Let $(X,\mathcal{B},m),(Y,\mathcal{C},p)$ be measure spaces and let $(X\times Y, \mathcal{B}\times \mathcal{C},m \times p)$ be the product space. Let $E \in \mathcal{B}\times \mathcal{C}$. For $x \in X$ and $y \in Y$, define the fibers $E_x=\{y \in Y:x \in X\}$ and $E^y=\{x \in X:(x,y) \in E\}$.
Our fact is that $E_x \in \mathcal{C}$ and $E^y \in \mathcal{B}$ for all $x \in X$ and $y\in Y$. This can be proven by defining the collection $\mathcal{M}=\{E \in \mathcal{A}\times \mathcal{B}:\forall x\in X, \forall y \in Y,E^y \in \mathcal{B},E_x \in \mathcal{C} \}$, and showing $\mathcal{M}$ is a $\sigma$-algebra and $\mathcal{R}=\{B\times C:B \in \mathcal{B}, C \in \mathcal{C}\} \subset \mathcal{M}$.
I was wondering if there was a more constructivist proof of this. If $E \in \mathcal{B}\times \mathcal{C}$, is there some constructive process to show $E_x \in \mathcal{C}$?