Hello and thanks for the time spend to read this :)
Consider $(\Omega,\mathcal{F},P)$
Consider $A=\{x_1,...,x_p\}$ a set of random variables and $\Theta=\sigma(A)$ be the sigma algebra generated by $\{x_1,...,x_p\}$.
Moreover consider $A'$ and $A''$ such that $A'\dot\cup A''=A$ (where $\dot\cup$ represent the disjoint union) and $\mathcal{A}=\sigma(A')$ and $\mathcal{B}=\sigma (A'')$ the sigma algebras generated by $A'$ and $A''$. Finally consider $\mathcal{G}=\{A\cap B:A\in\mathcal{A}\;\text{and}\;B\in\mathcal{B}\}$.
I am asking myself if the following question : is it true that for all $\theta \in \Theta$ we have $\theta\in \mathcal{A}$, or $\theta\in \mathcal{B}$ or $\theta \in \mathcal{G}$ ?
It seems intuitively true, but I am not 100% sure and unable to prove it.
Thanks in advance for the help.
Mathieu