Nonlinear path extension is a probability kernel

By [F2], both finite path spaces carry their stated product sigma-algebras and the target path space is standard Borel. Fix a prefix $z_{0:t-1}$. The append map $j_{z_{0:t-1}}(z_t)=(z_{0:t-1},z_t)$ is measurable: the inverse image of a generating rectangle is either empty or the final-coordinate measurable set, and [F3] generates the target sigma-algebra from these rectangles. Thus $A\mapsto M_t(z_{t-1},j_{z_{0:t-1}}^{-1}(A))$ is the pushforward of the probability measure $M_t(z_{t-1},\cdot )$ and is a probability measure.

The projection $z_{0:t-1}\mapsto z_{t-1}$ is measurable, so composing each set-evaluation map $z_{t-1}\mapsto M_t(z_{t-1},B)$ with this projection gives a probability kernel from the prefix space to ${\mathsf{Z}}_t$. The function $(z_{0:t-1},z_t)\mapsto {\mathbf{1}}_A(z_{0:t-1},z_t)$ is product-measurable. By [F5], its integral against this composed kernel is measurable in the prefix. Together with step 1.1, this proves that ${\bar{M}}_t$ has probability-measure sections and measurable set evaluations, hence is a probability kernel. Iterating the definition appends each state using exactly $M_t$ because at each step the indicator is evaluated on the already fixed prefix and the new coordinate.