Proof of Survival Identity, Measurability, Event-Count Recursion, and Normalization of the Likelihood of a Causal Intensity

Preliminaries. Write $\lambda_{[0,T]}$ for the restricted Lebesgue measure on $[0,T]$ (claim 1 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval); it is finite. Integrals of nonnegative functions are handled with claim 1 of Linearity and Monotonicity of the Lebesgue Integral (additivity, positive homogeneity, monotonicity) and integrable functions with claim 2 there; bounded measurable functions on a finite measure space are integrable. Sums, products and pointwise limits of measurable real functions are measurable by Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, continuous functions on compact intervals by claim 4 of Measurability of Countable Suprema, Bounded Pointwise Limits, Monotone Functions, and Continuous Functions, and compositions with sequentially continuous maps (such as $\exp$ and finite products) by Sequentially Continuous Functions of Measurable Euclidean Maps are Measurable. By Basic Properties of the Exponential Function, $\exp$ is positive, increasing, and $\exp(0)=1$, so $\exp(-x)\le1$ for $x\ge0$; and $\exp(u)=\sum_{m\ge0}u^{m}/m!$ by The Real Exponential Function, the series converging absolutely. Indicators $\mathbf{1}_{[0,s]}$, $\mathbf{1}_{(a,b]}$ are measurable (their sets are Borel), and $\mathbf{1}_{(a,b]}=\mathbf{1}_{[0,b]}-\mathbf{1}_{[0,a]}$ for $0\le a\le b\le T$.

Step 1: claim 1. The function $\mu$ is measurable and bounded, hence integrable over $[0,T]$, so part (i) of Integration by Parts for Indefinite Lebesgue Integrals on a Compact Interval with $f=\mu$, $u_0=0$ shows that $s\mapsto M_s=\int_{[0,s]}\mu(u)\,du$ is continuous on $[0,T]$; also $0\le M_s\le\bar\mu T$ by monotonicity, and $M$ is measurable on $[0,T]$ as a continuous function.

(a) Powers of $M$. We show by induction on $m\ge0$ that for every $s\in[0,T]$,

$$M_s^{m+1}=(m+1)\int_{[0,s]}\mu(u)M_u^{m}\,du .\tag{1.1}$$

For $m=0$ this is the definition of $M_s$. Let $m\ge1$ and assume (1.1) for $m-1$, i.e. $M_t^{m}=m\int_{[0,t]}\mu M^{m-1}$ for every $t\in[0,T]$. Fix $s\in(0,T]$ and apply part (ii) of Integration by Parts for Indefinite Lebesgue Integrals on a Compact Interval on the interval $[0,s]$ (the lemma holds for every positive horizon; the integrals $\int_{[0,t]}$ for $t\le s$ are the same whether formed on $[0,s]$ or on $[0,T]$, both being $\int_{\mathbb{R}}$ of the same zero extension by claim 2 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval) with $f=\mu$, $u_0=0$, so that $u_t=M_t$, and with $g=m\mu M^{m-1}$, $v_0=0$, so that $v_t=M_t^{m}$ by the induction hypothesis; $g$ is measurable and bounded by $m\bar\mu(\bar\mu T)^{m-1}$, hence integrable. The conclusion reads

$$M_s\,M_s^{m}=\int_{[0,s]}\bigl(\mu(u)M_u^{m}+M_u\,m\,\mu(u)M_u^{m-1}\bigr)\,du=(m+1)\int_{[0,s]}\mu(u)M_u^{m}\,du ,$$

which is (1.1) for $m$ at $s$; at $s=0$ both sides of (1.1) vanish. Consequently, for $0\le a\le b\le T$, by $\mathbf{1}_{(a,b]}=\mathbf{1}_{[0,b]}-\mathbf{1}_{[0,a]}$ and linearity,

$$\int_{(a,b]}\mu(u)M_u^{m}\,du=\frac{M_b^{m+1}-M_a^{m+1}}{m+1}.\tag{1.2}$$

(b) The series. For $u\in[0,T]$ put $S_n(u)=\sum_{m=0}^{n}\frac{(-1)^{m}}{m!}\mu(u)M_u^{m}$. Each $S_n$ is measurable, $S_n(u)\to\mu(u)\exp(-M_u)$ as $n\to\infty$ by the series for $\exp(-M_u)$, and $|S_n(u)|\le\bar\mu\sum_{m\ge0}(\bar\mu T)^{m}/m!=\bar\mu\exp(\bar\mu T)$, a constant, integrable on the finite measure space $([0,T],\lambda_{[0,T]})$. In particular $u\mapsto\mu(u)\exp(-M_u)$ is measurable (a pointwise limit), and by Dominated Convergence Theorem and (1.2),

$$\int_{(a,b]}\mu(u)\exp(-M_u)\,du=\lim_{n\to\infty}\int_{(a,b]}S_n(u)\,du=\lim_{n\to\infty}\sum_{m=0}^{n}\frac{(-1)^{m}}{(m+1)!}\bigl(M_b^{m+1}-M_a^{m+1}\bigr).$$

For real $x$, $\sum_{m=0}^{n}\frac{(-1)^{m}x^{m+1}}{(m+1)!}=-\sum_{j=1}^{n+1}\frac{(-x)^{j}}{j!}\to-(\exp(-x)-1)=1-\exp(-x)$ by the series for $\exp(-x)$. Applying this with $x=M_b$ and $x=M_a$ and subtracting (limits of sums, Arithmetic of Limits of Real Sequences) gives $\int_{(a,b]}\mu(u)\exp(-M_u)\,du=(1-\exp(-M_b))-(1-\exp(-M_a))=\exp(-M_a)-\exp(-M_b)$, which is claim 1.

Step 2: claim 2. Cells and transports. For $k\ge1$ and $v\in V^k$ let $\chi_{k,v}:D_k(T)\to C_{k,v}$, $\chi_{k,v}(t)=(k,t,v)$, be the bijection along which the cell $C_{k,v}$ carries the transport of the restriction of $(\mathbb{R}^k,\mathcal{B}_k,\lambda_k)$ to $D_k(T)$ (The Observation Record Space, with $\mathcal{B}_k$ and $\lambda_k$ as in Finite Products of Lebesgue Measure and Coordinate Integration on $\mathbb{R}^l$); write $\mathcal{D}_k$ for the $\sigma$-algebra of that restriction (the Borel subsets of $D_k(T)$, claim 1 of Assembly of Measure Spaces: Restriction, Transport, One-Point Spaces, and Countable Disjoint Unions) and $\lambda_{D_k}$ for its measure, which is finite with $\lambda_{D_k}(D_k(T))=T^k/k!$ by The Ordered Time Simplex: Borel Measurability and Volume. By claims 2, 3 and 4 of Assembly of Measure Spaces: Restriction, Transport, One-Point Spaces, and Countable Disjoint Unions, a function $h:\mathbf{R}\to[0,\infty]$ is $\mathcal{R}$-measurable if and only if $h\circ\chi_{k,v}$ is $\mathcal{D}_k$-measurable for every $k\ge1$ and $v\in V^k$ (there is no condition on the one-point cell $C_\emptyset$), and then, by claim 4(c) there and the change of variables of claim 2 of Image Measures, Measures with Densities, and Change of Variables along the transport,

$$\int_{\mathbf{R}}h\,d\rho=h(r_\emptyset)+\sum_{k\ge1}\sum_{v\in V^k}\int_{D_k(T)}h\circ\chi_{k,v}\,d\lambda_{D_k},\qquad \int_{\mathbf{R}^{(k)}}h\,d\rho=\sum_{v\in V^k}\int_{D_k(T)}h\circ\chi_{k,v}\,d\lambda_{D_k}\ (k\ge1),\qquad\int_{\mathbf{R}^{(0)}}h\,d\rho=h(r_\emptyset),\tag{2.1}$$

the first sum being over the countable cell family in the finite-partial-sum sense of that lemma (the second and third identities follow from the first applied to $h\mathbf{1}_{\mathbf{R}^{(k)}}$, whose restrictions to the other cells vanish). Moreover, by claim 1 of Assembly of Measure Spaces: Restriction, Transport, One-Point Spaces, and Countable Disjoint Unions, $\int_{D_k(T)}h\circ\chi_{k,v}\,d\lambda_{D_k}=\int_{\mathbb{R}^k}\widetilde{h\circ\chi_{k,v}}\,d\lambda_k$ with $\widetilde{\ }$ the zero extension.

Measurability on a cell. Fix $k\ge1$, $v\in V^k$, write $\chi=\chi_{k,v}$, and let $\Xi:[0,T]\times D_k(T)\to[0,T]\times\mathbf{R}$, $\Xi(s,t)=(s,\chi(t))$. The trace on $[0,T]\times D_k(T)$ of every set in $\mathcal{B}_{1+k}$ belongs to the product $\sigma$-algebra $\mathcal{B}_{[0,T]}\otimes\mathcal{D}_k$: the trace of a Borel rectangle $A_1\times A_2\times\dots\times A_{1+k}$ is $(A_1\cap[0,T])\times\bigl((A_2\times\dots\times A_{1+k})\cap D_k(T)\bigr)$, a measurable rectangle (the second factor being the trace on $D_k(T)$ of a member of $\mathcal{B}_k$), and the sets in $\mathcal{B}_{1+k}$ whose trace lies in the product $\sigma$-algebra form a $\sigma$-algebra containing the Borel rectangles, which generate $\mathcal{B}_{1+k}$ (claim 1 of Finite Products of Lebesgue Measure and Coordinate Integration on $\mathbb{R}^l$). The map $\Xi$ is measurable with respect to $\mathcal{B}_{[0,T]}\otimes\mathcal{D}_k$ and $\mathcal{B}_{[0,T]}\otimes\mathcal{R}$: the preimage of a rectangle $A\times B$ ($A\in\mathcal{B}_{[0,T]}$, $B\in\mathcal{R}$) is $A\times\chi^{-1}(B\cap C_{k,v})$, and $B\cap C_{k,v}$ belongs to the cell $\sigma$-algebra (claim 4(a) of Assembly of Measure Spaces: Restriction, Transport, One-Point Spaces, and Countable Disjoint Unions), whose members are the images $\chi(E)$, $E\in\mathcal{D}_k$ (claim 2 there); rectangles generate the product $\sigma$-algebra, so the generator criterion of Measurable Function and Real-Valued Measurable Function applies. Hence, for every $\upsilon\in V$, $(s,t)\mapsto\lambda^\upsilon_s(\chi(t))=\lambda^\upsilon\circ\Xi(s,t)$ is $\mathcal{B}_{[0,T]}\otimes\mathcal{D}_k$-measurable by condition (i) of Causal Intensity on the Observation Record Space, and so is $(s,t)\mapsto\lambda^{\mathrm{tot}}_s(\chi(t))$. Consequently: (a) for each $i\le k$, $t\mapsto\lambda^{v_i}_{t_i}(\chi(t))$ is $\mathcal{D}_k$-measurable, being the composition of the previous map with $t\mapsto(t_i,t)$, which is measurable from $\mathcal{D}_k$ to $\mathcal{B}_{[0,T]}\otimes\mathcal{D}_k$ by the generator criterion of Measurable Function and Real-Valued Measurable Function, the preimage of a rectangle $A\times B$ being $\{t\in B:t_i\in A\}\in\mathcal{D}_k$ (the coordinate projection $t\mapsto t_i$ is measurable 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); (b) the function $(s,t)\mapsto\mathbf{1}\{s\le t_k\}\lambda^{\mathrm{tot}}_s(\chi(t))$ is jointly measurable (the set $\{(s,t):s\le t_k\}$ is closed in $\mathbb{R}^{1+k}$, hence Borel 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 by the Tonelli theorem (Tonelli and Fubini Theorems, for the finite measures $\lambda_{[0,T]}$ and $\lambda_{D_k}$) the map $t\mapsto\Lambda_{t_k}(\chi(t))=\int_{[0,T]}\mathbf{1}\{s\le t_k\}\lambda^{\mathrm{tot}}_s(\chi(t))\,ds$ is $\mathcal{D}_k$-measurable, and likewise $t\mapsto\Lambda_s(\chi(t))$ for every fixed $s\in[0,T]$ (with the Borel set $\{(u,t):u\le s\}$ in place of $\{(u,t):u\le t_k\}$), in particular $t\mapsto\Lambda_T(\chi(t))$; as $\Lambda_s(r_\emptyset)$ imposes no condition, $r\mapsto\Lambda_s(r)$ is $\mathcal{R}$-measurable for every $s$. Therefore $q\circ\chi=\bigl(\prod_{i\le k}\lambda^{v_i}_{t_i}\circ\chi\bigr)\exp(-\Lambda_{t_k}\circ\chi)$ and $\ell_\lambda\circ\chi=\bigl(\prod_{i\le k}\lambda^{v_i}_{t_i}\circ\chi\bigr)\exp(-\Lambda_T\circ\chi)$ are $\mathcal{D}_k$-measurable, and $q$, $\ell_\lambda$ are $\mathcal{R}$-measurable.

Bounds. For $r=(k,t,v)\in\mathbf{R}^{(k)}$: each factor $\lambda^{v_i}_{t_i}(r)$ lies in $[0,\bar\lambda]$ and $\exp(-\Lambda_{t_k}(r))\le1$, so $0\le q(r)\le\bar\lambda^k$. Since $\mathbf{1}_{[0,T]}=\mathbf{1}_{[0,t_k]}+\mathbf{1}_{(t_k,T]}$ and $\lambda^{\mathrm{tot}}\ge0$, linearity gives $\Lambda_T(r)=\Lambda_{t_k}(r)+\int_{(t_k,T]}\lambda^{\mathrm{tot}}_u(r)\,du\ge\Lambda_{t_k}(r)$, and the functional equation of $\exp$ (claim 1 of Basic Properties of the Exponential Function) gives $\ell_\lambda(r)=q(r)\exp(-(\Lambda_T(r)-\Lambda_{t_k}(r)))\le q(r)$. Finally, by (2.1) and monotonicity, $Q_n\le\sum_{v\in V^n}\bar\lambda^n\lambda_{D_n}(D_n(T))=\tilde{l}^n\bar\lambda^nT^n/n!$ for $n\ge1$ ($V^n$ having $\tilde{l}^n$ members), and $Q_0=q(r_\emptyset)=1$.

Step 3: claim 3. Fix $n\ge0$ and $r=(n,t,v)\in\mathbf{R}^{(n)}$ (with $t_0=0$ and empty tuples when $n=0$). For $s\in(t_n,T]$ and $v'\in V$ define the appended record

r\sqcup(s,v')=\bigl(n+1,\ (t_1,\dots,t_n,s),\ (v_1,\dots,v_n,v')\bigr)\in\mathbf{R}^{(n)}+1)},

a record since $0<t_1<\dots<t_n<s\le T$. Causality. For $u\in[0,s]$ the strict prefix maps satisfy $\pi_{u-}(r\sqcup(s,v'))=\pi_{u-}(r)$: both keep exactly the events of $r$ with time $<u$, the appended time $s\ge u$ not being $<u$. Condition (ii) of Causal Intensity on the Observation Record Space therefore gives $\lambda^\upsilon_u(r\sqcup(s,v'))=\lambda^\upsilon_u(r)$ for all $u\in[0,s]$ and $\upsilon\in V$. In particular $\lambda^{v_i}_{t_i}(r\sqcup(s,v'))=\lambda^{v_i}_{t_i}(r)$ for $i\le n$ (as $t_i<s$), $\lambda^{v'}_s(r\sqcup(s,v'))=\lambda^{v'}_s(r)$, and $\Lambda_s(r\sqcup(s,v'))=\int_{[0,s]}\lambda^{\mathrm{tot}}_u(r\sqcup(s,v'))\,du=\Lambda_s(r)$. With $P(r)=\prod_{i\le n}\lambda^{v_i}_{t_i}(r)$ this yields

$$q\bigl(r\sqcup(s,v')\bigr)=P(r)\,\lambda^{v'}_s(r)\exp(-\Lambda_s(r)),\qquad \sum_{v'\in V}q\bigl(r\sqcup(s,v')\bigr)=P(r)\,\lambda^{\mathrm{tot}}_s(r)\exp(-\Lambda_s(r))\qquad(s\in(t_n,T]).$$

Applying claim 1 with $\mu=\lambda^{\mathrm{tot}}_\cdot(r)$ (measurable and bounded by $\tilde{l}\bar\lambda$, as noted in Likelihood of a Causal Intensity on the Observation Record Space), $M=\Lambda_\cdot(r)$, $a=t_n$ and $b=T$, and using $\ell_\lambda(r)=P(r)\exp(-\Lambda_T(r))$ and $q(r)=P(r)\exp(-\Lambda_{t_n}(r))$,

$$\sum_{v'\in V}\int_{(t_n,T]}q\bigl(r\sqcup(s,v')\bigr)\,ds=P(r)\bigl(\exp(-\Lambda_{t_n}(r))-\exp(-\Lambda_T(r))\bigr)=q(r)-\ell_\lambda(r).\tag{3.1}$$

Integration over $\mathbf{R}^{(n)}$. Fix $v\in V^n$ and $v'\in V$ and let $\tilde q$ be the zero extension to $\mathbb{R}^{n+1}$ of $q\circ\chi_{n+1,(v,v')}$, measurable by Step 2 and claim 1 of Assembly of Measure Spaces: Restriction, Transport, One-Point Spaces, and Countable Disjoint Unions. Under the identification $\mathbb{R}^{n+1}=\mathbb{R}^n\times\mathbb{R}$ of Finite Products of Lebesgue Measure and Coordinate Integration on $\mathbb{R}^l$, $\lambda_{n+1}=\lambda_n\otimes\lambda$ and $\mathbf{1}_{D_{n+1}(T)}(t,s)=\mathbf{1}_{D_n(T)}(t)\mathbf{1}_{(t_n,T]}(s)$ (for $n=0$ read $\mathbb{R}^0\times\mathbb{R}$ as $\mathbb{R}$, $\mathbf{1}_{D_0(T)}=1$, $t_0=0$, and $D_1(T)=(0,T]$), while $q(r\sqcup(s,v'))=\tilde q(t,s)$ for $t\in D_n(T)$, $s\in(t_n,T]$. The Tonelli theorem (Tonelli and Fubini Theorems) for $\lambda_n\otimes\lambda$ applied to the nonnegative measurable function $\mathbf{1}_{D_{n+1}(T)}\tilde q$ gives

$$\int_{\mathbb{R}^n}\mathbf{1}_{D_n(T)}(t)\Bigl(\int_{\mathbb{R}}\mathbf{1}_{(t_n,T]}(s)\,\tilde q(t,s)\,\lambda(ds)\Bigr)\lambda_n(dt)=\int_{\mathbb{R}^{n+1}}\mathbf{1}_{D_{n+1}(T)}\tilde q\,d\lambda_{n+1}=\int_{D_{n+1}(T)}q\circ\chi_{n+1,(v,v')}\,d\lambda_{D_{n+1}} ,$$

the inner integral being $\int_{(t_n,T]}q(r\sqcup(s,v'))\,ds$ for $r=\chi_{n,v}(t)$ (claim 2 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval, the integrand vanishing off $[0,T]$), and $t\mapsto$ (inner integral) being measurable by Tonelli. Summing over $v'\in V$ and $v\in V^n$, using (2.1) for $n$ and for $n+1$ (the pairs $(v,v')$ running through $V^{n+1}$), and inserting (3.1) pointwise on $\mathbf{R}^{(n)}$:

$$Q_{n+1}=\sum_{v\in V^n}\int_{D_n(T)}\Bigl(\sum_{v'\in V}\int_{(t_n,T]}q(\chi_{n,v}(t)\sqcup(s,v'))\,ds\Bigr)\lambda_{D_n}(dt)=\int_{\mathbf{R}^{(n)}}(q-\ell_\lambda)\,d\rho=Q_n-\int_{\mathbf{R}^{(n)}}\ell_\lambda\,d\rho ,$$

the last step by additivity, both $q$ and $\ell_\lambda$ being nonnegative with finite integrals over $\mathbf{R}^{(n)}$ (claim 2) and $q-\ell_\lambda\ge0$. For $n=0$ the same computation reads $Q_1=\sum_{v'}\int_{D_1(T)}q\circ\chi_{1,v'}\,d\lambda_{D_1}=\sum_{v'}\int_{(0,T]}q(r_\emptyset\sqcup(s,v'))\,ds=q(r_\emptyset)-\ell_\lambda(r_\emptyset)=Q_0-\int_{\mathbf{R}^{(0)}}\ell_\lambda\,d\rho$. This is claim 3.

Step 4: claim 4. By claim 3, $\int_{\mathbf{R}^{(k)}}\ell_\lambda\,d\rho=Q_k-Q_{k+1}$ for every $k\ge0$, so the sum over $k\le n$ telescopes to $Q_0-Q_{n+1}=1-Q_{n+1}$. By (2.1), $\int_{\mathbf{R}}\ell_\lambda\,d\rho$ is the least upper bound of the finite partial sums of the cell integrals of $\ell_\lambda$; every finite family of cells is contained in the family of cells with at most $n$ events for some $n$, whose (finite) partial sum is $\sum_{k\le n}\int_{\mathbf{R}^{(k)}}\ell_\lambda\,d\rho=1-Q_{n+1}\le1$, and conversely each of these is a finite partial sum. Hence $\int_{\mathbf{R}}\ell_\lambda\,d\rho=\sup_n(1-Q_{n+1})$. Since $0\le Q_{n+1}\le(\tilde{l}\bar\lambda T)^{n+1}/(n+1)!$ by claim 2, and the partial sums $s_n$ of the series $\sum_m(\tilde{l}\bar\lambda T)^{m}/m!=\exp(\tilde{l}\bar\lambda T)$ converge (The Real Exponential Function), so that their differences $s_{n+1}-s_n=(\tilde{l}\bar\lambda T)^{n+1}/(n+1)!$ tend to $0$ by claim 1 of Arithmetic of Limits of Real Sequences, we get $Q_{n+1}\to0$ and $\int_{\mathbf{R}}\ell_\lambda\,d\rho=1$. $\blacksquare$