Let $(X,\mathcal F,\phi)$ and $(Y,\mathcal G,\psi)$ be two probability spaces and $(Z,\mathcal H,\mu)$ be their product space: i.e. we have that $Z = X\times Y$. Given an arbitrary set $F\in\mathcal H$ and any point $x\in X$ let us define $ \pi_x(F) = \{y\in Y:(x,y)\in F\}. $ I wonder if the following formula is true: $ \mu(F) = \int\limits_X \psi(\pi_x(F))\phi(dx).\tag{1} $
I think it is true and the proof goes like this:
For any $A\in \mathcal F$ and $B\in\mathcal G$ we have $ \int\limits_X \psi(\pi_x(A\times B))\phi(dx) = \int\limits_X \psi(B)1_A(x)\phi(dx) = \phi(A)\psi(B) = \mu(A\times B) $ so $(1)$ is true for the class $\mathcal C$ of measurable rectangles $A\times B$.
If $(1)$ holds for $F\in \mathcal H$ it also holds for $F^c$.
If $(1)$ holds for the disjoint sequence $(F_n)_{n\geq 0}$ in $\mathcal H$ then it holds for $F = \bigcup\limits_{n\geq 0}F_n$.
As a result, if $\mathcal H'$ is the class of sets for which $(1)$ holds then $\mathcal C\subset \mathcal H'$ and it is closed under taking complements and countable unions, so $\mathcal H'$ is a $\sigma$-algebra and hence $\mathcal H\subset \mathcal H'$ hence $(1)$ is true for any $F\in \mathcal H$.
My question are the following: first, please tell me if the proof is correct, especially I am not sure if I need to show measurability of $\psi(\pi_x(F))$ or is it implicitly proved already. Second, if the result is true I'm pretty sure it is known - so I wonder about a reference.