Proof 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

Throughout, measurable for a real-valued map means measurable with respect to the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R})$ on the target; $\mathcal{B}_n$ and $\lambda_n$ denote the $n$-fold Borel $\sigma$-algebra and product Lebesgue measure on $\mathbb{R}^n$, $\lambda_{[0,T]}$ the restricted Lebesgue measure on $[0,T]$, $D_k(T)$ the ordered time simplex (the set of time tuples of the cell $C_{k,v}$), and $R_j(T)$ the record space of Observation-Driven Control Policy. Records are written $r=(k,\mathbf{t},v)$ with $\mathbf{t}=(t_1,\dots,t_k)$, as in the statement. The $\sigma$-algebra $\mathcal{B}_n$ equals the Borel $\sigma$-algebra generated by the open sets 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. For a $\sigma$-algebra $\mathcal{G}$ on a set $X$ and $X_0\in\mathcal{G}$, $\mathcal{G}|_{X_0}=\{A\in\mathcal{G}:A\subseteq X_0\}$ is the restriction (claim 1 there). For $t\in[0,T]$ and $\omega\in\Omega$ we write $\Sigma^{r}_t(\omega)=(\Sigma^{r,1}_t(\omega),\dots,\Sigma^{r,l}_t(\omega))$, and similarly for $\bar\Sigma$.

Preliminaries.

(P1) Generator criterion. Let $\phi:(X,\mathcal{G})\to Y$ be a map into a set $Y$ carrying the $\sigma$-algebra $\sigma(\mathcal{E})$ generated by a family $\mathcal{E}$ of subsets of $Y$. If $\phi^{-1}(E)\in\mathcal{G}$ for every $E\in\mathcal{E}$, then $\phi$ is measurable. Indeed, $\mathcal{D}=\{A\subseteq Y:\phi^{-1}(A)\in\mathcal{G}\}$ is a $\sigma$-algebra on $Y$ (preimages commute with complements and countable unions, and $\phi^{-1}(Y)=X$) containing $\mathcal{E}$, hence $\mathcal{D}\supseteq\sigma(\mathcal{E})$ by Intersections of Sigma-Algebras and Minimality of the Generated Sigma-Algebra. Applied to a product $\sigma$-algebra $\mathcal{G}_1\otimes\mathcal{G}_2$, generated by the measurable rectangles $A_1\times A_2$: a map $\phi=(\phi_1,\phi_2)$ into $X_1\times X_2$ is measurable as soon as $\phi^{-1}(A_1\times A_2)=\phi_1^{-1}(A_1)\cap\phi_2^{-1}(A_2)\in\mathcal{G}$ for all $A_1\in\mathcal{G}_1$, $A_2\in\mathcal{G}_2$; in particular the coordinate projections of a product are measurable, and the composition of measurable maps is measurable (preimages compose).

(P2) Sections. If $E\in\mathcal{G}_1\otimes\mathcal{G}_2$ and $x_2\in X_2$, then the section $\{x_1:(x_1,x_2)\in E\}$ lies in $\mathcal{G}_1$. Indeed, the family of all $E\subseteq X_1\times X_2$ whose section at $x_2$ lies in $\mathcal{G}_1$ is a $\sigma$-algebra (taking the section commutes with complements and countable unions) containing every rectangle $A_1\times A_2$ (its section is $A_1$ or $\emptyset$), hence contains $\mathcal{G}_1\otimes\mathcal{G}_2$ by Intersections of Sigma-Algebras and Minimality of the Generated Sigma-Algebra. Consequently, if $g:X_1\times X_2\to\mathbb{R}$ is $\mathcal{G}_1\otimes\mathcal{G}_2$-measurable, then every section $x_1\mapsto g(x_1,x_2)$ is $\mathcal{G}_1$-measurable, its preimage of a Borel set $D$ being the section of $g^{-1}(D)$.

(P3) Nested sections. Let $E\in\mathcal{B}_{[0,T]}\otimes(\mathcal{R}\otimes\mathcal{F})$ and $\omega\in\Omega$. Then $E^{\omega}=\{(t,r):(t,r,\omega)\in E\}\in\mathcal{B}_{[0,T]}\otimes\mathcal{R}$. Indeed, the family $\mathcal{D}$ of all $E\subseteq[0,T]\times\mathbf{R}\times\Omega$ with $E^{\omega}\in\mathcal{B}_{[0,T]}\otimes\mathcal{R}$ is a $\sigma$-algebra (taking the section at $\omega$ commutes with complements and countable unions), and it contains every rectangle $B\times E'$ with $B\in\mathcal{B}_{[0,T]}$ and $E'\in\mathcal{R}\otimes\mathcal{F}$, since $(B\times E')^{\omega}=B\times\{r:(r,\omega)\in E'\}$ and the section of $E'$ lies in $\mathcal{R}$ by (P2). Hence $\mathcal{D}$ contains the generated $\sigma$-algebra $\mathcal{B}_{[0,T]}\otimes(\mathcal{R}\otimes\mathcal{F})$ by Intersections of Sigma-Algebras and Minimality of the Generated Sigma-Algebra. Consequently the section at $\omega$ of a $\mathcal{B}_{[0,T]}\otimes(\mathcal{R}\otimes\mathcal{F})$-measurable real map is $\mathcal{B}_{[0,T]}\otimes\mathcal{R}$-measurable.

(P4) Piecewise criterion. Let $(X,\mathcal{G})$ be a measurable space, let $(X_n)_{n}$ be countably many pairwise disjoint members of $\mathcal{G}$ with union $X$, and let $g:X\to\mathbb{R}$ be such that each restriction $g|_{X_n}$ is measurable with respect to $\mathcal{G}|_{X_n}$. Then $g$ is measurable, because $g^{-1}(D)=\bigcup_n(g|_{X_n})^{-1}(D)$ is a countable union of members of $\mathcal{G}$.

(P5) Trace description. Let $j\ge0$ and let $S_j=[0,T]\times R_j(T)\subseteq\mathbb{R}^{1+j}$ for $j\ge1$ and $S_0=[0,T]\subseteq\mathbb{R}$, with $R_j(T)$ the record space of Observation-Driven Control Policy. The $\sigma$-algebra $\mathcal{O}_j$ generated by the relatively open subsets of $S_j$ (the sets $U\cap S_j$ with $U$ open in $\mathbb{R}^{1+j}$) equals the trace $\{A\cap S_j:A\in\mathcal{B}_{1+j}\}$. Indeed, the trace is a $\sigma$-algebra on $S_j$ (traces commute with complements and countable unions) containing the relatively open sets, since open sets lie in $\mathcal{B}_{1+j}$ 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 $\mathcal{O}_j$ is contained in the trace. Conversely, the family of $A\in\mathcal{B}_{1+j}$ with $A\cap S_j\in\mathcal{O}_j$ is a $\sigma$-algebra containing the open sets, hence equals $\mathcal{B}_{1+j}$ 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 and Intersections of Sigma-Algebras and Minimality of the Generated Sigma-Algebra. For $j=0$ the trace is the trace Borel $\sigma$-algebra $\mathcal{B}_{[0,T]}$.

Step 1 (claim 1: joint measurability of the record-frozen control). Event times. For $j\ge1$ define $\mathsf{t}_j:\mathbf{R}\to[0,\infty)$ by $\mathsf{t}_j(r)=t_j$ if $r=(k,\mathbf{t},v)$ has $k\ge j$ events and $\mathsf{t}_j(r)=0$ otherwise (this is not the jump-time notation $\tau_j(q)$ of Existence, Uniqueness, Causality, and Measurability of the Open-Loop Aggregate Solution). On a cell $C_{k,v}$ with $k\ge j$, $\mathsf{t}_j$ is the composition of the inverse of the transport bijection $t\mapsto(k,t,v)$ (measurable by claim 2 of Assembly of Measure Spaces: Restriction, Transport, One-Point Spaces, and Countable Disjoint Unions) with the coordinate projection $t\mapsto t_j$ restricted to $D_k(T)$, which is measurable with respect to the restriction of $\mathcal{B}_k$ to $D_k(T)$ by 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 (a preimage under the restricted map is the intersection of $D_k(T)$ with a preimage under the projection); on the other cells $\mathsf{t}_j$ is constant. By claim 4(c) of Assembly of Measure Spaces: Restriction, Transport, One-Point Spaces, and Countable Disjoint Unions (the $\sigma$-algebra $\mathcal{R}$ being that of the countable disjoint union of the cells), $\mathsf{t}_j$ is $\mathcal{R}$-measurable as a $[0,\infty]$-valued map, hence as a real-valued map: the preimage of a Borel set $D\subseteq\mathbb{R}$ is the preimage of $D\cap[0,\infty)$, a measurable subset of $[0,\infty]$. Put $\mathsf{t}_0=0$.

The partition. For a cell $C_{k,v}$ ($k\ge0$, with $C_{0,()}=C_\emptyset$) and $j\in\{0,\dots,k\}$ let $E_{k,v,j}=\{(s,r)\in[0,T]\times C_{k,v}:\ k_r(s)=j\}.$ Since $t_1<\dots<t_k$ for $r\in C_{k,v}$, one has $k_r(s)=j$ if and only if $\mathsf{t}_j(r)\le s$ and, when $j<k$, $s<\mathsf{t}_{j+1}(r)$. The maps $(s,r)\mapsto s$ and $(s,r)\mapsto\mathsf{t}_j(r)$ are $\mathcal{B}_{[0,T]}\otimes\mathcal{R}$-measurable (compositions of the coordinate projections of (P1) with measurable maps), so their difference is measurable by claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, and $E_{k,v,j}$ is the intersection of $[0,T]\times C_{k,v}$ (a measurable rectangle) with preimages of Borel sets under such differences. Hence $E_{k,v,j}\in\mathcal{B}_{[0,T]}\otimes\mathcal{R}$. The sets $E_{k,v,j}$ ($k\ge0$, $v\in V^k$, $0\le j\le k$) are pairwise disjoint (the cells are, and $k_r(s)$ takes one value), countably many (finitely many for each $k$), and their union is $[0,T]\times\mathbf{R}$ because $k_r(s)\in\{0,\dots,k\}$.

Measurability on a piece. Fix $k,v,j$ and a component index $i\in\{1,\dots,m\}$. For $(s,r)\in E_{k,v,j}$ the definition of the record-frozen control path gives $\mathsf{a}^{i}(s,r)=h^{i}_j(s,(\mathsf{t}_1(r),\dots,\mathsf{t}_j(r)),(v_1,\dots,v_j))$ for $j\ge1$, and $\mathsf{a}^{i}(s,r)=h^{i}_0(s)$ for $j=0$. Consider the map $\Psi:E_{k,v,j}\to S_j$, $\Psi(s,r)=(s,\mathsf{t}_1(r),\dots,\mathsf{t}_j(r))$ ($\Psi(s,r)=s$ for $j=0$); its values lie in $S_j$ because $0<\mathsf{t}_1(r)<\dots<\mathsf{t}_j(r)\le T$ for $r\in C_{k,v}$ (the time tuple lying in $D_k(T)$), so that $(\mathsf{t}_1(r),\dots,\mathsf{t}_j(r))$ satisfies the membership condition $0\le\tau_1\le\dots\le\tau_j\le T$ of $R_j(T)$. Each component of $\Psi$ is measurable with respect to $(\mathcal{B}_{[0,T]}\otimes\mathcal{R})|_{E_{k,v,j}}$, so $\Psi$ is measurable into $(\mathbb{R}^{1+j},\mathcal{B}_{1+j})$ by claim 2 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets (for $j=0$, into $(\mathbb{R},\mathcal{B}(\mathbb{R}))$). Let $H=h^{i}_j(\cdot,\cdot,(v_1,\dots,v_j))$ ($H=h^{i}_0$ for $j=0$), a real map on $S_j$ which is measurable with respect to $\mathcal{O}_j$ by Observation-Driven Control Policy. For a Borel set $D\subseteq\mathbb{R}$, (P5) gives $H^{-1}(D)=A\cap S_j$ with $A\in\mathcal{B}_{1+j}$, so $(\mathsf{a}^{i}|_{E_{k,v,j}})^{-1}(D)=(H\circ\Psi)^{-1}(D)=\Psi^{-1}(A)$ belongs to $(\mathcal{B}_{[0,T]}\otimes\mathcal{R})|_{E_{k,v,j}}$. By (P4), $\mathsf{a}^{i}$ is $\mathcal{B}_{[0,T]}\otimes\mathcal{R}$-measurable. Since $h$ takes values in $\mathcal{A}$, so does $\mathsf{a}$, and claim 4 of Existence, Uniqueness, Causality, and Measurability of the Open-Loop Aggregate Solution applies with $(\mathsf{R},\mathcal{R})=(\mathbf{R},\mathcal{R})$ (nonempty, as it contains $r_\emptyset$); this yields the recursion paths $\Sigma^{r}_t(\omega)$, the set $\mathsf{G}\in\mathcal{R}\otimes\mathcal{H}_T$, and the measurability statements quoted below.

Step 2 (claim 2: the regularised path). Fix $(r,\omega)$ and run the recursion for $(\mathsf{P}(\omega),a^r,x_0)$, with stopping index $K$, times $\theta_0=0<\theta_1<\dots<\theta_K\le T$ (strict by claim 1 of Existence, Uniqueness, Causality, and Measurability of the Open-Loop Aggregate Solution) and points $x^{(0)}=x_0,\dots,x^{(K)}$. By the definition of the recursion path, $\Sigma^{r}_t(\omega)=x^{(k)}$ for $t\in[\theta_k,\theta_{k+1})$ and $k<K$, and $\Sigma^{r}_t(\omega)=x^{(K)}$ for $t\in[\theta_K,T]$. Since the recursion stops at the first index $K$ with $\theta_K=T$ or $x^{(K)}\notin\mathbb{G}_N$, one has $x^{(k)}\in\mathbb{G}_N$ for every $k<K$.

(a) Hence $\bar\Sigma^{r}_t(\omega)=x^{(k)}$ on $[\theta_k,\theta_{k+1})$ for $k<K$, and on $[\theta_K,T]$ the map $\bar\Sigma^{r}(\omega)$ is constant, equal to $x^{(K)}$ if $x^{(K)}\in\mathbb{G}_N$ and to $x_0$ otherwise. All values lie in $\mathbb{G}_N$, and $\bar\Sigma^{r}(\omega)=\Sigma^{r}(\omega)$ on $[0,\theta_K)$.

(b) Let $t\in(0,T]$. If $t\in(\theta_k,\theta_{k+1}]$ for some $k<K$, put $\bar\Sigma^{r}_{t-}(\omega)=x^{(k)}$ and $\delta=t-\theta_k>0$: then $[t-\delta,t)=[\theta_k,t)\subseteq[\theta_k,\theta_{k+1})$, on which $\bar\Sigma^{r}(\omega)=x^{(k)}$. If $t\in(\theta_K,T]$, put $\bar\Sigma^{r}_{t-}(\omega)=\bar\Sigma^{r}_T(\omega)$ and $\delta=t-\theta_K$. These cases are exhaustive and exclusive since $0=\theta_0<\theta_1<\dots<\theta_K\le T$. The point is unique: if $y$ and $y'$ are constant values of $\bar\Sigma^{r}(\omega)$ on $[t-\delta,t)\cap[0,T]$ and on $[t-\delta',t)\cap[0,T]$ respectively, both intervals contain $[t-\min(\delta,\delta'),t)\cap[0,T]$, which is nonempty as $t>0$, so $y=y'$. In the first case $\bar\Sigma^{r}_{t-}(\omega)=\bar\Sigma^{r}_t(\omega)$ unless $t=\theta_{k+1}$; in the second case equality holds. With $\bar\Sigma^{r}_{0-}(\omega)=x_0=\bar\Sigma^{r}_0(\omega)$, the set of $t$ where the left limit differs from the value is contained in $\{\theta_1,\dots,\theta_K\}$. The left limit lies in $\mathbb{G}_N$ by (a).

(c) If $(r,\omega)\in\mathsf{G}$, then $x^{(K)}\in\mathbb{G}_N$, so $\bar\Sigma^{r}(\omega)=\Sigma^{r}(\omega)$ on all of $[0,T]$, and by claim 2 of Existence, Uniqueness, Causality, and Measurability of the Open-Loop Aggregate Solution the recursion path is the unique open-loop aggregate solution for the data.

(d) The path. By claim 4 of Existence, Uniqueness, Causality, and Measurability of the Open-Loop Aggregate Solution, each map $(t,r,\omega)\mapsto\Sigma^{r,\gamma}_t(\omega)$ is $\mathcal{B}_{[0,T]}\otimes(\mathcal{R}\otimes\mathcal{F})$-measurable. The indicator $\chi(t,r,\omega)$ of $\{\Sigma^{r}_t(\omega)\in\mathbb{G}_N\}$ equals $\sum_{x\in\mathbb{G}_N}\prod_{\gamma=1}^{l}\mathbf{1}\{\Sigma^{r,\gamma}_t(\omega)=x^{\gamma}\}$, a finite sum of finite products of indicators of preimages of one-point Borel sets, hence measurable by claims 1, 2 and 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions; and $\bar\Sigma^{r,\gamma}_t(\omega)=\chi\,\Sigma^{r,\gamma}_t(\omega)+(1-\chi)\,x_0^{\gamma}$ is measurable by the same claims.

The left limit. For $n\ge1$ let $\phi_n:[0,T]\to[0,T]$, $\phi_n(t)=\max(t-1/n,0)$, which satisfies $|\phi_n(t)-\phi_n(s)|\le|t-s|$, hence is continuous on $[0,T]$ and $\mathcal{B}_{[0,T]}$-measurable by claim 4 of Measurability of Countable Suprema, Bounded Pointwise Limits, Monotone Functions, and Continuous Functions. The map $\Phi_n(t,r,\omega)=(\phi_n(t),r,\omega)$ is measurable from $\mathcal{B}_{[0,T]}\otimes(\mathcal{R}\otimes\mathcal{F})$ to itself by (P1), the preimage of a rectangle $B\times E'$ being $\phi_n^{-1}(B)\times E'$. Hence $g_n(t,r,\omega)=\bar\Sigma^{r,\gamma}_{\phi_n(t)}(\omega)$ is measurable, with values in $[0,1]$. We claim $g_n(t,r,\omega)\to\bar\Sigma^{r,\gamma}_{t-}(\omega)$ for every $(t,r,\omega)$. For $t=0$, $\phi_n(0)=0$ and $g_n=\bar\Sigma^{r,\gamma}_0(\omega)=x_0^{\gamma}$. For $t>0$ take $\delta$ from (b): for $n>1/\delta$, $\phi_n(t)\in[t-\delta,t)\cap[0,T]$, so $g_n(t,r,\omega)=\bar\Sigma^{r,\gamma}_{t-}(\omega)$. Thus the sequence is eventually constant, and claim 2 of Measurability of Countable Suprema, Bounded Pointwise Limits, Monotone Functions, and Continuous Functions gives the measurability of $(t,r,\omega)\mapsto\bar\Sigma^{r,\gamma}_{t-}(\omega)$.

Sections. The statements for fixed $\omega$ follow from (P3).

Step 3 (claim 3: causality). Bounds. For $x\in\Delta^l$ and every $\upsilon$, $0\le\tilde{b}^\upsilon(x)=\sum_{\sigma}x^{\sigma}\tilde\beta(\sigma,\upsilon,x)\le\tilde{B}\sum_\sigma x^{\sigma}=\tilde{B}$ by condition 1 of Observation-Rate Family, since $x^{\sigma}\ge0$ and $\sum_\sigma x^{\sigma}=1$; if $\tilde{b}^\upsilon\ge\underline{b}$ on $\Delta^l$ then $\lambda^{\omega,\upsilon}\ge N\underline{b}$. As $\bar\Sigma^{r}_{t-}(\omega)\in\mathbb{G}_N\subseteq\Delta^l$, $\lambda^{\omega,\upsilon}$ takes values in $[0,N\tilde{B}]$, and its total intensity is $\sum_\upsilon N\tilde{b}^\upsilon(\bar\Sigma^{r}_{t-}(\omega))=N\tilde{b}^{\mathrm{tot}}(\bar\Sigma^{r}_{t-}(\omega))$.

Sequential continuity of $\tilde{b}^\upsilon$ on $\Delta^l$. Let $x_n\to x$ in $\Delta^l$ with respect to the Euclidean distance. Then $|x_n^{\sigma}-x^{\sigma}|\le d(x_n,x)\to0$ for every $\sigma$, and $\tilde\beta(\sigma,\upsilon,x_n)\to\tilde\beta(\sigma,\upsilon,x)$ by condition 2 of Observation-Rate Family; by Arithmetic of Limits of Real Sequences the products and the finite sum converge, so $\tilde{b}^\upsilon(x_n)\to\tilde{b}^\upsilon(x)$.

Measurability (condition (i)). Fix $\omega$. Each component of $(t,r)\mapsto\bar\Sigma^{r}_{t-}(\omega)$ is $\mathcal{B}_{[0,T]}\otimes\mathcal{R}$-measurable by claim 2(d), and the map takes values in $\Delta^l$; by Sequentially Continuous Functions of Measurable Euclidean Maps are Measurable (with $E=\Delta^l$) the composition $(t,r)\mapsto\tilde{b}^\upsilon(\bar\Sigma^{r}_{t-}(\omega))$ is measurable, hence so is its constant multiple $\lambda^{\omega,\upsilon}$ (claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions). The same argument on $[0,T]\times\mathbf{R}\times\Omega$ shows that $\Lambda^{\upsilon}(t,r,\omega)=\lambda^{\omega,\upsilon}_t(r)$ is $\mathcal{B}_{[0,T]}\otimes(\mathcal{R}\otimes\mathcal{F})$-measurable; this is used in Step 4.

The identity. Fix $t\in[0,T]$, $r=(k,\mathbf{t},v)\in\mathbf{R}$ and $\omega$, and put $r'=\pi_{t-}(r)$, which consists of the $\kappa$ events of $r$ with $t_j<t$, these being exactly the events with indices $1,\dots,\kappa$ by The Strict Prefix Map on the Observation Record Space. For $u\in[0,t)$, an index $j$ with $t_j\le u$ satisfies $t_j<t$, so $j\le\kappa$; hence $k_{r'}(u)=k_r(u)$, and the first $k_r(u)$ event times and marks of $r$ and $r'$ coincide, so $a^{r'}(u)=a^{r}(u)$ by the definition of the record-frozen control path. Let $t'\in[0,t)$. The control paths $a^{r}$ and $a^{r'}$ agree on all of $[0,t']$, so the exceptional set in claim 3 of Existence, Uniqueness, Causality, and Measurability of the Open-Loop Aggregate Solution is empty (contained in the null set $\emptyset$), and with $p'=p=\mathsf{P}(\omega)$ that claim gives $\Sigma^{r'}_u(\omega)=\Sigma^{r}_u(\omega)$ for all $u\in[0,t']$. Since $t'<t$ was arbitrary, $\Sigma^{r'}_u(\omega)=\Sigma^{r}_u(\omega)$ for all $u\in[0,t)$, and therefore $\bar\Sigma^{r'}_u(\omega)=\bar\Sigma^{r}_u(\omega)$ for $u\in[0,t)$, the regularisation being a function of the value at $u$. For $t=0$ both left limits are $x_0$. For $t>0$, the left limit at $t$ is by claim 2(b) the common value of the path on an interval $[t-\delta,t)$, which is determined by the values on $[0,t)$; hence $\bar\Sigma^{r'}_{t-}(\omega)=\bar\Sigma^{r}_{t-}(\omega)$.

Non-anticipation (condition (ii)). By the identity, $\lambda^{\omega,\upsilon}_t(\pi_{t-}(r))=N\tilde{b}^\upsilon(\bar\Sigma^{\pi_{t-}(r)}_{t-}(\omega))=N\tilde{b}^\upsilon(\bar\Sigma^{r}_{t-}(\omega))=\lambda^{\omega,\upsilon}_t(r)$. Thus $\lambda^{\omega}$ is a causal intensity with bound $N\tilde{B}$.

Step 4 (claim 4: the likelihood). Bounds and normalization. For every $\omega$, claims 2 and 4 of Survival Identity, Measurability, Event-Count Recursion, and Normalization of the Likelihood of a Causal Intensity applied to the causal intensity $\lambda^{\omega}$ with bound $\bar\lambda=N\tilde{B}$ give $0\le\ell^{\omega}\le(N\tilde{B})^{k}$ on $\mathbf{R}^{(k)}$ and $\int_{\mathbf{R}}\ell^{\omega}\,d\rho=1$.

The formula on $\mathsf{G}$. Let $(r,\omega)\in\mathsf{G}$, $r=(k,\mathbf{t},v)$. By Likelihood of a Causal Intensity on the Observation Record Space, $\ell^{\omega}(r)=\bigl(\prod_{i=1}^{k}\lambda^{\omega,v_i}_{t_i}(r)\bigr)\exp(-\int_{[0,T]}\lambda^{\omega,\mathrm{tot}}_s(r)\,ds)$ with $\lambda^{\omega,v_i}_{t_i}(r)=N\tilde{b}^{v_i}(\bar\Sigma^{r}_{t_i-}(\omega))$ and $\lambda^{\omega,\mathrm{tot}}_s(r)=N\tilde{b}^{\mathrm{tot}}(\bar\Sigma^{r}_{s-}(\omega))$. On $\mathsf{G}$, $\bar\Sigma^{r}(\omega)=\Sigma^{r}(\omega)$ by claim 2(c), and the integrands $s\mapsto\tilde{b}^{\mathrm{tot}}(\Sigma^{r}_{s-}(\omega))$ and $s\mapsto\tilde{b}^{\mathrm{tot}}(\Sigma^{r}_{s}(\omega))$ are bounded, measurable on $[0,T]$ (the first by condition (i) and (P2), the second likewise from claim 2(d)), and agree on $D=[0,T]\setminus\{\theta_1,\dots,\theta_K\}$ by claim 2(b); the complement of $D$ in $[0,T]$ is a finite set, hence a null set for the restricted Lebesgue measure, one-point sets having Lebesgue measure zero by claim 4 of Existence of Lebesgue Measure on the Real Line. By claim 6 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval the integral over $[0,T]$ of each integrand equals the integral of its product with the indicator of $D$, and these products coincide; hence the two integrals agree. This gives the displayed formula.

Joint measurability. We use the map $\Lambda^{\upsilon}$ of Step 3 and the measurability of $\Lambda^{\mathrm{tot}}=\sum_\upsilon\Lambda^{\upsilon}$ (claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions).

(i) The survival factor. The measure $\rho$ is $\sigma$-finite: $\mathbf{R}$ is the countable union of its cells, and $\rho(C_{k,v})=\lambda_k(D_k(T))$ is finite by The Ordered Time Simplex: Borel Measurability and Volume, while $\rho(C_\emptyset)=1$. Hence the product measure $\rho\otimes P$ on $\mathcal{R}\otimes\mathcal{F}$ exists and is $\sigma$-finite (the sets $C\times\Omega$, $C$ a cell, cover $\mathbf{R}\times\Omega$ and have finite measure), and $\lambda_{[0,T]}$ is finite. Apply the Tonelli theorem to the nonnegative $\mathcal{B}_{[0,T]}\otimes(\mathcal{R}\otimes\mathcal{F})$-measurable function $\Lambda^{\mathrm{tot}}$ on $[0,T]\times(\mathbf{R}\times\Omega)$: the map $(r,\omega)\mapsto\int_{[0,T]}\Lambda^{\mathrm{tot}}(s,r,\omega)\,ds$ is $\mathcal{R}\otimes\mathcal{F}$-measurable. For fixed $(r,\omega)$ the integrand is the section $s\mapsto\lambda^{\omega,\mathrm{tot}}_s(r)$, so this integral is the one appearing in Likelihood of a Causal Intensity on the Observation Record Space; it is real-valued, bounded by $\tilde{l}N\tilde{B}T$, since the integrand takes values in $[0,\tilde{l}N\tilde{B}]$. Composing with the map $y\mapsto\exp(-y)$, which is continuous on $\mathbb{R}$ by Basic Properties of the Exponential Function (Sequentially Continuous Functions of Measurable Euclidean Maps are Measurable with $E=\mathbb{R}$), shows that $(r,\omega)\mapsto\exp(-\int_{[0,T]}\lambda^{\omega,\mathrm{tot}}_s(r)\,ds)$ is $\mathcal{R}\otimes\mathcal{F}$-measurable.

(ii) The event factors. Fix a cell $C_{k,v}$ with $k\ge1$ and $i\in\{1,\dots,k\}$. The map $\Theta_i:C_{k,v}\times\Omega\to[0,T]\times\mathbf{R}\times\Omega$, $\Theta_i(r,\omega)=(\mathsf{t}_i(r),r,\omega)$, is measurable from $(\mathcal{R}\otimes\mathcal{F})|_{C_{k,v}\times\Omega}$ to $\mathcal{B}_{[0,T]}\otimes(\mathcal{R}\otimes\mathcal{F})$ by (P1): the preimage of a rectangle $B\times E'$ is $(\mathsf{t}_i^{-1}(B)\times\Omega)\cap E'\cap(C_{k,v}\times\Omega)$. Hence $(r,\omega)\mapsto\lambda^{\omega,v_i}_{t_i}(r)=\Lambda^{v_i}\circ\Theta_i(r,\omega)$ is measurable on $C_{k,v}\times\Omega$, and so is the finite product over $i$ (claim 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions). On $C_\emptyset\times\Omega$ the product is the constant $1$.

(iii) Assembly. On each piece $C_{k,v}\times\Omega$ of the countable measurable partition of $\mathbf{R}\times\Omega$ into the rectangles $C_{k,v}\times\Omega$, the map $(r,\omega)\mapsto\ell^{\omega}(r)$ is the product of the measurable maps of (i) and (ii), hence measurable with respect to the restricted $\sigma$-algebra; by (P4) it is $\mathcal{R}\otimes\mathcal{F}$-measurable. $\blacksquare$