Markov property for bounded future path functionals

Let $C=A_0\times \cdots \times A_r\times E\times E\times \cdots$ be a [F1, F2, F5] rectangular path cylinder. Backward kernel integration gives the measurable function $h_C(x)=1_{A_0}(x)K(1_{A_1}K(\cdots K{1}_{A_r}))(x)\mathrm{.}$ By [F5], it equals ${\mathbb{P}}_x(C)$ under the canonical chain started from $x$. Starting at time $n+r-1$ and applying [F1] backward $r$ times, with $1_{A_0}(X_n)$ and the already exposed factors left outside, gives $\mathbb{E}[1_C(X_n,X_{n+1},\dots )\mid {\mathcal{F}}_n]=h_C(X_n)\mathrm{.}$ The formula also covers $r=0$, an empty $A_j$, and all $A_j=E$.

Let $\mathcal{D}$ be the path events $D$ for which [F3, F4, step 1.1] $h_D(x)={\mathbb{P}}_x(D)$ is measurable and the conditional identity in step 1.1 holds with $D$. The whole path space belongs to $\mathcal{D}$, with $h=1$. Complements remain in $\mathcal{D}$ because $h_{D^c}=1-h_D$. For pairwise disjoint $D_j\in \mathcal{D}$, countable additivity gives $h_{\cup D_j}={\sum }_j{h}_{D_j}$; measurable partial sums increase to this function, and dominated convergence in the defining event integrals gives the conditional identity for the union. Hence $\mathcal{D}$ is a lambda-system. Rectangular cylinders form a pi-system and belong by step 1.1, so [F3] yields every product-measurable path event.

Finite real linear combinations of event indicators now satisfy both [F4, step 2.1] measurability and the identity. If $0\le H\le M$, choose simple $H_j\uparrow H$ by [F4]. Then $h_j(x)={\mathbb{E}}_x{H}_j\to {\mathbb{E}}_{x}H=h(x)$ by dominated convergence, making $h$ measurable. The same theorem passes the limit through all event tests for conditional expectation and proves the displayed identity. Apply this to the positive and negative parts of a general bounded real $H$ and subtract. Zero, one, constant, and degenerate one-point path functionals are included. Choice enters through the canonical laws ${\mathbb{P}}_x$ and conditional-expectation versions used by [F1].