Splitting of the Observation Record Space at an Intermediate Time

Let $\tilde{l}\ge1$ be a natural number and let $T>0$ and $s\in(0,T)$ be real numbers. For a real horizon $a>0$ write $(\mathbf{R}_a,\mathcal{R}_a,\rho_a)$ for the observation record space with horizon $a$ and $\tilde{l}$ channels, so that $\mathbf{R}_a=\mathbf{R}(a,\tilde{l})$, $\mathcal{R}_a=\mathcal{R}(a,\tilde{l})$, and $\rho_a=\rho(a,\tilde{l})$, records being triples $r=(k,t,v)$ as there, and write $k_r(u)$ for the number of event times of $r$ that are at most $u$, as in The Record-Frozen Control Path and Record-Frozen Policy. Each $\rho_a$ is a finite measure: by claims 1, 2, 3, and 4(c) of Assembly of Measure Spaces: Restriction, Transport, One-Point Spaces, and Countable Disjoint Unions, $\rho_a(\mathbf{R}_a)$ is the sum of the cell masses, which is $\sum_{k\ge0}\tilde{l}^k a^k/k!=e^{\tilde{l}a}$ by The Ordered Time Simplex: Borel Measurability and Volume and the series defining the real exponential function. Let $\mathcal{R}_s\otimes\mathcal{R}_{T-s}$ be the product $\sigma$-algebra on $\mathbf{R}_s\times\mathbf{R}_{T-s}$ and let $\rho_s\otimes\rho_{T-s}$ be the product measure.

Define the prefix map $\pi_s:\mathbf{R}_T\to\mathbf{R}_s$ and the increment map $\iota_s:\mathbf{R}_T\to\mathbf{R}_{T-s}$ by: for $r=(k,t,v)\in\mathbf{R}_T$, writing $\kappa=k_r(s)$, $\pi_s(r)=\bigl(\kappa,\ (t_1,\dots,t_\kappa),\ (v_1,\dots,v_\kappa)\bigr),\qquad \iota_s(r)=\bigl(k-\kappa,\ (t_{\kappa+1}-s,\dots,t_k-s),\ (v_{\kappa+1},\dots,v_k)\bigr),$ empty blocks being the empty tuple; and define the concatenation $\oplus_s:\mathbf{R}_s\times\mathbf{R}_{T-s}\to\mathbf{R}_T$ by: for $\varrho=(\kappa,\tau,w)\in\mathbf{R}_s$ and $r'=(k',t',v')\in\mathbf{R}_{T-s}$, $\varrho\oplus_s r'=\bigl(\kappa+k',\ (\tau_1,\dots,\tau_\kappa,\ s+t'_1,\dots,s+t'_{k'}),\ (w_1,\dots,w_\kappa,\ v'_1,\dots,v'_{k'})\bigr).$

1. (Bijection) For every $r\in\mathbf{R}_T$ the values $\pi_s(r)$ and $\iota_s(r)$ are observation records with horizons $s$ and $T-s$; for every $\varrho\in\mathbf{R}_s$ and $r'\in\mathbf{R}_{T-s}$ the value $\varrho\oplus_s r'$ is an observation record with horizon $T$; and $\oplus_s$ is a bijection from $\mathbf{R}_s\times\mathbf{R}_{T-s}$ onto $\mathbf{R}_T$ whose inverse is $r\mapsto(\pi_s(r),\iota_s(r))$.

2. (Measurability) The map $\oplus_s$ is measurable with respect to $\mathcal{R}_s\otimes\mathcal{R}_{T-s}$ and $\mathcal{R}_T$; the map $\pi_s$ is measurable with respect to $\mathcal{R}_T$ and $\mathcal{R}_s$; and the map $\iota_s$ is measurable with respect to $\mathcal{R}_T$ and $\mathcal{R}_{T-s}$.

3. (Measure splitting) The image measure of $\rho_s\otimes\rho_{T-s}$ under $\oplus_s$ is $\rho_T$. In particular, for every $\mathcal{R}_T$-measurable $g:\mathbf{R}_T\to[0,\infty]$, $\int_{\mathbf{R}_T}g\,d\rho_T=\int_{\mathbf{R}_s}\Bigl(\int_{\mathbf{R}_{T-s}}g(\varrho\oplus_s r')\,\rho_{T-s}(dr')\Bigr)\rho_s(d\varrho)\qquad\text{in }[0,\infty],$ the iterated form holding by the Tonelli theorem.