The Space of Piecewise Constant Paths in a Finite Set: Measurability of Evaluations, Restrictions, Left Limits, and Occupation Integrals

Let $T>0$ be a real number and let $E$ be a nonempty finite set. Let $\mathsf{Path}(E,T)$ be the set of maps $p:[0,T]\to E$ for which there are a count $k$, either $0$ or a natural number, and times $0<t_1<\dots<t_k\le T$ such that $p$ is constant on $[0,t_1)$, on $[t_j,t_{j+1})$ for each $j\in\{1,\dots,k-1\}$, and on $[t_k,T]$ (constant on $[0,T]$ when $k=0$). Let $\mathcal{C}_T$ be the $\sigma$-algebra generated on $\mathsf{Path}(E,T)$ by the sets $\{p:p(u)=y\}$ with $u\in[0,T]$ and $y\in E$; for any real $S>0$, $\mathcal{C}_S$ denotes the corresponding $\sigma$-algebra on $\mathsf{Path}(E,S)$. Write $\mathcal{B}_{[0,T]}$ for the trace Borel $\sigma$-algebra on $[0,T]$, $\int_{[0,T]}\cdot\,du$ for the Lebesgue integral over the compact interval $[0,T]$, $\otimes$ for the product $\sigma$-algebra, and $\mathbf{1}\{\cdots\}$ for the indicator of the condition in braces; measurability of a real-valued map is with respect to the Borel $\sigma$-algebra of the real line on the target. Then:

1. (Evaluations and restrictions.) For every $u\in[0,T]$ the evaluation map $p\mapsto p(u)$ is measurable with respect to $\mathcal{C}_T$ and the $\sigma$-algebra of all subsets of $E$, and for every function $\phi:E\to\mathbb{R}$ the map $p\mapsto\phi(p(u))$ is $\mathcal{C}_T$-measurable. For every real $T'$ with $0<T'\le T$, the restriction of every $p\in\mathsf{Path}(E,T)$ to $[0,T']$ belongs to $\mathsf{Path}(E,T')$, and the restriction map $p\mapsto p|_{[0,T']}$ is measurable with respect to $\mathcal{C}_T$ and $\mathcal{C}_{T'}$.

2. (Left limits and right-continuity.) For every $p\in\mathsf{Path}(E,T)$ and every $t\in(0,T]$ there is exactly one point $p(t-)\in E$, the left limit of $p$ at $t$, for which there is a real $\delta$ with $0<\delta\le t$ such that $p(s)=p(t-)$ for all $s\in[t-\delta,t)$; and for every $u\in[0,T)$ there is a real $\delta>0$ such that $p(s)=p(u)$ for all $s\in[u,u+\delta)\cap[0,T]$. For every $t\in(0,T]$ the map $p\mapsto p(t-)$ is measurable with respect to $\mathcal{C}_T$ and the $\sigma$-algebra of all subsets of $E$.

3. (Occupation integrals.) For every $y\in E$ the map $(u,p)\mapsto\mathbf{1}\{p(u)=y\}$ on $[0,T]\times\mathsf{Path}(E,T)$ is measurable with respect to $\mathcal{B}_{[0,T]}\otimes\mathcal{C}_T$. Consequently, for every function $\phi:E\to\mathbb{R}$ and every $p\in\mathsf{Path}(E,T)$, the map $u\mapsto\phi(p(u))$ is bounded and $\mathcal{B}_{[0,T]}$-measurable, hence integrable over $[0,T]$, the map

$$p\mapsto\int_{[0,T]}\phi(p(u))\,du$$

is $\mathcal{C}_T$-measurable, and its absolute value is at most $T\max_{y\in E}|\phi(y)|$.