I'm solving problems from my Functional Analysis course and I don't know how to prove one of the implications of this exercise:
Given $(X,\Sigma)$ a measure space and given $S\subset X$. Consider $\Omega\subset \mathcal{P}(X)$ defined as $$\Omega = \left\{ C\subset X\mid \exists A,B\in\Sigma \text{ s.t. }C=(A\cap S)\cup(B\cap S^c)\right\}.$$ Prove that the $\sigma$-algebra generated by $\Sigma$ and $S$ (I'll denote it $\sigma(\Sigma,S)$) equals $\Omega$.
I've tried proving that $\sigma(\Sigma,S)$ contains $\Omega$ and viceversa:
$\subseteq$ : This one was easy. Assume $E\in\Omega$, then $E=(A\cap S)\cup(B\cap S^c)$ for some $A,B\in\Sigma$. Since $\sigma$-algebras are closed by countable unions and intersections, it is so by finite unions and intersections; and also closed by complements, and clearly $A,B,S,S^c\in\sigma(\Sigma, S)$ so necessarily $(A\cap S)\cup(B\cap S^c)=E\in\sigma(\Sigma,S)$ and this gives as that $\Omega\subseteq\sigma(\Sigma,S)$.
$\supseteq$ : This is the one I don't know how to approach. I assume $E\in\sigma(\Sigma,S)$, but then I don't know how to prove it is implied that $E\in\Omega$. I've considered assuming $E\notin\Omega$ and see if I get a contradiction, but I get lost again. How can I prove this one? Is the raw definition of $\sigma$-algebra needed to do this?
Is my approach to the first contain correct? How can I prove the second one? Any help or hint will be appreciated, thanks in advance.
Hint: You have to first verify that $\Omega$ is a $\sigma-$ algebra. Then observe that it contains $\Sigma$ and the set $S$. It follows that it contains $\sigma (\Sigma, S)$.