Let
- $(E_i,\mathcal E_i)$ be a measurable space;
- $f:E_1\times E_2\to E_1\to E_1$ be $(\mathcal E_1\otimes\mathcal E_2,\mathcal E_1)$-measurable;
- $(\Omega,\mathcal A,\operatorname P)$ be a probability space;
- $X_0$ be an $E_1$-valued random variable on $(\Omega,\mathcal A,\operatorname P)$;
- $(Z_n)_{n\in\mathbb N}$ be an $E_2$-valued i.i.d. process on $(\Omega,\mathcal A,\operatorname P)$ independent of $X_0$ and $\mu$ denote the distribution of $Z_1$.
How do we show that $$X_n:=f(X_{n-1},Z_n)\;\;\;\text{for }n\in\mathbb N$$ is a time-homogeneous Markov chain with transition kernel $$(\kappa g)(x):=\operatorname E\left[(g\circ f)(x,Z_1)\right]=\int(g\circ f)(x,z)\:\mu({\rm d}z)\;\;\;\text{for }x\in E_1$$ for bounded $\mathcal E_1$-measurable $g:E_1\to\mathbb R$?
Remember the following result:
Lemma: Let $Y_i$ be an $(E_i,\mathcal E_i)$-valued random variable on $(\Omega,\mathcal A,\operatorname P)$, $h:E_1\times E_2\to\mathbb R$ is bounded and $(\mathcal E_1\otimes\mathcal E_2)$-measurable and $\mathcal F\subseteq\mathcal A$ be a $\sigma$-algebra. If $\mathcal F$ is independent of $Y_2$ and $Y_1$ is $\mathcal F$-measurable, then $$\operatorname E\left[h(Y_1,Y_2)\mid\mathcal F\right]=\left.\operatorname E\left[h(y_1,Y_2)\right]\right|_{y_1\:=\:Y_1}\tag1.$$
Using this lemma, it is easy to see that $$\operatorname E\left[g(X_{n+1})\mid\mathcal F^X_n\right]=(\kappa g)(X_n)\tag2$$ holds true for $n=0$.
But how do we need to proceed for $n\ge1$? Are we somehow able to show that $\mathcal F^X_n$ is independent of $Z$?