The Observation Record Space

Let $\tilde{l}\ge1$ be a natural number and let $T>0$ be a real number. Write $V=\{1,\dots,\tilde{l}\}$, whose members are called channels, and write $\mathcal{B}_k$ and $\lambda_k$ for the $k$-fold Borel $\sigma$-algebra and product Lebesgue measure on $\mathbb{R}^k$.

An observation record with horizon $T$ and $\tilde{l}$ channels is a triple $r=(k,t,v)$ in which $k$ is either $0$ or a natural number; for $k\ge1$, $t=(t_1,\dots,t_k)$ lies in the ordered time simplex $D_k(T)$ and the mark vector $v=(v_1,\dots,v_k)$ lies in the $k$-fold Cartesian product $V^k$; for $k=0$, both $t$ and $v$ are the empty tuple $()$, and $r_\emptyset=(0,(),())$ is called the empty record. The observation record space $\mathbf{R}(T,\tilde{l})$ is the set of all observation records with horizon $T$ and $\tilde{l}$ channels. (It is distinct from the set $R_k(T)$ of Observation-Driven Control Policy, which collects time tuples only.)

The cells of the observation record space are the set $C_\emptyset=\{r_\emptyset\}$ and, for each natural number $k\ge1$ and each mark vector $v\in V^k$, the set $C_{k,v}=\{(k,t,v):t\in D_k(T)\}$. The cells are pairwise disjoint (records in distinct cells differ in their first or third components), their union is $\mathbf{R}(T,\tilde{l})$, and the cell family is countable in the sense of Assembly of Measure Spaces: Restriction, Transport, One-Point Spaces, and Countable Disjoint Unions: each $V^k$ is finite, so the cells may be listed as a sequence by increasing $k$ and, within each $k$, by any fixed listing of $V^k$. The cell $C_\emptyset$ is equipped with the one-point measure space structure with unit mass at $r_\emptyset$. For $k\ge1$ and $v\in V^k$, the set $D_k(T)$ belongs to $\mathcal{B}_k$ by The Ordered Time Simplex: Borel Measurability and Volume, and the cell $C_{k,v}$ is equipped with the transport along the bijection $t\mapsto(k,t,v)$ of the restriction of $(\mathbb{R}^k,\mathcal{B}_k,\lambda_k)$ to $D_k(T)$.

The record $\sigma$-algebra $\mathcal{R}(T,\tilde{l})$ and the reference measure $\rho(T,\tilde{l})$ are the $\sigma$-algebra and the measure of the countable disjoint union of this family of cell measure spaces, so that $\mathbf{R}(T,\tilde{l})$, equipped with them, is a measure space.