Filtering and finite-horizon smoothing laws exist

If $T=0$ put ${\mathrm{\Omega }}_T:=E_0$; otherwise put ${\mathrm{\Omega }}_T:=E_0\times E_1\times F_1\times \cdots \times E_T\times F_T$. Let ${\mathcal{Z}}_T$ be its finite-product sigma-algebra. By [F1] and [A1], [F2] supplies a probability measure $P_T$ on $({\mathrm{\Omega }}_T,{\mathcal{Z}}_T)$ with the stated iterated-kernel law; the coordinate maps $X_t,Y_t$ and history maps $Y_{1:t}$ are measurable projections.

The initial probability makes $E_0$ nonempty; recursively, probability-kernel sections on each nonempty state space force the next state and observation spaces to be nonempty. By [F1] each factor is standard Borel, so [F3] makes $E_0\times \cdots \times E_T$, every $E_t$, and each history target ${\mathsf{F}}_{1:t}$ standard Borel, including the empty-history singleton.

For each $1\le t\le T$, apply [F4] under [A1] to $X_t$ given $\sigma (Y_{1:t})$ and given $\sigma (Y_{1:t-1})$; their state targets and history targets are standard Borel by step 2.1. The path law [F2] has $X_0$-marginal ${\mu }_0$, so the constant kernel ${\pi }_0((),\cdot )={\mu }_0$ satisfies the conditional identities for the trivial empty history. Apply [F4] under [A1] to $(X_0,\dots ,X_T)$ given $\sigma (Y_{1:T})$; its target is standard Borel by step 2.1.

Each nonempty observation-history target is standard Borel by step 2.1, so [F5] under [A1] factors the RCDs from step 3.1 through their history maps; [F6]–[F8] identify the resulting probability kernels as the required conditional laws. The initial kernel was checked directly in step 3.1. AC is used only through [F2], [F4], and [F5]. No density or pointwise uniqueness on null histories is inferred.