Shift operator and future-coordinate sigma-algebra

Let $(E,\mathcal{E})$ and $(\mathrm{\Omega },\mathcal{F})$ be measurable spaces. On $E^{{\mathbb{N}}_0}$ with the product sigma-algebra ${\mathcal{E}}^{\otimes {\mathbb{N}}_0}$, the left shift is $\theta (x_0,x_1,x_2,\dots )=(x_1,x_2,x_3,\dots )\mathrm{.}$ It is measurable because every coordinate of $\theta$ is a coordinate projection. Write ${\theta }^n$ for its $n$th iterate, including ${\theta }^0=\mathrm{id}$. For an $E$-valued process $X$, its future-coordinate sigma-algebra from time $n$ is ${\mathcal{T}}_n^{+}=\sigma (X_n,X_{n+1},\dots )\mathrm{.}$ Here an $E$-valued process means a sequence of $\mathcal{F}\mathrm{/}\mathcal{E}$-measurable maps $X_n:\mathrm{\Omega }\to E$. If $H:E^{{\mathbb{N}}_0}\to \mathbb{R}$ is ${\mathcal{E}}^{\otimes {\mathbb{N}}_0}\mathrm{/}\mathcal{B}(\mathbb{R})$-measurable, then $H(X_n,X_{n+1},\dots )$ is called a future path functional from time $n$. On canonical path space this is $H\circ {\theta }^n$. Constants, including zero and one, and $n=0$ are included. These definitions are choice-free; this notation does not itself assert existence of a canonical law or attach an expectation ${\mathbb{E}}_{x}H$ to the functional.