We've got the following definition
Let $\mathcal C$ be a class of subsets of $\Omega$. We say that $\sigma(\mathcal C)$ is the $\sigma$-algebra generated by $\mathcal C$ if satisfies that: 1. $\mathcal C\subseteq \sigma(\mathcal C)$. 2. If $\mathcal C\subseteq \mathcal A$, with $\mathcal A$ another $\sigma$-algebra, then $\sigma(\mathcal C)\subseteq \mathcal A$.
I was checking some old problems in my probability notes, and to be honest I really don't understand this, like why do we need this? when do we use this? I can see some things derived from the definition, like $\sigma(\mathcal C)$ is the smallest $\sigma$-algebra that contains $\mathcal C$ or the trivial fact that this is really a $\sigma$-algebra. Also, supposedly, the $\sigma(\mathcal C)$ is the intersection of all the $\sigma$-algebras that contain $\mathcal C$, I don't get how this is something easy to see.
(I was hesitant to add the [measure-theory] tag, since I haven't study that yet)