Finite-horizon state-space kernel law exists and is unique

If $T=0$, then ${\mathcal{Z}}_0=E_0$ and ${\mu }_0$ is the unique asserted law. For $T\ge 1$, interleave the coordinate spaces as $D_0=E_0$, $D_{2t-1}=E_t$, and $D_{2t}=F_t$ for $1\le t\le T$. At the transition to $E_t$, define the lifted kernel by $L_{2t-2}(h,A)=K_t({\pi }_{t-1}(h),A)$, where ${\pi }_{t-1}$ selects the $E_{t-1}$ coordinate of the preceding history $h$; at the next transition define $L_{2t-1}(h,B)=G_t({\pi }_t(h),B)$, where ${\pi }_t$ selects the newly adjoined $E_t$ coordinate. These projections are measurable because inverse images of measurable sets are product cylinders. Each lifted section is a probability measure, and each evaluation is measurable by composition, so [F2] makes every lift a probability kernel. After coordinate $D_{2T}$, append singleton spaces with their unit point-mass kernels.

By [A1] and [F3], Ionescu–Tulcea gives a unique probability measure on the countable interleaved product with these lifted kernels. Its marginal on the first $2T+1$ coordinates is a probability measure on ${\mathcal{Z}}_T$; the prescribed prefix formula is exactly the displayed iterated integral. The finite product is standard Borel by [F4].

Suppose $Q_T$ is another probability measure on ${\mathcal{Z}}_T$ with the same displayed rectangle probabilities. Append the singleton tail to the coordinate tuple by a measurable map and push $Q_T$ forward to the countable product. For any shorter prefix, set all later coordinate sets in the rectangle formula to their whole spaces; the later probability kernels integrate to one, so each shorter marginal is the prescribed iterated integral. Prefixes after $D_{2T}$ add only deterministic singleton coordinates. Ionescu–Tulcea uniqueness [F3] identifies this extension with the measure in step 2.1, so their finite-prefix marginals agree and $Q_T$ equals that marginal. For $T=0$, the formula already fixes $Q_0={\mu }_0$.