Proof of Almost Sure Tracking on the Synthetic Copy: Jump Times of the Deterministic-Count Clocks, Almost Sure Conflict-Freeness of Every Record, Almost Sure Null Mass of the Untracked Records, and Trimming an Event to the Tracked Set

Throughout, "measurable" for a real-valued map on a measurable space means measurable with respect to the given $\sigma$-algebra and the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R})$ of the real line, and for a $[0,\infty]$-valued map it is the notion of Lebesgue Integral of a Nonnegative Measurable Function (for real-valued maps the two agree, as stated there); integrals of $[0,\infty]$-valued measurable maps are those of Lebesgue Integral of a Nonnegative Measurable Function, and for the indicator $\mathbf{1}_A$ of a measurable set $A$ the integral is the measure of $A$, since $\mathbf{1}_A$ is a nonnegative simple function with that integral. We use claims 1 to 5 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions for measurability of constants, indicators, sums, products, minima and pointwise limits, and claims 2 and 4 of Basic Properties of a Measure for monotonicity and countable subadditivity of measures (a finite union being a sequence padded with the empty set). The recursion and its quantities $\theta_k$, $x^{(k)}$, $K$, $\kappa^{c}_k$, $\mathsf{C}^{c,(k)}$, $\lambda^{c}_k$, $\mathcal{J}_k$ and $\mathsf{C}^{\mathrm{rec},c}$ are those of Existence, Uniqueness, Causality, and Measurability of the Open-Loop Aggregate Solution (undecorated $K$ is the recursion stopping index; the decorated $K^{c}$ are the Poisson driving variables and $\mathsf{K}$ is the cell-count vector), and ties are those of Ties in the Aggregate Recursion: Consumed Levels and First Hitting Times, Conflict-Freeness in the Absence of Ties, and the Point-Deletion Identity. Generic Borel sets are written $B'$, the letter $B$ being the rate bound. Write $\lambda$ for Lebesgue measure on $\mathbb{R}$ (undecorated $\lambda$ always means Lebesgue measure; the decorated $\lambda^{c}_k$ are the next jump levels of the recursion and are unrelated to it) and $\#S$ for the number of elements of a finite set $S$. Whenever $\mathcal{G}$ is a sub-$\sigma$-algebra of $\mathcal{F}$, the restriction $P|_{\mathcal{G}}$ of $P$ to $\mathcal{G}$ is a probability measure on $(\Omega,\mathcal{G})$ (countable additivity on disjoint sequences in $\mathcal{G}$ and $P|_{\mathcal{G}}(\Omega)=1$ are inherited from $P$), and $P|_{\mathcal{G}}(A)=P(A)$ for $A\in\mathcal{G}$.

Step 0 (finite point configurations). Let $M\in\mathbb{N}_0$ and let $u_1,\dots,u_M>0$ be pairwise distinct real numbers; put $q(t)=\sum_{a=1}^{M}\mathbf{1}\{u_a\le t\}=\#\{a:u_a\le t\}$ for $t\ge0$ (the zero function when $M=0$). We show: (a) $q$ is a counting path and $q(t-)=\#\{a:u_a<t\}$ for every $t>0$; (b) the jump times of $q$ are exactly $u_1,\dots,u_M$; (c) if $n\ge1$ is a natural number with $\tau_n(q)<+\infty$, then $\tau_n(q)$ is a jump time of $q$, hence equals one of the $u_a$.

The values of $q$ lie in $\{0,\dots,M\}$ and $q(0)=0$ since every $u_a>0$; $q$ is nondecreasing since $\{a:u_a\le s\}\subseteq\{a:u_a\le t\}$ for $s\le t$. Fix $t\ge0$ and choose $\epsilon>0$ smaller than every $u_a-t$ with $u_a>t$ ($\epsilon=1$ if there is no such $a$); for $s\in(t,t+\epsilon)$ we have $\{a:u_a\le s\}=\{a:u_a\le t\}$, so $q(s)=q(t)$, and together with $q(s)\ge q(t)$ for all $s>t$ this shows that $q(t)$ is the greatest lower bound of $\{q(s):s>t\}$: condition 3 of Counting Path and Its Jump Times. Now let $t>0$ and $q^{-}=\#\{a:u_a<t\}$. For $0\le s<t$, $\{a:u_a\le s\}\subseteq\{a:u_a<t\}$, so $q(s)\le q^{-}$; and taking $s=0$ if no $u_a$ satisfies $u_a<t$, and otherwise $s$ the largest $u_a$ with $u_a<t$ (in both cases $s\in[0,t)$), we get $q(s)=q^{-}$. Hence the least upper bound $q(t-)$ of $\{q(s):0\le s<t\}$ equals $q^{-}$, and $q(t)-q(t-)=\#\{a:u_a=t\}\le1$ by distinctness; also $q(0)-q(0-)=0$. So $q$ is a counting path, (a) holds, and $q(t)>q(t-)$ holds exactly when $t=u_a$ for some $a$, which is (b). For (c), let $t_0=\tau_n(q)<+\infty$, the greatest lower bound of the nonempty set $S=\{t\ge0:q(t)\ge n\}$. If $s>t_0$, then $s$ is not a lower bound of $S$, so some $t\in S$ satisfies $t<s$, and $q(s)\ge q(t)\ge n$; by condition 3, $q(t_0)$ is the greatest lower bound of $\{q(s):s>t_0\}$, so $q(t_0)\ge n$. Since $q(0)=0<n$ we have $t_0\neq0$, so $t_0>0$. For $0\le s<t_0$ we have $s\notin S$, so $q(s)\le n-1$ ($q(s)$ being an integer below $n$), whence $q(t_0-)\le n-1<n\le q(t_0)$: $t_0$ is a jump time, and by (b) it is one of the $u_a$.

Step 1 (claim 1). Let $\omega\in\Omega^{U}_0$, $y\in\mathbb{N}_0^{\mathsf{L}}$ and $c$ a label. Since $\omega\in\Omega^{U}_0$, the factor $\mathbf{1}_{\Omega^{U}_0}(\omega)$ in the definition of the deterministic-count clocks in The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record equals $1$, so $\mathsf{P}^{(y),c}_u(\omega)=\sum_{j=1}^{J_c}\sum_{i=1}^{y_{c,j}}\mathbf{1}\{U^{c,j}_i(\omega)\le u\}$ for $u\ge0$; and on $\Omega^{U}_0$ the points $U^{c,j}_i(\omega)$ ($1\le j\le J_c$, $1\le i\le y_{c,j}$) are pairwise distinct and lie in $I_{c,j}\subseteq(0,R]$, so are positive. Thus $u\mapsto\mathsf{P}^{(y),c}_u(\omega)$ is the path $q$ of Step 0 for these finitely many points, and claim 1 is (b) and (c) of Step 0.

Step 2 (measurability of the bad sets). Fix $y\in\mathbb{N}_0^{\mathsf{L}}$. 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 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 applies to the probability space $(\Omega,\mathcal{U},P|_{\mathcal{U}})$ with the clocks $\mathsf{P}^{(y)}$, and its claim 1 gives $\mathsf{G}^{(y)}\in\mathcal{R}\otimes\mathcal{H}^{(y)}_T$, where $\mathcal{H}^{(y)}_T$ is the $\sigma$-algebra generated by the variables $\mathsf{P}^{(y),c}_u$, $u\in[0,NBT]$; these variables are $\mathcal{U}$-measurable (claim 2 of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record), so their preimages of Borel sets lie in the $\sigma$-algebra $\mathcal{U}$ and $\mathcal{H}^{(y)}_T\subseteq\mathcal{U}$. Every measurable rectangle of $\mathcal{R}\otimes\mathcal{H}^{(y)}_T$ is a measurable rectangle of $\mathcal{R}\otimes\mathcal{U}$ (Product Sigma-Algebra), and $\mathcal{R}\otimes\mathcal{U}$ is a $\sigma$-algebra containing them, so $\mathcal{R}\otimes\mathcal{H}^{(y)}_T\subseteq\mathcal{R}\otimes\mathcal{U}$ and $\mathsf{G}^{(y)}\in\mathcal{R}\otimes\mathcal{U}$. Hence $f_y=\mathbf{1}_{(\mathbf{R}\times\Omega)\setminus\mathsf{G}^{(y)}}$ is an $\mathcal{R}\otimes\mathcal{U}$-measurable map into $[0,\infty]$. The measure spaces $(\mathbf{R},\mathcal{R},\rho)$ and $(\Omega,\mathcal{U},P|_{\mathcal{U}})$ are $\sigma$-finite ($\rho$ by The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record, $P$ being finite), so the section statement of Tonelli and Fubini Theorems applies to $f_y$: for every $r\in\mathbf{R}$ the map $\omega\mapsto f_y(r,\omega)=\mathbf{1}_{\Omega^{r,y}}(\omega)$ is $\mathcal{U}$-measurable, so $\Omega^{r,y}=\{\omega:f_y(r,\omega)>0\}\in\mathcal{U}$; and for every $\omega\in\Omega$ the map $r\mapsto f_y(r,\omega)=\mathbf{1}_{B^{(y)}_\omega}(r)$ is $\mathcal{R}$-measurable, so $B^{(y)}_\omega\in\mathcal{R}$. This proves the first assertions of claims 2 and 3.

Step 3 (the deleted clocks and the hitting functional). Fix $y\in\mathbb{N}_0^{\mathsf{L}}$ and $r\in\mathbf{R}$. Call a pair $(\iota,\iota')$ of triples $\iota=(c_0,j,i)$, $\iota'=(c',j',i')$ admissible if $c_0$ and $c'$ are distinct labels, $1\le j\le J_{c_0}$, $1\le i\le y_{c_0,j}$, $1\le j'\le J_{c'}$ and $1\le i'\le y_{c',j'}$; there are finitely many admissible pairs. Fix one. Every object introduced in this step depends on $(y,r,\iota,\iota')$; this dependence is suppressed in the notation except where it is displayed. Let $\mathcal{G}_\iota$ be the $\sigma$-algebra generated by the family of all driving variables other than $U^{c_0,j}_i$ (a sub-$\sigma$-algebra of $\mathcal{F}$). Let $\tilde\Omega_\iota$ be the complement of the union of the countably many events $\{U^{c,j_1}_{a_1}=U^{c,j_2}_{a_2}\}$, with $c$ a label, $j_1,j_2\in\{1,\dots,J_c\}$, $a_1,a_2\in\mathbb{N}$, $(j_1,a_1)\neq(j_2,a_2)$ and neither $(c,j_1,a_1)$ nor $(c,j_2,a_2)$ equal to $\iota$, and $\{U^{c,j_1}_{a_1}\notin I_{c,j_1}\}$ with $c$ a label, $j_1\in\{1,\dots,J_c\}$, $a_1\in\mathbb{N}$ and $(c,j_1,a_1)\neq\iota$. Each of these events lies in $\mathcal{G}_\iota$: the first kind is the preimage of the Borel set $\{0\}$ under the $\mathcal{G}_\iota$-measurable difference of two $\mathcal{G}_\iota$-measurable variables, the second is the preimage of the Borel set $\mathbb{R}\setminus I_{c,j_1}$. Hence $\tilde\Omega_\iota\in\mathcal{G}_\iota$; and $\Omega^{U}_0\subseteq\tilde\Omega_\iota$, since the conditions defining $\tilde\Omega_\iota$ are among those defining $\Omega^{U}_0$. For every label $c$ and $u\ge0$ put $\mathsf{Q}^{c}_u=\mathbf{1}_{\tilde\Omega_\iota}\sum_{j''=1}^{J_c}\sum_{a=1}^{y_{c,j''}}\mathbf{1}\{(c,j'',a)\neq\iota\}\,\mathbf{1}\{U^{c,j''}_a\le u\},$ where $\mathbf{1}\{(c,j'',a)\neq\iota\}$ is the constant $0$ or $1$. Each $\mathsf{Q}^{c}_u$ is $\mathcal{G}_\iota$-measurable (the event $\{U^{c,j''}_a\le u\}$ is the preimage of a Borel set under a variable generating $\mathcal{G}_\iota$ whenever $(c,j'',a)\neq\iota$; then use claims 1 to 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions). Every path $u\mapsto\mathsf{Q}^{c}_u(\omega)$ is a counting path: for $\omega\notin\tilde\Omega_\iota$ it is the zero function, which satisfies conditions 1 to 4 of Counting Path and Its Jump Times (all greatest lower and least upper bounds involved being $0$); for $\omega\in\tilde\Omega_\iota$ it is the path $q$ of Step 0 for the finitely many points $U^{c,j''}_a(\omega)$ with $1\le j''\le J_c$, $1\le a\le y_{c,j''}$ and $(c,j'',a)\neq\iota$, which are pairwise distinct and lie in $I_{c,j''}\subseteq(0,R]$ by the definition of $\tilde\Omega_\iota$. Thus $\mathsf{Q}=(\mathsf{Q}^{c})_c$ is a family of stochastic processes on the probability space $(\Omega,\mathcal{G}_\iota,P|_{\mathcal{G}_\iota})$ all of whose paths are counting paths, and we write $\mathsf{Q}(\omega)$ for the clock family $(u\mapsto\mathsf{Q}^{c}_u(\omega))_c$. Moreover, for $\omega\in\Omega^{U}_0$ (so that $\mathbf{1}_{\Omega^{U}_0}(\omega)=\mathbf{1}_{\tilde\Omega_\iota}(\omega)=1$) the definition of $\mathsf{P}^{(y)}$ gives $\mathsf{Q}^{c}_u(\omega)=\mathsf{P}^{(y),c}_u(\omega)\ (c\neq c_0),\qquad \mathsf{Q}^{c_0}_u(\omega)=\mathsf{P}^{(y),c_0}_u(\omega)-\mathbf{1}\{u\ge U^{c_0,j}_i(\omega)\}\qquad(u\ge0):$ $\mathsf{Q}(\omega)$ is the family obtained from $\mathsf{P}^{(y)}(\omega)$ by deleting the jump of $\mathsf{P}^{(y),c_0}(\omega)$ at $U^{c_0,j}_i(\omega)$, in the sense of claim 3 of Ties in the Aggregate Recursion: Consumed Levels and First Hitting Times, Conflict-Freeness in the Absence of Ties, and the Point-Deletion Identity.

By claim 1 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 (available 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), each component of $\mathsf{a}(s,r')=a^{r'}(s)$ on $[0,T]\times\mathbf{R}$ is measurable with respect to $\mathcal{B}_{[0,T]}\otimes\mathcal{R}$. Apply claim 4 of Existence, Uniqueness, Causality, and Measurability of the Open-Loop Aggregate Solution on the probability space $(\Omega,\mathcal{G}_\iota,P|_{\mathcal{G}_\iota})$ with the clocks $\mathsf{Q}$, the measurable space $(\mathbf{R},\mathcal{R})$ (nonempty, as $r\in\mathbf{R}$) and the map $\mathsf{a}$: for $r'\in\mathbf{R}$ and $\omega\in\Omega$ run the recursion for the data $(\mathsf{Q}(\omega),a^{r'},x_0)$ and write $\mathsf{C}^{\mathsf{Q},r',c}_t(\omega)$ for its recursion consumed times; then for every $t\in[0,T]$ and every label $c$ the map $(r',\omega)\mapsto\mathsf{C}^{\mathsf{Q},r',c}_t(\omega)$ is $\mathcal{R}\otimes\mathcal{G}_\iota$-measurable. By claim 1 of Ties in the Aggregate Recursion: Consumed Levels and First Hitting Times, Conflict-Freeness in the Absence of Ties, and the Point-Deletion Identity applied to each data set $(\mathsf{Q}(\omega),a^{r'},x_0)$, this map is nonnegative, so the section statement of Tonelli and Fubini Theorems (for the $\sigma$-finite spaces $(\mathbf{R},\mathcal{R},\rho)$ and $(\Omega,\mathcal{G}_\iota,P|_{\mathcal{G}_\iota})$) shows that $\omega\mapsto\mathsf{C}^{c}_t(\omega):=\mathsf{C}^{\mathsf{Q},r,c}_t(\omega)$, for our fixed $r$, is a $\mathcal{G}_\iota$-measurable real-valued map, for every $t\in[0,T]$ and every $c$. By the same claim 1, for every $\omega$ the map $t\mapsto\mathsf{C}^{c}_t(\omega)$ is nondecreasing on $[0,T]$ with $\mathsf{C}^{c}_0(\omega)=0$ and $|\mathsf{C}^{c}_t(\omega)-\mathsf{C}^{c}_s(\omega)|\le NB(t-s)$ for $0\le s\le t\le T$.

Put $W=U^{c',j'}_{i'}$ (a symbol coined here; it is not one of the driving variables $V^{c}_i$), which is $\mathcal{G}_\iota$-measurable because $\iota'\neq\iota$ (as $c'\neq c_0$). For $\omega\in\Omega$ let $t^{*}(\omega)$ be the greatest lower bound of $S_\omega=\{t\in[0,T]:\mathsf{C}^{c'}_t(\omega)\ge W(\omega)\}$ ($+\infty$ if $S_\omega$ is empty), let $\bar{t}(\omega)=\min(t^{*}(\omega),T)\in[0,T]$, and let $F(\omega)=\mathsf{C}^{c_0}_{\bar{t}(\omega)}(\omega).$

(a) For every $s\in[0,T]$, $\{\omega:t^{*}(\omega)\le s\}=\{\omega:\mathsf{C}^{c'}_s(\omega)\ge W(\omega)\}$. Indeed, if $\mathsf{C}^{c'}_s(\omega)\ge W(\omega)$ then $s\in S_\omega$ and $t^{*}(\omega)\le s$. Conversely let $t^{*}(\omega)\le s$; then $S_\omega$ is nonempty and $t^{*}(\omega)\in[0,T]$. For every $\epsilon>0$ there is $t\in S_\omega$ with $t<t^{*}(\omega)+\epsilon$ (otherwise $t^{*}(\omega)+\epsilon$ would be a lower bound of $S_\omega$ exceeding its greatest lower bound), and $t\ge t^{*}(\omega)$, so the Lipschitz bound gives $\mathsf{C}^{c'}_{t^{*}(\omega)}(\omega)\ge\mathsf{C}^{c'}_t(\omega)-NB\epsilon\ge W(\omega)-NB\epsilon$; as $\epsilon>0$ is arbitrary, $\mathsf{C}^{c'}_{t^{*}(\omega)}(\omega)\ge W(\omega)$, and by monotonicity $\mathsf{C}^{c'}_s(\omega)\ge W(\omega)$. Consequently $\{t^{*}\le s\}=\{\mathsf{C}^{c'}_s-W\ge0\}\in\mathcal{G}_\iota$ (preimage of the Borel set $[0,\infty)$ under a $\mathcal{G}_\iota$-measurable difference), and $\{\bar{t}\le s\}\in\mathcal{G}_\iota$ for every $s\in[0,T]$, this set being $\{t^{*}\le s\}$ for $s<T$ and $\Omega$ for $s=T$.

(b) $F$ is $\mathcal{G}_\iota$-measurable. For $n\in\mathbb{N}$ and $1\le k\le n$ put $A_{n,k}=\{\bar{t}\le kT/n\}\setminus\{\bar{t}\le(k-1)T/n\}=\{\bar{t}\in((k-1)T/n,kT/n]\}\in\mathcal{G}_\iota$ and $F_n=\sum_{k=1}^{n}\mathbf{1}_{A_{n,k}}\mathsf{C}^{c_0}_{kT/n}$, a $\mathcal{G}_\iota$-measurable map. Let $\omega\in\Omega$. If $\bar{t}(\omega)=0$, then $F(\omega)=\mathsf{C}^{c_0}_0(\omega)=0$ and $\omega\notin A_{n,k}$ for all $k$, so $F_n(\omega)=0=F(\omega)$. Otherwise $\bar{t}(\omega)\in(0,T]$ lies in exactly one of the intervals $((k-1)T/n,kT/n]$, $1\le k\le n$, so $F_n(\omega)=\mathsf{C}^{c_0}_{kT/n}(\omega)$ for that $k$ and, by the Lipschitz bound, $|F_n(\omega)-F(\omega)|\le NB\,(kT/n-\bar{t}(\omega))\le NBT/n$. Hence $F_n(\omega)$ converges to $F(\omega)$ for every $\omega$, and $F$ is $\mathcal{G}_\iota$-measurable by claim 5 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions. We write $F_{\iota,\iota'}$ for $F$ when the admissible pair needs to be displayed.

Step 4 (the bad set is covered by tie events). Keep $y$ and $r$ fixed and put, for each admissible pair, $E_{\iota,\iota'}=\{\omega\in\Omega:\ U^{c_0,j}_i(\omega)=F_{\iota,\iota'}(\omega)\}$. We show $\Omega^{r,y}\subseteq(\Omega\setminus\Omega^{U}_0)\cup\bigcup_{(\iota,\iota')\text{ admissible}}E_{\iota,\iota'} .$ Let $\omega\in\Omega^{r,y}\cap\Omega^{U}_0$ and run the recursion for the data $(p,a,x_0)$ with $p=\mathsf{P}^{(y)}(\omega)$ and $a=a^{r}$. These data are not conflict-free, so by claim 2 of Ties in the Aggregate Recursion: Consumed Levels and First Hitting Times, Conflict-Freeness in the Absence of Ties, and the Point-Deletion Identity some step $k<K$ is a tie: $\mathcal{J}_k$ contains two distinct labels $c_0$ and $c'$. By the definition of $\mathcal{J}_k$, $\mathsf{C}^{c_0,(k)}_{\theta_{k+1}}\ge\lambda^{c_0}_k$ with a real left side, so $\lambda^{c_0}_k=\tau_{n_0}(p^{c_0})$ (with $n_0=p^{c_0}(\kappa^{c_0}_k)+1$) is finite; by claim 1 of the present lemma (Step 1), $u:=\lambda^{c_0}_k$ is a jump time of $p^{c_0}$ and $u=U^{c_0,j}_i(\omega)$ for some $1\le j\le J_{c_0}$ and $1\le i\le y_{c_0,j}$. In the same way $v:=\lambda^{c'}_k=U^{c',j'}_{i'}(\omega)$ for some $1\le j'\le J_{c'}$ and $1\le i'\le y_{c',j'}$. Then $\iota=(c_0,j,i)$, $\iota'=(c',j',i')$ is an admissible pair. Apply claim 3 of Ties in the Aggregate Recursion: Consumed Levels and First Hitting Times, Conflict-Freeness in the Absence of Ties, and the Point-Deletion Identity to $p$, the label $c_0$, the jump time $u$, the step $k$ and the label $c'$ (so that its $v$ is our $v$): the deleted family $p^{-}$ there is exactly $\mathsf{Q}(\omega)$ for this $\iota$ by the identity displayed in Step 3, so the quantities marked by a minus sign there are those of the recursion for $(\mathsf{Q}(\omega),a^{r},x_0)$, that is, $\mathsf{C}^{-,\mathrm{rec},c}_t=\mathsf{C}^{c}_t(\omega)$ in the notation of Step 3 for this admissible pair $(\iota,\iota')$ (which also fixes $W=U^{c',j'}_{i'}$, $t^{*}$ and $\bar{t}$ there). Its conclusion gives $\theta_{k+1}=\inf\{t\in[0,T]:\mathsf{C}^{c'}_t(\omega)\ge v\}$, which is $t^{*}(\omega)$ because $v=W(\omega)$; thus $t^{*}(\omega)=\theta_{k+1}\in(0,T]$, $\bar{t}(\omega)=\theta_{k+1}$, and the conclusion $u=\mathsf{C}^{-,\mathrm{rec},c_0}_{\theta_{k+1}}$ reads $U^{c_0,j}_i(\omega)=\mathsf{C}^{c_0}_{\bar{t}(\omega)}(\omega)=F_{\iota,\iota'}(\omega)$, i.e. $\omega\in E_{\iota,\iota'}$.

Step 5 (each tie event is null). Fix an admissible pair and let $X=U^{c_0,j}_i$, a random variable, that is, measurable from $\mathcal{F}$ to $\mathcal{B}(\mathbb{R})$, which is the $\sigma$-algebra $\mathcal{B}_1$ of Finite Products of Lebesgue Measure and Coordinate Integration on $\mathbb{R}^l$. The driving variables form an independent family indexed by a countable set; enumerating this index set by a bijection with $\mathbb{N}$ (independence of a family is a condition on its finite subfamilies and is therefore unaffected by reindexing) and applying Grouping Lemma for Independent Random Variables to the two disjoint nonempty index sets consisting of the index of $X$ and of all other indices, the $\sigma$-algebras $\sigma(X)$ and $\mathcal{G}_\iota$ are independent. The image measure $\mu_X$ of $P$ under $X$, $\mu_X(B)=P(X^{-1}(B))$, is the distribution of $X$, which by The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record is the uniform law $\nu_{I}$ on $I=I_{c_0,j}$ 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: $\mu_X(B')=\lambda(I\cap B')/\lambda(I)$ for Borel $B'$, with $\lambda(I)=b^{c_0}_j-b^{c_0}_{j-1}=|I_{c_0,j}|$ by claim 4 of Existence of Lebesgue Measure on the Real Line, which is $>0$ since $b^{c_0}_{j-1}<b^{c_0}_j$.

Define $\Psi:\Omega\times\mathbb{R}\to[0,\infty]$ by $\Psi(\omega,u)=\mathbf{1}\{u=F(\omega)\}$. It is $\mathcal{G}_\iota\otimes\mathcal{B}(\mathbb{R})$-measurable: the maps $(\omega,u)\mapsto u$ and $(\omega,u)\mapsto F(\omega)$ are measurable with respect to $\mathcal{G}_\iota\otimes\mathcal{B}(\mathbb{R})$, since for a Borel set $B'$ their preimages $\Omega\times B'$ and $F^{-1}(B')\times\mathbb{R}$ are measurable rectangles; hence their difference is measurable, the set $Z=\{(\omega,u):u-F(\omega)=0\}$ is the preimage of the Borel set $\{0\}$, and $\Psi=\mathbf{1}_Z$. By claim 3 of Independence Fubini: Integration in an Independent Random Vector Given a Sub-Sigma-Algebra (with $n=1$, $\mathcal{G}=\mathcal{G}_\iota$ and this $X$), $\mathbb{E}\bigl[\Psi(\cdot,X(\cdot))\bigr]=\mathbb{E}\Bigl[\int_{\mathbb{R}}\Psi(\cdot,u)\,d\mu_X(u)\Bigr].$ The left side is $P(E_{\iota,\iota'})$, because $\Psi(\omega,X(\omega))=\mathbf{1}_{E_{\iota,\iota'}}(\omega)$, and this map is $\mathcal{F}$-measurable by claim 1 of the same lemma, so that $E_{\iota,\iota'}$, the preimage of the Borel set $\{1\}$ under it, lies in $\mathcal{F}$. On the right, for every $\omega$ the section $u\mapsto\Psi(\omega,u)$ is the indicator of the Borel set $\{F(\omega)\}$, so the inner integral equals $\mu_X(\{F(\omega)\})=\lambda(I\cap\{F(\omega)\})/\lambda(I)\le\lambda(\{F(\omega)\})/\lambda(I)=0$, since the interval $\{F(\omega)\}=[F(\omega),F(\omega)]$ has Lebesgue measure $0$ by claim 4 of Existence of Lebesgue Measure on the Real Line. Hence the right side is the expectation of the zero function, which is $0$, and $P(E_{\iota,\iota'})=0$.

Step 6 (claim 2). By Step 4, monotonicity and subadditivity, $P(\Omega^{r,y})\le P(\Omega\setminus\Omega^{U}_0)+\sum_{(\iota,\iota')}P(E_{\iota,\iota'})$, the sum running over the finitely many admissible pairs. Each summand is $0$ by Step 5, and $P(\Omega\setminus\Omega^{U}_0)=1-P(\Omega^{U}_0)=0$ by claim 3 of Basic Properties of a Measure and claim 1 of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record. So $P(\Omega^{r,y})=0$, which with Step 2 proves claim 2.

Step 7 (claim 3). Fix $y$ and let $f_y$ be as in Step 2. In this paragraph we work on the probability space $(\Omega,\mathcal{U},P|_{\mathcal{U}})$, writing $P'=P|_{\mathcal{U}}$ and $\mathbb{E}'$ for the expectation with respect to $P'$. By Tonelli and Fubini Theorems on $(\mathbf{R},\mathcal{R},\rho)$ and $(\Omega,\mathcal{U},P')$, the map $\phi_y(\omega)=\int_{\mathbf{R}}f_y(r,\omega)\,d\rho(r)$ is $\mathcal{U}$-measurable with values in $[0,\infty]$, and $\int_{\Omega}\phi_y\,dP'=\int_{\mathbf{R}}\Bigl(\int_{\Omega}f_y(r,\omega)\,dP'(\omega)\Bigr)d\rho(r).$ For fixed $r$ the inner integrand is $\mathbf{1}_{\Omega^{r,y}}$ with $\Omega^{r,y}\in\mathcal{U}$, whose integral is $P'(\Omega^{r,y})=P(\Omega^{r,y})=0$ by claim 2; so the inner integrals vanish identically and $\int_{\Omega}\phi_y\,dP'=0$. Also, for fixed $\omega$, $f_y(r,\omega)=\mathbf{1}_{B^{(y)}_\omega}(r)$ with $B^{(y)}_\omega\in\mathcal{R}$ (Step 2), so $\phi_y(\omega)=\rho(B^{(y)}_\omega)$ and $\Omega_y=\{\phi_y=0\}=\Omega\setminus\{\phi_y>0\}\in\mathcal{U}$. Put $\psi=\min(\phi_y,1)$; then $\{\psi>\alpha\}=\{\phi_y>\alpha\}$ for real $\alpha<1$ and $\{\psi>\alpha\}=\emptyset$ for $\alpha\ge1$, so $\psi$ is a $\mathcal{U}$-measurable map with values in $[0,1]$, a nonnegative random variable on $(\Omega,\mathcal{U},P')$, with $\psi\le\phi_y$ and therefore $\mathbb{E}'[\psi]\le\int_{\Omega}\phi_y\,dP'=0$ by claim 1 of Linearity and Monotonicity of the Lebesgue Integral. Markov's inequality (Markov's and Chebyshev's Inequalities) on $(\Omega,\mathcal{U},P')$ gives $P'(\psi\ge1/n)\le n\,\mathbb{E}'[\psi]=0$ for every $n\in\mathbb{N}$. Since $\{\phi_y>0\}=\bigcup_{n\in\mathbb{N}}\{\psi\ge1/n\}$ (if $\phi_y(\omega)>0$ then $\psi(\omega)>0$, so $\psi(\omega)\ge1/n$ for some $n$; conversely $\psi\ge1/n$ forces $\phi_y>0$), countable subadditivity gives $P'(\phi_y>0)=0$, hence $P'(\Omega_y)=1$ by claim 3 of Basic Properties of a Measure; as $\Omega_y\in\mathcal{U}$, this says $P(\Omega_y)=1$.

The set $\mathbb{N}_0^{\mathsf{L}}$ is countable: $y\mapsto y/\sqrt{N}$ is a bijection of it onto the parameter lattice $\mathsf{S}$, which is countable by claim 5 of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record, in the sense of Assembly of Measure Spaces: Restriction, Transport, One-Point Spaces, and Countable Disjoint Unions: finite or the set of values of a sequence, and in either case (padding a finite list by repeating its last term; $\mathsf{S}$ is nonempty as $0\in\mathsf{S}$) the set of values of a sequence, which composed with the inverse bijection exhibits $\mathbb{N}_0^{\mathsf{L}}$ as the set of values of a sequence $(y_n)_{n\in\mathbb{N}}$. Hence $\Omega'=\bigcap_{n\in\mathbb{N}}\Omega_{y_n}$ is a countable intersection of members of $\mathcal{U}$, so $\Omega'\in\mathcal{U}$, and $P(\Omega\setminus\Omega')=P\bigl(\bigcup_{n\in\mathbb{N}}(\Omega\setminus\Omega_{y_n})\bigr)\le\sum_{n\in\mathbb{N}}P(\Omega\setminus\Omega_{y_n})=0$, so $P(\Omega')=1$.

Finally let $\omega\in\Omega'$, put $y_0=\mathsf{K}(\omega)$ and let $Y(\omega)$ be the finite set consisting of $y_0$ and of the vectors $y_0-\mathsf{m}e_{c,j}$ for the $(c,j)\in\mathsf{L}$ with $\mathsf{K}_{c,j}(\omega)\ge\mathsf{m}$; each member of $Y(\omega)$ lies in $\mathbb{N}_0^{\mathsf{L}}$, the $(c,j)$-coordinate of $y_0-\mathsf{m}e_{c,j}$ being $\mathsf{K}_{c,j}(\omega)-\mathsf{m}\ge0$ and the others unchanged. 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, $(r,\omega)\in\mathsf{G}^{\sharp}$ if and only if $(r,\omega)\in\mathsf{G}^{(y_0)}$; so by the definition of $\mathsf{T}_\omega$, $\mathbf{R}\setminus\mathsf{T}_\omega=\bigcup_{y\in Y(\omega)}B^{(y)}_\omega ,$ a finite union of members of $\mathcal{R}$ (Step 2), hence a member of $\mathcal{R}$, and $\rho(\mathbf{R}\setminus\mathsf{T}_\omega)\le\sum_{y\in Y(\omega)}\rho(B^{(y)}_\omega)=0$ because $\omega\in\Omega_y$ for every $y\in Y(\omega)$. This proves claim 3.

Step 8 (claim 4). Let $G_0\in\mathcal{F}$ and $G=G_0\cap\Omega'$. Since $\Omega'\in\mathcal{U}\subseteq\mathcal{F}$, $G\in\mathcal{F}$, and $\rho(\mathbf{R}\setminus\mathsf{T}_\omega)=0$ for $\omega\in G\subseteq\Omega'$ by claim 3. As $\Omega\setminus G=(\Omega\setminus G_0)\cup(\Omega\setminus\Omega')$, monotonicity gives $P(\Omega\setminus G)\ge P(\Omega\setminus G_0)$ and subadditivity gives $P(\Omega\setminus G)\le P(\Omega\setminus G_0)+P(\Omega\setminus\Omega')=P(\Omega\setminus G_0)$.