Proof of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record

Throughout, measurable for a real-valued map means measurable with respect to the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R})$ on the target, and $\mathcal{B}_d=\mathcal{B}(\mathbb{R}^d)$ by claim 5 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets. We write $\mathcal{G}_c$ for the $\sigma$-algebra generated by $K^{c}$ and the $V^{c}_i$ ($i\in\mathbb{N}$), and $\mathcal{U}_c$ for the one generated by the $U^{c,j}_i$ ($1\le j\le J_c$, $i\in\mathbb{N}$); thus $\mathcal{G}_c\subseteq\mathcal{V}$ and $\mathcal{U}_c\subseteq\mathcal{U}$.

Preliminaries.

(Q1) Generator criterion and projections. If $\phi:(X,\mathcal{G})\to Y$ is a map into a set carrying the $\sigma$-algebra generated by a family $\mathcal{E}$, and $\phi^{-1}(E)\in\mathcal{G}$ for all $E\in\mathcal{E}$, then $\phi$ is measurable: the sets $A\subseteq Y$ with $\phi^{-1}(A)\in\mathcal{G}$ form a $\sigma$-algebra containing $\mathcal{E}$, hence containing the generated $\sigma$-algebra by Intersections of Sigma-Algebras and Minimality of the Generated Sigma-Algebra. A product $\sigma$-algebra is generated by its measurable rectangles, so the coordinate projections of a product are measurable, and a map into a product is measurable as soon as the preimages of rectangles are measurable. In particular the maps $(\omega,\theta,r)\mapsto\omega$, $\mapsto\theta$, $\mapsto r$, $\mapsto(\omega,\theta)$, $\mapsto(\theta,r)$ and $\mapsto(r,\omega)$ on $\Omega^{\sharp}$ are measurable from $\mathcal{F}^{\sharp}$ to $\mathcal{F}$, $\mathcal{B}_d$, $\mathcal{R}$, $\mathcal{F}\otimes\mathcal{B}_d$, $\mathcal{B}_d\otimes\mathcal{R}$ and $\mathcal{R}\otimes\mathcal{F}$ respectively: the preimage of a rectangle is in each case a set of the form $A\times B\times C'$ with $A\in\mathcal{F}$, $B\in\mathcal{B}_d$, $C'\in\mathcal{R}$, which lies in $\mathcal{F}^{\sharp}=(\mathcal{F}\otimes\mathcal{B}_d)\otimes\mathcal{R}$ as the rectangle $(A\times B)\times C'$. Compositions of measurable maps are measurable.

(Q2) Countable sums under the expectation. If $Z_n:\Omega\to[0,\infty]$ ($n\in\mathbb{N}$) are $\mathcal{F}$-measurable, then $\sum_nZ_n$ (a pointwise least upper bound of the partial sums) is measurable and $\mathbb{E}[\sum_nZ_n]=\sum_n\mathbb{E}[Z_n]$: the partial sums increase to the sum, their expectations are the partial sums of the right side by Linearity and Monotonicity of the Lebesgue Integral, and Monotone Convergence Theorem gives the identity. The same holds for integrals with respect to any measure.

(Q3) Countability. $\mathbb{N}_0^{\mathsf{L}}$ is countable in the sense of Assembly of Measure Spaces: Restriction, Transport, One-Point Spaces, and Countable Disjoint Unions: for each $n\in\mathbb{N}_0$ the set of $y\in\mathbb{N}_0^{\mathsf{L}}$ with $\sum_{(c,j)}y_{c,j}=n$ is finite, and listing these finite sets one after the other yields a sequence whose set of values is $\mathbb{N}_0^{\mathsf{L}}$. Sums of nonnegative terms over $\mathbb{N}_0^{\mathsf{L}}$, or over $\mathsf{S}$, are least upper bounds of finite partial sums (Sum of a Nonnegative Function over an Arbitrary Set); they are unchanged by bijective reindexing, and by the preamble of Assembly of Measure Spaces: Restriction, Transport, One-Point Spaces, and Countable Disjoint Unions they agree with the sequential sums along any enumeration.

Step 1 (claim 1). Independence. Enumerate the driving family by a bijection with $\mathbb{N}$ (independence of a family is unaffected by reindexing, being a condition on finite subfamilies by Independence of Events and of Random Variables). The $\sigma$-algebras $\mathcal{V}$ and $\mathcal{U}$ are generated by disjoint subfamilies, so they are independent by Grouping Lemma for Independent Random Variables; likewise the family $(\mathcal{G}_c)_c$ is independent, and so is $(\mathcal{U}_c)_c$.

The event $\Omega^{U}_0$. It is the complement of the union of the countably many events $\{U^{c,j}_i=U^{c,j'}_{i'}\}$ ($(j,i)\neq(j',i')$) and $\{U^{c,j}_i\notin I_{c,j}\}$, over all labels $c$; each of these is in $\mathcal{U}$ (the first is the zero set of the measurable difference $U^{c,j}_i-U^{c,j'}_{i'}$, by claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, the second a preimage of a Borel set), so $\Omega^{U}_0\in\mathcal{U}$. For each label $c$ the event $\Omega^{c}_0$ of Uniform Representation of the Rate-One Poisson Counting Path on a Bounded Interval: Distinct Points, Poisson Increments, Conditional Law Given the Cell Counts, and Point Insertion is contained in the complement of the union of those of these events that concern the label $c$, so $\bigcap_c\Omega^{c}_0\subseteq\Omega^{U}_0$; each $\Omega^{c}_0$ has probability one by claim 1 of that lemma, hence so has their finite intersection (the complement is a finite union of null events, of measure at most the sum of their measures by Basic Properties of a Measure), and $P(\Omega^{U}_0)=1$.

The cell counts. Write $\mathsf{K}_{c,\cdot}=(\mathsf{K}_{c,1},\dots,\mathsf{K}_{c,J_c})$, the cell-count vector of the label $c$ (the vector $C$ of Uniform Representation of the Rate-One Poisson Counting Path on a Bounded Interval: Distinct Points, Poisson Increments, Conditional Law Given the Cell Counts, and Point Insertion for that label). By claim 2 of that lemma applied to the label $c$, each $\mathsf{K}_{c,j}$ is a random variable with values in $\mathbb{N}_0$, the counts $\mathsf{K}_{c,1},\dots,\mathsf{K}_{c,J_c}$ are independent with $\mathsf{K}_{c,j}$ Poisson with parameter $|I_{c,j}|$, and $P(\mathsf{K}_{c,\cdot}=y_{c,\cdot})=\prod_{j}\mathrm{poi}_{|I_{c,j}|}(y_{c,j})>0$ for every $y_{c,\cdot}\in\mathbb{N}_0^{J_c}$. Each $\mathsf{K}_{c,j}=\sum_{i\in\mathbb{N}}\mathbf{1}\{i\le\widetilde{K}^{c}\}\mathbf{1}\{V^{c}_i\in I_{c,j}\}$ is the pointwise limit of its partial sums, which are $\mathcal{G}_c$-measurable ($\widetilde{K}^{c}$ is a function of $K^{c}$), hence $\mathsf{K}_{c,j}$ is $\mathcal{G}_c$-measurable by claim 5 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, and a fortiori $\mathcal{V}$-measurable. Its expectation is $|I_{c,j}|$ by claim (a) of Factorial Moments and Moments of Every Order of the Poisson Distribution with the order there equal to $1$. For $y\in\mathbb{N}_0^{\mathsf{L}}$, the event $\{\mathsf{K}=y\}$ is the intersection over $c$ of the events $\{\mathsf{K}_{c,\cdot}=y_{c,\cdot}\}\in\mathcal{G}_c$; by the independence of $(\mathcal{G}_c)_c$, $P(\mathsf{K}=y)=\prod_{c}P(\mathsf{K}_{c,\cdot}=y_{c,\cdot})=\prod_{(c,j)\in\mathsf{L}}\mathrm{poi}_{|I_{c,j}|}(y_{c,j})>0 .$ The events $\{\mathsf{K}=y\}$, $y\in\mathbb{N}_0^{\mathsf{L}}$, are pairwise disjoint with union $\Omega$ (as $\mathsf{K}$ takes values in $\mathbb{N}_0^{\mathsf{L}}$), and countably many by (Q3); countable additivity gives $\sum_yP(\mathsf{K}=y)=1$.

Factorisation. Let $Z:\Omega\to[0,\infty]$ be $\mathcal{U}$-measurable. For $n\in\mathbb{N}$ put $Z_n=\min(Z,n)$, a bounded $\mathcal{U}$-measurable random variable: for a real $a$, $\{Z_n>a\}=\{Z>a\}\in\mathcal{U}$ if $a<n$ and $\{Z_n>a\}=\emptyset$ if $a\ge n$, and the sets $(a,\infty)$ generate the Borel $\sigma$-algebra (Borel Sigma-Algebra on the Real Line). The indicator $\mathbf{1}\{\mathsf{K}=y\}$ is $\mathcal{V}$-measurable, so $\mathbf{1}\{\mathsf{K}=y\}$ and $Z_n$ are independent random variables by Grouping Lemma for Independent Random Variables (its final assertion, applied to the generating subfamilies of $\mathcal{V}$ and $\mathcal{U}$), both with finite expectation; Expectation of a Product of Independent Random Variables gives $\mathbb{E}[\mathbf{1}\{\mathsf{K}=y\}Z_n]=P(\mathsf{K}=y)\mathbb{E}[Z_n]$. As $n\to\infty$, $Z_n$ and $\mathbf{1}\{\mathsf{K}=y\}Z_n$ increase pointwise to $Z$ and $\mathbf{1}\{\mathsf{K}=y\}Z$, and Monotone Convergence Theorem applied to both sides yields the identity in $[0,\infty]$.

Step 2 (claim 2). Deterministic-count clocks. Fix $y$ and $c$. Off $\Omega^{U}_0$ the path of $\mathsf{P}^{(y),c}$ is identically $0$, a counting path. On $\Omega^{U}_0$ it is $u\mapsto\sum_{q=1}^{n}\mathbf{1}\{W_q\le u\}$, where $W_1,\dots,W_n$ ($n=\sum_jy_{c,j}$) are the points $U^{c,j}_i$ with $1\le j\le J_c$, $1\le i\le y_{c,j}$, which are pairwise distinct and lie in $(0,R]$. We verify Counting Path and Its Jump Times: the value at $0$ is $0$ because every $W_q>0$; the values are in $\mathbb{N}_0$; the path is nondecreasing because each summand is; for right-continuity at $u\ge0$, let $\delta>0$ be smaller than every $W_q-u$ with $W_q>u$ (or $\delta=1$ if there is none), then every summand, hence the sum, is constant on $[u,u+\delta)$, so the value at $u$ is the greatest lower bound over $(u,\infty)$ of a nondecreasing path; for the unit-jump condition at $t>0$, the least upper bound over $[0,t)$ is the number of $W_q<t$, the value at $t$ is the number of $W_q\le t$, and their difference is the number of $W_q$ equal to $t$, at most $1$ by distinctness; at $t=0$ the difference is $0-0=0$. Each $\mathsf{P}^{(y),c}_u$ is a finite sum of products of indicators of members of $\mathcal{U}$ (claims 1 to 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions), hence $\mathcal{U}$-measurable. On $\Omega^{U}_0$ and for $u\in[0,R]$ the defining sum is exactly the deterministic-count path $p^{c,(y_{c,\cdot})}_u$ of Uniform Representation of the Rate-One Poisson Counting Path on a Bounded Interval: Distinct Points, Poisson Increments, Conditional Law Given the Cell Counts, and Point Insertion.

Insertion. For $c\neq c_0$ the defining sums for $y$ and $y+\mathsf{m}e_{c_0,j_0}$ coincide. For $c=c_0$, the sum for $y+\mathsf{m}e_{c_0,j_0}$ contains the additional terms $i=y_{c_0,j_0}+1,\dots,y_{c_0,j_0}+\mathsf{m}$ of the cell $j_0$, which gives the displayed identity; on $\Omega^{U}_0$ the inserted points lie in $I_{c_0,j_0}$ and are distinct from each other and from all other points $U^{c_0,j}_{i'}$ by the definition of $\Omega^{U}_0$.

Copy clocks. For every $\omega$ and $u$, $\mathsf{P}^{\sharp,c}_u(\omega)=\sum_{y\in\mathbb{N}_0^{\mathsf{L}}}\mathbf{1}\{\mathsf{K}(\omega)=y\}\mathsf{P}^{(y),c}_u(\omega)$, a countable sum with exactly one nonzero term, hence the pointwise limit of $\mathcal{F}$-measurable partial sums along the enumeration of (Q3); so $\mathsf{P}^{\sharp,c}_u$ is $\mathcal{F}$-measurable by claim 5 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions. Its path at $\omega$ is the path of $\mathsf{P}^{(\mathsf{K}(\omega)),c}$ at $\omega$, a counting path.

Step 3 (claim 3). The triple $(\Omega,\mathcal{U},P|_{\mathcal{U}})$ is a probability space ($P|_{\mathcal{U}}$ is countably additive on $\mathcal{U}\subseteq\mathcal{F}$ with total mass $1$), and by Step 2 each $\mathsf{P}^{(y),c}$ is a family of $\mathcal{U}$-measurable random variables all of whose paths are counting paths, i.e. a stochastic process on this space with counting paths. Likewise $\mathsf{P}^{\sharp}$ on $(\Omega,\mathcal{F},P)$. All remaining data 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 ($N,l,m,\mathcal{A},B,T,\beta,\tilde{l},\tilde\beta,\tilde{B},h,x_0$) are those fixed here, so that lemma applies in both cases. Its claim 4 gives the measurability (with respect to $\mathcal{R}\otimes\mathcal{U}$, respectively $\mathcal{R}\otimes\mathcal{F}$), the bounds on $\mathbf{R}^{(k)}$, and the normalization for every $\omega$.

Fix $\omega$ and put $y=\mathsf{K}(\omega)$. The clock families $\mathsf{P}^{\sharp}(\omega)$ and $\mathsf{P}^{(y)}(\omega)$ are the same family of counting paths. For each $r$, the recursion of Existence, Uniqueness, Causality, and Measurability of the Open-Loop Aggregate Solution for the data $(\mathsf{P}^{\sharp}(\omega),a^r,x_0)$ is therefore the recursion for $(\mathsf{P}^{(y)}(\omega),a^r,x_0)$: the same stopping index, times, points, recursion path and conflict-freeness. The regularised path, the left limits, the intensity and the 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 are defined pointwise in $(r,\omega)$ from the recursion path alone, so $\bar\Sigma^{\sharp,r}_t(\omega)=\bar\Sigma^{(y),r}_t(\omega)$, $\ell^{\sharp,\omega}=\ell^{(y),\omega}$, and $(r,\omega)\in\mathsf{G}^{\sharp}$ if and only if $(r,\omega)\in\mathsf{G}^{(y)}$.

Step 4 (claim 4). Measurability of the density. The map $(\omega,\theta,r)\mapsto\theta_{c,j}-\mathsf{K}_{c,j}(\omega)/\sqrt{N}$ is $\mathcal{F}^{\sharp}$-measurable for every $(c,j)$: $\theta_{c,j}$ is the composition of the projection to $\theta$ (Q1) with a coordinate projection of $\mathbb{R}^d$ (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), $\mathsf{K}_{c,j}(\omega)$ is the composition of the projection to $\omega$ with the random variable $\mathsf{K}_{c,j}$, and differences and scalar multiples are measurable by claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions. Since $\varphi_\eta$ is sequentially continuous on $\mathbb{R}^d$ by claim 1 of The Gaussian Smoothing Weight: Normalization, Derivatives, Exponential Tilting, Moments, and First-Order Remainder, the composition $(\omega,\theta,r)\mapsto\varphi_\eta(\theta-\mathsf{K}(\omega)/\sqrt{N})$ is measurable by Sequentially Continuous Functions of Measurable Euclidean Maps are Measurable (with $E=\mathbb{R}^d$). The map $(\omega,\theta,r)\mapsto\ell^{\sharp,\omega}(r)$ is the composition of the projection $(\omega,\theta,r)\mapsto(r,\omega)$ of (Q1) with the $\mathcal{R}\otimes\mathcal{F}$-measurable map of claim 3. The product $\mathsf{q}^{\sharp}$ is measurable by claim 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, with values in $[0,\infty)$.

The reference measure. $P$ is finite and $\lambda_d$ is $\sigma$-finite, so $P\otimes\lambda_d$ exists by Existence and Uniqueness of the Product Measure and is $\sigma$-finite; $\rho$ is $\sigma$-finite (Step 5(b) below, which uses nothing from this step), so $(P\otimes\lambda_d)\otimes\rho$ exists and is $\sigma$-finite. By claim 3 of Image Measures, Measures with Densities, and Change of Variables (the density $\mathsf{q}^{\sharp}$ being finite-valued), $\mu^{\sharp}$ is a measure on $\mathcal{F}^{\sharp}$ and $\int_{\Omega^{\sharp}}H\,d\mu^{\sharp}=\int_{\Omega^{\sharp}}H\mathsf{q}^{\sharp}\,d((P\otimes\lambda_d)\otimes\rho)$ for every $\mathcal{F}^{\sharp}$-measurable $H:\Omega^{\sharp}\to[0,\infty]$.

Total mass. By the Tonelli theorem on $(\Omega\times\mathbb{R}^d)\times\mathbf{R}$, $\mu^{\sharp}(\Omega^{\sharp})=\int_{\Omega\times\mathbb{R}^d}\Bigl(\int_{\mathbf{R}}\varphi_\eta(\theta-\mathsf{K}(\omega)/\sqrt{N})\,\ell^{\sharp,\omega}(r)\,\rho(dr)\Bigr)d(P\otimes\lambda_d)(\omega,\theta)=\int_{\Omega\times\mathbb{R}^d}\varphi_\eta(\theta-\mathsf{K}(\omega)/\sqrt{N})\,d(P\otimes\lambda_d)(\omega,\theta),$ using $\int_{\mathbf{R}}\ell^{\sharp,\omega}\,d\rho=1$ (claim 3) and the linearity of Linearity and Monotonicity of the Lebesgue Integral for the constant factor. By Tonelli on $\Omega\times\mathbb{R}^d$, the last integral equals $\int_{\Omega}\bigl(\int_{\mathbb{R}^d}\varphi_\eta(\theta-\mathsf{K}(\omega)/\sqrt{N})\,\lambda_d(d\theta)\bigr)P(d\omega)=\int_\Omega1\,dP=1$ by claim 1 of The Gaussian Smoothing Weight: Normalization, Derivatives, Exponential Tilting, Moments, and First-Order Remainder with $a=\mathsf{K}(\omega)/\sqrt{N}$. Hence $\mu^{\sharp}$ is a probability measure. The measurability of $\Theta$ and $\mathsf{D}$ is (Q1).

Step 5 (claim 5). (a) The count mass function. $\mathsf{S}$ is the image of $\mathbb{N}_0^{\mathsf{L}}$ under the injective map $y\mapsto y/\sqrt{N}$, hence countable by (Q3). $\mathsf{p}$ takes values in $[0,1]$ and is positive exactly on $\mathsf{S}$ (Step 1). Since $\mathsf{p}$ vanishes off $\mathsf{S}$, every finite partial sum of $\mathsf{p}$ over a finite set $F\subseteq\mathbb{R}^d$ equals the partial sum over $F\cap\mathsf{S}$, so the two least upper bounds in Sum of a Nonnegative Function over an Arbitrary Set agree: $\sum_{x\in\mathbb{R}^d}\mathsf{p}(x)=\sum_{x\in\mathsf{S}}\mathsf{p}(x)=\sum_{y}P(\mathsf{K}=y)=1$ by (Q3) and Step 1; so $\mathsf{p}$ is a discrete probability mass function with support $\mathsf{S}$. For $x\in\mathbb{R}^d$, $\lVert x\rVert^{2}=\sum_{(c,j)}x_{c,j}^{2}\le(\sum_{(c,j)}|x_{c,j}|)^{2}$, so $\lVert x\rVert\le\sum_{(c,j)}|x_{c,j}|$, and for $x=y/\sqrt{N}\in\mathsf{S}$ all coordinates are nonnegative. Hence, by (Q3) and (Q2), $\sum_{x\in\mathsf{S}}\mathsf{p}(x)\lVert x\rVert\le\frac{1}{\sqrt{N}}\sum_{(c,j)\in\mathsf{L}}\sum_{y}P(\mathsf{K}=y)\,y_{c,j}=\frac{1}{\sqrt{N}}\sum_{(c,j)\in\mathsf{L}}\mathbb{E}\Bigl[\sum_{y}\mathbf{1}\{\mathsf{K}=y\}\,y_{c,j}\Bigr]=\frac{1}{\sqrt{N}}\sum_{(c,j)\in\mathsf{L}}\mathbb{E}[\mathsf{K}_{c,j}]=\frac{1}{\sqrt{N}}\sum_{c}\sum_{j=1}^{J_c}|I_{c,j}|=\frac{l(l-1)R}{\sqrt{N}},$ since $\sum_j|I_{c,j}|=b^{c}_{J_c}-b^{c}_0=R$ for each of the $l(l-1)$ labels.

(b) $\sigma$-finiteness of $\rho$. By The Observation Record Space, $\mathbf{R}$ is the countable disjoint union of its cells, $\rho(C_\emptyset)=1$, and $\rho(C_{k,v})=\lambda_k(D_k(T))$ (the cell being the transport of the restriction of $\lambda_k$ to $D_k(T)$, and the measure of the disjoint union agreeing with the cell measures by claim 4(a) of Assembly of Measure Spaces: Restriction, Transport, One-Point Spaces, and Countable Disjoint Unions), where $\lambda_k$ is the $k$-fold product Lebesgue measure and $D_k(T)$ the ordered time simplex, whose volume $\lambda_k(D_k(T))$ is finite by that lemma.

(c) The kernel. Fix $x=y/\sqrt{N}\in\mathsf{S}$. The map $(r,\omega)\mapsto\ell^{(y),\omega}(r)$ is $\mathcal{R}\otimes\mathcal{U}$-measurable, hence $\mathcal{R}\otimes\mathcal{F}$-measurable, with values in $[0,(N\tilde{B})^{k}]$ on $\mathbf{R}^{(k)}\times\Omega$ (claim 3). By Tonelli on $\mathbf{R}\times\Omega$ with the $\sigma$-finite measures $\rho$ and $P$, the map $r\mapsto f(x,r)=\int_\Omega\ell^{(y),\omega}(r)\,P(d\omega)$ is $\mathcal{R}$-measurable, and $\int_{\mathbf{R}}f(x,r)\,\rho(dr)=\int_{\Omega}\Bigl(\int_{\mathbf{R}}\ell^{(y),\omega}(r)\,\rho(dr)\Bigr)P(d\omega)=\int_\Omega1\,dP=1$ by claim 3; this is (K1). By the monotonicity of Linearity and Monotonicity of the Lebesgue Integral, $f(x,r)\le(N\tilde{B})^{k}=M(r)$ for $r\in\mathbf{R}^{(k)}$, and $M$ is $\mathcal{R}$-measurable, each $\mathbf{R}^{(k)}$ being a finite union of cells (claim 4(a) of Assembly of Measure Spaces: Restriction, Transport, One-Point Spaces, and Countable Disjoint Unions) and $M$ being constant on each; this is (K2), and $f$ is finite. Together with $d\ge1$, $0<\eta\le1$, (a) and (b), all hypotheses of Gaussian Smoothing of a Kernel-Weighted Discrete Family with Countable Support: Joint Density, van Trees Regularity, and the Symmetrised Directional Score Bound on $(d,\eta,\mathsf{p},f)$ hold; for moves of the displayed form, $x+a_q=(y+\mathsf{m}_qe_{c_q,j_q})/\sqrt{N}\in\mathsf{S}$.

(d) The identity for $g$. Fix $(\theta,r)$. For every $\omega$, exactly one term of $\sum_{y}\mathbf{1}\{\mathsf{K}(\omega)=y\}\varphi_\eta(\theta-y/\sqrt{N})\ell^{(y),\omega}(r)$ is nonzero, namely the one with $y=\mathsf{K}(\omega)$, and it equals $\varphi_\eta(\theta-\mathsf{K}(\omega)/\sqrt{N})\ell^{\sharp,\omega}(r)$ by claim 3. For each $y$ the map $\omega\mapsto\ell^{(y),\omega}(r)$ is $\mathcal{U}$-measurable, being the section at $r$ of an $\mathcal{R}\otimes\mathcal{U}$-measurable map (the sets $E\subseteq\mathbf{R}\times\Omega$ whose section at $r$ lies in $\mathcal{U}$ form a $\sigma$-algebra containing the measurable rectangles, hence containing $\mathcal{R}\otimes\mathcal{U}$ by Intersections of Sigma-Algebras and Minimality of the Generated Sigma-Algebra, and the preimage of a Borel set under the section is the section of the preimage). Hence, by (Q2) and the factorisation of claim 1, $\mathbb{E}\bigl[\varphi_\eta(\theta-\mathsf{K}/\sqrt{N})\,\ell^{\sharp,\cdot}(r)\bigr]=\sum_{y}\varphi_\eta(\theta-y/\sqrt{N})\,\mathbb{E}\bigl[\mathbf{1}\{\mathsf{K}=y\}\ell^{(y),\cdot}(r)\bigr]=\sum_{y}P(\mathsf{K}=y)\,\varphi_\eta(\theta-y/\sqrt{N})\,\mathbb{E}\bigl[\ell^{(y),\cdot}(r)\bigr]=\sum_{x\in\mathsf{S}}\mathsf{p}(x)\varphi_\eta(\theta-x)f(x,r)=g(\theta,r),$ the last step being the reindexing $x=y/\sqrt{N}$ of (Q3).

(e) The joint density. Let $F:\mathbb{R}^d\times\mathbf{R}\to[0,\infty]$ be $\mathcal{B}_d\otimes\mathcal{R}$-measurable. Then $F(\Theta,\mathsf{D})=F\circ\pi$ with $\pi(\omega,\theta,r)=(\theta,r)$ is $\mathcal{F}^{\sharp}$-measurable by (Q1), and by Step 4, $\int_{\Omega^{\sharp}}F(\Theta,\mathsf{D})\,d\mu^{\sharp}=\int_{\Omega^{\sharp}}F(\theta,r)\mathsf{q}^{\sharp}(\omega,\theta,r)\,d((P\otimes\lambda_d)\otimes\rho)$. By Tonelli on $(\Omega\times\mathbb{R}^d)\times\mathbf{R}$ this equals $\int_{\mathbf{R}}\bigl(\int_{\Omega\times\mathbb{R}^d}F(\theta,r)\mathsf{q}^{\sharp}(\omega,\theta,r)\,d(P\otimes\lambda_d)(\omega,\theta)\bigr)\rho(dr)$, and for fixed $r$ the inner integrand is $\mathcal{F}\otimes\mathcal{B}_d$-measurable (section clause), so by Tonelli on $\Omega\times\mathbb{R}^d$, integrating first over $\omega$, the inner integral equals $\int_{\mathbb{R}^d}F(\theta,r)\Bigl(\int_{\Omega}\varphi_\eta(\theta-\mathsf{K}(\omega)/\sqrt{N})\,\ell^{\sharp,\omega}(r)\,P(d\omega)\Bigr)\lambda_d(d\theta)=\int_{\mathbb{R}^d}F(\theta,r)\,g(\theta,r)\,\lambda_d(d\theta)$ by (d) (for fixed $\theta$ the constant $F(\theta,r)$ comes out of the $\omega$-integral by Linearity and Monotonicity of the Lebesgue Integral, or trivially if $F(\theta,r)=\infty$ and both sides are read in $[0,\infty]$ with $0\cdot\infty=0$). Finally, $g$ is $\mathcal{B}(\mathbb{R}^d)\otimes\mathcal{R}$-measurable by claim 1 of Gaussian Smoothing of a Kernel-Weighted Discrete Family with Countable Support: Joint Density, van Trees Regularity, and the Symmetrised Directional Score Bound, so $Fg$ is measurable and Tonelli on $\mathbb{R}^d\times\mathbf{R}$ turns the iterated integral $\int_{\mathbf{R}}\int_{\mathbb{R}^d}Fg\,d\lambda_d\,d\rho$ into $\int_{\mathbb{R}^d\times\mathbf{R}}Fg\,d(\lambda_d\otimes\rho)$. $\blacksquare$