$\sigma$-field of sets of measure $0$ and $1$ - independence

Question

$(\Omega, \mathcal{F}, P)$ - probability space and $\mathcal{N}=\sigma\{A:P(A)=0\vee P(A)=1\}$. Where I can find theorem with proof which states that $\mathcal{N}$ is independent from any other $\sigma$-field on this probability space?

Answer 1

Let $\mathcal{G}$ be any sub-$\sigma$-field on $\mathcal{F}$. Let $A \in \mathcal{N}$ and $B \in \mathcal{G}$. What can you say about the relationship between $P(A \cap B)$ and $P(A)P(B)$?