In my application, I essentially sample random numbers $U^{(m)}\in[0,1)^{\mathbb N_0}$, $m\in\mathbb N_0$, such that $$\operatorname P\left[U^{(m)}\in B\mid U^{(m-1)}\right]=\kappa\left(U^{(m-1)},B\right)\;\;\;\text{almost surely for all }B\in\mathcal B([0,1))^{\otimes\mathbb N_0},\tag1$$ where $\kappa$ is a Markov kernel on $\left([0,1)^{\mathbb N_0},\mathcal B([0,1))^{\otimes\mathbb N_0}\right)$ and $U^{(0)}$ is uniformly distributed on $[0,1)^{\mathbb N_0}$. Now let $k\in\mathbb N$ and $I:=\{0,\ldots,k-1\}$. I don't use the $U^{(m)}$ as they are, but transform them according to $$\left(S^{(m)},X^{(m)}\right):=\left(\lfloor kU^{(m)}_0\rfloor,\left(U^{(m)}_n\right)_{n\in\mathbb N}\right).$$ In order to apply a theoretically established result, I need that the $\left(S^{(m)},X^{(m)}\right)$ evolve according to a Markov kernel $\Sigma$ on $\left(I\times[0,1)^{\mathbb N},2^I\otimes\mathcal B([0,1))^{\otimes\mathbb N}\right)$.
Are we able to define $\Sigma$ so that the application is actually in the situation of the theory?
I need that $\Sigma$ has a density $\sigma:\left(I\times[0,1)^{\mathbb N}\right)^2\to[0,\infty)$ with respect to $\zeta\otimes\left.\lambda\right|_{[0,\:1)}^{\otimes\mathbb N}$, where $\zeta$ denotes the counting measure on $(I,2^I)$.
You may assume that $\kappa$ has a symmetric density $\rho:[0,1)^{\mathbb N_0}\times[0,1)^{\mathbb N_0}\to[0,\infty)$ with respect to $\left.\lambda\right|_{[0,\:1)}^{\otimes\mathbb N_0}$.
Remark: Please take note of my related question: What is the modified transition kernel induced by this sample truncation?.