This is a continuation of my previous question and inspired by Arturo Margidin's reply.
Suppose there are a collection of measurable spaces $(X_i, \mathbb{S}_i), i \in I$. Let $\mathbf{X}=\prod_{i\in I}X_i$, and let $\mathbb{S}=\prod_{i\in I} \mathbb{S}_i$ be the product $\sigma$-algebra.
Suppose $A_{i_0}\subseteq X_{i_0}$ is such that $A_{i_0}\times\prod_{j\neq i_0}X_j\in\mathbb{S}$. Does it follow that $A_{i_0}\in\mathbb{S}_{i_0}$?
More generally, suppose we have a family of subsets, $A_i\subseteq X_i$, and we know that $A_j\in\mathbb{S}_j$ for all $j\neq i_0$ and that $\prod_{i\in I}A_i\in\mathbb{S}$. Does it follow that $A_{i_0}\in \mathbb{S}_{i_0}$?
Basically, I would like to reconstruct each $\mathbb{S}_i$ from $\prod_{i \in I} \mathbb{S}_i$. If you have other approaches, please don't hesitate to reply. In particular, Part 1 and Part 2 are attempts based on projection of measurable rectangles in the product $\sigma$-algebra. I was wondering if it is possible to go from sectioning a measurable set in the product $\sigma$-algebra?
Thanks and regards!