Let $k_i$ be a stochastic transition kernel from $(\times_{j=0}^{i-1} \Omega_j,\times_{j=0}^{i-1} \mathcal{A}_j)$ to $(\Omega_i,\mathcal{A}_i)$.
$\times_{j=0}^{i-1} \Omega_j$ is for the cartesian product,
$\times_{j=0}^{i-1} \mathcal{A}_j$ is for the product of sigma-algebras.
Let's define the probability measures $P_i=P_0\otimes\bigotimes^i_{j=1}k_j$ on $(\times_{j=0}^{i} \Omega_j,\times_{j=0}^{i} \mathcal{A}_j)$.
Then why do we have that $P_i(A\times \Omega_{k+1} \cdots \times \Omega_{i})=P_j(A\times \Omega_{k+1} \cdots \times \Omega_{j})$, for any $A \in \times_{j=0}^{k} \mathcal{A}_j$ with $j,i\geq k$?
$$P_i(A\times \Omega_{k+1} \cdots \times \Omega_{i})=\int_{X_0(A)}\bigotimes_{j=1}^i k_j(\omega_0,X_K(A)\times \Omega_{k+1} \times \cdots \times \Omega_{i}) P_0(d\omega_0),$$ and
by the definition of product of transition kernels
$$=\int_{X_0(A)}\int_{\times_{j=1}^{i-1} \Omega_j}\int_{\Omega_i}\mathbf{1}_{B}(\omega)k_i((\omega_0,\omega_1,\cdots, \omega_{i-1}),d\omega_{i})\bigotimes_{j=1}^{i-1} k_j(\omega_0,d(\omega_1,\ldots, \omega_{i-1})) P_0(d\omega_0) $$
, where $X_K$ is the canonical projection from $\times_{j=0}^{i} \Omega_j$ to $\times_{j \in K=\{1,\ldots,k\}} \Omega_j$ and similarly for $X_0$. Also $B=X_K(A)\times \Omega_{k+1}\times \cdots\times \Omega_{i}$
and because $\mathbf{1}_{B}(\omega)=\mathbf{1}_{X_K(A)}(X_K(\omega))\cdots \mathbf{1}_{\Omega_i}(\omega_i)$, we have
$$=\int_{X_0(A)}\int_{\times_{j=1}^{i-1} \Omega_j}\mathbf{1}_{X_K(A)}(X_K(\omega))\cdots \mathbf{1}_{\Omega_{i-1}}(\omega_{i-1})k_i((\omega_0,\omega_1,\ldots, \omega_{i-1}),\Omega_i)\bigotimes_{j=1}^{i-1} k_j(\omega_0,d(\omega_1,\ldots, \omega_{i-1})) P_0(d\omega_0) $$
and because the transition kernels are stochastic we have $k_i((\omega_0,\omega_1,\ldots, \omega_{i-1}),\Omega_{i})=1$, so
$$=\int_{X_0(A)}\int_{\times_{j=1}^{i-1} \Omega_j}\mathbf{1}_{X_K(A)}(X_K(\omega))\cdots \mathbf{1}_{\Omega_{i-1}}(\omega_{i-1}) \bigotimes_{j=1}^{i-1} k_j(\omega_0,d(\omega_1,\ldots, \omega_{i-1})) P_0(d\omega_0) $$
Here, I get stuck. I would say that we would get $$\int_{X_0(A)}\int_{\times_{j=1}^k \Omega_j}\mathbf{1}_{X_K(A)}(X_K(\omega))\bigotimes_{j=1}^k k_j(\omega_0,d(\omega_1,\ldots, \omega_{k}) ) P_0(d\omega_0)$$
Since intuitively I would say we may have something like $\int_{X_0(A)}\bigotimes_{j=1}^{k} k_j(\omega_0,X_K(A),\Omega_{k+1},\ldots, \Omega_k) )P_0(d\omega_0)$. But I'm not sure how to deduce it, nor if I'm using correctly the definition...
Any help would be appreciated.