Countable generation of $\sigma$-algebras I feel that a positive answer to the following question would be helpful in solving some exercises in introductory measure theory:
Suppose $\cal A$ is a collection of subsets of a set $X$ and let $\cal \Sigma$ be the smallest $\sigma$-algebra containing $\cal A$, that is to say, the intersection of all $\sigma$-algebras containing $\cal A$.
Suppose $S \in \cal \Sigma$. Then, can we state that there is a countable collection of sets $\cal B \subset \cal A$ such that, $S$ can be obtained from $\cal B$ with countably many operations of complement, intersection and union?
Does the above query have a positive answer and if so where can a proof be found?

Hint: Let $\Sigma'$ be the collection of sets that can be obtained from countable subsets of $\mathcal A$ by countably many operations.  Show that $\Sigma'$ is a 
$\sigma$-algebra.
