Parameter-measurable statistical models as kernels

Suppose an experiment on $(\mathcal{X},\mathcal{A})$ also has a measurable parameter space $(\mathrm{\Theta },\mathcal{T})$ (Measurable spaces and measurable sets). A Markov kernel (or probability kernel) representation is a function

$$K:\mathrm{\Theta }\times \mathcal{A}\to [0,1]$$

such that, for every $\theta \in \mathrm{\Theta }$, the set function $A\mapsto K(\theta ,A)$ is a probability measure on $(\mathcal{X},\mathcal{A})$, and, for every $A\in \mathcal{A}$ and every Borel set $C\subseteq [0,1]$,

$$\{\theta \in \mathrm{\Theta }:K(\theta ,A)\in C\}\in \mathcal{T}\mathrm{.}$$

Here the Borel sigma-algebra on $[0,1]$ is generated by its relatively open subsets $U\cap [0,1]$, where $U$ is open in $\mathbb{R}$. Thus the second condition says that each event-probability map $\theta \mapsto K(\theta ,A)$ is measurable. The kernel represents the given experiment exactly when $K(\theta ,A)=P_{\theta }(A)$ for every $\theta$ and $A$ (Statistical experiments, Probability measures and probability spaces).

This is an optional parameter-measurable version of a model. An arbitrary experiment does not include a measurable structure on $\mathrm{\Theta }$ or imply this kernel condition.