Normalized record measure and absence of explosion

The component Borel sigma-algebras agree with the time-and-mark product sigma-algebras: rational open boxes form a countable basis for each Euclidean time space, so open sets are countable unions of basic rectangles; the finite discrete mark space adds only finitely many pieces. Thus the ordered simplexes are measurable, and [F6] supplies the finite component base measures ${\mu }_n$. The matrix densities are continuous in the time coordinates for each mark word and nonnegative, hence Borel measurable by [F5]. For $n\ge 1$, define $q_n(t)$ by integrating $\mathrm{Tr}(W_n\rho W_n^{†})$ over the ordered $n$ event times up to $t$ and summing marks; let $q_0(t)=1$. For any positive $X$, ${\sum }_a\mathrm{Tr}(L_{a}X{L}_a^{†})=\mathrm{Tr}(QX)\le r\mathrm{Tr}X$, while each intervening $S_v$ contracts trace. Summing marks successively from the last mark backwards bounds the summed integrand by $r^n$. The ordered simplex has volume $t^n\mathrm{/}n!$, by iterating the elementary integral of $t^{n-1}\mathrm{/}(n-1)!$. Hence $q_n(t)\le (rt{)}^n\mathrm{/}n!$. An available bound always exists: take $r=\mathrm{\parallel }Q\mathrm{\parallel }$, the largest eigenvalue of the positive matrix $Q$. If $M$ is empty, every positive-event component is empty and $q_n(t)=0$ for $n\ge 1$.

Write $\lambda$ for Borel Lebesgue measure on $\mathbb{R}$. Translation is a Borel bijection with Borel inverse, and [F11] gives $\lambda (E+b)=\lambda (E)$ for every Borel $E$. For $\mathrm{\ell }\ge 1$ and $b\in {\mathbb{R}}^{\mathrm{\ell }}$, the translated product measure and ${\lambda }^{\otimes \mathrm{\ell }}$ agree on rectangles by the one-dimensional identity, and are sigma-finite; the uniqueness clause in [F6], iterated through the finite factors, makes them equal on all product-Borel sets. For a Borel $D\subseteq {\mathbb{R}}^{\mathrm{\ell }}$ and nonnegative Borel $g$ on $D+b$, it follows first for indicators, then for nonnegative simple functions by their coefficient-sum definition, and finally for $g$ by the supremum definition in [F12], that ${\int }_{D+b}g(v)d{\lambda }^{\otimes \mathrm{\ell }}(v)={\int }_{D}g(u+b)d{\lambda }^{\otimes \mathrm{\ell }}(u)\mathrm{.}$ The last passage uses the bijection between simple minorants given by composition with the translation and its inverse. Restricting to intervals or ordered simplexes preserves this identity, and summing unchanged discrete marks preserves it as well. With no time coordinates the identity is immediate. When $b$ depends on a fixed prefix, apply this identity separately for each prefix and then use [F3] for the nonnegative outer integral; the translations used below depend continuously on the prefix time, so the composed integrands remain product-measurable.

The family ${\mathcal{A}}_U$ is a sigma-algebra: intersections with any component commute with complements relative to ${\mathrm{\Omega }}_U$ and with countable unions. By step 1.1 and [F7], each ${\nu }_n(A_n)={\int }_{A_n}\mathrm{Tr}(K_U\rho K_U^{†})d{\mu }_n$ is a measure on $E_n(U)$, including the singleton component. Define ${\mathbb{P}}_U(A)={\sum }_n{\nu }_n(A\cap E_n(U))$. It vanishes at the empty set. For pairwise disjoint $A_j\in {\mathcal{A}}_U$, component countable additivity gives ${\mathbb{P}}_U({\bigcup }_j{A}_j)={\sum }_n{\sum }_j{\nu }_n(A_j\cap E_n(U))={\sum }_j{\mathbb{P}}_U(A_j)$. The interchange holds for nonnegative double series, since either order is the supremum of the sums over finite rectangles of indices, or by [F3] on counting spaces. Thus the proposed law is a countably additive measure before normalization.

Let $p_n(U)={\nu }_n(E_n(U))$. For a prefix ending at $t_n$, set $X=W_n\rho W_n^{†}$, with $X=\rho$ and $t_n=0$ when $n=0$. The waiting-kernel identity, multiplied by $\mathrm{Tr}X$, says $\mathrm{Tr}(S_{U-t_n}(X))+{\sum }_a{\int }_0^{U-t_n}\mathrm{Tr}(L_a{S}_u(X)L_a^{†})du=\mathrm{Tr}X$. It also holds when $\mathrm{Tr}X=0$, since positivity then gives $X=0$. For each fixed prefix, step 1.2 permits the change from the absolute next-event time $v\in (t_n,U)$ to $u=v-t_n$. Integrating prefixes by [F3] therefore gives $p_n(U)=q_n(U)-q_{n+1}(U)$, including $n=0$. Products of successive normalized waiting densities and terminal survival factors cancel their trace denominators, leaving exactly $\mathrm{Tr}(K_U\rho K_U^{†})$; zero-density prefixes contribute zero. Hence ${\sum }_{n=0}^{k-1}{p}_n(U)=1-q_k(U)\to 1$: for $rU>0$, the bound $(rU{)}^k\mathrm{/}k!$ has successive ratio $rU\mathrm{/}(k+1)\le 1\mathrm{/}2$ eventually, and the case $rU=0$ is immediate. Therefore step 2.1 has total mass one and is a probability measure by [F8]. This argument holds for every density matrix and finite remaining horizon, without presupposing a stochastic existence theorem.

For fixed $0<t<U$, partition each $E_n(U)$ according to $k=\mathrm{\#}\{j:t_j<t\}$. These pieces are Borel, as the cut conditions are coordinate inequalities. On each piece $R_{U,t}$ is the coordinate projection onto the first $k$ time-and-mark pairs, with the constant empty record for $k=0$. Inverse images of Borel subsets of $E_k(t)$ are Borel by continuity of this projection, and taking the finite union over $k$ and the countable union over $n$ proves measurability for ${\mathcal{A}}_t$. The endpoint restrictions are constant at zero and the identity at $U$. Also $N_t={\sum }_{j=1}^n{\mathbf{1}}_{\{t_j\le t\}}$ on $E_n(U)$, so each $N_t$ is measurable into the discrete space ${\mathbb{N}}_0$. By [F9], this family on $({\mathrm{\Omega }}_U,{\mathcal{A}}_U,{\mathbb{P}}_U)$ is a stochastic process. For each fixed $t$, the coordinate sets $\{t_j=t\}$ have base measure zero by the product rectangle formula and Tonelli, because a singleton time has Lebesgue measure zero. Their finite union on each component, then countable union over components, has ${\mathbb{P}}_U$-measure zero by [F7] and step 2.1. Thus the strict and non-strict count conventions agree almost surely at each fixed time.

Fix $0<t<U$ and a prefix of $k$ events before $t$. Its unnormalized state at $t$ is $X_t=K_t\rho K_t^{†}$ with weight $w=\mathrm{Tr}{X}_t$. If $w>0$, every completed continuation after $t$ factors into the prefix weight times the same record density on horizon $U-t$ with initial state $X_t\mathrm{/}w$; this follows from multiplication of the matrices defining $K_U$. Step 1.2 sends the absolute continuation times $v_j\in (t,U)$ to $v_j-t\in (0,U-t)$ without changing their product-time/counting-mark measure. By step 3.1 those continuation densities have total mass one. If $w=0$, positivity gives $X_t=0$ and every continuation density vanishes. The state maps are measurable by [F2]. Multiplying by the indicator of any Borel prefix event and using [F3] to integrate and sum continuations therefore leaves exactly its ${\mathbb{P}}_t$ weight. The null event at the cut, from step 3.2, is harmless. This proves $(R_{U,t}{)}_{\ast }{\mathbb{P}}_U={\mathbb{P}}_t$; the endpoint cases are immediate. Finally $\mathrm{Pr}(N_t\ge n)={\sum }_{k\ge n}{p}_k(t)=q_n(t)\le (rt{)}^n\mathrm{/}n!$, using the telescoping identity and vanishing remainder; at $t=0$ both sides are zero for $n\ge 1$. Every record contains finitely many strictly ordered times, so its count path is right-continuous with unit jumps and finitely many jumps on the horizon. The process is therefore simple and non-explosive.

To specify successive conditioning, fix an integer $k\ge 0$ and give ${\mathcal{P}}_k=E_k(U)\bigsqcup \{\mathrm{\partial }\}$ the component Borel sigma-algebra, with $\mathrm{\partial }$ isolated. The random prefix $P_k$ is the first $k$ time-and-mark pairs when $N_U\ge k$, and is $\mathrm{\partial }$ otherwise; $P_0$ is always the empty record. Give $D=((0,U)\times M)\bigsqcup \{†\}$ the analogous sigma-algebra, with $†$ the censored next-outcome symbol. Define $Y_k=(t_{k+1}-t_k,a_{k+1})$ when $N_U\ge k+1$, with $t_0=0$, and $Y_k=†$ otherwise. On every record component these maps are either coordinate projections, continuous differences of coordinates, or constants on measurable pieces, so they are measurable by [F5]. For $p\in E_k(U)$ write $s(p)=t_k$ (zero for $k=0$), $w_k(p)=\mathrm{Tr}(W_k\rho W_k^{†})$ and $\sigma (p)=W_k\rho W_k^{†}\mathrm{/}{w}_k(p)$ on $w_k>0$. Integrating all continuations after $s(p)$ gives prefix-law density $w_k(p)$ on $E_k(U)$: indeed the matrix product factors as $W_k$ followed by a record on horizon $U-s(p)$, and step 1.2 shifts every continuation time by $-s(p)$ without changing its base measure. Step 3.1 then normalizes that continuation for every positive-weight state. Precisely, for every Borel $B\subseteq E_k(U)$, ${\mathbb{P}}_U(P_k\in B)={\int }_B{w}_k(p)d{\mu }_k(p)$. Zero-weight prefixes have zero mass; on them the matrix vanishes and all continuation weights vanish. This argument uses prefix densities, not probabilities of individual continuous prefixes.

For $p\in E_k(U)$ with $w_k(p)>0$, put $h(p)=U-s(p)$ and define the candidate next-outcome kernel by $C_k(p,A)={\sum }_{a\in M}{\int }_0^{h(p)}{\mathbf{1}}_A((u,a))f_{\sigma (p),a}(u)du+{\mathbf{1}}_A(†)s_{\sigma (p)}(h(p))\mathrm{.}$ Here $f_{\sigma ,a}$ and $s_{\sigma }$ are the marked density and survivor in [F2]. For $p=\mathrm{\partial }$ or $w_k(p)=0$ set $C_k(p,\cdot )={\delta }_{†}$. Every section is a probability measure: it is the [F2] waiting law with waiting times at or beyond $h(p)$, including the infinite no-event outcome, sent to $†$; a finite endpoint has zero waiting-law mass. For each measurable $A\subseteq D$, its evaluation is Borel in $p$. Indeed $w_k$ is continuous on each mark component, the positive-weight set is open, and $\sigma (p)$ is continuous there. The integrand on the fixed interval $(0,U)$, multiplied by ${\mathbf{1}}_{\{u<h(p)\}}$, is nonnegative and product-measurable, so its parameter integral is measurable by [F3]. The prefix base measure ${\mu }_k$ and interval Lebesgue measure are finite, as required there; the singleton case $k=0$ is immediate. The survivor term is continuous on each positive-weight component, and the complementary assignment is constant on a Borel set. Thus $C_k$ is a probability kernel as required in [F10].

For any measurable $A\subseteq D$ and Borel $B\subseteq E_k(U)$, separating the next waiting time and mark from its later continuation gives ${\mathbb{P}}_U(P_k\in B,Y_k\in A)={\int }_B{w}_k(p)C_k(p,A)d{\mu }_k(p)\mathrm{.}$ To see this directly, the branch with no further event has density $w_k(p)s_{\sigma (p)}(U-s(p))$ and outcome $†$. A next event $(u,a)$ with $0<u<U-s(p)$ has density $w_k(p)f_{\sigma (p),a}(u)$ after all later records have been integrated out by step 3.1 from the post-event state in [F2]. At zero next-event density its entire contribution is zero. Tonelli permits these nonnegative sums and iterated integrals; step 1.2 justifies the translation from the next absolute time $v$ to $u=v-s(p)$ and the shifts of later continuation times, for each fixed prefix, preserving the relevant Lebesgue integrals. Step 4.2 then identifies the displayed right side with ${\int }_{P_k^{-1}(B)}{C}_k(P_k(\omega ),A)d{\mathbb{P}}_U$. On $\{P_k=\mathrm{\partial }\}$ one has $Y_k=†$ and $C_k={\delta }_{†}$, so the same identity holds for every measurable $B\subseteq {\mathcal{P}}_k$. By [F10] this is exactly the conditional-law identity for $Y_k$ given $P_k$. It establishes each successive censored waiting law, including $k=0$ and the empty mark family, without invoking a general conditional-distribution existence theorem or selecting versions. Together with the process and tail arguments above, this proves all assertions.