Why $\bigcup_{i\in I}\pi_i^{-1}(\mathcal F_i)$ is not a $\cap-$stable generator of $\bigotimes_{i\in I}\mathcal F_i$? Let $(\Omega _i,\mathcal F_i)$ measurable spaces. We denote de product space $(\prod_{i\in I}\Omega _i,\bigotimes_{i\in I}\mathcal F_i)$ where $\bigotimes_{i\in I}\mathcal  F_i$ is the smallest $\sigma -$field that make the projection $\pi_j$ being $\mathcal F_j-\otimes_{i\in I}\mathcal F_i$ measurable. Why $$\mathcal D=\bigcup_{i\in I}\pi_i^{-1}(\mathcal F_i),$$
is not a $\cap-$stable generator of $\bigotimes_{i\in I}\mathcal F_i$ ? Because for me, if $A,B\in \mathcal D$, there are $A_i\in \mathcal F_i$ and $B_i\in \mathcal F_i$ s.t. $$A=\bigcup_{i\in I}\pi_i^{-1}(A_i)\quad \text{and}\quad B=\bigcup_{i\in I}\pi_i^{-1}(B_i).$$ Then $$A\cap B=\bigcup_{i\in I}\pi_i^{-1}(A_i\cap B_i)\in \mathcal D,$$
and thus is $\cap-$stable, no ?
 A: Observe that the following statements are equivalent:

*

*$A\in\mathcal D=\bigcup_{i\in I}\pi_i^{-1}(\mathcal F_i)$.

*$A\in\pi_i^{-1}(\mathcal F_i)$ for some $i\in I$.

*$A=\pi_i^{-1}(A_i)$ for some $i\in I$ and some $A_i\in\mathcal F_i$
If also $B\in\mathcal D$ so that $B=\pi_j^{-1}(B_i)$ for some $j\in I$ and some $B_i\in\mathcal F_j$ then can we conclude that also $A\cap B\in\mathcal D$?
Not in general.
For observing that it is enough to look at the special case where $I$ contains exactly two elements.
If $A_1$ is a non-trivial subset of $\Omega_1$ and $B_2$ is a non-trivial subset of $\Omega_2$ then we cannot write subset:$$\pi_1^{-1}(A_1)\cap\pi_2^{-1}(B_2)=(A_1\times Y)\cap(X\times B_2)$$as $\pi_1^{-1}(U)$ or as $\pi_2^{-1}(V)$.
A: First and foremost, the notation $\bigcup_{i\in I}\pi_i^{-1}(\mathcal F_i)$ does not indicate what you claim. $\bigcup_{i\in I}\pi_i^{-1}(\mathcal F_i)$ is the family of sets $A\subseteq \prod_{i\in I}\Omega_i$ such that there is some $i$ and some $S \in \mathcal F_i$ such that $A=\pi^{-1}(S)$. I.e. $A=\prod_{j\in I}H_j$, where $H_j=\begin{cases}\Omega_j&\text{if }j\ne i\\ S&\text{if }j=i\end{cases}$.
Therefore you really need the family of sets that are finite intersections of elements of $\bigcup_{i\in I}\pi_i^{-1}(\mathcal F_i)$.
For the record (but its importance is lost due to the previous remark), $\left(\bigcup_{i\in I} X_i\right)\cap\left(\bigcup_{i\in I} Y_i\right)=\bigcup_{i\in I}X_i\cap Y_i$ is not an identity: $\left(\bigcup_{i\in I} X_i\right)\cap\left(\bigcup_{i\in I} Y_i\right)=\bigcup_{(i,j)\in I\times I}X_i\cap Y_j$ is.
