Proof of Law Identity between the N-Agent Aggregate Path with Its Observation Record and the Synthetic Copy's Regularised Path with Its Record

Step 0 (Conventions, the path functional, and two elementary facts). Write $E=\{1,\dots,l\}^N$ and $\Sigma:E\to\mathbb{G}_N$, $\Sigma(x)^\gamma=\frac1N\#\{i:x^i=\gamma\}$, as in Forward Equation on the Aggregate Lattice for the Reconstructed Record-Frozen N-Agent Dynamics, so that the empirical state measure of any family of $\{1,\dots,l\}$-valued agent states is its image under $\Sigma$ and lies in $\mathbb{G}_N$ (we write $\sigma_t=(\sigma^1_t,\dots,\sigma^N_t)$, $\sigma^r_u=(\sigma^{r,1}_u,\dots,\sigma^{r,N}_u)$ and $\varsigma_0=(\varsigma^1_0,\dots,\varsigma^N_0)$ for the $N$-tuples of agent states); in particular, by the derived notation of Solution of the Controlled N-Agent Dynamics, $\Sigma_t(\omega)=\Sigma(\sigma_t(\omega))\in\mathbb{G}_N$ for every $\omega\in\Omega$ and $t\in[0,T]$, so that $F'(\Sigma_s,W)$ in claim 3 is defined on all of $\Omega$. For $r=(k,\mathbf{t},v)\in\mathbf{R}$ define the path functional $\mathsf{f}_r:\mathsf{Path}\to[0,\infty)$ by

with $p(t-)$ the left limit of claim 2 of that lemma, the empty product equal to $1$, $\exp$ the exponential function, and the integral the Lebesgue integral over the compact interval $[0,T]$ of the bounded measurable map $u\mapsto\tilde{b}^{\mathrm{tot}}(p(u))$ (claim 3 of The Space of Piecewise Constant Paths in a Finite Set: Measurability of Evaluations, Restrictions, Left Limits, and Occupation Integrals applied to the restriction of $\tilde{b}^{\mathrm{tot}}$ to the finite set $\mathbb{G}_N$). The map $\mathsf{f}_r$ is $\mathcal{C}$-measurable: each factor $p\mapsto\tilde{b}^{v_j}(p(t_j-))$ takes finitely many values and equals $\sum_{y\in\mathbb{G}_N}\tilde{b}^{v_j}(y)\mathbf{1}\{p(t_j-)=y\}$, a linear combination of indicators of members of $\mathcal{C}$ (claim 2 of The Space of Piecewise Constant Paths in a Finite Set: Measurability of Evaluations, Restrictions, Left Limits, and Occupation Integrals); the map $p\mapsto\int_{[0,T]}\tilde{b}^{\mathrm{tot}}(p(u))\,du$ is $\mathcal{C}$-measurable by claim 3 of that lemma; the map $t\mapsto\exp(-Nt)$ is continuous on the real line, hence sequentially continuous (given $x_n\to x$ and $\varepsilon>0$, choose $\delta$ from the continuity at $x$ and then $n_0$ with $|x_n-x|<\delta$ for $n\ge n_0$), so its composition with the latter map is measurable by Sequentially Continuous Functions of Measurable Euclidean Maps are Measurable; and finite products of measurable real-valued maps are measurable by claim 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions. Two elementary facts: (N) if $\xi$ is a $[0,\infty]$-valued measurable map on a probability space and $Z_0$ is an event of probability zero, then $\int\xi\,\mathbf{1}_{Z_0}\le\sup_n\int n\,\mathbf{1}_{Z_0}=0$ by the monotone convergence theorem and monotonicity, so by additivity (both from claim 1 of Linearity and Monotonicity of the Lebesgue Integral) $\int\xi=\int\xi\,\mathbf{1}_{\Omega'}$ for the complement $\Omega'$ of $Z_0$; consequently two $[0,\infty]$-valued measurable maps that agree outside an event of probability zero have the same integral. (M) The product of two $[0,\infty]$-valued measurable maps $\xi,\zeta$ is measurable, since $\{\xi\zeta>a\}=\bigcup_q(\{\xi>q\}\cap\{\zeta>a/q\})$ over the positive rationals $q$ when $a\ge0$, and is the whole space when $a<0$; and a map into a generated $\sigma$-algebra is measurable as soon as the preimages of the generators are measurable, the sets with measurable preimage forming a $\sigma$-algebra. In particular a pair map $\omega\mapsto(\Phi_1(\omega),\Phi_2(\omega))$ into a product is measurable for the product $\sigma$-algebra as soon as its components are, the preimage of a measurable rectangle being the intersection of the two preimages.

Step 1 (Claim 1). The $\Sigma^\gamma_u$ are $\mathcal{F}^{\mathrm{sys}}_u$-measurable by clause (iv) of Existence, Uniqueness, and Regularity for the Controlled N-Agent Dynamics, so $\{\Sigma_u=y\}\in\mathcal{F}$ for all $u\in[0,T]$ and $y\in\mathbb{R}^l$; in particular $D_0\in\mathcal{F}$. Let $\omega\in\Omega_0$. By condition 1 of Solution of the Controlled N-Agent Dynamics, for each $i$ the path $t\mapsto\sigma^i_t(\omega)$ is constant on the intervals determined by finitely many times $0<t^{(i)}_1<\dots<t^{(i)}_{K^{(i)}}\le T$; let $0<s_1<\dots<s_M\le T$ enumerate the union of these finite sets. Each of the intervals $[0,s_1)$, $[s_j,s_{j+1})$, $[s_M,T]$ (or $[0,T]$ if $M=0$) contains no $t^{(i)}_{j'}$ except possibly at its left endpoint, so it is contained in one of the constancy intervals of each $\sigma^i$; hence all $\sigma^i$, and therefore $\Sigma_\cdot(\omega)=\Sigma(\sigma_\cdot(\omega))$, are constant on it. Thus $\Pi(\omega)\in\mathsf{Path}$, with values in $\mathbb{G}_N$ by Step 0; for $\omega\notin\Omega_0$, $\Pi(\omega)$ is constant. For a generator $\{p:p(u)=y\}$ of $\mathcal{C}$, $\{\Pi\in\{p:p(u)=y\}\}=(\Omega_0\cap\{\Sigma_u=y\})\cup((\Omega\setminus\Omega_0)\cap\{\omega:x_0=y\})\in\mathcal{F}$, the last set being $\Omega\setminus\Omega_0$ or empty; so $\Pi$ is measurable by (M), $\Omega_0$ being an event by Solution of the Controlled N-Agent Dynamics. For $\Pi^\sharp$: by claim 3 of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record, the paths $\bar{\Sigma}^{\sharp,r}(\omega^\flat)$ are the regularised recursion paths of The Record-Driven Causal Intensity of the Open-Loop Aggregate Solution: Joint Measurability of the Record-Frozen Control, the Regularised Recursion Path, Non-Anticipation, and Measurability of the Likelihood for the clocks $\mathsf{P}^\sharp$; by claim 2(a) of that lemma each such path takes values in $\mathbb{G}_N$ and is constant on $[\theta_k,\theta_{k+1})$ ($k<K$) and on $[\theta_K,T]$ for the times $0=\vartheta_0<\vartheta_1<\dots<\vartheta_K\le T$ of the recursion (written $\theta_k$ there; here $\theta$ is reserved for the $\mathbb{R}^d$-coordinate of $\Omega^\sharp$), so it lies in $\mathsf{Path}$ with the times $\vartheta_1,\dots,\vartheta_K$. By claim 2(d) of that lemma, for each $\gamma$ the map $(t,r,\omega^\flat)\mapsto\bar{\Sigma}^{\sharp,r,\gamma}_t(\omega^\flat)$ is $\mathcal{B}_{[0,T]}\otimes(\mathcal{R}\otimes\mathcal{F}^\flat)$-measurable; its section at $t=u$ is $\mathcal{R}\otimes\mathcal{F}^\flat$-measurable by claim 3 of Sections of Product-Measurable Sets and Maps Are Measurable, and Insertion Maps into Products Are Measurable, and its composition with the rearrangement $((\omega^\flat,\theta),r)\mapsto(r,\omega^\flat)$, measurable from $\mathcal{F}^\sharp$ to $\mathcal{R}\otimes\mathcal{F}^\flat$ by claim 5 of that lemma, is $\mathcal{F}^\sharp$-measurable. Hence $\{\Pi^\sharp\in\{p:p(u)=y\}\}=\bigcap_{\gamma}\{(\omega^\flat,\theta,r):\bar{\Sigma}^{\sharp,r,\gamma}_u(\omega^\flat)=y^\gamma\}\in\mathcal{F}^\sharp$, and $\Pi^\sharp$ is measurable by (M). Finally $W$ is measurable by clause (f) of Measurable Reconstruction of the Controlled N-Agent Dynamics from Observation Records, and $\mathsf{D}$ by claim 4 of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record.

Step 2 (The $N$-agent side). Let $\mathcal{T}$ be the $\sigma$-algebra generated by the initial states and the transition-clock variables, as in Measurable Reconstruction of the Controlled N-Agent Dynamics from Observation Records, and define $\Pi^G$ on $\mathbf{R}\times\Omega$ by $\Pi^G(r,\omega)(u)=\Sigma^r_u(\omega)$ for $(r,\omega)\in G$ and $\Pi^G(r,\omega)(u)=x_0$ otherwise. For $(r,\omega)\in G$, clause (c) of Measurable Reconstruction of the Controlled N-Agent Dynamics from Observation Records gives $\eta^{r,i,\gamma}_u(\omega)=\mathbf{1}\{\sigma^{r,i}_u(\omega)=\gamma\}$, so $\Sigma^r_u(\omega)=\Sigma(\sigma^r_u(\omega))\in\mathbb{G}_N$, and each path $u\mapsto\sigma^{r,i}_u(\omega)$ is piecewise constant in the sense of condition 1 of Solution of the Controlled N-Agent Dynamics; the union-of-times argument of Step 1 shows $\Pi^G(r,\omega)\in\mathsf{Path}$. Off $G$ the path is constant. For a generator $\{p:p(u)=y\}$, $\{\Pi^G\in\{p:p(u)=y\}\}=(G\cap\{(r,\omega):\Sigma^r_u(\omega)=y\})\cup(((\mathbf{R}\times\Omega)\setminus G)\cap\{(r,\omega):x_0=y\})$; here $G\in\mathcal{R}\otimes\mathcal{T}$, and $(r,\omega)\mapsto\eta^{r,i,\gamma}_u(\omega)$ is $\mathcal{R}\otimes\mathcal{T}$-measurable by clause (a) of Measurable Reconstruction of the Controlled N-Agent Dynamics from Observation Records and claim 4 of Sections of Product-Measurable Sets and Maps Are Measurable, and Insertion Maps into Products Are Measurable (the insertion $(r,\omega)\mapsto((r,u),\omega)$ with the middle coordinate fixed), so $\{(r,\omega):\Sigma^r_u(\omega)=y\}=\bigcap_\gamma\{(r,\omega):\frac1N\sum_i\eta^{r,i,\gamma}_u(\omega)=y^\gamma\}\in\mathcal{R}\otimes\mathcal{T}$. Thus $\Pi^G$ is measurable with respect to $\mathcal{R}\otimes\mathcal{T}$ and $\mathcal{C}$ by (M). Put $D'_0=\{\omega:\Sigma(\varsigma^1_0(\omega),\dots,\varsigma^N_0(\omega))=x_0\}$, a member of $\mathcal{T}$ since the $\varsigma^i_0$ are among the generators of $\mathcal{T}$, and define

which is $\mathcal{R}\otimes\mathcal{T}$-measurable and $[0,\infty]$-valued: the pair map $(r,\omega)\mapsto(\Pi^G(r,\omega),r)$ is measurable into $\mathcal{C}\otimes\mathcal{R}$ by (M), so $F$ composed with it is measurable, and the factor $\mathbf{1}_{D'_0}(\omega)$ is the indicator of the rectangle $\mathbf{R}\times D'_0$.

Reduction to $H$. Let $\Omega_1$ be the intersection of $\Omega_0$ with the event of probability one of clause (f) of Measurable Reconstruction of the Controlled N-Agent Dynamics from Observation Records; $P(\Omega_1)=1$. For $\omega\in\Omega_1$: $(W(\omega),\omega)\in G$ and $\eta^{i,\gamma}_u(\omega)=\eta^{W(\omega),i,\gamma}_u(\omega)$ for all $u,i,\gamma$, so $\Sigma_u(\omega)=\Sigma^{W(\omega)}_u(\omega)$ for all $u$ and $\Pi(\omega)=\Pi^G(W(\omega),\omega)$; and $\sigma^i_0(\omega)=\varsigma^i_0(\omega)$ by condition 1 of Solution of the Controlled N-Agent Dynamics, so $\Sigma_0(\omega)=\Sigma(\varsigma_0(\omega))$ and $\mathbf{1}_{D_0}(\omega)=\mathbf{1}_{D'_0}(\omega)$. Hence $\mathbf{1}_{D_0}F(\Pi,W)=H(W,\cdot)$ on $\Omega_1$, and by (N)

The record density. Let $f:\mathbf{R}\times\Omega\to[0,\infty)$ be the record density kernel of Conditional Density of the Observation Record Given the Initial States and Transition Clocks for the fixed reconstruction data. By claim 1 of that lemma $\rho$ is finite, hence $\sigma$-finite; by claim 2, $f$ is $\mathcal{R}\otimes\mathcal{T}$-measurable; and claim 3 (with $Z=\mathbf{1}_C$ and $g=\mathbf{1}_A$) gives the hypothesis of A Conditional Density Identity for Products Extends to Jointly Measurable Integrands for the sub-$\sigma$-algebra $\mathcal{T}$, the map $W$ and the kernel $f$. That lemma yields

the second equality by the Tonelli theorem on the product of $(\mathbf{R},\mathcal{R},\rho)$ and $(\Omega,\mathcal{F},P)$, the map $Hf$ being measurable for $\mathcal{R}\otimes\mathcal{T}\subseteq\mathcal{R}\otimes\mathcal{F}$ (every measurable rectangle of the former is one of the latter) and by (M); both iterated integrals equal the integral against the product measure, and Tonelli also gives the $\mathcal{R}$-measurability of $r\mapsto\mathbb{E}[H(r,\cdot)f(r,\cdot)]$.

The integrand at a fixed record. Fix $r=(k,\mathbf{t},v)\in\mathbf{R}$ and let $\hat{\Sigma}^r$, $\Omega^r$ and the map $\Pi^r$ be those of Forward Equation on the Aggregate Lattice for the Reconstructed Record-Frozen N-Agent Dynamics and of claim 2 of Law Identity between the Reconstructed Record-Frozen Aggregate and the Aggregate Recursion Driven by the Same Control Path with $T'=T$ and the point $x_0$ ($\Pi^r$ is defined on the $N$-agent side alone: it is the path $u\mapsto\hat{\Sigma}^r_u$ on $\Omega^r$ and the constant path $x_0$ off $\Omega^r$). For $\omega\in\Omega^r$ we have $(r,\omega)\in G$ (clause (d) of Measurable Reconstruction of the Controlled N-Agent Dynamics from Observation Records), hence: $\Pi^G(r,\omega)$ is the path $u\mapsto\Sigma^r_u(\omega)=\hat{\Sigma}^r_u(\omega)$ (claim 1 of Forward Equation on the Aggregate Lattice for the Reconstructed Record-Frozen N-Agent Dynamics), which is $\Pi^r(\omega)$; $\mathbf{1}_{D'_0}(\omega)=\mathbf{1}\{\hat{\Sigma}^r_0(\omega)=x_0\}$, because $\sigma^{r,i}_0(\omega)=\varsigma^i_0(\omega)$ by clause (c); and, by the definition of the kernel on $G$,

since the left limit $\Sigma^r_{t_j-}(\omega)$ of the path (the common value on some $[t_j-\delta,t_j)$) is the left limit $p(t_j-)$ of claim 2 of The Space of Piecewise Constant Paths in a Finite Set: Measurability of Evaluations, Restrictions, Left Limits, and Occupation Integrals for $p=\Pi^G(r,\omega)$. Put $\Phi_r(p)=F(p,r)\,\mathsf{f}_r(p)$ for $p\in\mathsf{Path}$; $p\mapsto F(p,r)$ is $\mathcal{C}$-measurable by claim 3 of Sections of Product-Measurable Sets and Maps Are Measurable, and Insertion Maps into Products Are Measurable (section at $r$), so $\Phi_r$ is $\mathcal{C}$-measurable by Step 0 and (M). Since $P(\Omega^r)=1$, (N) gives

Transfer to the copy clocks. Apply Law Identity between the Reconstructed Record-Frozen Aggregate and the Aggregate Recursion Driven by the Same Control Path with $T'=T$, the point $x_0$, and the copy setting consisting of the probability space $(\Omega^\flat,\mathcal{F}^\flat,P^\flat)$ carrying the copy clocks $\mathsf{P}^\sharp$, that is, the space called $(\Omega^\sharp,\mathcal{F}^\sharp,P^\sharp)$ in that lemma is here $(\Omega^\flat,\mathcal{F}^\flat,P^\flat)$, its conflict-free event $\Omega^\sharp_0$ is here $\Omega^\flat_{0,r}$, its regularised path $\bar{\Sigma}^\sharp$ is here $\bar{\Sigma}^{\sharp,r}$, and its map $\Pi^\sharp$ is here written $\Pi^{\flat,r}$, while the symbols $\Omega^\sharp$, $\mathcal{F}^\sharp$ and $\Pi^\sharp$ keep the meanings fixed in the statement: the copy clocks are Poisson clocks with horizon $R$ with independent generated $\sigma$-algebras by claims 2 and 3 of The Copy Clocks Are Independent Poisson Clocks with Horizon R, and Every Record Is Almost Surely Conflict-Free for Them, $R>NBT$ by hypothesis, the aggregate recursion for $(\mathsf{P}^\sharp(\omega^\flat),a^r,x_0)$ has the regularised recursion path $u\mapsto\bar{\Sigma}^{\sharp,r}_u(\omega^\flat)$ (the same regularisation, replacing values outside $\mathbb{G}_N$ by $x_0$, is used in Forward Equation for the Aggregate Recursion Driven by Independent Poisson Clocks with a Horizon and in claim 2 of The Record-Driven Causal Intensity of the Open-Loop Aggregate Solution: Joint Measurability of the Record-Frozen Control, the Regularised Recursion Path, Non-Anticipation, and Measurability of the Likelihood), and its conflict-free event $\Omega^\flat_{0,r}=\{\omega^\flat:(r,\omega^\flat)\in\mathsf{G}^\sharp\}$ has $P^\flat(\Omega^\flat_{0,r})=1$ by claim 4 of The Copy Clocks Are Independent Poisson Clocks with Horizon R, and Every Record Is Almost Surely Conflict-Free for Them. Claim 2 of Law Identity between the Reconstructed Record-Frozen Aggregate and the Aggregate Recursion Driven by the Same Control Path gives, with $\Pi^{\flat,r}$ (equal to $u\mapsto\bar{\Sigma}^{\sharp,r}_u(\omega^\flat)$ on $\Omega^\flat_{0,r}$),

where $\bar{\Sigma}^{\sharp,r}(\omega^\flat)$ denotes the path $u\mapsto\bar{\Sigma}^{\sharp,r}_u(\omega^\flat)$, using (N) on $\Omega^\flat$ (the integrands agree on $\Omega^\flat_{0,r}$; the second integrand is $\mathcal{F}^\flat$-measurable as $\Phi_r$ composed with the section at $\theta$ and $r$ of the measurable map $\Pi^\sharp$ of Step 1, claims 2 and 4 of Sections of Product-Measurable Sets and Maps Are Measurable, and Insertion Maps into Products Are Measurable), and using $P(\hat{\Sigma}^r_0=x_0)=P(D'_0)=P(D_0)$, the three events agreeing on $\Omega^r\cap\Omega_1$. The map $(r,\omega^\flat)\mapsto F(\bar{\Sigma}^{\sharp,r}(\omega^\flat),r)\,\mathsf{f}_r(\bar{\Sigma}^{\sharp,r}(\omega^\flat))$ is $\mathcal{R}\otimes\mathcal{F}^\flat$-measurable: the pair map $(r,\omega^\flat)\mapsto(\bar{\Sigma}^{\sharp,r}(\omega^\flat),r)$ is measurable into $\mathcal{C}\otimes\mathcal{R}$ by (M), the preimage of a generator $\{p:p(u)=y\}$ being $\{(r,\omega^\flat):\bar{\Sigma}^{\sharp,r}_u(\omega^\flat)=y\}\in\mathcal{R}\otimes\mathcal{F}^\flat$ as shown in Step 1, and the second factor equals $\ell^{\sharp,\omega^\flat}(r)$ by the identification proved in Step 3 below (which does not depend on this step), a map that is $\mathcal{R}\otimes\mathcal{F}^\flat$-measurable by claim 3 of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record; hence, by the Tonelli theorem, $r\mapsto\int_{\Omega^\flat}F(\bar{\Sigma}^{\sharp,r}(\omega^\flat),r)\,\mathsf{f}_r(\bar{\Sigma}^{\sharp,r}(\omega^\flat))\,P^\flat(d\omega^\flat)$ is $\mathcal{R}$-measurable. Combining (1)--(4) and pulling the constant $P(D_0)\in[0,1]$ out of the $\rho$-integral (claim 1 of Linearity and Monotonicity of the Lebesgue Integral; when $P(D_0)=0$ the left side of (1) vanishes by (N) and the right side of the display below vanishes by the convention $0\cdot\infty=0$),

Step 3 (The copy side). The map $(\omega^\flat,\theta,r)\mapsto(\Pi^\sharp(\omega^\flat,\theta,r),r)$ is measurable from $\mathcal{F}^\sharp$ to $\mathcal{C}\otimes\mathcal{R}$ by Step 1 and (M), so $F(\Pi^\sharp,\mathsf{D})$ is $\mathcal{F}^\sharp$-measurable. By claim 4 of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record, $\mu^\sharp$ is the measure with density $\mathsf{q}^\sharp(\omega^\flat,\theta,r)=\varphi_\eta(\theta-\mathsf{K}(\omega^\flat)/\sqrt{N})\,\ell^{\sharp,\omega^\flat}(r)$ with respect to $(P^\flat\otimes\lambda_d)\otimes\rho$, so by claim 3 of Image Measures, Measures with Densities, and Change of Variables and the Tonelli theorem on the product of the $\sigma$-finite spaces $(\Omega^\flat\times\mathbb{R}^d,\mathcal{F}^\flat\otimes\mathcal{B}(\mathbb{R}^d),P^\flat\otimes\lambda_d)$ (the product measure of the finite $P^\flat$ and the $\sigma$-finite $\lambda_d$, itself $\sigma$-finite) and $(\mathbf{R},\mathcal{R},\rho)$ ($\rho$ finite),

For fixed $r$ the inner integrand is $\mathcal{F}^\flat\otimes\mathcal{B}(\mathbb{R}^d)$-measurable (claim 3 of Sections of Product-Measurable Sets and Maps Are Measurable, and Insertion Maps into Products Are Measurable), and Tonelli on the product of $(\Omega^\flat,\mathcal{F}^\flat,P^\flat)$ and $(\mathbb{R}^d,\mathcal{B}(\mathbb{R}^d),\lambda_d)$ writes the inner integral as $\int_{\Omega^\flat}\bigl(\int_{\mathbb{R}^d}\kappa(\omega^\flat)\,\varphi_\eta(\theta-\mathsf{K}(\omega^\flat)/\sqrt{N})\,\lambda_d(d\theta)\bigr)P^\flat(d\omega^\flat)$ with $\kappa(\omega^\flat)=F(\bar{\Sigma}^{\sharp,r}(\omega^\flat),r)\,\ell^{\sharp,\omega^\flat}(r)\in[0,\infty]$. For every $a\in\mathbb{R}^d$, $\int_{\mathbb{R}^d}\varphi_\eta(\theta-a)\,\lambda_d(d\theta)=1$ by claim 1 of The Gaussian Smoothing Weight: Normalization, Derivatives, Exponential Tilting, Moments, and First-Order Remainder; hence $\int_{\mathbb{R}^d}\kappa\,\varphi_\eta(\theta-a)\,\lambda_d(d\theta)=\kappa$ for $\kappa\in[0,\infty)$ by claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and also for $\kappa=\infty$, since then the integrand is the supremum of the nondecreasing sequence $n\varphi_\eta(\theta-a)$ ($\varphi_\eta>0$ everywhere by claim 1 of The Gaussian Smoothing Weight: Normalization, Derivatives, Exponential Tilting, Moments, and First-Order Remainder) and the monotone convergence theorem gives $\sup_nn=\infty$. Therefore

The likelihood is the path functional. Fix $r=(k,\mathbf{t},v)$ and $\omega^\flat$, and write $p$ for the path $u\mapsto\bar{\Sigma}^{\sharp,r}_u(\omega^\flat)$. By claim 3 of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record, $\ell^{\sharp,\omega^\flat}$ is the record-driven likelihood of The Record-Driven Causal Intensity of the Open-Loop Aggregate Solution: Joint Measurability of the Record-Frozen Control, the Regularised Recursion Path, Non-Anticipation, and Measurability of the Likelihood for the clocks $\mathsf{P}^\sharp$, that is, by claim 4 of that lemma and Likelihood of a Causal Intensity on the Observation Record Space, the likelihood of the causal intensity $\lambda^{\upsilon}_t(r)=N\tilde{b}^\upsilon(\bar{\Sigma}^{\sharp,r}_{t-}(\omega^\flat))$ of its claim 3, with total intensity $\lambda^{\mathrm{tot}}_u(r)=N\tilde{b}^{\mathrm{tot}}(\bar{\Sigma}^{\sharp,r}_{u-}(\omega^\flat))$:

The left limit $\bar{\Sigma}^{\sharp,r}_{t-}(\omega^\flat)$ of claim 2(b) of The Record-Driven Causal Intensity of the Open-Loop Aggregate Solution: Joint Measurability of the Record-Frozen Control, the Regularised Recursion Path, Non-Anticipation, and Measurability of the Likelihood is defined by the same property as the left limit $p(t-)$ of claim 2 of The Space of Piecewise Constant Paths in a Finite Set: Measurability of Evaluations, Restrictions, Left Limits, and Occupation Integrals, so they coincide; and by claim 2(b) the maps $u\mapsto\tilde{b}^{\mathrm{tot}}(p(u-))$ and $u\mapsto\tilde{b}^{\mathrm{tot}}(p(u))$ on $[0,T]$ (the first read with $p(0-)=x_0$, which is $p(0)$: the recursion has $\vartheta_0=0<\vartheta_1$ and $x^{(0)}=x_0\in\mathbb{G}_N$, so $\bar{\Sigma}^{\sharp,r}_0(\omega^\flat)=x_0$) agree outside the finite set $\{\vartheta_1,\dots,\vartheta_K\}$, which is a $\lambda_{[0,T]}$-null set by Existence of Lebesgue Measure on the Real Line (points have Lebesgue measure zero); both maps are nonnegative and $\mathcal{B}_{[0,T]}$-measurable (the first as the integrand of the likelihood, the second by claim 3 of The Space of Piecewise Constant Paths in a Finite Set: Measurability of Evaluations, Restrictions, Left Limits, and Occupation Integrals), so their integrals over $[0,T]$ agree by claim 6 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval, applied to each with the co-null set $D$ equal to the complement of $\{\vartheta_1,\dots,\vartheta_K\}$, on which the two products with $\mathbf{1}_D$ coincide. Pulling out the constant $N$ (claim 1 of Linearity and Monotonicity of the Lebesgue Integral), we conclude $\ell^{\sharp,\omega^\flat}(r)=\mathsf{f}_r(p)$ for every $r$ and $\omega^\flat$.

Step 4 (Conclusion). Substituting $\ell^{\sharp,\omega^\flat}(r)=\mathsf{f}_r(\bar{\Sigma}^{\sharp,r}(\omega^\flat))$ into (6) shows that the right side of (6), multiplied by $P(D_0)$, is the right side of (5). This proves claim 2. For claim 3, let $s\in[0,T]$ and $F'$ be as stated and put $F(p,r)=F'(p(s),r)$; the map $(p,r)\mapsto(p(s),r)$ is measurable from $\mathcal{C}\otimes\mathcal{R}$ to the product of the $\sigma$-algebra of all subsets of $\mathbb{G}_N$ and $\mathcal{R}$ (its components are the evaluation at $s$, measurable by claim 1 of The Space of Piecewise Constant Paths in a Finite Set: Measurability of Evaluations, Restrictions, Left Limits, and Occupation Integrals composed with the coordinate projection of claim 5 of Sections of Product-Measurable Sets and Maps Are Measurable, and Insertion Maps into Products Are Measurable, and the other coordinate projection), so $F$ is $\mathcal{C}\otimes\mathcal{R}$-measurable. On $\Omega_0$, $F(\Pi,W)=F'(\Sigma_s,W)$, so $\mathbb{E}[\mathbf{1}_{D_0}F(\Pi,W)]=\mathbb{E}[\mathbf{1}_{D_0}F'(\Sigma_s,W)]$ by (N) (the map $\omega\mapsto F'(\Sigma_s(\omega),W(\omega))$ being $\mathcal{F}$-measurable by (M), as $\{\Sigma_s=y\}\in\mathcal{F}$); and $F(\Pi^\sharp,\mathsf{D})=F'(\bar{\Sigma}^{\sharp,\mathsf{D}}_s,\mathsf{D})$ everywhere. Claim 2 for this $F$ is the identity of claim 3. If $P(D_0)=1$ then $\mathbf{1}_{D_0}=1$ outside an event of probability zero, so, taking $F'=\mathbf{1}_S$ for $S$ in the product $\sigma$-algebra and using (N), $P((\Sigma_s,W)\in S)=\mu^\sharp((\bar{\Sigma}^{\sharp,\mathsf{D}}_s,\mathsf{D})\in S)$, the two pair maps being measurable by (M); this is the equality of the two image measures.