Proof of Bayes Disintegration and Filtering Formula for the Observation Record

Claim 1. The pairing $\Phi(\omega)=(W(\omega),\omega)$ is measurable: for a rectangle $A\times C$ with $A\in\mathcal{R}$ and $C\in\mathcal{T}$, $\Phi^{-1}(A\times C)=W^{-1}(A)\cap C\in\mathcal{F}$ by claim (f) of Measurable Reconstruction of the Controlled N-Agent Dynamics from Observation Records, and the class of sets with measurable preimage is a $\sigma$-algebra containing the rectangles, which generate $\mathcal{R}\otimes\mathcal{T}$. Let $P_\Phi$ be the image measure and let $\nu$ be the measure with density $f$ with respect to $\pi=\rho\otimes P|_{\mathcal{T}}$ (the product of a finite and a probability measure, both $\sigma$-finite; $P|_{\mathcal{T}}$ is a probability measure as in the proof pattern of Independence Fubini: Integration in an Independent Random Vector Given a Sub-Sigma-Algebra). Both are probability measures on $\mathcal{R}\otimes\mathcal{T}$: $P_\Phi$ as an image of $P$, and $\nu$ because, by the Tonelli theorem and claim 4 of Conditional Density of the Observation Record Given the Initial States and Transition Clocks, $\nu(\mathbf{R}\times\Omega)=\mathbb{E}[\int_{\mathbf{R}}f(r,\cdot)\,\rho(dr)]=1$. They agree on the rectangles: by claim 3 of Conditional Density of the Observation Record Given the Initial States and Transition Clocks with $Z=\mathbf{1}_C$ and $g=\mathbf{1}_A$, and then Tonelli, $P_\Phi(A\times C)=\mathbb{E}\bigl[\mathbf{1}_C\,\mathbf{1}_A(W)\bigr]=\mathbb{E}\Bigl[\mathbf{1}_C\int_A f(r,\cdot)\,\rho(dr)\Bigr]=\int_{\mathbf{R}\times\Omega}\mathbf{1}_{A\times C}\,f\,d\pi=\nu(A\times C).$ The rectangles form a $\pi$-system generating $\mathcal{R}\otimes\mathcal{T}$, and the total masses agree, so $P_\Phi=\nu$ by claim 1 of Uniqueness of Finite Measures on a Generating Pi-System and the Density of the Exponential Law. The displayed integral identity follows for indicators by this equality (using claim 3 of Image Measures, Measures with Densities, and Change of Variables to write $\nu$-values as $\pi$-integrals against $f$), for simple functions by linearity, and in general by the Monotone Convergence Theorem, together with the change of variables for the image measure on the left side.

Claim 2. The events $W^{-1}(A)$ lie in $\mathcal{G}_T$ by claim (f) of Measurable Reconstruction of the Controlled N-Agent Dynamics from Observation Records, and $\mathcal{G}_T$ contains every event of probability zero by its definition in Solution of the Controlled N-Agent Dynamics; this gives one inclusion. Conversely, $\mathcal{G}_T$ is generated by the variables $\Upsilon^\upsilon_s$ ($s\le T$) and the null events. On $\Omega_0$, condition 4 of Solution of the Controlled N-Agent Dynamics and condition 5 (the unique-channel identification of the events) give $\Upsilon^\upsilon_s=\frac{1}{N}\,\#\{j\le K_s:\upsilon_j=\upsilon\}$, which equals $\psi_{\upsilon,s}(W)$ for the $\mathcal{R}$-measurable counting map $\psi_{\upsilon,s}(r)=\frac{1}{N}\,\#\{j:t_j\le s,\ v_j=\upsilon\}$ (cellwise, a finite sum of indicators of coordinate conditions). Hence each $\Upsilon^\upsilon_s$ agrees off a null event with a measurable function of $W$, and is therefore measurable with respect to the $\sigma$-algebra generated by the events $W^{-1}(A)$ and the null events; this gives the other inclusion.

Claim 3. By claim 5 of Conditional Density of the Observation Record Given the Initial States and Transition Clocks, $P(p(W)=0)=\int_{\{p=0\}}p\,d\rho=0$, so $p(W)>0$ almost surely. The numerator $r\mapsto\mathbb{E}[\Psi(r,\cdot)f(r,\cdot)]$ is $\mathcal{R}$-measurable for bounded measurable $\Psi\ge0$ by the Tonelli theorem, hence for bounded real $\Psi$ by splitting into positive and negative parts, and it is bounded in absolute value by $(\sup|\Psi|)\,p(r)$ by monotonicity; so $\varphi_\Psi$ is $\mathcal{R}$-measurable with $|\varphi_\Psi|\le\sup|\Psi|$, and $\varphi_\Psi(W)$ is a bounded, hence square-integrable, $\mathcal{G}_T$-measurable random variable. It remains to verify the defining property of the conditional expectation: $\mathbb{E}[\Psi(W,\cdot)\,\mathbf{1}_{C'}]=\mathbb{E}[\varphi_\Psi(W)\,\mathbf{1}_{C'}]$ for every $C'\in\mathcal{G}_T$. The class of $C'\in\mathcal{F}$ whose symmetric difference with some $W^{-1}(A)$, $A\in\mathcal{R}$, is a null event forms a $\sigma$-algebra containing the generators of claim 2 and every null event, hence contains $\mathcal{G}_T$; both sides being unchanged by altering $C'$ on a null event, it suffices to take $C'=W^{-1}(A)$ with $A\in\mathcal{R}$, and by linearity to take $\Psi\ge0$. By claim 1 and Tonelli, $\mathbb{E}\bigl[\Psi(W,\cdot)\,\mathbf{1}_A(W)\bigr]=\int_A\mathbb{E}\bigl[\Psi(r,\cdot)f(r,\cdot)\bigr]\,\rho(dr)=\int_A\varphi_\Psi(r)\,p(r)\,\rho(dr),$ the last equality because on $\{p=0\}$ the integrand $\mathbb{E}[\Psi(r,\cdot)f(r,\cdot)]\le(\sup\Psi)\,p(r)$ vanishes; and by claim 5 of Conditional Density of the Observation Record Given the Initial States and Transition Clocks with the density $p$ and claim 3 of Image Measures, Measures with Densities, and Change of Variables, $\int_A\varphi_\Psi\,p\,d\rho=\mathbb{E}[\varphi_\Psi(W)\mathbf{1}_A(W)]$.

Claim 4. Restriction. The restricted policy is an observation-driven control policy with horizon $t$: the required measurability holds because $[0,t]\times R_j(t)$ is a Borel-trace subset of $[0,T]\times R_j(T)$, restrictions of measurable maps to measurable subsets remain measurable for the trace $\sigma$-algebras, and the relatively-open-generated and Borel-trace $\sigma$-algebras agree as in The Record-Frozen Control Path and Record-Frozen Policy. The restricted processes with the regular event $\Omega_0$ satisfy conditions 1--6 of Solution of the Controlled N-Agent Dynamics on $[0,t]$: conditions 1, 3, 4, and 6 restrict directly (the restriction of a counting path restriction is again one); condition 2 restricts since the consumed clock times on $[0,t]$ are the restrictions of the originals; and condition 5 restricts because for $s\le t$ the counts $K_s$, times $\tau_j$, and channels $\upsilon_j$ of the restricted observation total are those of the original, so the control identity is inherited with the restricted policy members. The observation filtration of the restriction at time $s\le t$ is generated by the same variables and null events as $\mathcal{G}_s$; and $W_t$ is $\mathcal{G}_t$-measurable by claim (f) of Measurable Reconstruction of the Controlled N-Agent Dynamics from Observation Records applied to the horizon-$t$ solution.

The formula. Apply claims 1--3 to the horizon-$t$ solution, with its record $W_t$, kernel $f^{(t)}$, and marginal density $p^{(t)}$; note that the $\sigma$-algebra $\mathcal{T}$ is the same (it is generated by the initial states and the full transition-clock paths, independently of the horizon). By claim (f) of Measurable Reconstruction of the Controlled N-Agent Dynamics from Observation Records for the horizon-$t$ solution, almost surely $\Sigma_t=\Sigma^{W_t,(t)}_t$, so $g(\Sigma_t)$ agrees almost surely with $\Psi(W_t,\cdot)$ for $\Psi(r,\omega)=g(\Sigma^{r,(t)}_t(\omega))$, which is bounded and $\mathcal{R}(t,\tilde{l})\otimes\mathcal{T}$-measurable: $(r,\omega)\mapsto\Sigma^{r,(t)}_t(\omega)$ is measurable by claim (a) of Measurable Reconstruction of the Controlled N-Agent Dynamics from Observation Records (evaluation at the fixed time $t$), and $g$ is Borel on $\mathbb{R}^l$, the empirical measures taking values there. Conditional expectations of almost surely equal square-integrable random variables coincide almost surely (apply the monotonicity of conditional expectation in both directions to the almost sure inequalities between the two variables), so, by claim 3 at horizon $t$, $\mathbb{E}\bigl[g(\Sigma_t)\bigm|\mathcal{G}_t\bigr]=\mathbb{E}\bigl[\Psi(W_t,\cdot)\bigm|\mathcal{G}_t\bigr]=\varphi^{(t)}_\Psi(W_t)$ almost surely, which is the displayed ratio, the denominator being positive almost surely by claim 3 at horizon $t$.