Given a probability space $(\Omega, \Sigma, \pi)$, a measurable space $(X, \chi)$ and a measurable function $f : \Omega \rightarrow X$.

We can define a regular conditional probability $v(x, A)$, such that for all $A \in \Sigma$ and all $C \in \chi$,

$\int_C v(x, A) \pi(f^{-1}(dx)) = \pi(A \cap f^{-1}(C))$.

Let $g$ be a function such that $g(x, A) = A_x = \{y \vert y \in A \wedge f(y) = x \}$ (Throwing away all the elements which don't map to $x$).

We define a measurable space for each $x$, $(\Omega_x, \Sigma_x)$, such that $\Omega_x = g(x, \Omega)$, $\Sigma_x = \{g(x, A) \vert A \in \Sigma\}$. There exists a probability measure $\pi_x$ on $(\Omega_x, \Sigma_x)$ such that $v(x, A) = \pi_x(g(x, A))$.

This existence would require the following to be true about $v$, $\forall A_1, A_2 \in \Sigma$,

$g(x, A_1) = g(x, A_2) \implies v(x, A_1) = v(x, A_2)$.

Are their any restrictions other than existence of regular conditional probability distribution for me to construct spaces $(\Omega_x, \Sigma_x, \pi_x)$ for each $x \in X$ and guarantee the above condition?

This formulation seems to be trivially true for product spaces i.e. $\Omega = X \times Y$.

Can anyone point me to a paper/book exploring something similar ?


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