Past $\cap$ future probability, disintegrated over the past

Let $P$ be a Markov kernel, $\lambda$ an initial distribution, and $\mathbb{P}_\lambda$ the resulting law on path space. Fix $k, n$ and measurable sets $A_0 \subseteq \prod_{i \le k}\mathsf{X}$ and $B_0 \subseteq \mathsf{X}^{\mathbb{N}}$. Then

What it is. This is the joint law of a past event and a future event, disintegrated over the past. By the identification of the two $\sigma$-algebras of the coordinate process, every event in $\sigma(X_0,\dots,X_k)$ has the form $\mathrm{fr}_k^{-1}(A_0)$ and every event in $\sigma(X_{k+n},X_{k+n+1},\dots)$ has the form $(\sigma^{k+n})^{-1}(B_0)$, so the left-hand side is the completely general "past $\cap$ future" probability appearing in the definition of the mixing coefficients $\alpha(n)$, $\rho(n)$, $\phi(n)$.

Why the right-hand side is the useful form. It exhibits the joint probability as an average, over the past, of the conditional probability of the future — and that conditional probability is written explicitly as $\int \mathbb{P}_y(B_0)\,P^n(u_k,\mathrm{d}y)$, a quantity depending on the past only through the single state $u_k$. Comparing it with the stationary value $\int \mathbb{P}_y(B_0)\,\pi(\mathrm{d}y)$ and applying the total-variation bound for $[0,1]$-valued integrands converts a rate $\|P^n(x,\cdot)-\pi\| \le C$ directly into $\phi(n) \le C$, uniformly in the split point $k$. Everything after this point is arithmetic of averages: no measure-theoretic or Markov input remains.

Proof. Disintegrate $\mathbb{P}_\lambda$ over its first $k+1$ coordinates, writing it as $\mathrm{traj}_k$ composed with the marginal $(\mathrm{fr}_k)_*\mathbb{P}_\lambda$, and expand the resulting bind. It then suffices to evaluate $\mathrm{traj}_k(u)$ on the intersection, for each $u$.

Two facts do this. First, $\mathrm{traj}_k(u)$ reproduces its own initial segment: pushing it forward along $\mathrm{fr}_k$ gives $\mathrm{partialTraj}(k,k) = \mathrm{id}$, i.e. the Dirac mass at $u$. Hence the past event has $\mathrm{traj}_k(u)$-measure $\mathbf{1}_{A_0}(u)$ — it is deterministic given the conditioning, which is why it becomes an indicator and the integral restricts to $A_0$. When $u \in A_0$ its complement is null, so the intersection has the same measure as the future event alone; when $u \notin A_0$ the intersection is null.

Second, on the future event, the conditional restart theorem gives $(\sigma^{k+n})_*[\mathrm{traj}_k(u)] = \int \mathbb{P}_y\,P^n(u_k,\mathrm{d}y)$, which is exactly the inner integrand.