Prove or disprove the identity on $\sigma$-algebras

Question

I encountered a problem about the containing relations about the $\sigma$-algebras. Let $<\mathcal{A}>$ denote the $\sigma$-algebra generated by $\mathcal{A}$. Suppose for a collection of sets $\mathcal{C}$, and a set $Z$ we define $$\mathcal{C}\times Z := \{C\times Z; C\in \mathcal{C} \}, $$ where "$\times$" described in the set is the Cartesian product. My problem is that do we have: $$<\mathcal{C}\times Z> = <\mathcal{C}> \times Z$$ holds? I strongly believe that they are the same, and proved the direction "$\subset$" above. So how to verify the other direction? Or is there anything wrong? Thanks.

Answer 1

Had a huge misunderstanding. You're wanting to show something different than what I wrote in the comments. It's interesting.

The goal is to show that $\sigma(A \times Z) = \sigma(A) \times Z$, where $Z$ is just some set and $A$ is a collection of sets. First question is whether $\sigma(A) \times Z$ is a $\sigma$-algbera. The answer is yes, by properties of Cartesian product. Naturally we have $\sigma(A \times Z) \subseteq \sigma(A) \times Z$. Now let $\mathcal{M}$ be any other $\sigma$-algebra containing $A \times Z$. Then again just using some nice properties about products and unions/intersections/complements we have $\sigma(A) \times Z \subseteq \mathcal{M}$. Thus by minimality we get the result.

https://proofwiki.org/wiki/Cartesian_Product_of_Unions

https://proofwiki.org/wiki/Cartesian_Product_of_Intersections

https://proofwiki.org/wiki/Cartesian_Product_with_Complement