Let $(\Omega,\mathscr A)$ be a measurable space. If $\varnothing \subset X \subset \Omega$, let $\mathscr F = \{ F \subseteq \Omega, F = X \cap Y, Y \in \mathscr A\} \;. $
I need to prove that $\mathscr F$ is a $ \sigma$-Algebra on $X$.
So, I have to show that
- $\varnothing \in \mathscr F$
- If $F \in \mathscr F$, then $F^C \in \mathscr F $
- If $F_i \in \mathscr F$, then $\bigcup_{i=1}^\infty F_i \in \mathscr F $
I have trouble in showing 2 and 3 conditions.