We have two probability spaces $(\Omega_1,\mathcal{F_1},P_1)$ and $(\Omega_2,\mathcal{F_2},P_2)$. Is it possible to construct probability space $(\Omega=\Omega_1\times\Omega_2,\mathcal{F},P)$ such that:

$\forall A_1\in\mathcal{F_1}, A_2\in\mathcal{F_2}$ set $A_1\times A_2\in\mathcal{F}$ and its measure equals $P_1(A_1)P_2(A_2)$

$\mathcal{F}$ contains all sets $A\subseteq\Omega$ such that $\forall \omega_1\ \{\omega_2:(\omega_1,\omega_2)\in A\}\in\mathcal{F_2}$ and $P_2(\{\omega_2:(\omega_1,\omega_2)\in A\})$ is constant for all $\omega_1$

$P$-measure of such sets from (2) equals $P_2(\{\omega_2:(\omega_1,\omega_2)\in A\})$ (does not depend on $\omega_1)$

In other words, how to add to ($\Omega_1\times\Omega_2,\mathcal{F_1}\times\mathcal{F_2},P_1P_2)$ all sets from $\Omega$ with constant cross-$\Omega_2$ measure?

One can assume $\Omega_1=[0,1]$ with Lebesgue measure and $\Omega_2=[0,1]^\Gamma$ (infinite-dimensional cube).

This is needed for my research in Economics. Any advice on this problem will be greatly appreciated. Links to papers would be the best help. I am struggling with this question for several months already. Please, share any thoughts.