I have been trying to get my head around this question. Any help greatly appreciated.
Let $\mathscr{C}$ be any class of subsets $\Omega$ with $\emptyset,\Omega\in \mathscr{C}$. Define $\mathscr{C}_0=\mathscr{C}$ and for any ordinal $\alpha>0$ write inductively, $$ \mathscr{C}_\alpha=\Big(\bigcup\left\{ \mathscr{C}_\beta:\beta<\alpha\right\} \Big)^{\prime} $$ where $\mathscr{D}^{\prime}$ denotes the class of all countable unions of differences of sets in $\mathscr{D}$. Let $\mathscr{S}=\bigcup\left\{\mathscr{C}_\alpha:\alpha<\omega_1 \right\}$ where $\omega_1$ is the first uncountable ordinal and let $\mathscr{F}$ be the minimal $\sigma$-field over $\mathscr{C}$.
Show $\mathscr{S}\subset \mathscr{F}$ and $\mathscr{C}\subset\mathscr{C}_\alpha$. Also, $\mathscr{C}_\alpha$ increases with $\alpha$.