G}_{\mathcal{N}}$ for the $\sigma$-algebra generated by $\sigma(W^{(s)})\cup\mathcal{N}$. We show the two inclusions.

First, $\mathcal{G}_{\mathcal{N}}\subseteq\mathcal{G}_s$: by Step 1, $\sigma(W^{(s)})\subseteq\mathcal{G}_s$, and $\mathcal{N}\subseteq\mathcal{G}_s$ by the definition of the observation filtration just recalled; a $\sigma$-algebra containing a family contains the $\sigma$-algebra it generates (Generated Sigma-Algebra).

Second, $\mathcal{G}_s\subseteq\mathcal{G}_{\mathcal{N}}$: it suffices that $\mathcal{C}\cup\mathcal{N}\subseteq\mathcal{G}_{\mathcal{N}}$, and $\mathcal{N}\subseteq\mathcal{G}_{\mathcal{N}}$ holds by construction. Let $u\in[0,s]$, let $\upsilon$ be a channel and let $B\in\mathcal{B}(\mathbb{R})$. Put $A'=(\chi^{\upsilon}_u\circ W^{(s)})^{-1}(B)$; since $\chi^{\upsilon}_u$ is $\mathcal{R}(s,\tilde{l})$-measurable by claim 3, $A'=(W^{(s)})^{-1}\bigl((\chi^{\upsilon}_u)^{-1}(B)\bigr)\in\sigma(W^{(s)})$. By claim 3 the maps $\Upsilon^{\upsilon}_u$ and $\chi^{\upsilon}_u\circ W^{(s)}$ agree at every point of $\Omega_0$, so

where $E\triangle E'=(E\setminus E')\cup(E'\setminus E)$. Both $(\Upsilon^{\upsilon}_u)^{-1}(B)$ and $A'$ lie in $\mathcal{F}$, so their symmetric difference does, and it has probability $0$ by monotonicity of $P$ (Basic Properties of a Measure), $P(\Omega\setminus\Omega_0)=0$. Hence that symmetric difference lies in $\mathcal{N}\subseteq\mathcal{G}_{\mathcal{N}}$, and

where $A'\in\mathcal{G}_{\mathcal{N}}$ and the two subtracted or adjoined sets are subsets of the symmetric difference lying in $\mathcal{F}$, hence of probability $0$, hence in $\mathcal{N}\subseteq\mathcal{G}_{\mathcal{N}}$. As $\mathcal{G}_{\mathcal{N}}$ is a $\sigma$-algebra, $(\Upsilon^{\upsilon}_u)^{-1}(B)\in\mathcal{G}_{\mathcal{N}}$. Thus $\mathcal{C}\subseteq\mathcal{G}_{\mathcal{N}}$ and $\mathcal{G}_s\subseteq\mathcal{G}_{\mathcal{N}}$.

So $\mathcal{G}_s=\mathcal{G}_{\mathcal{N}}$. By claim 1 of Factorisation of Random Variables Through a Measurable Map, the Variational Form of the Mean-Square Filtering Error, and Its Invariance Under the Joint Law, applied on $(\Omega,\mathcal{F},P)$ with the measurable map $W^{(s)}$ into $(\mathbf{R}(s,\tilde{l}),\mathcal{R}(s,\tilde{l}))$, the $\sigma$-algebra generated by $\sigma(W^{(s)})\cup\mathcal{N}$ is $W^{(s)}$-generated up to null sets; hence so is $\mathcal{G}_s$. The final display is then claim 3 of that lemma applied with $\mathcal{G}=\mathcal{G}_s$. $\blacksquare$