I am quite confused by this equation. Let $P_{x|\xi}$ denote the conditional distribution of $x$ with respect to $\xi$. Similarly, $\pi_{\theta|\xi}$ denote the conditional distribution of $\theta$ with respect to $\xi$. Then
\begin{align*} P_{x|\xi}(A) = \int_{\Theta}P_{x|\theta}(A)d\Pi_{\theta|\xi} \end{align*}
I understand the proof of $P_x(A)=\int_\theta P_{x|\theta}(A)d\Pi_{\theta}$. However, I am not sure this is true.
If I write $X,\Theta, \xi$ as random variable from sample space $\Omega$ to $\mathbb{R}^n$. Then \begin{align*} P_{x|\xi}(A) = E[1_{X\in A}|\xi], \quad \int_{\Theta}P_{x|\theta}(A)d\Pi_{\theta|\xi} = E[P_{x|\theta}(A)|\xi] = E[E[1_{X\in A}|\Theta]|\xi] \end{align*} So this is literally saying \begin{align*} E[1_{x\in A}|\xi] = E[E[1_{X\in A}|\Theta]|\xi] \end{align*} However, this is true provided $\sigma(\xi)\subseteq \sigma(\Theta)$. But this is not necessarily true. i.e. This equation seems to not hold in general.
Is there any comments
Thanks in advance!