I would like to prove, that for any Markov kernel $K$ on a Polish space $(F,\mathcal{F})$ (with a $\sigma$-field) you can find a measurable space $(S,\mathcal{S})$, a random element $Z$ on $S$ and an update function $\alpha:F\times S\to F$ such that for $A\in\mathcal{F}$ it holds $$K(x,A) = P(\alpha(x,Z)\in A) \qquad \mbox{for all }x\in F. $$ Does anyone have any hints on how this can be accomplished?
If the Polish space is $\mathbb{R}$, one can take the cdf $F_y(x) = K(y,(-\infty,x])$ and define $(S,\mathcal{S})=([0,1],\mathcal{B}([0,1])$, $Z=U$ for a uniform $U$ and $\alpha(y,u):= F_y^{\leftarrow}(u)$ (the generalized inverse). This can be done for any $y\in\mathbb{R}$. But the general case of a Polish state space?
Edit: In fact, finding a proof for the case where the Polish space is $\mathbb{R}^n$ would be satisfying.