Problem in proving a set is a Sigma Algebra 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.

HINTS: For both (2) and (3), note that $F\in\mathscr{F}$ iff there is a $Y_F\in\mathscr{A}$ such that $F=X\cap Y_F$.
(2) What is $X\cap(\Omega\setminus Y_F)$?
(3) What is $X\cap\bigcup_iY_{F_i}$?
