Measure Theory : Probability Space given a measurable function

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 ?