The future $\sigma$-algebra is a single pullback along the shift

For the coordinate process $Y_i(\omega) = \omega_i$ on the path space $\mathsf{X}^{\mathbb{N}}$, the $\sigma$-algebra generated by the coordinates from time $m$ onwards coincides with the pullback of the full product $\sigma$-algebra along the $m$-shift:

Why this reformulation is the useful one. The mixing coefficients $\alpha(n)$, $\rho(n)$, $\phi(n)$ are all defined by quantifying over sets $B$ in the future $\sigma$-algebra $\sigma(Y_i : i \ge k+n)$, which as written is a supremum of pullbacks along individual coordinate maps — a form that supports no computation at all. This theorem replaces it by a single pullback: every future event is $\{\omega : \sigma^m\omega \in B_0\}$ for one measurable subset $B_0$ of path space.

That is exactly what makes the restart property usable. Knowing that the shifted trajectory has the law of a chain restarted from $P^m(x,\cdot)$ says nothing about a supremum of coordinate pullbacks, but says everything about a single pullback: the probability of a future event becomes the probability of $B_0$ under the restarted chain, and a total-variation rate on $P^m(x,\cdot)$ then bounds it directly. Without this identification the two halves of a mixing estimate — the probabilistic input (restart) and the combinatorial input (the shape of the future $\sigma$-algebra) — cannot be connected.

Proof. The product $\sigma$-algebra is by definition $\bigsqcup_a \mathrm{eval}_a^{-1}$, and pullback commutes with suprema (comap_iSup) and with composition (comap_comp), so the right-hand side is $\bigsqcup_{a} (\omega \mapsto \omega_{m+a})^{-1}$. It remains to see that this indexed family agrees with $\bigsqcup_{i \ge m}(\omega \mapsto \omega_i)^{-1}$, which is the reindexing bijection $a \mapsto m+a$ between $\mathbb{N}$ and $\{i : i \ge m\}$: one inclusion uses $m \le m+a$, the other writes $i = m + (i-m)$, valid because $m \le i$.