Proof of Conditional Restart of the Record Channel at an Intermediate Time

Throughout, an observation clock index is a pair $d=(i,\upsilon)$, listed once and for all as a sequence of length $n=N\tilde{l}$; $\tilde{Y}^d$ and $\hat{\tilde{Y}}^d$ denote the corresponding observation clock and residual observation clock, and $\tilde{A}^{r,d}=\tilde{A}^{r,i,\upsilon}$ for records $r$. We use the agent-grouping identities of the preamble of the proof-independent statement of Conditional Density of the Observation Record Given the Initial States and Transition Clocks: for $(r,\omega)\in G$ and $u\in[0,T]$, summing the rates $\tilde{\beta}(\sigma^{r,i}_u(\omega),\upsilon,\Sigma^r_u(\omega))$ over all indices $d$ gives $N\tilde{b}^{\mathrm{tot}}(\Sigma^r_u(\omega))$, and summing over $i$ for fixed $\upsilon$ gives $N\tilde{b}^{\upsilon}(\Sigma^r_u(\omega))$, by grouping the agents by their state and using the aggregate observation drift; the same holds for left limits.

Claim 1. The restarted system is a driving system by the final assertion of Fresh-Start Property of the Controlled N-Agent Dynamics. For the prefix-frozen policy: fix $\varrho=(\kappa,\mathbf{t},w)$ and $j\ge0$. The concatenated tuple in the display lies in the tuple set of Observation-Driven Control Policy with horizon $T$: its entries are strictly increasing, since $\mathbf{t}_\kappa\le s<s+\tau'_1$, and at most $T$. The map $(u,\tau')\mapsto(s+u,(\tau,\ s+\tau'))$ is a coordinatewise translation, hence continuous, so preimages of relatively open sets are relatively open; by the identification, used in The Record-Frozen Control Path and Record-Frozen Policy, of the $\sigma$-algebra generated by the relatively open subsets with the trace of the generated $\sigma$-algebra of the ambient space, the composition $(h^{s,\varrho})_j$ of this map with $h_{\kappa+j}$ has the measurability required by Observation-Driven Control Policy, and its values lie in $\mathbb{R}^m$. Hence $h^{s,\varrho}$ is an observation-driven control policy with horizon $T-s$. For $W_s$: by claim (iv) of Existence, Uniqueness, and Regularity for the Controlled N-Agent Dynamics, $K_s$ and the times $\tau_j$ and channels $\upsilon_j$ with $j\le K_s$ are $\mathcal{G}_s$-measurable; on $\Omega_0$, $W_s=\pi_s(W)$ is the record with count $K_s$, times $(\tau_1,\dots,\tau_{K_s})$, and marks $(\upsilon_1,\dots,\upsilon_{K_s})$, since $k_W(s)=K_s$ (an event time is counted at $s$ exactly when the right-continuous count does); off $\Omega_0$ it is the empty record. Decomposing the preimage of a member of $\mathcal{R}_s$ over the cells and using that $\mathcal{G}_s$ contains all events of probability zero, $W_s^{-1}(A)\in\mathcal{G}_s$ for every $A\in\mathcal{R}_s$ by claim 4(b) of Assembly of Measure Spaces: Restriction, Transport, One-Point Spaces, and Countable Disjoint Unions. The horizon-$s$ restricted solution of claim 4 of Bayes Disintegration and Filtering Formula for the Observation Record has, at each $\omega\in\Omega_0$, observation-event count $K_s$ at its horizon with the same event times and channels, its observation total being the restriction of the original; so its observation record agrees with $W_s$ on $\Omega_0$.

Claim 2. Fix a transition clock $Y=Y^{i,\sigma\gamma}$ with consumed times $A_t=A^{i,\sigma\gamma}_t$, counters $N_t=N^{i,\sigma\gamma}_t$, and residual clock $\hat{Y}$. Let $a\ge0$ be a real. On the event where $A_s\ge a$: put $t(a)=\inf\{t\in[0,s]:A_t\ge a\}$; the set is nonempty and closed, $t\mapsto A_t$ being continuous and nondecreasing with $A_0=0$, so the infimum is attained, $A_{t(a)}\ge a$, and $A_t<a$ for $t<t(a)$, whence $A_{t(a)}=a$ by continuity (for $t(a)=0$, $a=0$). Then $Y_a=Y_{A_{t(a)}}=N_{t(a)}$. Measurability: for $c\in[0,s]$ the event where $t(a)\le c$ is the event where $A_c\ge a$, which lies in $\mathcal{F}^{\mathrm{sys}}_s$ by claim (iv) of Existence, Uniqueness, and Regularity for the Controlled N-Agent Dynamics; and $N_{t(a)}$ is the pointwise limit of the variables $N_{t_k}$ with $t_k$ the upper dyadic staircase approximations of $t(a)$ capped at $s$, by right-continuity of $t\mapsto N_t$ (a right-continuous clock path composed with a continuous nondecreasing time change), each $N_{t_k}$ being a finite sum of products of indicators of dyadic events of $\mathcal{F}^{\mathrm{sys}}_s$ with the variables $N_{q}$, $q$ dyadic in $[0,s]$, which are $\mathcal{F}^{\mathrm{sys}}_s$-measurable by claim (iv). On the complementary event where $A_s<a$: $Y_a=Y_{A_s}+\hat{Y}_{a-A_s}=N_s+\hat{Y}_{a-A_s}$ by the definition of the residual clocks, and $\hat{Y}_{a-A_s}$ is measurable with respect to the $\sigma$-algebra generated by $\mathcal{F}^{\mathrm{sys}}_s$ and the variables of $\hat{Y}$, again by the dyadic staircase argument, the paths of $\hat{Y}$ being counting paths by claim (a) of Fresh-Start Property of the Controlled N-Agent Dynamics, hence right-continuous, and $A_s$, $N_s$ being $\mathcal{F}^{\mathrm{sys}}_s$-measurable. Both events lie in $\mathcal{F}^{\mathrm{sys}}_s$, so $Y_a$ is $\mathcal{T}_s$-measurable. The initial states are $\mathcal{F}^{\mathrm{sys}}_s$-measurable (claim (iv) with $\sigma^i_0=\varsigma^i_0$). Hence every generator of $\mathcal{T}$ is $\mathcal{T}_s$-measurable and $\mathcal{T}\subseteq\mathcal{T}_s$.

For the independence: by claim (b) of Fresh-Start Property of the Controlled N-Agent Dynamics, the finite family consisting of $\mathcal{F}^{\mathrm{sys}}_s$ and the $\sigma$-algebras of the individual residual clocks is independent. Group it into the block consisting of $\mathcal{F}^{\mathrm{sys}}_s$ and the residual transition clocks and the block of the residual observation clocks: the finite intersections of members of a block form a $\pi$-system generating the block $\sigma$-algebra, the product rule holds on these $\pi$-systems by the family independence, and it extends to the generated $\sigma$-algebras one block at a time by Dynkin's lemma via claim 1 of Uniqueness of Finite Measures on a Generating Pi-System and the Density of the Exponential Law applied to the two finite measures obtained by fixing the other block's event; adjoining the events of probability zero changes no probabilities. The first block generates $\mathcal{T}_s$, proving claim 2.

Claim 3. First, let $E_s$ be the union over the finitely many counters of the intersections, over the naturals $k$ with $1/k<s$, of the events where the counter increment from time $s-1/k$ to time $s$ is at least $1$; each such increment is a random variable by claim (iv) of Existence, Uniqueness, and Regularity for the Controlled N-Agent Dynamics, so $E_s\in\mathcal{F}$; $E_s$ contains every $\omega$ at which some counter jumps at time exactly $s$, and off $E_s$ every counter is continuous at $s$, a nondecreasing right-continuous map being continuous at $s$ exactly when some increment from $s-1/k$ to $s$ vanishes. The event $E_s$ is null: for $u<s$ and a transition counter, $N_s-N_u=Y_{A_s}-Y_{A_u}\le Y_{A_u+B(s-u)}-Y_{A_u}=\hat{Y}^{(u)}_{B(s-u)}$, where $\hat{Y}^{(u)}$ is the corresponding residual clock at time $u$ (Fresh-Start Property of the Controlled N-Agent Dynamics applied at $u$), using $A_s-A_u\le B(s-u)$ from condition 2 of Solution of the Controlled N-Agent Dynamics and monotonicity of the clock path; by claim (a) of Fresh-Start Property of the Controlled N-Agent Dynamics and Moments of the Poisson Distribution the right side has expectation $B(s-u)$. A jump at $s$ implies $N_s-N_u\ge1$ for every $u<s$, and the probability of the latter is at most its expectation, at most $B(s-u)$, by monotonicity; letting $u\uparrow s$ along a sequence gives probability zero, and likewise for observation counters with $\tilde{B}$. Summing over the finitely many counters, $P(E_s)=0$. Put $\Omega^*_s=\Omega_0\setminus E_s$; this proves (i).

(ii) Fix $\omega\in\Omega^*_s$. Condition 1: $\hat{\sigma}^i_0=\sigma^i_s$, the restarted initial state; $u\mapsto\hat{\sigma}^i_u$ is the shift of the restriction to $[s,T]$ of a piecewise constant right-continuous path, hence again of that form, and its jump times are strictly positive, since a state change of some agent at time exactly $s$ would force a counter jump at $s$ by condition 6 for the original solution, which is excluded on $\Omega^*_s$. Condition 2: the segment rate fields are the compositions of the original fields with $(u,\omega)\mapsto(s+u,\omega)$, which is measurable (its components are), multiplied by the indicator of $\Omega^*_s\in\mathcal{F}$, so they are jointly measurable with the required bounds; the resulting consumed times are, on $\Omega^*_s$, the increments $\hat{A}^{i,\sigma\gamma}_u=A^{i,\sigma\gamma}_{s+u}-A^{i,\sigma\gamma}_s$ and $\hat{\tilde{A}}^{i,\upsilon}_u=\tilde{A}^{i,\upsilon}_{s+u}-\tilde{A}^{i,\upsilon}_s$, by additivity of the integral over $[0,s]$ and $[s,s+u]$ (single points being immaterial, Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval) and Translation Invariance of Lebesgue Measure and the Lebesgue Integral applied to the shifted integrand. Condition 3: on $\Omega^*_s$, $\hat{Y}^{i,\sigma\gamma}_{\hat{A}^{i,\sigma\gamma}_u}=Y^{i,\sigma\gamma}_{A^{i,\sigma\gamma}_s+\hat{A}^{i,\sigma\gamma}_u}-Y^{i,\sigma\gamma}_{A^{i,\sigma\gamma}_s}=N^{i,\sigma\gamma}_{s+u}-N^{i,\sigma\gamma}_s=\hat{N}^{i,\sigma\gamma}_u$ by the definition of the residual clocks, and likewise for observation clocks; each map $u\mapsto\hat{N}^{i,\sigma\gamma}_u$ coincides on $[0,T-s]$ with the restriction of a counting path: taking a counting path $\Phi$ agreeing with $t\mapsto N^{i,\sigma\gamma}_t$ on $[0,T]$ (condition 3 for the original solution), the shifted path $v\mapsto\Phi(s+v)-\Phi(s)$ is nondecreasing, right-continuous, integer-valued, starts at $0$, and has jumps of size one at strictly increasing, strictly positive times, since $\Phi$ does not jump at $s$ on $\Omega^*_s$; the same argument applies to the observation counters, the observation total, and the grand total. Condition 4: $\hat{\Upsilon}^\upsilon_u=\Upsilon^\upsilon_{s+u}-\Upsilon^\upsilon_s=\frac{1}{N}\sum_i\hat{\tilde{N}}^{i,\upsilon}_u$ by condition 4 for the original solution. Condition 6: subtract the state identity at $s$ from the one at $s+u$.

(iii) On $\Omega^*_s$ the jump times of the segment observation total in $[0,T-s]$ are exactly the $\hat{\tau}_j=\tau_{K_s+j}-s$ (no jump occurs at $s$), with channels $\hat{\upsilon}_j=\upsilon_{K_s+j}$, and the segment count is $\hat{K}_u=K_{s+u}-K_s$. By condition 5 for the original solution at time $s+u$, $\hat{\alpha}_u=\alpha_{s+u}=h_{K_{s+u}}\bigl(s+u,\ \tau_1,\dots,\tau_{K_{s+u}},\ \upsilon_1,\dots,\upsilon_{K_{s+u}}\bigr),$ and the argument tuple splits into the first $K_s$ events, which are exactly the events of $W_s(\omega)$ by claim 1, followed by the times $s+\hat{\tau}_j$ with channels $\hat{\upsilon}_j$ for $j\le\hat{K}_u$; by the definition of the prefix-frozen policy this value is $(h^{s,W_s(\omega)})_{\hat{K}_u}(u,\hat{\tau}_1,\dots,\hat{\tau}_{\hat{K}_u},\hat{\upsilon}_1,\dots,\hat{\upsilon}_{\hat{K}_u})$, as asserted. For fixed $\varrho$, on the set of $\omega\in\Omega^*_s$ with $W_s(\omega)=\varrho$ this is the control identity for the fixed policy $h^{s,\varrho}$, and conditions 1--4 and 6 hold at every point of $\Omega^*_s$, so all pathwise requirements hold there with policy $h^{s,\varrho}$.

(iv) The segment count at horizon $T-s$ is $K_T-K_s$, with the times $\hat{\tau}_j\in(0,T-s]$ strictly increasing and channels $\hat{\upsilon}_j$; comparing with the definition of $\iota_s$ and $k_W(s)=K_s$ gives that the record formed from the segment events is $\iota_s(W)=\hat{W}$.

Claim 4. Measurability: the map $(r',\omega)\mapsto(W_s(\omega)\oplus_s r',\omega)$ is measurable from $\mathcal{R}_{T-s}\otimes\mathcal{T}_s$ to $\mathcal{R}_T\otimes\mathcal{T}$: its first component is the composition of the pairing $(r',\omega)\mapsto(W_s(\omega),r')$ (measurable since $W_s$ is $\mathcal{G}_s$-measurable, $\mathcal{G}_s\subseteq\mathcal{F}^{\mathrm{sys}}_s\subseteq\mathcal{T}_s$, and rectangles generate the product) with $\oplus_s$, measurable by claim 2 of Splitting of the Observation Record Space at an Intermediate Time; its second component is the identity, measurable into $\mathcal{T}$ since $\mathcal{T}\subseteq\mathcal{T}_s$ by claim 2. The set $G$ lies in $\mathcal{R}_T\otimes\mathcal{T}$, so its pullback lies in $\mathcal{R}_{T-s}\otimes\mathcal{T}_s$. Decompose $\mathbf{R}_{T-s}\times\Omega$ over the cells $C'_{k',v'}$ and the events where $K_s=\kappa$: on each piece, the event-position evaluations $(r',\omega)\mapsto\Sigma^{q,\gamma}_{(s+t'_j)-}(\omega)$ (the $(\kappa+j)$-th event-time evaluations of $q=W_s(\omega)\oplus_s r'$) are compositions of the measurable map above with the fields of claim (b) of Measurable Reconstruction of the Controlled N-Agent Dynamics from Observation Records, and $(r',\omega)\mapsto\int_{[s,T]}\tilde{b}^{\mathrm{tot}}(\Sigma^{q}_u(\omega))du$ is measurable by claim (a) there, the composition arguments of claim 2 of Conditional Density of the Observation Record Given the Initial States and Transition Clocks, and the Tonelli theorem; finite products, $\exp$, and the case split over the pullback of $G$ preserve measurability (Sequentially Continuous Functions of Measurable Euclidean Maps are Measurable). The bound on cells holds since each event factor is at most $N\tilde{B}$ (the aggregate drift is bounded by $\tilde{B}$) and the exponential factor is at most $1$.

Factorization: let $\Omega_G$ be an event of probability one with $\mathbf{R}_T\times\Omega_G\subseteq G$ and let $\Omega_f$ be the probability-one event of claim (f) of Measurable Reconstruction of the Controlled N-Agent Dynamics from Observation Records on which $(W(\omega),\omega)\in G$ and the reconstruction at $W(\omega)$ reproduces the occupation indicators and consumed times of the solution on $[0,T]$. Fix $\omega\in\Omega_0\cap\Omega_G\cap\Omega_f$ and $r'\in\mathbf{R}_{T-s}$, and put $q=W_s(\omega)\oplus_s r'$. The records $W(\omega)$ and $q$ have $k_{W(\omega)}(s)=k_q(s)=K_s(\omega)$ and identical first $K_s(\omega)$ events (both prefixes are $W_s(\omega)$, claim 1 of Splitting of the Observation Record Space at an Intermediate Time), and both $(W(\omega),\omega)$ and $(q,\omega)$ lie in $G$; by causality (claim (e) of Measurable Reconstruction of the Controlled N-Agent Dynamics from Observation Records) the reconstructed occupation fields at $q$ and at $W(\omega)$ agree on $[0,s]$, and by claim (f) the latter agree with the solution's on $[0,T]$. Hence $\Sigma^{q}_u(\omega)=\Sigma_u(\omega)$ for $u\in[0,s]$, and the paths being piecewise constant, the left limits agree at every point of $(0,s]$, in particular at the prefix event times. Splitting the product in the definition of $f(q,\omega)$ at index $K_s(\omega)$ and the integral over $[0,T]$ into $[0,s]$ and $[s,T]$ (additivity; the point $s$ is immaterial), and using that the exponential of a sum is the product of the exponentials (The Real Exponential Function), gives $f(q,\omega)=f_s(\omega)\hat{f}(r',\omega)$, the prefix block reducing to $f_s(\omega)$ by the identities just established. This holds simultaneously for every $r'$ on the displayed probability-one event.

Claim 5. Let $\hat{\Omega}_1$ be the almost-sure event on which every residual observation clock has finite, strictly increasing, unbounded jump times, provided by part (a) of Jump Times of the Homogeneous Poisson Process: Finiteness and Exponential Interarrival Law, applicable since each $\hat{\tilde{Y}}^d$ is a homogeneous Poisson process of rate $1$ with counting paths by claim (a) of Fresh-Start Property of the Controlled N-Agent Dynamics; let $\hat{\xi}^d_1,\hat{\xi}^d_2,\dots$ be its interarrival times as in part (b) there, set to $0$ off $\hat{\Omega}_1$, each measurable over the $\sigma$-algebra generated by the variables of $\hat{\tilde{Y}}^d$ and the null events.

Step 1: the threshold vector. Fix $k\ge0$ and let $\hat{\Xi}$ be the $\mathbb{R}^{n(k+1)}$-valued map with components $\hat{\xi}^d_j$, $1\le j\le k+1$. Its components are independent: grouping events by clock, the product rule follows from the independence of the family of claim (b) of Fresh-Start Property of the Controlled N-Agent Dynamics by the block argument of claim 2, and within each clock from part (b) of Jump Times of the Homogeneous Poisson Process: Finiteness and Exponential Interarrival Law. By claim 1 of Joint Distribution, Expectations, and Block Independence for Independent Random Variables, the distribution of $\hat{\Xi}$ is the product of its marginals, each the exponential law with density $h(u)=e^{-u}\mathbf{1}_{(0,\infty)}(u)$; as in the identification of product densities via claims 1 and 2 of Uniqueness of Finite Measures on a Generating Pi-System and the Density of the Exponential Law, claim 3 of Finite Products of Lebesgue Measure and Coordinate Integration on $\mathbb{R}^l$, and Image Measures, Measures with Densities, and Change of Variables, this distribution is the measure with density $e(u)=\prod_{d,j}h(u_{d,j})$ with respect to $\lambda_{n(k+1)}$. Moreover $\sigma(\hat{\Xi})$ is independent of $\mathcal{T}'_s$, the enlargement of $\mathcal{T}_s$ by the null events, by claim 2.

Step 2: the segment recursion. For $r'\in\mathbf{R}_{T-s}$, $u\in[0,T-s]$, and an index $d$, define the segment consumed fields $\hat{A}^{r',d}_u(\omega)=\tilde{A}^{W_s(\omega)\oplus_s r',d}_{s+u}(\omega)-\tilde{A}^{W_s(\omega)\oplus_s r',d}_{s}(\omega)$. By claim (a) of Measurable Reconstruction of the Controlled N-Agent Dynamics from Observation Records, the measurable map of claim 4, and the measurable time shift, the field $((r',u),\omega)\mapsto\hat{A}^{r',d}_u(\omega)$ is measurable for the product of $\mathcal{R}_{T-s}\otimes\mathcal{B}_{[0,T-s]}$ with $\mathcal{T}'_s$; for $(W_s(\omega)\oplus_s r',\omega)\in G$, $u\mapsto\hat{A}^{r',d}_u$ is continuous and nondecreasing by claim (c) there and claim 1 of Cumulative-Rate Time Change: Regularity, Substitution, and Crossing Times. Fix $k\ge0$, a mark vector $v$ with $k$ entries, and an $\mathcal{R}_{T-s}$-measurable $g\ge0$. Define $\hat{F}(\omega,u)$ for $u\in\mathbb{R}^{n(k+1)}$ by the following recursion: $\hat{F}=0$ unless all $u_{d,j}>0$; start from the empty record $\hat{R}_0$ of horizon $T-s$, time $\hat{T}_0=0$, and zero fired counts; at stage $j$, the threshold of clock $d$ is the sum of its first (fired count $+1$) coordinates, its crossing time is the least $x\in[\hat{T}_j,T-s]$ with $\hat{A}^{\hat{R}_j,d}_x(\omega)$ at least the threshold if such $x$ exists and $+\infty$ otherwise; the next event time is the minimum of the crossing times, fired by the first minimizing clock in the listing, with its channel appended to the record; outcomes with a repeated event time are declared degenerate and stop the recursion; and $\hat{F}(\omega,u)=g(\hat{R}_k)$ if the recursion stops after exactly $k$ nondegenerate events with mark vector $v$, and $0$ otherwise. Measurability of $\hat{F}$ with respect to $\mathcal{T}'_s\otimes\mathcal{B}_{n(k+1)}$ follows by induction over the finitely many stages: the pairing of the stage record and time with $\omega$, composed with the jointly measurable segment consumed fields, makes the crossing comparisons measurable via the characterization that a crossing time is at most $x$ exactly when the consumed field at $x$ has reached the threshold and the stage time is at most $x$, valid by continuity; finitely many minima and comparisons then produce the next stage data, and $g$ enters through its cell restriction under the transport.

Step 3: identification. Let $\hat{\Omega}_2$ be the intersection of $\Omega^*_s$, $\hat{\Omega}_1$, $\Omega_G$, and the factorization event of claim 4 (which contains the claim (f) event); $P(\hat{\Omega}_2)=1$. Fix $\omega\in\hat{\Omega}_2$. By claim (f) and causality (claim (e)), applied to the pairs of records $W(\omega)$ and $W_s(\omega)\oplus_s\hat{R}_j$, which share their first $K_s(\omega)+j$ events whenever $\hat{R}_j$ is the record of the first $j$ segment events, the actual consumed observation increments $\tilde{A}^{d}_{s+x}-\tilde{A}^{d}_{s}$ coincide with $\hat{A}^{\hat{R}_j,d}_x$ for all $x$ up to and including the $(j+1)$-th segment event time, the consumed times being continuous. By claim 3, the segment counters are $\hat{\tilde{Y}}^d$ evaluated along the actual consumed increments, and on $\hat{\Omega}_1$ the clock $\hat{\tilde{Y}}^d$ jumps exactly when its argument crosses the next partial sum of its interarrivals. It follows by induction over the segment observation events --- at each stage, the identification of the consumed fields up to and including the next event time shows that the next crossing of the recursion is exactly the next segment observation event, with the same firing clock and channel --- that the segment record satisfies $\hat{W}\in\hat{C}_{k,v}$ (the horizon-$(T-s)$ cell) if and only if the recursion at $(\omega,\hat{\Xi}(\omega))$ stops after exactly $k$ nondegenerate events with marks $v$, in which case the segment event times are the recursion's event times; the crossing clock at each event is unique, by condition 3 for the segment solution (claim 3: the segment observation total jumps by exactly one), so no degenerate outcome occurs; and the recursion stops, the segment count being finite. Therefore $\mathbf{1}_{\hat{C}_{k,v}}(\hat{W})g(\hat{W})=\hat{F}(\cdot,\hat{\Xi}(\cdot))$ on $\hat{\Omega}_2$. For $\mathcal{T}_s$-measurable $Z\ge0$, the map $(\omega,u)\mapsto Z(\omega)\hat{F}(\omega,u)$ is $\mathcal{T}'_s\otimes\mathcal{B}_{n(k+1)}$-measurable, $\sigma(\hat{\Xi})$ is independent of $\mathcal{T}'_s$, and the two nonnegative maps just identified agree off a null event, so Independence Fubini: Integration in an Independent Random Vector Given a Sub-Sigma-Algebra, with the density form of the law of $\hat{\Xi}$ from Step 1, gives $\mathbb{E}\bigl[Z\,g(\hat{W})\,\mathbf{1}_{\hat{C}_{k,v}}(\hat{W})\bigr]=\mathbb{E}\Bigl[Z(\cdot)\int_{\mathbb{R}^{n(k+1)}}\hat{F}(\cdot,u)\,e(u)\,d\lambda_{n(k+1)}(u)\Bigr].$

Step 4: evaluation. Fix $\omega\in\Omega_G$ and abbreviate $q(t')=W_s(\omega)\oplus_s(k,t',v)$. We claim $\int_{\mathbb{R}^{n(k+1)}}\hat{F}(\omega,u)\,e(u)\,d\lambda_{n(k+1)}(u)=\int_{\mathbb{R}^k}\mathbf{1}_{D_k(T-s)}(t')\,g\bigl((k,t',v)\bigr)\,\hat{f}\bigl((k,t',v),\omega\bigr)\,d\lambda_k(t').$ We evaluate the threshold integral directly: decompose $\hat{F}$ over the firing assignments of clocks to the $k$ events with the prescribed channels, the assignment events being disjoint up to $\lambda$-null ties; for each assignment, integrate the unconstrained coordinates to $1$; integrate each survival coordinate, obtaining for clock $d$ the factor $e^{-x}$ evaluated at $x=\hat{A}^{r',d}_{T-s}-\ell_d$ with $\ell_d$ the consumed level at its last firing, by the crossing identity of claim 3 of Cumulative-Rate Time Change: Regularity, Substitution, and Crossing Times; the product over $d$ of all exponential factors telescopes to $\exp(-\sum_d\hat{A}^{r',d}_{T-s})=\exp(-N\int_{[s,T]}\tilde{b}^{\mathrm{tot}}(\Sigma^{q(t')}_u(\omega))du)$ by the agent-grouping identity, additivity, and Translation Invariance of Lebesgue Measure and the Lebesgue Integral; integrate the firing coordinates backwards, converting each by claim 4 of Cumulative-Rate Time Change: Regularity, Substitution, and Crossing Times (with the threshold shift removed by Translation Invariance of Lebesgue Measure and the Lebesgue Integral and the Monotone Convergence Theorem) into a time integral over the interval from the previous event to $T-s$ against the firing clock's rate along the stage reconstruction; identify, by causality and piecewise constancy (all but finitely many points), the stage rates with the rates along the final record's reconstruction evaluated at left limits, finitely many points being immaterial (Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval); sum over the assignments with the prescribed channels, turning the per-event factors into $\prod_{j}N\tilde{b}^{v_j}(\Sigma^{q(t')}_{(s+t'_j)-}(\omega))$ by the agent-grouping identity and linearity; and convert the resulting time-ordered iterated integral into the $\lambda_k$-integral over $D_k(T-s)$ by the Tonelli theorem as in The Ordered Time Simplex: Borel Measurability and Volume. The result is the displayed identity, the case $k=0$ being the survival computation alone.

Conclusion. Combining Steps 3 and 4 (the evaluation holding on the probability-one event $\Omega_G$), unwinding $\int_{\hat{C}_{k,v}}g\,\hat{f}\,d\rho_{T-s}$ into the $\lambda_k$-integral by claims 1, 2, and 4(c) of Assembly of Measure Spaces: Restriction, Transport, One-Point Spaces, and Countable Disjoint Unions, and summing over the countably many cells by the Monotone Convergence Theorem and additivity, gives the identity of claim 5. For the normalization, take $g=1$: then $\mathbb{E}[Z]=\mathbb{E}[Z\hat{I}]$ for every $\mathcal{T}_s$-measurable $Z\ge0$, where $\hat{I}=\int_{\mathbf{R}_{T-s}}\hat{f}(r',\cdot)\rho_{T-s}(dr')$ is $\mathcal{T}_s$-measurable by claim 4 and the Tonelli theorem, both measures being finite. With $Z$ the indicator that $\hat{I}<1$, the nonnegative variable $\mathbf{1}_{\{\hat{I}<1\}}(1-\hat{I})$ has expectation $0$, hence vanishes almost surely; with $Z$ the indicator that $\hat{I}>1$, the truncations $\min(\hat{I},n)-1$ and the Monotone Convergence Theorem give the reverse, so $\hat{I}=1$ almost surely.